jmlr jmlr2010 jmlr2010-30 knowledge-graph by maker-knowledge-mining

30 jmlr-2010-Dimensionality Estimation, Manifold Learning and Function Approximation using Tensor Voting

Source: pdf

Author: Philippos Mordohai, Gérard Medioni

Abstract: We address instance-based learning from a perceptual organization standpoint and present methods for dimensionality estimation, manifold learning and function approximation. Under our approach, manifolds in high-dimensional spaces are inferred by estimating geometric relationships among the input instances. Unlike conventional manifold learning, we do not perform dimensionality reduction, but instead perform all operations in the original input space. For this purpose we employ a novel formulation of tensor voting, which allows an N-D implementation. Tensor voting is a perceptual organization framework that has mostly been applied to computer vision problems. Analyzing the estimated local structure at the inputs, we are able to obtain reliable dimensionality estimates at each instance, instead of a global estimate for the entire data set. Moreover, these local dimensionality and structure estimates enable us to measure geodesic distances and perform nonlinear interpolation for data sets with varying density, outliers, perturbation and intersections, that cannot be handled by state-of-the-art methods. Quantitative results on the estimation of local manifold structure using ground truth data are presented. In addition, we compare our approach with several leading methods for manifold learning at the task of measuring geodesic distances. Finally, we show competitive function approximation results on real data. Keywords: dimensionality estimation, manifold learning, geodesic distance, function approximation, high-dimensional processing, tensor voting

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Keywords: dimensionality estimation, manifold learning, geodesic distance, function approximation, high-dimensional processing, tensor voting 1. [sent-15, score-1.308]

2 Dimensionality estimation and structure inference are accomplished simultaneously by encoding the observations as symmetric, second order, non-negative deﬁnite tensors and analyzing the outputs of tensor voting (Medioni et al. [sent-33, score-0.965]

3 We compute local a dimensionality estimates, but instead of performing dimensionality reduction, we perform all operations in the original input space, taking into account the estimated dimensionality of the data. [sent-44, score-0.577]

4 All processing in our method is performed using tensor voting, which is a computational framework for perceptual organization based on the Gestalt principles of proximity and good continuation (Medioni et al. [sent-65, score-0.571]

5 Each vote conveys the orientation the receiver would have if the voter and receiver were in the same structure. [sent-70, score-0.654]

6 In Section 3, we present a new implementation of tensor voting that is not limited to low-dimensional spaces as the original one of Medioni et al. [sent-71, score-0.839]

7 Using tensor voting, we are able to reliably estimate the normal and tangent space at each sample, as described in the following section. [sent-220, score-0.639]

8 Tensor Voting in High-Dimensional Spaces The tensor voting framework, in its preliminary version (Guy and Medioni, 1996), is an implementation of two Gestalt principles, namely proximity and good continuation, for grouping generic tokens in 2-D. [sent-223, score-0.839]

9 Guy and Medioni (1997) extended tensor voting to three dimensions. [sent-226, score-0.839]

10 Tensor voting is a pairwise operation in which elements cast and collect votes in local neighborhoods. [sent-230, score-0.701]

11 Each vote is a tensor and encodes the orientation the receiver would have according to the voter, if the voter and receiver belonged in the same structure. [sent-231, score-1.137]

12 The evidence in tensor voting are the position and preferred orientation, if available, of the voter and the position of the receiver. [sent-233, score-1.03]

13 The simplest curve, the one with minimum total curvature, that passes through an oriented voter and 418 D IMENSIONALITY E STIMATION , M ANIFOLD L EARNING AND F UNCTION A PPROXIMATION Figure 1: Illustration of tensor voting in 2-D. [sent-239, score-1.03]

14 Therefore, we connect the voter and receiver by a circular arc which is tangent at the voter and passes through the receiver and deﬁne the vote cast as the normal to this arc at the location of the receiver. [sent-246, score-0.978]

15 Even though the applicability of tensor voting as a manifold learning technique seems to have merit, the generalization of the implementation of Medioni et al. [sent-265, score-1.05]

16 The bottleneck is the generation and storage of voting ﬁelds, the number of which is equal to the dimensionality of the space. [sent-267, score-0.552]

17 In this section, we describe the tensor voting framework beginning with data representation and proceeding to the voting mechanism and vote analysis. [sent-268, score-1.367]

18 The novelty of 419 M ORDOHAI AND M EDIONI (a) Oriented or stick tensor (b) Unoriented or ball tensor (c) Generic tensor Figure 2: Examples of tensors in 3-D. [sent-271, score-1.793]

19 The tensor on the left has only one non-zero eigenvalue and encodes a preference for an orientation parallel to the eigenvector corresponding to that eigenvalue. [sent-272, score-0.633]

20 The eigenvalues of the tensor in the middle are all equal, and thus the tensor does not encode a preference for a particular orientation. [sent-273, score-1.009]

21 our work is a new formulation of the voting process that is practical for spaces of dimensionality up to a few hundreds. [sent-275, score-0.525]

22 All tensors in this paper are second order, symmetric and non-negative deﬁnite, so any reference to a tensor automatically implies these properties. [sent-279, score-0.609]

23 A tensor represents the structure of a manifold going through the point by encoding the normals to the manifold as eigenvectors of the tensor that correspond to non-zero eigenvalues, and the tangents as eigenvectors that correspond to zero eigenvalues. [sent-281, score-1.433]

24 ) For example, a point in an N-D hyperplane has one normal and N − 1 tangents, and thus is represented by a tensor with one non-zero eigenvalue associated with an eigenvector parallel to the hyperplane’s normal. [sent-283, score-0.558]

25 A point belonging to a 2-D manifold in N-D is represented by two tangents and N − 2 normals, and thus is represented by a tensor with two zero eigenvalues associated with the eigenvectors that span the tangent space of the manifold. [sent-285, score-0.845]

26 The tensor also has N − 2 non-zero, equal eigenvalues whose corresponding eigenvectors span the manifold’s normal space. [sent-286, score-0.574]

27 The tensor at a point on a manifold of dimensionality d, with ni being the unit vectors that span the normal space, can be computed as follows: d T = ∑ ni nT . [sent-289, score-0.911]

28 i i=1 420 (1) D IMENSIONALITY E STIMATION , M ANIFOLD L EARNING AND F UNCTION A PPROXIMATION An unoriented point can be represented by a ball tensor which contains all possible normals and is encoded as the N × N identity matrix. [sent-290, score-0.615]

29 (a) Generic 3-D tensor (b) Elementary 3-D tensors Figure 3: Tensor decomposition in 3-D. [sent-310, score-0.609]

30 A generic tensor can be decomposed into the stick, plate and ball components that have a normal subspace of rank one, two and three respectively. [sent-311, score-0.591]

31 Given a tensor at A and a tensor at B, the vote the token at A (the voter) casts to B (the receiver) has the orientation the receiver would have, if both the voter and receiver belong to the same structure. [sent-314, score-1.62]

32 The vote is also a stick tensor and is generated as described in Section 3 according to the following equation: 2 +cκ2 ) σ2 −( s Svote (s, θ, κ) = e θ = arcsin( s = κ = − sin(2θ) cos(2θ) [− sin(2θ) cos(2θ)], (3) vT e1 ˆ ), v θ v , sin(θ) 2 sin(θ) . [sent-324, score-0.813]

33 Regardless of the dimensionality of the space, stick vote generation always takes place in a 2-D subspace deﬁned by the position of the voter and the receiver and the orientation of the voter. [sent-336, score-0.897]

34 Moreover, no votes are cast if the receiver is at an angle larger than 45◦ with respect to the tangent of the osculating circle at the voter. [sent-346, score-0.556]

35 For all experiments shown here, we limited voting neighborhoods to the extent in which the magnitude of a vote is more than 3% of the magnitude of the voter. [sent-351, score-0.561]

36 , 2000) these votes can be computed by integrating the votes cast by a rotating stick tensor that spans the normal space of the voting tensor. [sent-358, score-1.609]

37 Since the resulting integral has no closed form solution, the integration is approximated numerically by taking sample positions of the rotating stick tensor and adding the votes it generates at each point within the voting neighborhood. [sent-359, score-1.254]

38 As a result, votes that cover the voting neighborhood are pre-computed and stored in voting ﬁelds. [sent-360, score-0.969]

39 The advantage of this scheme is that all votes are generated based on the stick voting ﬁeld. [sent-361, score-0.771]

40 Thus, the use of pre-computed voting ﬁelds becomes impractical as dimensionality grows. [sent-365, score-0.525]

41 Here, we present a simpliﬁed vote generation scheme that allows the direct computation of votes from arbitrary tensors in arbitrary dimensions. [sent-367, score-0.555]

42 As in the original formulation, the eigenstructure of the vote represents the normal and tangent spaces that the receiver would have, if the voter and receiver belong in the same smooth structure. [sent-370, score-0.687]

43 Thus, the vote generated by a ball voter is a tensor that spans the (N − 1)-D normal space of the line and has one zero eigenvalue associated with the eigenvector that is parallel to the line. [sent-376, score-0.981]

44 Taking these observations into account, the ball vote can be constructed by subtracting the direct product of the unit vector in the direction from the voter to the receiver from a full rank tensor with equal eigenvalues (i. [sent-378, score-1.033]

45 The resulting tensor is attenuated by the same Gaussian weight according to the distance between the voter and the receiver. [sent-381, score-0.714]

46 4 VOTING BY E LEMENTARY T ENSORS To complete the description of vote generation, we need to describe the case of a tensor that has d equal eigenvalues, where d is not equal to 1 or N. [sent-387, score-0.655]

47 (An example of such a tensor would be a curve element in 3-D, which has a rank-two normal subspace and a rank-one tangent subspace. [sent-388, score-0.639]

48 3 is not a factor when θ is zero, or, in other words, if the voting stick is orthogonal to vn . [sent-393, score-0.546]

49 The vote is then constructed as the tensor addition of the votes cast by stick tensors parallel to the new basis vectors. [sent-396, score-1.246]

50 Among those votes, only the one generated by the stick tensor parallel to vn is not parallel to the normal space of the voter and curvature has to be considered. [sent-397, score-0.969]

51 Analytically, the vote is computed as the summation of d stick votes cast by the new basis of the normal space. [sent-400, score-0.685]

52 If Svote (A, B, b) is the function that generates the stick vote from a (a) Stick voting (b) Ball voting Figure 4: Vote generation for a stick and a ball voter. [sent-402, score-1.26]

53 The votes are functions of the position of the voter A and receiver B and the tensor of the voter. [sent-403, score-1.015]

54 The voter here is a tensor with a 2-D normal subspace in 3-D. [sent-405, score-0.722]

55 unit stick tensor at A parallel to b to the receiver B, then the vote from a generic tensor with normal space N is given by: Vvote (A, B, Te,d ) = Svote (A, B, b1 ) + ∑ Svote (A, B, bi ). [sent-410, score-1.428]

56 (7) i∈[2,d] In the above equation, Te,d denotes the elementary voting tensor with d equal non-zero eigenvalues. [sent-411, score-0.839]

57 On the right-hand side, all the terms are pure stick tensors parallel to the voters, except the ﬁrst one which is affected by the curvature of the path connecting the voter and receiver. [sent-412, score-0.532]

58 5 T HE VOTING P ROCESS During the voting process (Algorithm 1), each point casts a vote to all its neighbors within the voting neighborhood. [sent-418, score-0.94]

59 Then, each component votes separately and the vote is weighted by λd − λd+1 , except the ball component whose vote is weighted by λN . [sent-421, score-0.661]

60 Besides the voting tensor T, points also have a receiving tensor R that acts as vote accumulator. [sent-422, score-1.549]

61 The eigensystem of the receiving tensor R is computed once at the end of the voting process and the tensor is decomposed as in Eq. [sent-427, score-1.357]

62 Our past and current research has demonstrated that tensor voting is very robust against outliers. [sent-451, score-0.839]

63 If one restricts the analysis to the estimates of orientation, tensor voting can be viewed as a method for maximizing an objective at each point. [sent-455, score-0.839]

64 In all subsequent sections, the eigensystem of the accumulator tensor is used as the voter during subsequent processing steps described in the following sections. [sent-460, score-0.709]

65 3, the intrinsic dimensionality at each point can be found as the maximum gap in the eigenvalues of the tensor after votes from its neighboring points have been collected. [sent-465, score-1.07]

66 The percentage of correct point-wise dimensionality estimates after tensor voting can be seen in Table 2. [sent-588, score-1.008]

67 The error reported is the (unsigned) angle between the eigenvector corresponding to the largest eigenvalue of the estimated tensor at each point and the ground truth surface normal. [sent-624, score-0.546]

68 The error reported is the (unsigned) angle between the eigenvector corresponding to the largest eigenvalue of the estimated tensor at each point and the ground truth surface normal. [sent-630, score-0.546]

69 As for tensor voting, orientation accuracy was measured on 10 instances of each data set. [sent-634, score-0.606]

70 It should be noted, that, unlike tensor voting, local PCA cannot be used to reﬁne these estimates or take advantage of existing orientation estimates that may be available at the inputs. [sent-640, score-0.644]

71 We observe that accuracy using tensor voting is very high for a large range of scales and improves, as expected, with higher data density. [sent-641, score-0.839]

72 Also note that one could reject the outliers by thresholding, since they have smaller eigenvalues than the inliers, and perform tensor voting again to obtain even better estimates of structure and dimensionality. [sent-671, score-0.993]

73 Shown in the ﬁrst column is σ, in the second is the average number of neighbors that cast votes to each point, in the third the average error in degrees of the estimated normals, and in the fourth the accuracy of dimensionality estimation. [sent-685, score-0.532]

74 The key concept is that the intrinsic distance between any two points on a manifold can be approximated by taking small steps on the manifold, collecting votes, estimating the local tangent space and advancing on it until the destination is reached. [sent-790, score-0.551]

75 The tangent space of the new point is computed by collecting votes from the neighboring points, as in regular tensor voting. [sent-797, score-0.848]

76 Note that the tensors used here are no longer balls, but the ones resulting from the previous pass of tensor voting, according to Algorithm 1, step 3. [sent-798, score-0.609]

77 For the results reported in Table 8, we learn the local structure of the cylinder and sphere manifolds of the previous section using tensor voting. [sent-818, score-0.721]

78 We randomly select 5000 pairs of points on each manifold and attempt to measure the geodesic distance between the points of each pair in the input 435 M ORDOHAI AND M EDIONI space using tensor voting and in the embedding space using the other ﬁve methods. [sent-845, score-1.377]

79 In the case of tensor voting, the same scale was used for both learning the manifold and computing distances. [sent-857, score-0.694]

80 The error rates using tensor voting for the sphere are 0. [sent-860, score-0.892]

81 The best result of each method is reported along with the number of neighbors used for the embedding (K), or the scale σ in the case of tensor voting (TV). [sent-886, score-0.951]

82 The best result of each method is reported along with the number of neighbors used for the embedding (K), or the scale σ in the case of tensor voting (TV). [sent-898, score-0.951]

83 Tensor voting is performed in the 50-D space and a path from point A on the line to point B on the cone is interpolated as in the previous experiment, making sure that it belongs to the local tangent space, which changes dimensionality from one to two. [sent-911, score-0.701]

84 The difference with the examples of the previous sections is that the queries are given as input vectors with unknown output values, and thus are of lower dimensionality than the voting space. [sent-927, score-0.557]

85 439 M ORDOHAI AND M EDIONI (a) Noise free inputs (b) Inputs with outliers (c) Interpolated points with outliers and perturbation (d) Interpolated points with outliers and perturbation in 60-D Figure 13: Inputs and interpolated points for Eq. [sent-957, score-0.562]

86 If the perturbation is zero-mean, its effects on individual votes are essentially canceled out, because they only contribute to the ball component of the accumulated tensor at each point, causing small errors in orientation estimation. [sent-973, score-0.923]

87 We split the data in training and test sets and perform tensor voting on the training data to learn the structure of the manifold. [sent-998, score-0.839]

88 Tensor voting may on the surface look similar to other local, instance-based learning algorithms that propagate information from point to point, but the fact that the votes are tensors and not scalars allows them to convey considerably more information. [sent-1079, score-0.775]

89 The properties of the tensor representation, which can handle the simultaneous presence of multiple orientations and structure types, allow the reliable inference of the normal and tangent space at each point. [sent-1080, score-0.639]

90 In addition, tensor voting is very robust against outliers, as demonstrated for the 2-D and 3-D case in numerous publications including Tang and Medioni (1998) and Medioni et al. [sent-1081, score-0.839]

91 For both tensor voting and the methods presented in Section 2, however, the distance metric in the input space has to be meaningful. [sent-1087, score-0.911]

92 The scale can be selected automatically by randomly sampling a few points before voting and making sure that enough points are included in their voting neighborhoods. [sent-1091, score-0.822]

93 The number of sufﬁcient samples for tensor voting cannot be easily predicted from the dimensionality of the space, since it also depends on the complexity (curvature) of the underlying manifolds. [sent-1102, score-1.008]

94 Another important advantage of tensor voting is the absence of global computations, which makes time complexity O(NMlogM), where N is the dimensionality of the space and M is the number of points. [sent-1107, score-1.034]

95 Moreover, since the dimensionality is found as the maximum gap in the eigenvalues of the tensor at each point, no thresholds are needed. [sent-1121, score-0.695]

96 The beneﬁts of this process are in the form of noise robustness and smooth component identiﬁcation, with respect to both dimensionality and orientation, via tensor voting followed by memory savings via dimensionality reduction. [sent-1132, score-1.209]

97 We have also presented, in Section 7, a local nonparametric approach for function approximation that combines the advantages of local methods with the efﬁcient representation and information propagation of tensor voting. [sent-1133, score-0.559]

98 Unsupervised dimensionality estimation and manifold learning in high-dimensional spaces by tensor voting. [sent-1363, score-0.863]

99 Stereo using monocular cues within the tensor voting framework. [sent-1368, score-0.839]

100 N-dimensional tensor voting and application to epipolar geometry estimation. [sent-1510, score-0.839]

similar papers computed by tfidf model

tfidf for this paper:

wordName wordTfidf (topN-words)

[('tensor', 0.483), ('voting', 0.356), ('votes', 0.257), ('manifold', 0.211), ('voter', 0.191), ('vote', 0.172), ('dimensionality', 0.169), ('stick', 0.158), ('medioni', 0.149), ('edioni', 0.142), ('ordohai', 0.142), ('tensors', 0.126), ('orientation', 0.123), ('anifold', 0.115), ('outliers', 0.111), ('tangent', 0.108), ('imensionality', 0.104), ('stimation', 0.104), ('geodesic', 0.089), ('receiver', 0.084), ('pproximation', 0.084), ('isomap', 0.08), ('manifolds', 0.08), ('unction', 0.076), ('mordohai', 0.071), ('roll', 0.067), ('cylinder', 0.067), ('intrinsic', 0.063), ('swiss', 0.06), ('lle', 0.06), ('ball', 0.06), ('curvature', 0.057), ('doll', 0.057), ('inliers', 0.057), ('saliency', 0.057), ('embedding', 0.056), ('neighbors', 0.056), ('points', 0.055), ('sphere', 0.053), ('perceptual', 0.052), ('eigenmaps', 0.05), ('cast', 0.05), ('brand', 0.05), ('gestalt', 0.05), ('gpr', 0.05), ('svote', 0.05), ('sde', 0.049), ('normal', 0.048), ('pj', 0.047), ('tenenbaum', 0.047), ('embeddings', 0.046), ('distances', 0.046), ('earning', 0.046), ('normals', 0.045), ('eigenvalues', 0.043), ('weinberger', 0.042), ('chu', 0.04), ('distance', 0.04), ('saul', 0.039), ('quantitative', 0.038), ('tresp', 0.038), ('local', 0.038), ('housing', 0.036), ('destination', 0.036), ('hlle', 0.036), ('schwaighofer', 0.036), ('tang', 0.036), ('organization', 0.036), ('surface', 0.036), ('bvote', 0.035), ('eigensystem', 0.035), ('laplacian', 0.034), ('inputs', 0.034), ('interpolation', 0.033), ('neighborhoods', 0.033), ('noise', 0.032), ('input', 0.032), ('vn', 0.032), ('salient', 0.031), ('fuzzy', 0.03), ('isometric', 0.03), ('interpolated', 0.03), ('circle', 0.029), ('nearest', 0.029), ('bruske', 0.028), ('costa', 0.028), ('osculating', 0.028), ('receivers', 0.028), ('correspondences', 0.028), ('intersecting', 0.027), ('unoriented', 0.027), ('storage', 0.027), ('eigenvalue', 0.027), ('ti', 0.027), ('global', 0.026), ('requirements', 0.026), ('nonlinear', 0.026), ('arc', 0.025), ('abalone', 0.025), ('guy', 0.025)]

similar papers list:

simIndex simValue paperId paperTitle

same-paper 1 0.99999917 30 jmlr-2010-Dimensionality Estimation, Manifold Learning and Function Approximation using Tensor Voting

Author: Philippos Mordohai, Gérard Medioni

2 0.10080943 54 jmlr-2010-Information Retrieval Perspective to Nonlinear Dimensionality Reduction for Data Visualization

Author: Jarkko Venna, Jaakko Peltonen, Kristian Nybo, Helena Aidos, Samuel Kaski

Abstract: Nonlinear dimensionality reduction methods are often used to visualize high-dimensional data, although the existing methods have been designed for other related tasks such as manifold learning. It has been difﬁcult to assess the quality of visualizations since the task has not been well-deﬁned. We give a rigorous deﬁnition for a speciﬁc visualization task, resulting in quantiﬁable goodness measures and new visualization methods. The task is information retrieval given the visualization: to ﬁnd similar data based on the similarities shown on the display. The fundamental tradeoff between precision and recall of information retrieval can then be quantiﬁed in visualizations as well. The user needs to give the relative cost of missing similar points vs. retrieving dissimilar points, after which the total cost can be measured. We then introduce a new method NeRV (neighbor retrieval visualizer) which produces an optimal visualization by minimizing the cost. We further derive a variant for supervised visualization; class information is taken rigorously into account when computing the similarity relationships. We show empirically that the unsupervised version outperforms existing unsupervised dimensionality reduction methods in the visualization task, and the supervised version outperforms existing supervised methods. Keywords: information retrieval, manifold learning, multidimensional scaling, nonlinear dimensionality reduction, visualization

3 0.089316122 62 jmlr-2010-Learning Gradients: Predictive Models that Infer Geometry and Statistical Dependence

Author: Qiang Wu, Justin Guinney, Mauro Maggioni, Sayan Mukherjee

Abstract: The problems of dimension reduction and inference of statistical dependence are addressed by the modeling framework of learning gradients. The models we propose hold for Euclidean spaces as well as the manifold setting. The central quantity in this approach is an estimate of the gradient of the regression or classiﬁcation function. Two quadratic forms are constructed from gradient estimates: the gradient outer product and gradient based diffusion maps. The ﬁrst quantity can be used for supervised dimension reduction on manifolds as well as inference of a graphical model encoding dependencies that are predictive of a response variable. The second quantity can be used for nonlinear projections that incorporate both the geometric structure of the manifold as well as variation of the response variable on the manifold. We relate the gradient outer product to standard statistical quantities such as covariances and provide a simple and precise comparison of a variety of supervised dimensionality reduction methods. We provide rates of convergence for both inference of informative directions as well as inference of a graphical model of variable dependencies. Keywords: gradient estimates, manifold learning, graphical models, inverse regression, dimension reduction, gradient diffusion maps

4 0.07683821 49 jmlr-2010-Hubs in Space: Popular Nearest Neighbors in High-Dimensional Data

Author: Miloš Radovanović, Alexandros Nanopoulos, Mirjana Ivanović

Abstract: Different aspects of the curse of dimensionality are known to present serious challenges to various machine-learning methods and tasks. This paper explores a new aspect of the dimensionality curse, referred to as hubness, that affects the distribution of k-occurrences: the number of times a point appears among the k nearest neighbors of other points in a data set. Through theoretical and empirical analysis involving synthetic and real data sets we show that under commonly used assumptions this distribution becomes considerably skewed as dimensionality increases, causing the emergence of hubs, that is, points with very high k-occurrences which effectively represent “popular” nearest neighbors. We examine the origins of this phenomenon, showing that it is an inherent property of data distributions in high-dimensional vector space, discuss its interaction with dimensionality reduction, and explore its inﬂuence on a wide range of machine-learning tasks directly or indirectly based on measuring distances, belonging to supervised, semi-supervised, and unsupervised learning families. Keywords: nearest neighbors, curse of dimensionality, classiﬁcation, semi-supervised learning, clustering

5 0.06827306 61 jmlr-2010-Learning From Crowds

Author: Vikas C. Raykar, Shipeng Yu, Linda H. Zhao, Gerardo Hermosillo Valadez, Charles Florin, Luca Bogoni, Linda Moy

Abstract: For many supervised learning tasks it may be infeasible (or very expensive) to obtain objective and reliable labels. Instead, we can collect subjective (possibly noisy) labels from multiple experts or annotators. In practice, there is a substantial amount of disagreement among the annotators, and hence it is of great practical interest to address conventional supervised learning problems in this scenario. In this paper we describe a probabilistic approach for supervised learning when we have multiple annotators providing (possibly noisy) labels but no absolute gold standard. The proposed algorithm evaluates the different experts and also gives an estimate of the actual hidden labels. Experimental results indicate that the proposed method is superior to the commonly used majority voting baseline. Keywords: multiple annotators, multiple experts, multiple teachers, crowdsourcing 1. Supervised Learning From Multiple Annotators/Experts A typical supervised learning scenario consists of a training set D = {(xi , yi )}N containing N i=1 instances, where xi ∈ X is an instance (typically a d-dimensional feature vector) and yi ∈ Y is the corresponding known label. The task is to learn a function f : X → Y which generalizes well on unseen data. Speciﬁcally for binary classiﬁcation the supervision is from the set Y = {0, 1}, for multi-class classiﬁcation Y = {1, . . . , K}, for ordinal regression Y = {1, . . . , K} (with an ordering 1 < . . . < K), and Y = R for regression. c 2010 Vikas C. Raykar, Shipeng Yu, Linda H. Zhao, Gerardo H. Valadez, Charles Florin, Luca Bogoni and Linda Moy. R AYKAR , Y U , Z HAO , VALADEZ , F LORIN , B OGONI AND M OY However, for many real life tasks, it may not be possible, or may be too expensive (or tedious) to acquire the actual label yi for training—which we refer to as the gold standard or the objective ground truth. Instead, we may have multiple (possibly noisy) labels y1 , . . . , yR provided by R i i different experts or annotators. In practice, there is a substantial amount of disagreement among the experts, and hence it is of great practical interest to address conventional supervised learning algorithms in this scenario. Our motivation for this work comes from the area of computer-aided diagnosis1 (CAD), where the task is to build a classiﬁer to predict whether a suspicious region on a medical image (like a X-ray, CT scan, or MRI) is malignant (cancerous) or benign. In order to train such a classiﬁer, a set of images is collected from hospitals. The actual gold standard (whether it is cancer or not) can only be obtained from a biopsy of the tissue. Since it is an expensive, invasive, and potentially dangerous process, often CAD systems are built from labels assigned by multiple radiologists who identify the locations of malignant lesions. Each radiologist visually examines the medical images and provides a subjective (possibly noisy) version of the gold standard.2 The radiologist also annotates various descriptors of the potentially malignant lesion, like the size (a regression problem), shape (a multiclass classiﬁcation problem), and also degree of malignancy (an ordinal regression problem). The radiologists come from a diverse pool including luminaries, experts, residents, and novices. Very often there is lot of disagreement among the annotations. For a lot of tasks the labels provided by the annotators are inherently subjective and there will be substantial variation among different annotators. The domain of text classiﬁcation offers such a scenario. In this context the task is to predict the category for a token of text. The labels for training are assigned by human annotators who read the text and attribute their subjective category. With the advent of crowdsourcing (Howe, 2008) services like Amazon’s Mechanical Turk,3 Games with a Purpose,4 and reCAPTCHA5 it is quite inexpensive to acquire labels from a large number of annotators (possibly thousands) in a short time (Sheng et al., 2008; Snow et al., 2008; Sorokin and Forsyth, 2008). Websites such as Galaxy Zoo6 allow the public to label astronomical images over the internet. In situations like these, the performance of different annotators can vary widely (some may even be malicious), and without the actual gold standard, it may not be possible to evaluate the annotators. In this work, we provide principled probabilistic solutions to the following questions: 1. How to adapt conventional supervised learning algorithms when we have multiple annotators providing subjective labels but no objective gold standard? 2. How to evaluate systems when we do not have absolute gold-standard? 3. A closely related problem—particularly relevant when there are a large number of annotators— is to estimate how reliable/trustworthy is each annotator. 1. See Fung et al. (2009) for an overview of the data mining issues in this area. 2. Sometimes even a biopsy cannot conﬁrm whether it is cancer or not and hence all we can hope to get is subjective ground truth. 3. Mechanical Turk found at https://www.mturk.com. 4. Games with a Purpose found at http://www.gwap.com. 5. reCAPTCHA found at http://recaptcha.net/. 6. Galaxy Zoo found at http://galaxyzoo.org. 1298 L EARNING F ROM C ROWDS 1.1 The Problem With Majority Voting When we have multiple labels a commonly used strategy is to use the labels on which the majority of them agree (or average for regression problem) as an estimate of the actual gold standard. For binary classiﬁcation problems this amounts to using the majority label,7 that is, j yi = ˆ 1 if (1/R) ∑R yi > 0.5 j=1 , j 0 if (1/R) ∑R yi < 0.5 j=1 as an estimate of the hidden true label and use this estimate to learn and evaluate classiﬁers/annotators. Another strategy is that of considering every pair (instance, label) provided by each expert as a separate example. Note that this amounts to using a soft probabilistic estimate of the actual ground truth to learn the classiﬁer, that is, R Pr[yi = 1|y1 , . . . , yR ] = (1/R) ∑ yi . i i j j=1 Majority voting assumes all experts are equally good. However, for example, if there is only one true expert and the majority are novices, and if novices give the same incorrect label to a speciﬁc instance, then the majority voting method would favor the novices since they are in a majority. One could address this problem by introducing a weight capturing how good each expert is. But how would one measure the performance of an expert when there is no gold standard available? 1.2 Proposed Approach and Organization To address the apparent chicken-and-egg problem, we present a maximum-likelihood estimator that jointly learns the classiﬁer/regressor, the annotator accuracy, and the actual true label. For ease of exposition we start with binary classiﬁcation problem in § 2. The performance of each annotator is measured in terms of the sensitivity and speciﬁcity with respect to the unknown gold standard (§ 2.1). The proposed algorithm automatically discovers the best experts and assigns a higher weight to them. In order to incorporate prior knowledge about each annotator, we impose a beta prior on the sensitivity and speciﬁcity and derive the maximum-a-posteriori estimate (§ 2.6). The ﬁnal estimation is performed by an Expectation Maximization (EM) algorithm that iteratively establishes a particular gold standard, measures the performance of the experts given that gold standard, and reﬁnes the gold standard based on the performance measures. While the proposed approach is described using logistic regression as the base classiﬁer (§ 2.2), it is quite general, and can be used with any black-box classiﬁer (§ 2.7), and can also handle missing labels (that is, each expert is not required to label all the instances). Furthermore, we extend the proposed algorithm to handle categorical (§ 3), ordinal (§ 4), and regression problems (§ 5). In § 6 section we extensively validate our approach using both simulated data and real data from different domains. 1.3 Related Work and Novel Contributions We ﬁrst summarize the novel contributions of this work in context of other related work in this emerging new area. There has been a long line of work in the biostatistics and epidemiology literature on latent variable models where the task is to get an estimate of the observer error rates based 7. When there is no clear majority among the multiple experts (that is, yi = 0.5) in CAD domain the ﬁnal decision is ˆ often made by an adjudicator or a super-expert. When there is no adjudicator a fair coin toss is used. 1299 R AYKAR , Y U , Z HAO , VALADEZ , F LORIN , B OGONI AND M OY on the results from multiple diagnostic tests without a gold standard (see Dawid and Skene, 1979, Hui and Walter, 1980, Hui and Zhou, 1998, Albert and Dodd, 2004 and references therein). In the machine learning community Smyth et al. (1995) ﬁrst addressed the same problem in the context of labeling volcanoes in satellite images of Venus. We differ from this previous body of work in the following aspects: 1. Unlike Dawid and Skene (1979) and Smyth et al. (1995) which just focused on estimating the ground truth from multiple noisy labels, we speciﬁcally address the issue of learning a classiﬁer. Estimating the ground truth and the annotator/classifer performance is a byproduct of our proposed algorithm. 2. In order to learn a classiﬁer Smyth (1995) proposed to ﬁrst estimate the ground truth (without using the features) and then use the probabilistic ground truth to learn a classiﬁer. In contrast, our proposed algorithm learns the classiﬁer and the ground truth jointly. Our experiments (§ 6.1.1) show that the classiﬁer learnt and ground truth obtained by the proposed algorithm is superior to that obtained by other procedures which ﬁrst estimates the ground truth and then learns the classiﬁer. 3. Our solution is more general and can be easily extended to categorical(§ 3), ordinal(§ 4), and continuous data(§ 5). It can also be used in conjunction with any supervised learning algorithm. A preliminary version of this paper (Raykar et al., 2009) mainly discussed the binary classiﬁcation problem. 4. Our proposed algorithm is also Bayesian—we impose a prior on the experts. The priors can potential capture the skill of different annotators. In this paper we refrain from doing a full Bayesian inference and use the mode of the posterior as a point estimate. A recent complete Bayesian generalization of these kind of models has been developed by Carpenter (2008). 5. The EM approach used in this paper is similar to that proposed by Jin and Ghahramani (2003). However their motivation is somewhat different. In their setting, each training example is annotated with a set of possible labels, only one of which is correct. There has been recent interest in the natural language processing (Sheng et al., 2008; Snow et al., 2008) and computer vision (Sorokin and Forsyth, 2008) communities where they use Amazon’s Mechanical Turk to collect annotations from many people. They show that it can be potentially as good as that provided by an expert. Sheng et al. (2008) analyzed when it is worthwhile to acquire new labels for some of the training examples. There is also some theoretical work (see Lugosi, 1992 and Dekel and Shamir, 2009a) dealing with multiple experts. Recently Dekel and Shamir (2009b) presented an algorithm which does not resort to repeated labeling, that is, each example does not have to be labeled by multiple teachers. Donmez et al. (2009) address the issue of active learning in this scenario—How to jointly learn the accuracy of labeling sources and obtain the most informative labels for the active learning task? There has also been some work in the medical imaging community (Warﬁeld et al., 2004; Cholleti et al., 2008). 2. Binary Classiﬁcation We ﬁrst describe our proposed noise model for the annotators. The performance of each annotator is measured in terms of the sensitivity and speciﬁcity with respect to the unknown gold standard. 1300 L EARNING F ROM C ROWDS 2.1 A Two-coin Model for Annotators Let y j ∈ {0, 1} be the label assigned to the instance x by the jth annotator/expert. Let y be the actual (unobserved) label for this instance. Each annotator provides a version of this hidden true label based on two biased coins. If the true label is one, she ﬂips a coin with bias α j (sensitivity). If the true label is zero, she ﬂips a coin with bias β j (speciﬁcity). In each case, if she gets heads she keeps the original label, otherwise she ﬂips the label. If the true label is one, the sensitivity (true positive rate) for the jth annotator is deﬁned as the probability that she labels it as one. α j := Pr[y j = 1|y = 1]. (1) On the other hand, if the true label is zero, the speciﬁcity (1−false positive rate) is deﬁned as the probability that she labels it as zero. β j := Pr[y j = 0|y = 0]. (2) The assumption introduced is that α j and β j do not depend on the instance x. For example, in the CAD domain, this means that the radiologist’s performance is consistent across different sub-groups of data.8 2.2 Classiﬁcation Model While the proposed method can be used for any classiﬁer, for ease of exposition, we consider the family of linear discriminating functions: F = { fw}, where for any x, w ∈ Rd , fw(x) = w⊤ x. The ﬁnal classiﬁer can be written in the following form: y = 1 if w⊤ x ≥ γ and 0 otherwise. The ˆ threshold γ determines the operating point of the classiﬁer. The Receiver Operating Characteristic (ROC) curve is obtained as γ is swept from −∞ to ∞. The probability for the positive class is modeled as a logistic sigmoid acting on fw, that is, Pr[y = 1|x, w] = σ(w⊤ x), where the logistic sigmoid function is deﬁned as σ(z) = 1/(1 + e−z ). This classiﬁcation model is known as logistic regression. 2.3 Estimation/Learning Problem Given the training data D consisting of N instances with annotations from R annotators, that is, D = {xi , y1 , . . . , yR }N , the task is to estimate the weight vector w and also the sensitivity α = i i i=1 [α1 , . . . , αR ] and the speciﬁcity β = [β1 , . . . , βR ] of the R annotators. It is also of interest to get an estimate of the unknown gold standard y1 , . . . , yN . 2.4 Maximum Likelihood Estimator Assuming the training instances are independently sampled, the likelihood function of the parameters θ = {w, α, β} given the observations D can be factored as N Pr[D |θ] = ∏ Pr[y1 , . . . , yR |xi , θ]. i i i=1 8. While this is a reasonable assumption, it is not entirely true. It is known that some radiologists are good at detecting certain kinds of malignant lesions based on their training and experience. 1301 R AYKAR , Y U , Z HAO , VALADEZ , F LORIN , B OGONI AND M OY j Conditioning on the true label yi , and also using the assumption yi is conditionally independent (of everything else) given α j , β j and yi , the likelihood can be decomposed as N Pr[D |θ] = ∏ Pr[y1 , . . . , yR |yi = 1, α]Pr[yi = 1|xi, w] i i i=1 + Pr[y1 , . . . , yR |yi = 0, β]Pr[yi = 0|xi, w] . i i Given the true label yi , we assume that y1 , . . . , yR are independent, that is, the annotators make their i i decisions independently.9 Hence, R R Pr[y1 , . . . , yR |yi = 1, α] = ∏ Pr[yi |yi = 1, α j ] = ∏ [α j ]yi [1 − α j ]1−yi . i i j j=1 j j j=1 Similarly, we have R Pr[y1 , . . . , yR |yi = 0, β] = ∏ [β j ]1−yi [1 − β j ]yi . i i j j j=1 Hence the likelihood can be written as N Pr[D |θ] = ∏ ai pi + bi (1 − pi ) , i=1 where we have deﬁned pi := σ(w⊤ xi ). R ai := ∏[α j ]y [1 − α j ]1−y . j i j i j=1 R bi := ∏[β j ]1−y [1 − β j ]y . j i j i j=1 The maximum-likelihood estimator is found by maximizing the log-likelihood, that is, ˆ θML = {α, β, w} = arg max{ln Pr[D |θ]}. ˆ ˆ ˆ θ 2.5 The EM Algorithm This maximization problem can be simpliﬁed a lot if we use the Expectation-Maximization (EM) algorithm (Dempster et al., 1977). The EM algorithm is an efﬁcient iterative procedure to compute the maximum-likelihood solution in presence of missing/hidden data. We will use the unknown hidden true label yi as the missing data. If we know the missing data y = [y1 , . . . , yN ] then the complete likelihood can be written as N ln Pr[D , y|θ] = ∑ yi ln pi ai + (1 − yi ) ln(1 − pi )bi . i=1 9. This assumption is not true in general and there is some correlations among the labels assigned by multiple annotators. For example in the CAD domain if the cancer is in advanced stage (which is very easy to detect) almost all the radiologists assign the same label. 1302 L EARNING F ROM C ROWDS Each iteration of the EM algorithm consists of two steps: an Expectation(E)-step and a Maximization(M)step. The M-step involves maximization of a lower bound on the log-likelihood that is reﬁned in each iteration by the E-step. 1. E-step. Given the observation D and the current estimate of the model parameters θ, the conditional expectation (which is a lower bound on the true likelihood) is computed as N E {ln Pr[D , y|θ]} = ∑ µi ln pi ai + (1 − µi ) ln(1 − pi )bi , (3) i=1 where the expectation is with respect to Pr[y|D , θ], and µi = Pr[yi = 1|y1 , . . . , yR , xi , θ]. Using i i Bayes’ theorem we can compute µi ∝ Pr[y1 , . . . , yR |yi = 1, θ] · Pr[yi = 1|xi, θ] i i ai pi = . ai pi + bi (1 − pi ) 2. M-step. Based on the current estimate µi and the observations D , the model parameters θ are then estimated by maximizing the conditional expectation. By equating the gradient of (3) to zero we obtain the following estimates for the sensitivity and speciﬁcity: j j αj = ∑N µi yi i=1 , ∑N µi i=1 βj = ∑N (1 − µi )(1 − yi ) i=1 . ∑N (1 − µi ) i=1 Due to the non-linearity of the sigmoid, we do not have a closed form solution for w and we have to use gradient ascent based optimization methods. We use the Newton-Raphson update given by wt+1 = wt − ηH −1 g, where g is the gradient vector, H is the Hessian matrix, and η is the step length. The gradient vector is given by N g(w) = ∑ µi − σ(w⊤ xi ) xi . i=1 The Hessian matrix is given by N H(w) = − ∑ σ(w⊤ xi ) 1 − σ(w⊤ xi ) xi x⊤ . i i=1 Essentially, we are estimating a logistic regression model with probabilistic labels µi . These two steps (the E- and the M-step) can be iterated till convergence. The log-likelihood increases monotonically after every iteration, which in practice implies convergence to a local maximum. The EM algorithm is only guaranteed to converge to a local maximum. In practice multiple restarts with different initializations can potentially mitigate the local maximum problem. In this j paper we use majority voting µi = 1/R ∑R yi as the initialization for µi to start the EM-algorithm. j=1 1303 R AYKAR , Y U , Z HAO , VALADEZ , F LORIN , B OGONI AND M OY 2.6 A Bayesian Approach In some applications we may want to trust a particular expert more than the others. One way to achieve this is by imposing priors on the sensitivity and speciﬁcity of the experts. Since α j and β j represent the probability of a binary event, a natural choice of prior is the beta prior. The beta prior is also conjugate to the binomial distribution. For any a > 0, b > 0, and δ ∈ [0, 1] the beta distribution is given by δa−1 (1 − δ)b−1 , Beta(δ|a, b) = B(a, b) where B(a, b) = 01 δa−1 (1 − δ)b−1 dδ is the beta function.We assume a beta prior10 for both the sensitivity and the speciﬁcity as R j j j j j j j j Pr[α j |a1 , a2 ] = Beta(α j |a1 , a2 ). Pr[β j |b1 , b2 ] = Beta(β j |b1 , b2 ). For sake of completeness we also assume a zero mean Gaussian prior on the weights w with inverse covariance matrix Γ, that is, Pr[w] = N (w|0, Γ−1 ). Assuming that {α j }, {β j }, and w have independent priors, the maximum-a-posteriori (MAP) estimator is found by maximizing the logposterior, that is, ˆ θMAP = arg max{ln Pr[D |θ] + ln Pr[θ]}. θ An EM algorithm can be derived in a similar fashion for MAP estimation by relying on the interpretation of Neal and Hinton (1998). The ﬁnal algorithm is summarized below: j 1. Initialize µi = (1/R) ∑R yi based on majority voting. j=1 2. Given µi , estimate the sensitivity and speciﬁcity of each annotator/expert as follows. j αj = j a1 − 1 + ∑N µi yi i=1 j j a1 + a2 − 2 + ∑N µi i=1 j β j = . j b1 − 1 + ∑N (1 − µi )(1 − yi ) i=1 j j b1 + b2 − 2 + ∑N (1 − µi ) i=1 . (4) The Newton-Raphson update for optimizing w is given by wt+1 = wt − ηH −1 g, with step length η, gradient vector N g(w) = ∑ µi − σ(w⊤ xi ) xi − Γw, i=1 and Hessian matrix N H(w) = − ∑ σ(w⊤ xi ) 1 − σ(w⊤ xi ) xi x⊤ − Γ. i i=1 10. It may be convenient to specify a prior in terms of the mean µ and variance σ2 . The mean and the variance for a beta prior are given by µ = a/(a + b) and σ2 = ab/((a + b)2 (a + b + 1)). Solving for a and b we get a = (−µ3 + µ2 − µσ2 )/σ2 and b = a(1 − µ)/µ. 1304 L EARNING F ROM C ROWDS 3. Given the sensitivity and speciﬁcity of each annotator and the model parameters, update µi as µi = ai pi , ai pi + bi (1 − pi ) (5) where pi = σ(w⊤ xi ). R ai = ∏[α j ]y [1 − α j ]1−y . j i j i j=1 R bi = ∏[β j ]1−y [1 − β j ]y . j i j i (6) j=1 Iterate (2) and (3) till convergence. 2.7 Discussions 1. Estimate of the gold standard The value of the posterior probability µi is a soft probabilistic estimate of the actual ground truth yi , that is, µi = Pr[yi = 1|y1 , . . . , yR , xi , θ]. The actual i i hidden label yi can be estimated by applying a threshold on µi , that is, yi = 1 if µi ≥ γ and zero otherwise. We can use γ = 0.5 as the threshold. By varying γ we can change the misclassiﬁcation costs and obtain a ground truth with large sensitivity or large speciﬁcity. Because of this in our experimental validation we can actually draw an ROC curve for the estimated ground truth. 2. Log-odds of µ A particularly revealing insight can be obtained in terms of the log-odds or the logit of the posterior probability µi . From (5) the logit of µi can be written as logit(µi ) = ln Pr[yi = 1|y1 , . . . , yR , xi , θ] µi i i = ln 1 − µi Pr[yi = 0|y1 , . . . , yR , xi , θ] i i R = w⊤ xi + c + ∑ yi [logit(α j ) + logit(β j )]. j j=1 j where c = ∑R log 1−α is a constant term which does not depend on i. This indicates that j=1 βj the estimated ground truth (in the logit form of the posterior probability) is a weighted linear combination of the labels from all the experts. The weight of each expert is the sum of the logit of the sensitivity and speciﬁcity. 3. Using any other classiﬁer For ease of exposition we used logistic regression. However, the proposed algorithm can be used with any generalized linear model or in fact with any classiﬁer that can be trained with soft probabilistic labels. In each step of the EM-algorithm, the classiﬁer is trained with instances sampled from µi . This modiﬁcation is easy for most probabilistic classiﬁers. For general black-box classiﬁers where we cannot tweak the training algorithm an alternate approach is to replicate the training examples according to the soft label. For example a probabilistic label µi = 0.8 can be effectively simulated by adding 8 training examples with deterministic label 1 and 2 examples with label 0. 1305 R AYKAR , Y U , Z HAO , VALADEZ , F LORIN , B OGONI AND M OY 4. Obtaining ground truth with no features In some scenarios we may not have features xi and we wish to obtain an estimate of the actual ground truth based only on the labels from multiple annotators. Here instead of learning a classiﬁer we estimate p which is the prevalence of the positive class, that is, p = Pr[yi = 1]. We further assume a beta prior for the prevalence, that is, Beta(p|p1 , p2 ). The algorithm simpliﬁes as follows. j (a) Initialize µi = (1/R) ∑R yi based on majority voting. j=1 (b) Given µi , estimate the sensitivity and speciﬁcity of each annotator using (4). The prevalence of the positive class is estimated as follows. p = p1 − 1 + ∑N µi i=1 . p1 + p2 − 2 + N (c) Given the sensitivity and speciﬁcity of each annotator and prevalence, reﬁne µi as follows. ai p µi = . ai p + bi (1 − p) Iterate (2) and (3) till convergence. This algorithm is similar to the one proposed by Dawid and Skene (1979) and Smyth et al. (1995). 5. Handling missing labels The proposed approach can easily handle missing labels, that is, when the labels from some experts are missing for some instances. Let Ri be the number of radiologists labeling the ith instance, and let N j be the number of instances labeled by the jth radiologist. Then in the EM algorithm, we just need to replace N by N j for estimating the sensitivity and speciﬁcity in (4), and replace R by Ri for updating µi in (6). 6. Evaluating a classiﬁer We can use the probability scores µi directly to evaluate classiﬁers. If zi are the labels obtained from any other classiﬁer, then sensitivity and speciﬁcity can be estimated as ∑N (1 − µi )(1 − zi ) ∑N µi zi , β = i=1 N . α = i=1 ∑N µi ∑i=1 (1 − µi ) i=1 7. Posterior approximation At the end of each EM iteration a crude approximation to the posterior is obtained as N N α j ∼ Beta α j |a1 + ∑ µi yi , a2 + ∑ µi (1 − yi ) , j j i=1 N j j i=1 N β j ∼ Beta β j |b1 + ∑ (1 − µi )(1 − yi ), b2 + ∑ (1 − µi )yi j j i=1 j j . i=1 3. Multi-class Classiﬁcation In this section we describe how the proposed approach for binary classiﬁcation can be extended to categorical data. Suppose there are K ≥ 2 categories. An example for categorical data from the CAD domain is in LungCAD, where the radiologist needs to label whether a nodule (known to be precursors of cancer) is a solid, a part-solid, or a ground glass opacity—which are three 1306 L EARNING F ROM C ROWDS different kinds on nodules. We can extend the previous model and introduce a vector of multinomial j j j parameters αc = (αc1 , . . . , αcK ) for each annotator, where αck := Pr[y j = k|y = c] j and ∑K αck = 1. Here αck denotes the probability that the annotator j assigns class k to an instance k=1 j j given the true class is c. When K = 2, α11 and α00 are sensitivity and speciﬁcity, respectively. A similar EM algorithm can be derived. In the E-step, we estimate j j R K Pr[yi = c|Y , α] ∝ Pr[yi = c|xi ] ∏ ∏ (αck )δ(yi ,k) , j j j=1 k=1 where δ(u, v) = 1 if u = v and 0 otherwise and in the M-step we learn a multi-class classiﬁer and update the multinomial parameter as j αck = j ∑N Pr[yi = c|Y , α]δ(yi , k) i=1 . ∑N Pr[yi = c|Y , α] i=1 One can also assign a Dirichlet prior for the multinomial parameters, and this results in a smoothing term in the above updates in the MAP estimate. 4. Ordinal Regression We now consider the situation where the outputs are categorical and have an ordering among the labels. In the CAD domain the radiologist often gives a score (for example, 1 to 5 from lowest to highest) to indicate how likely she thinks it is malignant. This is different from a multi-class setting in which we do not have any preference among the multiple class labels. j Let yi ∈ {1, . . . , K} be the label assigned to the ith instance by the jth expert. Note that there is an ordering in the labels 1 < . . . < K. A simple approach is to convert the ordinal data into a series of binary data (Frank and Hall, 2001). Speciﬁcally the K class ordinal labels are transformed into K − 1 binary class labels as follows: jc yi = j 1 if yi > c 0 otherwise c = 1, . . . , K − 1. Applying the same procedure used for binary labels we can estimate Pr[yi > c] for c = 1, . . . , K − 1. The probability of the actual class values can then be obtained as Pr[yi = c] = Pr[yi > c − 1 and yi ≤ c] = Pr[yi > c − 1] − Pr[yi > c]. The class with the maximum probability is assigned to the instance. 5. Regression In this section we develop a similar algorithm to learn a regression function using annotations from multiple experts. In the CAD domain as a part of the annotation process a common task for a radiologist is to measure the diameter of a suspicious lesion. 1307 R AYKAR , Y U , Z HAO , VALADEZ , F LORIN , B OGONI AND M OY 5.1 Model for Annotators j Let yi ∈ R be the continuous target value assigned to the ith instance by the jth annotator. Our model is that the annotator provides a noisy version of the actual true value yi . For the jth annotator we will assume a Gaussian noise model with mean yi (the true unknown value) and inverse-variance (precision) τ j , that is, j j Pr[yi |yi , τ j ] = N (yi |yi , 1/τ j ), (7) where the Gaussian distribution is deﬁned as N (z|m, σ2 ) = (2πσ2 )−1/2 exp(−(z − m)2 /2σ2 ). The unknown inverse-variance τ j measures the accuracy of each annotator—the larger the value of τ j the more accurate the annotator. We have assumed that τ j does not depend on the instance xi . For example, in the CAD domain, this means that the radiologist’s accuracy does not depend on the nodule she is measuring. While this a practical assumption, it is not entirely true. It is known that some nodules are harder to measure than others. 5.2 Linear Regression Model for Features As before we consider the family of linear regression functions: F = { fw}, where for any x, w ∈ Rd , fw(x) = w⊤ x. We assume that the actual target response yi is given by the deterministic regression function fw with additive Gaussian noise, that is, yi = w⊤ xi + ε, where ε is a zero-mean Gaussian random variable with inverse-variance (precision) γ. Hence Pr[yi |xi , w, γ] = N (yi |w⊤ xi , 1/γ). (8) 5.3 Combined Model Combining both the annotator (7) and the regressor (8) model we have Pr[yi |xi , w, τ j , γ] = j Z Pr[yi |yi , τ j ]Pr[yi |xi , w, γ]dyi = N (yi |w⊤ xi , 1/γ + 1/τ j ). j j Since the two precision terms (γ and τ j ) are grouped together they are not uniquely identiﬁable. Hence we will deﬁne a new precision term λ j as 1 1 1 = + j. j λ γ τ So we have the following model Pr[yi |xi , w, λ j ] = N (yi |w⊤ xi , 1/λ j ). j j (9) 5.4 Estimation/Learning Problem Given the training data D consisting of N instances with annotations from R experts, that is, D = {xi , y1 , . . . , yR }N , the task is to estimate the weight vector w and the precision λ = [λ1 , . . . , λR ] of i i=1 i all the annotators. 1308 L EARNING F ROM C ROWDS 5.5 Maximum-likelihood Estimator Assuming the instances are independent the likelihood of the parameters θ = {w, λ} given the observations D can be factored as N Pr[D |θ] = ∏ Pr[y1 , . . . , yR |xi , θ]. i i i=1 Conditional on the instance xi we assume that y1 , . . . , yR are independent, that is, the annotators i i provide their responses independently. Hence from (9) the likelihood can be written as N R Pr[D |θ] = ∏ ∏ N (yi |w⊤ xi , 1/λ j ). j i=1 j=1 The maximum-likelihood estimator is found by maximizing the log-likelihood θML = {λ, w} = arg max{ln Pr[D |θ]}. θ By equating the gradient of the log-likelihood to zero we obtain the following update equations for the precision and the weight vector. 1 λj = 1 N ∑ yij − w⊤ xi N i=1 N w = ∑ −1 2 . (10) N ∑R λ j yi j=1 i=1 ∑R λ j j=1 ∑ xi xi x⊤ i i=1 j . (11) As the parameters w and λ are coupled together we iterate these two steps till convergence. 5.6 Discussions 1. Is this standard least-squares? Deﬁne the design matrix X = [x1 , . . . , xN ]⊤ and the rej j sponse vector for each annotator as y j = [y1 , . . . , yN ]⊤ . Using matrix notation Equation 11 can be written as w = (X ⊤ X)−1 X ⊤ y where y = ∑R λ j y j j=1 ∑R λ j j=1 . (12) Equation 12 is essentially the solution to a standard linear regression model, except that we are training a linear regression model with y as the ground truth, which is a precision weighted mean of the response vectors from all the annotators. The variance of each annotator is estimated using (10). The ﬁnal algorithm iteratively establishes a particular gold standard (y), measures the performance of the annotators and learns a regressor given that gold standard, and reﬁnes the gold standard based on the performance measures. 2. Are we better than the best annotator? If we assume λ is ﬁxed (i.e., we ignore the variability and assume that it is well estimated) then w is an unbiased estimator of w and the ˆ covariance matrix is given by Cov(w) = Cov(y) X ⊤ X ˆ 1309 −1 = 1 ∑R λ j j=1 X ⊤X −1 . R AYKAR , Y U , Z HAO , VALADEZ , F LORIN , B OGONI AND M OY Since ∑R λ j > max j (λ j ) the proposed method has a lower variance than the regressor learnt j=1 with the best annotator (i.e., the one with the minimum variance). 3. Are we better than the average? For a ﬁxed X the error in w depends only on the variance of y j . If we know the true λ j then yi is the best linear unbiased estimator for yi which minij mizes the variance. To see this consider any linear estimator of the form yi = ∑ j a j (yi − b j ). The variance is given by Var[yi ] = ∑ j (a j )2 /λ j . Since E[yi ] = yi ∑ j a j , for the bias of this estimator to be zero we require that ∑ j a j = 1. Solving the constrained minimization problem we see that a j = λ j / ∑ j λ j minimizes the variance. 4. Obtaining a consensus without features When no features are available the same algorithm can be simpliﬁed to get a consensus estimate of the actual ground truth and also evaluate the annotators. Essentially we have to iterate the following two updates till convergence ∑R λ j yi j=1 1 ∑R λ j j=1 λj j yi = = 1 N ∑ y j − yi N i=1 i 2 . 6. Experimental Validation We now experimentally validate the proposed algorithms on both simulated and real data. 6.1 Classiﬁcation Experiments We use two CAD and one text data set in our experiments. The CAD data sets include a digital mammography data set and a breast MRI data set, both of which are biopsy proven, that is, the gold standard is available. For the digital mammography data set we simulate the radiologists in order to validate our methods. The breast MRI data has annotations from four radiologists. We also report results on a Recognizing Textual Entailment data collected by Snow et al. (2008) using the Amazon’s Mechanical Turk which has annotations from 164 annotators. 6.1.1 D IGITAL M AMMOGRAPHY WITH S IMULATED R ADIOLOGISTS Mammograms are used as a screening tool to detect early breast cancer. CAD systems search for abnormal areas (lesions) in a digitized mammographic image. These lesions generally indicate the presence of malignant cancer. The CAD system then highlights these areas on the images, alerting the radiologist to the need for a further diagnostic mammogram or a biopsy. In classiﬁcation terms, given a set of descriptive morphological features for a region on a image, the task is to predict whether it is potentially malignant (1) or not (0). In order to train such a classiﬁer, a set of mammograms is collected from hospitals. The ground truth (whether it is cancer or not) is obtained from biopsy. Since biopsy is an expensive, tedious, and an invasive process, very often CAD systems are built from labels collected from multiple expert radiologists who visually examine the mammograms and mark the lesion locations—this constitutes our ground truth (multiple labels) for learning. In this experiment we use a proprietary biopsy-proven data set (Krishnapuram et al., 2008) containing 497 positive and 1618 negative examples. Each instance is described by a set of 27 morphological features. In order to validate our proposed algorithm, we simulate multiple radiologists according to the two-coin model described in § 2.1. Based on the labels from multiple radiologists, 1310 L EARNING F ROM C ROWDS we can simultaneously (1) learn a logistic-regression classiﬁer, (2) estimate the sensitivity and speciﬁcity of each radiologist, and (3) estimate the golden ground truth. We compare the results with the classiﬁer trained using the biopsy proved ground truth as well as the majority-voting baseline. For the ﬁrst set of experiments we use 5 radiologists with sensitivity α = [0.90 0.80 0.57 0.60 0.55] and speciﬁcity β = [0.95 0.85 0.62 0.65 0.58]. This corresponds to a scenario where the ﬁrst two radiologists are experts and the last three are novices. Figure 1 summarizes the results. We compare on three different aspects: (1) How good is the learnt classiﬁer? (2) How well can we estimate the sensitivity and speciﬁcity of each radiologist? (3) How good is the estimated ground truth? The following observations can be made. 1. Classiﬁer performance Figure 1(a) plots the ROC curve of the learnt classiﬁer on the training set. The dotted (black) line is the ROC curve for the classiﬁer learnt using the actual ground truth. The solid (red) line is the ROC curve for the proposed algorithm and the dashed (blue) line is for the classiﬁer learnt using the majority-voting scheme. The classiﬁer learnt using the proposed method is as good as the one learnt using the golden ground truth. The area under the ROC curve (AUC) for the proposed algorithm is around 3.5% greater than that learnt using the majority-voting scheme. 2. Radiologist performance The actual sensitivity and speciﬁcity of each radiologist is marked as a black × in Figure 1(b). The end of the solid red line shows the estimates of the sensitivity and speciﬁcity from the proposed method. We used a uniform prior on all the parameters. The ellipse plots the contour of one standard deviation as obtained from the beta posterior estimates. The end of the dashed blue line shows the estimate obtained from the majorityvoting algorithm. We see that the proposed method is much closer to the actual values of sensitivity and speciﬁcity. 3. Actual ground truth Since the estimates of the actual ground truth are probabilistic scores, we can also plot the ROC curves of the estimated ground truth. From Figure 1(b) we can see that the ROC curve for the proposed method dominates the majority voting ROC curve. Furthermore, the area under the ROC curve (AUC) is around 3% higher. The estimate obtained by majority voting is closer to the novices since they form a majority (3/5). It does not have an idea of who is an expert and who is a novice. The proposed algorithm appropriately weights each radiologist based on their estimated sensitivity and speciﬁcity. The improvement obtained is quite large in Figure 2 which corresponds a situation where we have only one expert and 7 novices. 4. Joint Estimation To learn a classiﬁer, Smyth et al. (1995) proposed to ﬁrst estimate the golden ground truth and then use the probabilistic ground truth to learn a classiﬁer. In contrast, our proposed algorithm learns the classiﬁer and the ground truth jointly as a part of the EM algorithm. Figure 3 shows that the classiﬁer and the ground truth learnt obtained by the proposed algorithm is superior than that obtained by other procedures which ﬁrst estimates the ground truth and then learns the classiﬁer. 6.1.2 B REAST MRI In this example, each radiologist reviews the breast MRI data and assesses the malignancy of each lesion on a BIRADS scale of 1 to 5. The BIRADS scale is deﬁned as follows: 1 Negative, 2 Benign, 1311 R AYKAR , Y U , Z HAO , VALADEZ , F LORIN , B OGONI AND M OY Majority Voting Estimated 1 Estimated 2 Estimated 3 Estimated 4 Estimated 5 EM algorithm Estimated 1 Estimated 2 Estimated 3 Estimated 4 Estimated 5 True 1 True 2 True 3 True 4 True 5 x x x x x 0.0217 0.5869 0.2391 0.1521 0.0000 0 0 0 1 0 x x x x x 0.0000 0.1785 0.1071 0.2500 0.4642 True 1 True 2 True 3 True 4 True 5 x x x x x 0.0000 0.6957 0.1304 0.1739 0.0000 0 0 0 1 0 x x x x x 0.0000 0.1428 0.0000 0.3214 0.5357 Table 1: The confusion matrix for the estimate obtained using majority voting and the proposed EM algorithm. The x indicates that there was no such category in the true labels (the gold standard). The gold-standard is obtained by the biopsy which can conﬁrm whether it is benign (BIRADS=2) or malignant (BIRADS=5). 3 Probably Benign, 4 Suspicious abnormality, and 5 Highly suggestive of malignancy. Our data set comprises of 75 lesions with annotations from four radiologists, and the true labels from biopsy. Based on eight morphological features, we have to predict whether a lesion is malignant or not. For the ﬁrst experiment we reduce the BIRADS scale to a binary one: any lesion with a BIRADS > 3 is considered malignant and benign otherwise. The set included 28 malignant and 47 benign lesions. Figure 4 summarizes the results. We show the leave-one-out cross validated ROC for the classiﬁer. The cross-validated AUC of the proposed method is approximately 6% better than the majority voting baseline. We also consider the BIRADS labels as a set of ordinal measurements since there is an ordering among the BIRADS label. The confusion matrix in Table 1 shows that the EM algorithm is signiﬁcantly superior than the majority voting in estimating the true BIRADS. 6.1.3 R ECOGNIZING T EXTUAL E NTAILMENT Finally we report results on Recognizing Textual Entailment data collected by Snow et al. (2008) using the Amazon’s Mechanical Turk. In this task, the annotator is presented with two sentences and given a choice of whether the second sentence can be inferred from the ﬁrst. The data has 800 tasks and 164 distinct readers, with 10 annotations per task along with the golden ground truth. The majority of the entries (94 %) in the 800x164 matrix are missing. There is one annotator who has labeled all the tasks. We use this data set to obtain an estimate of the actual ground truth. Figure 5 plots the accuracy of the estimated ground truth as a function of the number of annotators. The proposed EM algorithm achieves a higher accuracy than majority voting. In other words to achieve a desired accuracy the proposed algorithm needs fewer annotators than the majority voting scheme. 1312 L EARNING F ROM C ROWDS ROC Curve for the classifier 1 True Positive Rate (sensitivity) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 Golden ground truth AUC=0.915 Proposed EM algorithm AUC=0.913 Majority voting baseline AUC=0.882 0.1 0 0 0.2 0.4 0.6 False Positive Rate (1−specifcity) 0.8 1 (a) ROC Curve for the estimated true labels 1 True Positive Rate (sensitivity) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 Proposed EM algorithm AUC=0.991 Majority voting baseline AUC=0.962 0.2 0.1 0 0 0.2 0.4 0.6 False Positive Rate (1−specifcity) 0.8 1 (b) Figure 1: Results for the digital mammography data set with annotations from 5 simulated radiologists. (a) The ROC curve of the learnt classiﬁer using the golden ground truth (dotted black line), the majority voting scheme (dashed blue line), and the proposed EM algorithm (solid red line). (b) The ROC curve for the estimated ground truth. The actual sensitivity and speciﬁcity of each of the radiologists is marked as a ×. The end of the dashed blue line shows the estimates of the sensitivity and speciﬁcity obtained from the majority voting algorithm. The end of the solid red line shows the estimates from the proposed method. The ellipse plots the contour of one standard deviation. 1313 R AYKAR , Y U , Z HAO , VALADEZ , F LORIN , B OGONI AND M OY ROC Curve for the classifier 1 True Positive Rate (sensitivity) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 Golden ground truth AUC=0.915 Proposed EM algorithm AUC=0.906 Majority voting baseline AUC=0.884 0.1 0 0 0.2 0.4 0.6 False Positive Rate (1−specifcity) 0.8 1 (a) ROC Curve for the estimated true labels 1 True Positive Rate (sensitivity) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 Proposed EM algorithm AUC=0.967 Majority voting baseline AUC=0.872 0.2 0.4 0.6 False Positive Rate (1−specifcity) 0.8 1 (b) Figure 2: Same as Figure 1 except with 8 different radiologist annotations. 1314 L EARNING F ROM C ROWDS ROC Curve for the classifier 1 True Positive Rate (sensitivity) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 Proposed EM algorithm [Joint Estimation] AUC=0.905 Decoupled Estimation AUC=0.884 0.2 0.4 0.6 False Positive Rate (1−specifcity) 0.8 1 (a) ROC Curve for the estimated true labels 1 True Positive Rate (sensitivity) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 Proposed EM algorithm [Joint Estimation] AUC=0.972 Decoupled Estimation AUC=0.921 0.2 0.4 0.6 False Positive Rate (1−specifcity) 0.8 1 (b) Figure 3: ROC curves comparing the proposed algorithm (solid red line) with the Decoupled Estimation procedure (dotted blue line), which refers to the algorithm where the ground truth is ﬁrst estimated using just the labels from the ﬁve radiologists and then a logistic regression classiﬁer is trained using the soft probabilistic labels. In contrast the proposed EM algorithm estimates the ground truth and learns the classiﬁer simultaneously during the EM algorithm. 1315 R AYKAR , Y U , Z HAO , VALADEZ , F LORIN , B OGONI AND M OY Leave−One−Out ROC Curve for the classifier 1 True Positive Rate (sensitivity) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 Golden ground truth AUC=0.909 Majority voting baseline AUC=0.828 Proposed EM algorithm AUC=0.879 0.2 0.1 0 0 0.2 0.4 0.6 False Positive Rate (1−specifcity) 0.8 1 (a) ROC Curve for the estimated true labels 1 0.9 True Positive Rate (sensitivity) 0.8 0.7 0.6 0.5 0.4 Proposed EM algorithm AUC=0.944 Majority voting baseline AUC=0.937 0.3 0.2 0.1 0 0 0.2 0.4 0.6 False Positive Rate (1−specifcity) 0.8 1 (b) Figure 4: Breast MRI results. (a) The leave-one-out cross validated ROC. (b) ROC for the estimated ground truth. 1316 L EARNING F ROM C ROWDS 0.95 0.9 0.85 Accuracy 0.8 0.75 0.7 0.65 Majority Voting EM Algorithm 0.6 0.55 0.5 20 40 60 80 100 120 Number of Annotators 140 160 Figure 5: The mean and the one standard deviation error bars for the accuracy of the estimated ground truth for the Recognizing Textual Entailment task as a function of the number of annotators. The plot was generated by randomly sampling the annotators 100 times. 6.2 Regression Experiments We ﬁrst illustrate the algorithm on a toy dataset and then present a case study for automated polyp measurements. 6.2.1 I LLUSTRATION Figure 6 illustrates the the proposed algorithm for regression on a one-dimensional toy data set with three annotators. The actual regression model (shown as a blue dotted line) is given by y = 5x − 2. We simulate 20 samples from three annotators with precisions 0.01, 0.1, and 1.0. The data are shown by the annotators’s number. While we can ﬁt a regression model using each annotators’s response, we see that only the model for annotator three (with highest precision) is close to the true regression model. The green dashed line shows the model learnt using the average response from all the three annotators. The red line shows the model learnt by the proposed algorithm. 6.2.2 AUTOMATED P OLYP M EASUREMENTS Colorectal polyps are small colonic ﬁndings that may develop into cancer at a later stage. The diameter of the polyp is one of the key factors which decides the malignancy of a suspicious polyp. Hence accurate size estimation is crucial to decide the action to be taken on a polyp. We have developed various algorithms to segment a polyp. Multiple segmentation algorithms give rise to a set of features which are correlated with the diameter of the polyp. We want to learn a regression function which can predict the diameter of a polyp as a function of these features. In order to learn a regression function we collect our ground truth by asking many radiologists to manually measure the the diameter of the polyps from the three-dimensional images. In practice there is a lot of disagreement among the radiologists as to the actual size of the polyp. 1317 R AYKAR , Y U , Z HAO , VALADEZ , F LORIN , B OGONI AND M OY 2 1 N=50 examples R=3 annotators 2 2 5 Actual regression model Model with annotator 1 (Precision=0.01) 2 1 Model with annotator 2 (Precision=0.10) 2 Model with annotator 3 (Precision=1.00) 3 2 3 2 Model with average response 2 Proposed algorithm 2 3 1 1 4 3 2 2 y (response) 2 2 1 3 2 3 3 2 3 1 −1 3 3 2 2 3 3 1 2 2 2 3 33 3 1 1 3 3 3 3 2 3 3 2 33 33 2 2 2 3 3 3 32 1 2 3 2 1 3 3 2 1 2 1 2 2 1 1 2 3 2 3 2 3 1 3 3 −3 2 3 3 2 3 3 3 2 3 3 2 1 3 3 2 0 −2 2 2 3 2 3 2 3 1 2 2 1 2 3 −4 1 2 2 1 −5 0 2 0.1 0.2 1 0.3 0.4 0.5 0.6 10.8 2 0.7 x (feature) 0.9 1 1 Figure 6: Illustration of the proposed algorithm on a one-dimensional toy data set. The actual regression model (shown as a blue dotted line) is given by y = 5x − 2. We simulate 50 samples from three annotators with precisions 0.01, 0.1, and 1.0. The data are shown by the annotators’s number. While we can ﬁt a regression model using each annotators’s response, we see that only the model for annotator three (with highest precision) is close to the true regression model. The green dashed line shows the model learnt using the average response from all the three annotators. The red line shows the model learnt by the proposed algorithm. 5 5 10 Actual Polyp Diameter (mm) (a) Gold standard model 15 10 5 0 0 Pearson Correlation Coefficient=0.554966 RMSE=2.887740 15 Estimated Polyp Diameter (mm) 10 0 0 Pearson Correlation Coefficient=0.706558 RMSE=1.991576 15 Estimated Polyp Diameter (mm) Estimated Polyp Diameter (mm) Pearson Correlation Coefficient=0.714720 RMSE=1.970815 15 5 10 Actual Polyp Diameter (mm) (b) Proposed model 15 10 5 0 0 5 10 Actual Polyp Diameter (mm) 15 (c) Average model Figure 7: Scatter plot of the actual polyp diameter vs the diameter predicted by the models learnt using (a) the actual gold standard, (b) the proposed algorithm with annotations from ﬁve radiologists, and (c) the average of the radiologist’s annotations. (See § 6.2.2 for a description of the experimental setup.) 1318 L EARNING F ROM C ROWDS We use a proprietary data set containing 393 examples (which point to 285 distinct polyps— the segmentation algorithms generally return multiple marks on the same polyp.) along with the measured diameter (ranging from 2mm to 15mm) as our training set. Each example is described by a set of 60 morphological features which are correlated to the diameter of the polyp. In order to validate the feasibility of our proposed algorithm, we simulate ﬁve radiologists according to the noisy model described in § 5.1 with τ = [0.001 0.01 0.1 1 10]. This corresponds to a situation where the ﬁrst three radiologists are extremely noisy and the last two are quite accurate. Based on the measurements from multiple radiologists, we can simultaneously (1) learn a linear regressor and (2) estimate the precision of each radiologist. We compare the results with the classiﬁer trained using the actual golden ground truth as well as the regressor learnt using the average of the radiologists measurements. The results are validated on an independent test set containing 397 examples (which point to 298 distinct polyps). Figure 7 shows the scatter plot of the actual polyp diameter vs the diameter predicted by the three different models. We compare the performance based on the root mean squared error (RMSE) and also the Pearson’s correlation coefﬁcient. The regressor learnt using the proposed iterative algorithm (Figure 7(b)) is almost as good as the one learnt using the golden ground truth (Figure 7(a)). The correlation coefﬁcient for the proposed algorithm is signiﬁcantly larger than that learnt using the average of the radiologists response. The estimate obtained by averaging is closer to the novices since they form a majority (3/5). The proposed algorithm appropriately weights each radiologist based on their estimated precisions. 7. Conclusions and Future Work In this paper we proposed a probabilistic framework for supervised learning with multiple annotators providing labels but no absolute gold standard. The proposed algorithm iteratively establishes a particular gold standard, measures the performance of the annotators given that gold standard, and then reﬁnes the gold standard based on the performance measures. We speciﬁcally discussed binary/categorical/ordinal classiﬁcation and regression problems. We made two key assumptions: (1) the performance of each annotator does not depend on the feature vector for a given instance and (2) conditional on the truth the experts are independent, that is, they make their errors independently. As we pointed out earlier these assumptions are not true in practice. The annotator performance depends on the instance he is labeling and there is some degree of correlation among the annotators. We brieﬂy discuss some strategies to relax these two assumptions. 7.1 Instance Difﬁculty One drawback of the current model is that it doesn’t estimate difﬁculty of items. It is often observed that for the easy instances all the annotators agree on the labels—thus violating our conditional independence assumption. The difﬁculty of annotating an item can be captured by another latent variable γi for each instance—which modulates the annotators performance. Models for this have been developed in the area of item-response theory (Baker and Kim, 2004) and also in epidemiology (Uebersax and Grove, 1993)—see also Whitehill et al. (2009) for a recent paper in the machine learning community. While these models do not take into account the available features our pro1319 R AYKAR , Y U , Z HAO , VALADEZ , F LORIN , B OGONI AND M OY posed model for sensitivity and speciﬁcity can be extended as follows (in place of (1) and (2)): α j (γi ) := Pr[yi = 1|yi = 1, γi ] = σ(a j1 + b j1 γi ). j β j (γi ) := Pr[yi = 0|yi = 0, γi ] = σ(a j0 + b j0 γi ). j Here the parameters a j1 and a j0 are related to the sensitivity and speciﬁcity of the jth annotator, while the latent term γi captures the difﬁculty of the instance. The key assumption here is that the annotators are independent conditional on both yi and γi . Various assumptions can be made on two parameters b j1 and b j0 to simplify these models further—for example we could set b j1 = b1 and b j0 = b0 for all the annotators. 7.2 Annotators Actually Look at the Data In our model we made the assumption that the sensitivity α j and the speciﬁcity β j of the jth annotator does not depend on the feature vector xi . For example, in the CAD domain, this meant that the radiologist’s performance is consistent across different sub-groups of data—which is not entirely true. It is known that some radiologists are good at detecting certain kinds of malignant lesions based on their training and experience. We can extend the previous model such that the sensitivity and the speciﬁcity depends on the feature vector xi explicitly as follows j⊤ α j (γi , xi ) := Pr[yi = 1|yi = 1, γi , xi ] = σ(a j1 + b j1 γi + wα xi ). j j⊤ α j (γi , xi ) := Pr[yi = 0|yi = 0, γi , xi ] = σ(a j0 + b j0 γi + wβ xi ). j However this change increases the number of parameters to be learned. References P. S. Albert and L. E. Dodd. A cautionary note on the robustness of latent class models for estimating diagnostic error without a gold standard. Biometrics, 60:427–435, 2004. F. B. Baker and S. Kim. Item Response Theory: Parameter Estimation Techniques. CRC Press, 2 edition, 2004. B. Carpenter. Multilevel bayesian models of categorical data annotation. Technical Report available at http://lingpipe-blog.com/lingpipe-white-papers/, 2008. S. R. Cholleti, S. A. Goldman, A. Blum, D. G. Politte, and S. Don. Veritas: Combining expert opinions without labeled data. In Proceedings of the 2008 20th IEEE international Conference on Tools with Artiﬁcial intelligence, 2008. A. P. Dawid and A. M. Skene. Maximum likeihood estimation of observer error-rates using the EM algorithm. Applied Statistics, 28(1):20–28, 1979. O. Dekel and O. Shamir. Vox Populi: Collecting high-quality labels from a crowd. In COLT 2009: Proceedings of the 22nd Annual Conference on Learning Theory, 2009a. O. Dekel and O. Shamir. Good learners for evil teachers. In ICML 2009: Proceedings of the 26th International Conference on Machine Learning, pages 233–240, 2009b. 1320 L EARNING F ROM C ROWDS A. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society: Series B, 39(1):1–38, 1977. P. Donmez, J. G. Carbonell, and J. Schneider. Efﬁciently learning the accuracy of labeling sources for selective sampling. In KDD 2009: Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining, pages 259–268, 2009. E. Frank and M. Hall. A simple approach to ordinal classiﬁcation. Lecture Notes in Computer Science, pages 145–156, 2001. G. Fung, B. Krishnapuram, J. Bi, M. Dundar, V. C. Raykar, S. Yu, R. Rosales, S. Krishnan, and R. B. Rao. Mining medical images. In Fifteenth Annual SIGKDD International Conference on Knowledge Discovery and Data Mining: Third Workshop on Data Mining Case Studies and Practice Prize, 2009. J. Howe. Crowd sourcing: Why the Power of the Crowd Is Driving the Future of Business. 2008. S. L. Hui and S. D. Walter. Estimating the error rates of diagnostic tests. Biometrics, 36:167–171, 1980. S. L. Hui and X. H. Zhou. Evaluation of diagnostic tests without gold standards. Statistical Methods in Medical Research, 7:354–370, 1998. R. Jin and Z. Ghahramani. Learning with multiple labels. In Advances in Neural Information Processing Systems 15, pages 897–904. 2003. B. Krishnapuram, J. Stoeckel, V. C. Raykar, R. B. Rao, P. Bamberger, E. Ratner, N. Merlet, I. Stainvas, M. Abramov, and A. Manevitch. Multiple-instance learning improves CAD detection of masses in digital mammography. In IWDM 2008: Proceedings of the 9th international workshop on Digital Mammography, pages 350–357. 2008. G. Lugosi. Learning with an unreliable teacher. Pattern Recognition, 25(1):79–87, 1992. R. M. Neal and G. E. Hinton. A view of the EM algorithm that justiﬁes incremental, sparse, and other variants. In Learning in Graphical Models, pages 355–368. Kluwer Academic Publishers, 1998. V. C. Raykar, S. Yu, L .H. Zhao, A. Jerebko, C. Florin, G. H. Valadez, L. Bogoni, and L. Moy. Supervised learning from multiple experts: Whom to trust when everyone lies a bit. In ICML 2009: Proceedings of the 26th International Conference on Machine Learning, pages 889–896, 2009. V. S. Sheng, F. Provost, and P. G. Ipeirotis. Get another label? improving data quality and data mining using multiple, noisy labelers. In Proceedings of the 14th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 614–622, 2008. P. Smyth. Learning with probabilistic supervision. In Computational Learning Theory and Natural Learning Systems 3, pages 163–182. MIT Press, 1995. 1321 R AYKAR , Y U , Z HAO , VALADEZ , F LORIN , B OGONI AND M OY P. Smyth, U. Fayyad, M. Burl, P. Perona, and P. Baldi. Inferring ground truth from subjective labelling of venus images. In Advances in Neural Information Processing Systems 7, pages 1085– 1092. 1995. R. Snow, B. O’Connor, D. Jurafsky, and A. Ng. Cheap and fast - But is it good? Evaluating nonexpert annotations for natural language tasks. In Proceedings of the Conference on Empirical Methods in Natural Language Processing, pages 254–263, 2008. A. Sorokin and D. Forsyth. Utility data annotation with Amazon Mechanical Turk. In Proceedings of the First IEEE Workshop on Internet Vision at CVPR 08, pages 1–8, 2008. J. S. Uebersax and W. M. Grove. A latent trait ﬁnite mixture model for the analysis of rating agreement. Biometrics, 49:823–835, 1993. S. K. Warﬁeld, K. H. Zou, and W. M. Wells. Simultaneous truth and performance level estimation (STAPLE): an algorithm for the validation of image segmentation. IEEE Transactions on Medical Imaging, 23(7):903–921, 2004. J. Whitehill, P. Ruvolo, T. Wu, J. Bergsma, and J. Movellan. Whose vote should count more: Optimal integration of labels from labelers of unknown expertise. In Advances in Neural Information Processing Systems 22, pages 2035–2043. 2009. 1322

6 0.048586581 22 jmlr-2010-Classification Using Geometric Level Sets

7 0.048564091 90 jmlr-2010-Permutation Tests for Studying Classifier Performance

8 0.047869042 46 jmlr-2010-High Dimensional Inverse Covariance Matrix Estimation via Linear Programming

9 0.044380624 40 jmlr-2010-Fast and Scalable Local Kernel Machines

10 0.044148665 82 jmlr-2010-On Learning with Integral Operators

11 0.040928282 44 jmlr-2010-Graph Kernels

12 0.040868759 69 jmlr-2010-Lp-Nested Symmetric Distributions

13 0.03867605 78 jmlr-2010-Model Selection: Beyond the Bayesian Frequentist Divide

14 0.036945976 72 jmlr-2010-Matrix Completion from Noisy Entries

15 0.034293063 117 jmlr-2010-Why Does Unsupervised Pre-training Help Deep Learning?

16 0.033578847 48 jmlr-2010-How to Explain Individual Classification Decisions

17 0.033334069 50 jmlr-2010-Image Denoising with Kernels Based on Natural Image Relations

18 0.033300404 112 jmlr-2010-Training and Testing Low-degree Polynomial Data Mappings via Linear SVM

19 0.0319479 14 jmlr-2010-Approximate Riemannian Conjugate Gradient Learning for Fixed-Form Variational Bayes

20 0.031661168 3 jmlr-2010-A Fast Hybrid Algorithm for Large-Scalel1-Regularized Logistic Regression

similar papers computed by lsi model

lsi for this paper:

topicId topicWeight

[(0, -0.166), (1, 0.006), (2, -0.021), (3, 0.051), (4, -0.064), (5, 0.082), (6, 0.0), (7, -0.071), (8, 0.037), (9, -0.038), (10, -0.028), (11, 0.125), (12, 0.029), (13, 0.034), (14, 0.07), (15, 0.035), (16, -0.22), (17, 0.159), (18, -0.051), (19, -0.176), (20, 0.095), (21, 0.009), (22, 0.188), (23, 0.225), (24, -0.111), (25, 0.177), (26, -0.1), (27, -0.019), (28, 0.097), (29, 0.023), (30, 0.03), (31, 0.066), (32, 0.143), (33, 0.071), (34, -0.115), (35, -0.073), (36, -0.114), (37, -0.124), (38, -0.044), (39, 0.105), (40, -0.109), (41, -0.11), (42, -0.021), (43, -0.041), (44, -0.048), (45, -0.087), (46, -0.015), (47, -0.125), (48, -0.059), (49, -0.035)]

similar papers list:

simIndex simValue paperId paperTitle

same-paper 1 0.95043784 30 jmlr-2010-Dimensionality Estimation, Manifold Learning and Function Approximation using Tensor Voting

Author: Philippos Mordohai, Gérard Medioni

2 0.68300426 54 jmlr-2010-Information Retrieval Perspective to Nonlinear Dimensionality Reduction for Data Visualization

Author: Jarkko Venna, Jaakko Peltonen, Kristian Nybo, Helena Aidos, Samuel Kaski

3 0.62546641 49 jmlr-2010-Hubs in Space: Popular Nearest Neighbors in High-Dimensional Data

Author: Miloš Radovanović, Alexandros Nanopoulos, Mirjana Ivanović

4 0.58620769 62 jmlr-2010-Learning Gradients: Predictive Models that Infer Geometry and Statistical Dependence

Author: Qiang Wu, Justin Guinney, Mauro Maggioni, Sayan Mukherjee

5 0.35087296 61 jmlr-2010-Learning From Crowds

Author: Vikas C. Raykar, Shipeng Yu, Linda H. Zhao, Gerardo Hermosillo Valadez, Charles Florin, Luca Bogoni, Linda Moy

6 0.28256431 40 jmlr-2010-Fast and Scalable Local Kernel Machines

7 0.26525882 59 jmlr-2010-Large Scale Online Learning of Image Similarity Through Ranking

8 0.25074506 77 jmlr-2010-Model-based Boosting 2.0

9 0.24421623 22 jmlr-2010-Classification Using Geometric Level Sets

10 0.22896574 37 jmlr-2010-Evolving Static Representations for Task Transfer

11 0.22217789 14 jmlr-2010-Approximate Riemannian Conjugate Gradient Learning for Fixed-Form Variational Bayes

12 0.18294558 112 jmlr-2010-Training and Testing Low-degree Polynomial Data Mappings via Linear SVM

13 0.18189035 36 jmlr-2010-Estimation of a Structural Vector Autoregression Model Using Non-Gaussianity

14 0.18050399 63 jmlr-2010-Learning Instance-Specific Predictive Models

15 0.18001133 46 jmlr-2010-High Dimensional Inverse Covariance Matrix Estimation via Linear Programming

16 0.17735487 82 jmlr-2010-On Learning with Integral Operators

17 0.17535658 50 jmlr-2010-Image Denoising with Kernels Based on Natural Image Relations

18 0.17246379 48 jmlr-2010-How to Explain Individual Classification Decisions

19 0.16931206 26 jmlr-2010-Consensus-Based Distributed Support Vector Machines

20 0.16606507 104 jmlr-2010-Sparse Spectrum Gaussian Process Regression

similar papers computed by lda model

lda for this paper:

topicId topicWeight

[(4, 0.021), (8, 0.013), (21, 0.022), (32, 0.051), (36, 0.031), (37, 0.046), (75, 0.122), (81, 0.5), (85, 0.051), (96, 0.012)]

similar papers list:

simIndex simValue paperId paperTitle

same-paper 1 0.77630532 30 jmlr-2010-Dimensionality Estimation, Manifold Learning and Function Approximation using Tensor Voting

Author: Philippos Mordohai, Gérard Medioni

2 0.7450242 10 jmlr-2010-An Exponential Model for Infinite Rankings

Author: Marina Meilă, Le Bao

Abstract: This paper presents a statistical model for expressing preferences through rankings, when the number of alternatives (items to rank) is large. A human ranker will then typically rank only the most preferred items, and may not even examine the whole set of items, or know how many they are. Similarly, a user presented with the ranked output of a search engine, will only consider the highest ranked items. We model such situations by introducing a stagewise ranking model that operates with ﬁnite ordered lists called top-t orderings over an inﬁnite space of items. We give algorithms to estimate this model from data, and demonstrate that it has sufﬁcient statistics, being thus an exponential family model with continuous and discrete parameters. We describe its conjugate prior and other statistical properties. Then, we extend the estimation problem to multimodal data by introducing an Exponential-Blurring-Mean-Shift nonparametric clustering algorithm. The experiments highlight the properties of our model and demonstrate that inﬁnite models over permutations can be simple, elegant and practical. Keywords: permutations, partial orderings, Mallows model, distance based ranking model, exponential family, non-parametric clustering, branch-and-bound

3 0.43334812 49 jmlr-2010-Hubs in Space: Popular Nearest Neighbors in High-Dimensional Data

Author: Miloš Radovanović, Alexandros Nanopoulos, Mirjana Ivanović

4 0.40244019 55 jmlr-2010-Information Theoretic Measures for Clusterings Comparison: Variants, Properties, Normalization and Correction for Chance

Author: Nguyen Xuan Vinh, Julien Epps, James Bailey

Abstract: Information theoretic measures form a fundamental class of measures for comparing clusterings, and have recently received increasing interest. Nevertheless, a number of questions concerning their properties and inter-relationships remain unresolved. In this paper, we perform an organized study of information theoretic measures for clustering comparison, including several existing popular measures in the literature, as well as some newly proposed ones. We discuss and prove their important properties, such as the metric property and the normalization property. We then highlight to the clustering community the importance of correcting information theoretic measures for chance, especially when the data size is small compared to the number of clusters present therein. Of the available information theoretic based measures, we advocate the normalized information distance (NID) as a general measure of choice, for it possesses concurrently several important properties, such as being both a metric and a normalized measure, admitting an exact analytical adjusted-for-chance form, and using the nominal [0, 1] range better than other normalized variants. Keywords: clustering comparison, information theory, adjustment for chance, normalized information distance

5 0.39570379 54 jmlr-2010-Information Retrieval Perspective to Nonlinear Dimensionality Reduction for Data Visualization

Author: Jarkko Venna, Jaakko Peltonen, Kristian Nybo, Helena Aidos, Samuel Kaski

6 0.36563444 59 jmlr-2010-Large Scale Online Learning of Image Similarity Through Ranking

7 0.35350883 89 jmlr-2010-PAC-Bayesian Analysis of Co-clustering and Beyond

8 0.35339332 114 jmlr-2010-Unsupervised Supervised Learning I: Estimating Classification and Regression Errors without Labels

9 0.34742785 46 jmlr-2010-High Dimensional Inverse Covariance Matrix Estimation via Linear Programming

10 0.34069645 111 jmlr-2010-Topology Selection in Graphical Models of Autoregressive Processes

11 0.33550799 109 jmlr-2010-Stochastic Composite Likelihood

12 0.32987475 105 jmlr-2010-Spectral Regularization Algorithms for Learning Large Incomplete Matrices

13 0.32815835 22 jmlr-2010-Classification Using Geometric Level Sets

14 0.32705998 74 jmlr-2010-Maximum Relative Margin and Data-Dependent Regularization

15 0.32682225 19 jmlr-2010-Characterization, Stability and Convergence of Hierarchical Clustering Methods

16 0.32646677 17 jmlr-2010-Bayesian Learning in Sparse Graphical Factor Models via Variational Mean-Field Annealing

17 0.32640186 69 jmlr-2010-Lp-Nested Symmetric Distributions

18 0.32623991 61 jmlr-2010-Learning From Crowds

19 0.32618782 92 jmlr-2010-Practical Approaches to Principal Component Analysis in the Presence of Missing Values

20 0.32330817 23 jmlr-2010-Classification with Incomplete Data Using Dirichlet Process Priors