nips nips2002 nips2002-36 knowledge-graph by maker-knowledge-mining

36 nips-2002-Automatic Alignment of Local Representations

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Author: Yee W. Teh, Sam T. Roweis

Abstract: We present an automatic alignment procedure which maps the disparate internal representations learned by several local dimensionality reduction experts into a single, coherent global coordinate system for the original data space. Our algorithm can be applied to any set of experts, each of which produces a low-dimensional local representation of a highdimensional input. Unlike recent efforts to coordinate such models by modifying their objective functions [1, 2], our algorithm is invoked after training and applies an efﬁcient eigensolver to post-process the trained models. The post-processing has no local optima and the size of the system it must solve scales with the number of local models rather than the number of original data points, making it more efﬁcient than model-free algorithms such as Isomap [3] or LLE [4]. 1 Introduction: Local vs. Global Dimensionality Reduction Beyond density modelling, an important goal of unsupervised learning is to discover compact, informative representations of high-dimensional data. If the data lie on a smooth low dimensional manifold, then an excellent encoding is the coordinates internal to that manifold. The process of determining such coordinates is dimensionality reduction. Linear dimensionality reduction methods such as principal component analysis and factor analysis are easy to train but cannot capture the structure of curved manifolds. Mixtures of these simple unsupervised models [5, 6, 7, 8] have been used to perform local dimensionality reduction, and can provide good density models for curved manifolds, but unfortunately such mixtures cannot do dimensionality reduction. They do not describe a single, coherent low-dimensional coordinate system for the data since there is no pressure for the local coordinates of each component to agree. Roweis et al [1] recently proposed a model which performs global coordination of local coordinate systems in a mixture of factor analyzers (MFA). Their model is trained by maximizing the likelihood of the data, with an additional variational penalty term to encourage the internal coordinates of the factor analyzers to agree. While their model can trade off modelling the data and having consistent local coordinate systems, it requires a user given trade-off parameter, training is quite inefﬁcient (although [2] describes an improved training algorithm for a more constrained model), and it has quite serious local minima problems (methods like LLE [4] or Isomap [3] have to be used for initialization). In this paper we describe a novel, automatic way to align the hidden representations used by each component of a mixture of dimensionality reducers into a single global representation of the data throughout space. Given an already trained mixture, the alignment is achieved by applying an eigensolver to a matrix constructed from the internal representations of the mixture components. Our method is efﬁcient, simple to implement, and has no local optima in its optimization nor any learning rates or annealing schedules. 2 The Locally Linear Coordination Algorithm H 9¥ EI¡ CD66B9 ©9B 766 % G F 5 #

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu ¡ Abstract We present an automatic alignment procedure which maps the disparate internal representations learned by several local dimensionality reduction experts into a single, coherent global coordinate system for the original data space. [sent-3, score-1.278]

2 Our algorithm can be applied to any set of experts, each of which produces a low-dimensional local representation of a highdimensional input. [sent-4, score-0.183]

3 Unlike recent efforts to coordinate such models by modifying their objective functions [1, 2], our algorithm is invoked after training and applies an efﬁcient eigensolver to post-process the trained models. [sent-5, score-0.289]

4 The post-processing has no local optima and the size of the system it must solve scales with the number of local models rather than the number of original data points, making it more efﬁcient than model-free algorithms such as Isomap [3] or LLE [4]. [sent-6, score-0.584]

5 Global Dimensionality Reduction Beyond density modelling, an important goal of unsupervised learning is to discover compact, informative representations of high-dimensional data. [sent-8, score-0.127]

6 If the data lie on a smooth low dimensional manifold, then an excellent encoding is the coordinates internal to that manifold. [sent-9, score-0.476]

7 The process of determining such coordinates is dimensionality reduction. [sent-10, score-0.494]

8 Linear dimensionality reduction methods such as principal component analysis and factor analysis are easy to train but cannot capture the structure of curved manifolds. [sent-11, score-0.422]

9 Mixtures of these simple unsupervised models [5, 6, 7, 8] have been used to perform local dimensionality reduction, and can provide good density models for curved manifolds, but unfortunately such mixtures cannot do dimensionality reduction. [sent-12, score-0.666]

10 They do not describe a single, coherent low-dimensional coordinate system for the data since there is no pressure for the local coordinates of each component to agree. [sent-13, score-0.746]

11 Roweis et al [1] recently proposed a model which performs global coordination of local coordinate systems in a mixture of factor analyzers (MFA). [sent-14, score-1.046]

12 Their model is trained by maximizing the likelihood of the data, with an additional variational penalty term to encourage the internal coordinates of the factor analyzers to agree. [sent-15, score-0.745]

13 In this paper we describe a novel, automatic way to align the hidden representations used by each component of a mixture of dimensionality reducers into a single global representation of the data throughout space. [sent-17, score-0.832]

14 Given an already trained mixture, the alignment is achieved by applying an eigensolver to a matrix constructed from the internal representations of the mixture components. [sent-18, score-0.615]

15 Our method is efﬁcient, simple to implement, and has no local optima in its optimization nor any learning rates or annealing schedules. [sent-19, score-0.234]

16 ¥¨¨¥©¨(¤¢ $§¥ £ '¡% ¦ & Suppose we have a set of data points given by the rows of from a -dimensional space, which we assume are sampled from a dimensional manifold. [sent-21, score-0.125]

17 We approximate the manifold coordinates using images in a dimensional embedding space. [sent-22, score-0.571]

18 Suppose also that we have already trained, or have been given, a mixture of local dimensionality reducers. [sent-23, score-0.45]

19 The th reducer produces a dimensional internal representation for data point as well as a “responsibility” describing how reliable the th reducer’s representation of is. [sent-24, score-0.457]

20 These satisfy and can be obtained, for example, using a gating network in a mixture of experts, or the posterior probabilities in a probabilistic network. [sent-25, score-0.147]

21 Notice that the manifold coordinates and internal representations need not have the same number of dimensions. [sent-26, score-0.605]

22 $ 9 A¥ % 4 6@8 5 9 Given the data, internal representations, and responsibilities, our algorithm automatically aligns the various hidden representations into a single global coordinate system. [sent-27, score-0.525]

23 First, to use a convex cost function whose unique minimum is attained at the desired global coordinates. [sent-29, score-0.316]

24 Second, to restrict the global coordinates to depend on the data only through the local representations and responsibilities , thereby leveraging the structure of the mixture model to regularize and reduce the effective size of the optimization problem. [sent-30, score-1.045]

25 In effect, rather than working with individual data points, we work with large groups of points belonging to particular submodels. [sent-31, score-0.115]

26 Given an input We ﬁrst parameterize the global coordinates , each local model infers its internal coordinates and then applies a linear projection and offset to these to obtain its guess at the global coordinates. [sent-33, score-1.267]

27 The ﬁnal global is obtained by averaging the guesses using the responsibilities as weights: coordinates 6TSR 9 2 (1) (2) 69 ¥Q a 5¨ vuW a 5¨ vuW u 6R f s ¡ Vp) 6Q V¡ 9 2 U where is the th column of , is the th entry of , and is a bias. [sent-34, score-0.85]

28 the domain of Now deﬁne the matrices and as and the th row of as . [sent-38, score-0.177]

29 9 29 2 The key assumption, which we have emphasized by re-expressing above, is that the mapping between the local representations and the global coordinates is linear in each of , and the unknown parameters . [sent-41, score-0.851]

30 Crucially, however, the mapping between the original data and the images is highly non-linear since it depends on the multiplication of responsibilities and internal coordinates which are in turn non-linearly related to the data through the inference procedure of the mixture model. [sent-42, score-0.853]

31 c ¥ " c s £ "Q s " Q c Q §) £ " Q) s £" s ¡ £ ¢ ) E ¡ a W) ¡ a aW) a Q W W) Q f 6R 92 Q 9 ¥ 9 ¥ 6 B9 69 8 We now consider determining according to some given cost function . [sent-43, score-0.124]

32 The case where the cost and constraints are both quadratic is particularly appealing since we can use an eigensolver to ﬁnd the optimal . [sent-47, score-0.198]

33 In particular suppose and are matrices deﬁning the cost and constraints, and let and . [sent-48, score-0.118]

34 The matrices and are typically obtained from the original data and summarize the essential geometries among them. [sent-50, score-0.191]

35 The solution to the constrained minimization above is given by the smallest generalized eigenvectors with . [sent-51, score-0.165]

36 The idea of LLE is to preserve the same locally linear relationships between the original data points and their counterparts . [sent-55, score-0.23]

37 The weights summarize the local geometries relating the data points to their neighbours, hence to preserve these relationships among the coordinates we arrange to minimize the same cost (5) but with respect to instead. [sent-60, score-0.814]

38 For this choice, the cost function and constraints above become: (7) (8) with cost and constraint matrices (9) 1 In the unusual case where the number of neighbours is larger than the dimensionality of the data , simple regularization of the norm of the weights once again makes them unique. [sent-63, score-0.579]

39 To satisfy , we need to ﬁnd eigenvectors eigenvectors with that are orthogonal to the vector . [sent-65, score-0.132]

40 Fortunately, is the smallest generalized eigenvector, corresponding to an eigenvalue of 0. [sent-66, score-0.163]

41 Hence the solution to the problem is given by the to smallest generalized eigenvectors instead. [sent-67, score-0.165]

42 Apply this mixture to , obtaining a local representation and responsibility for each submodel and each data point . [sent-69, score-0.468]

43 ¤ Form the matrix ¤ Find the eigenvectors corresponding to the smallest of the generalized eigenvalue system . [sent-70, score-0.257]

44 ¤ Let be a matrix with columns formed by the Return the th row of as alignment weight . [sent-71, score-0.3]

45 Return the global manifold coordinates as , compute local linear reconstruction weights with and calculate nd and using (4). [sent-72, score-0.829]

46 Note that the edge size of the matrices and whose generalized eigenvectors we seek which scales with the number of components and dimensions of the local is representations but not with the number of data points . [sent-76, score-0.544]

47 As a result, solving for the alignment weights is much more efﬁcient than the original LLE computation (or those of Isomap) which requires solving an eigenvalue system of edge size . [sent-77, score-0.339]

48 In effect, we have leveraged the mixture of local models to collapse large groups of points together and worked only with those groups rather than the original data points. [sent-78, score-0.516]

49 Notice however that the computation of the weights still requires determining the neighbours of the original in the worse case. [sent-79, score-0.242]

50 data points, which scales as C 52 a &C; W ¥ Coordination with LLC also yields a mixture of noiseless factor analyzers over the global coordinate space , with the th factor analyzer having mean and factor loading . [sent-80, score-1.137]

51 Given any global coordinates , we can infer the responsibilities and the posterior means over the latent space of each factor analyzer. [sent-81, score-0.839]

52 If our original local dimensionality reducers also supports computing from and , we can now infer the high dimensional mean data point which corresponds to the global coordinates . [sent-82, score-1.022]

53 This allows us to perform operations like visualization and interpolation using the global coordinate system. [sent-83, score-0.338]

54 6Q 66 RS B 68 2 B6 ¥ 2 68 ¥ 3 Experimental Results using Mixtures of Factor Analyzers The alignment computation we have described is applicable to any mixture of local dimensionality reducers. [sent-85, score-0.606]

55 In our experiments, we have used the most basic such model: a mixture of factor analyzers (MFA) [8]. [sent-86, score-0.389]

56 The th factor analyzer in the mixture describes a probabilistic linear mapping from a latent variable to the data with additive Gaussian noise. [sent-87, score-0.636]

57 The model assumes that the data manifold is locally linear and it is this local structure that is captured by each factor analyzer. [sent-88, score-0.492]

58 The non-linearity in the data manifold is handled by patching multiple factor analyzers together, each handling a locally linear region. [sent-89, score-0.487]

59 2 In our experiments, we initialized the parameters by drawing the means from the global covariance of the data and setting the factors to small random values. [sent-91, score-0.22]

60 We also simpliﬁed the factor analyzers to share the same spherical noise covariance although this is not essential to the process. [sent-92, score-0.274]

61 There are 14 factor analyzers in the mixture (B), each with 2 latent dimensions. [sent-94, score-0.48]

62 After alignment by LLC (C), the curve is successfully unrolled; it is also possible to retroactively align the original data space models (D). [sent-96, score-0.438]

63 Then we ﬁt an MFA with 30 components with 1 latent dimension each (A), but aligned them in a 2D space (B). [sent-99, score-0.155]

64 We used 10 neighbours to reconstruct each data point. [sent-100, score-0.172]

65 68 Since there is no constraint relating the various hidden variables , a MFA trained only to maximize likelihood cannot learn a global coordinate system for the manifold that is consistent across every factor analyzer. [sent-101, score-0.627]

66 Naturally, we use the mean of conditioned on the data (assuming the th factor analyzer generated ) as the th local representation of , while we use the posterior probability that the th factor analyzer generated as the responsibility. [sent-103, score-0.893]

67 An MFA trained on 1200 points sampled uniformly from the manifold with added noise (B) is able to model the linear structure of the curve locally, however the internal coordinates of the factor analyzers are not aligned properly. [sent-106, score-1.014]

68 We applied LLC to the local representations and aligned them in a 2D space (C). [sent-107, score-0.343]

69 When solving for local weights, we used 12 neighbours to reconstruct each data point. [sent-108, score-0.355]

70 Further, given the coordinate transforms produced by LLC, we can retroactively align the latent spaces of the MFAs (D). [sent-110, score-0.371]

71 This is done by determining directions in the various latent spaces which get transformed to the same direction in the global space. [sent-111, score-0.353]

72 To emphasize the topological advantages of aligning representations into a space of higher dimensionality than the local coordinates used by each submodel, we also trained a MFA on data sampled from a trefoil curve, as shown in ﬁgure 3(A). [sent-112, score-0.903]

73 As ﬁgure 3(B) shows, LLC connects these models into a ring of local topology faithful to the original data. [sent-114, score-0.254]

74 We applied LLC to MFAs trained on sets of real images believed to come from a complex manifold with few degrees of freedom. [sent-115, score-0.226]

75 We studied face images of a single person under varying pose and expression changes and handwritten digits from the MNIST database. [sent-116, score-0.142]

76 The ﬁrst dimension appears to describe Figure 4: A map of reconstructions of the faces when the global coordinates are speciﬁed. [sent-119, score-0.582]

77 Note that some reconstructions around the edge of the map are not good because the model is extrapolating from the training images to regions of low likelihood. [sent-121, score-0.115]

78 In particular, the ﬁrst row interpolates between left and right slanting digits, the second between fat and thin digits, the third between thick and thin line strokes, and the fourth between having a larger bottom loop and larger top loop. [sent-130, score-0.101]

79 Compared to the global coordination model [1], the closest parametric approach to ours, our algorithm can be understood as post coordination, in which the latent spaces are coordinated after they have been ﬁt to data. [sent-132, score-0.575]

80 By decoupling the data ﬁtting and coordination problems we gain efﬁciency and avoid local optima in the coordination phase. [sent-133, score-0.609]

81 Further, since we are just maximizing likelihood when ﬁtting the original mixture to data, we can use a whole range of known techniques to escape local minima, and improve efﬁciency in the ﬁrst phase as well. [sent-134, score-0.341]

82 On the nonparametric side, our approach can be compared to two recent algorithms, LLE Figure 5: Top: maps of reconstructions of digits when two global coordinates are speciﬁed, and the third integrated out. [sent-135, score-0.645]

83 Left: st and nd coordinates speciﬁed; right: nd and rd . [sent-136, score-0.303]

84 The LLC interpolants are spread out evenly along a line connecting the global coordinates of the two digits. [sent-139, score-0.492]

85 For comparison, we show the 20 training images whose coordinates are closest to the line segment connecting those of the two digits at each side. [sent-140, score-0.445]

86 It is trained on 6000 randomly chosen digits from the combined training and test sets of 8’s in MNIST. [sent-142, score-0.158]

87 The cost functions of LLE and Isomap are convex, so they do not suffer from the local minima problems of earlier methods [9, 10], but these methods must solve eigenvalue systems of size equal to the number of data points. [sent-145, score-0.4]

88 ) Another problem is neither LLE nor Isomap yield a probabilistic model or even a mapping between the data and embedding spaces. [sent-147, score-0.181]

89 Compared to these models (which are run on individual data points) LLC uses as its primitives descriptions of the data provided by the individual local models. [sent-148, score-0.273]

90 This makes the eigenvalue system to be solved much smaller and as a result the computational cost of the coordination phase of LLC is much less than that for LLE or Isomap. [sent-149, score-0.349]

91 (Note that the construction of the eigenvalue system still requires ﬁnding nearest neighbours for each point, which is costly. [sent-150, score-0.204]

92 ) Furthermore, if each local model describes a complete (probabilistic) mapping from data space to its latent space, the ﬁnal coordinated model will also describe a (probabilistic) mapping from the whole data space to the coordinated embedding space. [sent-151, score-0.684]

93 Our alignment algorithm improves upon local embedding or density models by elevating their status to full global dimensionality reduction algorithms without requiring any modiﬁcations to their training procedures or cost functions. [sent-152, score-0.962]

94 For example, using mixtures of factor analyzers (MFAs) as a test case, we show how our alignment method can allow a model previously suited only for density estimation to do complex operations on high dimensional data such as visualization and interpolation. [sent-153, score-0.622]

95 Brand [11] has recently proposed an approach, similar to ours, that coordinates local parametric models to obtain a globally valid nonlinear embedding function. [sent-154, score-0.612]

96 Like LLC, his “charting” method deﬁnes a quadratic cost function and ﬁnds the optimal coordination directly and efﬁciently. [sent-155, score-0.257]

97 However, charting is based on a cost function much closer in spirit to the original global coordination model and it instantiates one local model centred on each training point, so its scaling is the same as that of LLE and Isomap. [sent-156, score-0.757]

98 In principle, Brand’s method can be extended to work with fewer local models and our alignment procedure can be applied using the charting cost rather than the LLE cost we have pursued here. [sent-157, score-0.622]

99 Automatic alignment procedures emphasizes a powerful but often overlooked interpretation of local mixture models. [sent-158, score-0.454]

100 As we have shown, this view can lead to efﬁcient and powerful algorithms which allow separate local models to learn consistent global representations. [sent-160, score-0.4]

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same-paper 1 1.0000005 36 nips-2002-Automatic Alignment of Local Representations

Author: Yee W. Teh, Sam T. Roweis

2 0.26622611 49 nips-2002-Charting a Manifold

Author: Matthew Brand

Abstract: We construct a nonlinear mapping from a high-dimensional sample space to a low-dimensional vector space, effectively recovering a Cartesian coordinate system for the manifold from which the data is sampled. The mapping preserves local geometric relations in the manifold and is pseudo-invertible. We show how to estimate the intrinsic dimensionality of the manifold from samples, decompose the sample data into locally linear low-dimensional patches, merge these patches into a single lowdimensional coordinate system, and compute forward and reverse mappings between the sample and coordinate spaces. The objective functions are convex and their solutions are given in closed form. 1 Nonlinear dimensionality reduction (NLDR) by charting Charting is the problem of assigning a low-dimensional coordinate system to data points in a high-dimensional sample space. It is presumed that the data lies on or near a lowdimensional manifold embedded in the sample space, and that there exists a 1-to-1 smooth nonlinear transform between the manifold and a low-dimensional vector space. The datamodeler’s goal is to estimate smooth continuous mappings between the sample and coordinate spaces. Often this analysis will shed light on the intrinsic variables of the datagenerating phenomenon, for example, revealing perceptual or conﬁguration spaces. Our goal is to ﬁnd a mapping—expressed as a kernel-based mixture of linear projections— that minimizes information loss about the density and relative locations of sample points. This constraint is expressed in a posterior that combines a standard gaussian mixture model (GMM) likelihood function with a prior that penalizes uncertainty due to inconsistent projections in the mixture. Section 3 develops a special case where this posterior is unimodal and maximizable in closed form, yielding a GMM whose covariances reveal a patchwork of overlapping locally linear subspaces that cover the manifold. Section 4 shows that for this (or any) GMM and a choice of reduced dimension d, there is a unique, closed-form solution for a minimally distorting merger of the subspaces into a d-dimensional coordinate space, as well as an reverse mapping deﬁning the surface of the manifold in the sample space. The intrinsic dimensionality d of the data manifold can be estimated from the growth process of point-to-point distances. In analogy to differential geometry, we call the subspaces “charts” and their merger the “connection.” Section 5 considers example problems where these methods are used to untie knots, unroll and untwist sheets, and visualize video data. 1.1 Background Topology-neutral NLDR algorithms can be divided into those that compute mappings, and those that directly compute low-dimensional embeddings. The ﬁ has its roots in mapeld ping algorithms: DeMers and Cottrell [3] proposed using auto-encoding neural networks with a hidden layer “ bottleneck,” effectively casting dimensionality reduction as a compression problem. Hastie deﬁ principal curves [5] as nonparametric 1 D curves that pass ned through the center of “ nearby” data points. A rich literature has grown up around properly regularizing this approach and extending it to surfaces. Smola and colleagues [10] analyzed the NLDR problem in the broader framework of regularized quantization methods. More recent advances aim for embeddings: Gomes and Mojsilovic [4] treat manifold completion as an anisotropic diffusion problem, iteratively expanding points until they connect to their neighbors. The I SO M AP algorithm [12] represents remote distances as sums of a trusted set of distances between immediate neighbors, then uses multidimensional scaling to compute a low-dimensional embedding that minimally distorts all distances. The locally linear embedding algorithm (LLE) [9] represents each point as a weighted combination of a trusted set of nearest neighbors, then computes a minimally distorting low-dimensional barycentric embedding. They have complementary strengths: I SO M AP handles holes well but can fail if the data hull is nonconvex [12]; and vice versa for LLE [9]. Both offer embeddings without mappings. It has been noted that trusted-set methods are vulnerable to noise because they consider the subset of point-to-point relationships that has the lowest signal-to-noise ratio; small changes to the trusted set can induce large changes in the set of constraints on the embedding, making solutions unstable [1]. In a return to mapping, Roweis and colleagues [8] proposed global coordination— learning a mixture of locally linear projections from sample to coordinate space. They constructed a posterior that penalizes distortions in the mapping, and gave a expectation-maximization (EM) training rule. Innovative use of variational methods highlighted the difﬁ culty of even hill-climbing their multimodal posterior. Like [2, 7, 6, 8], the method we develop below is a decomposition of the manifold into locally linear neighborhoods. It bears closest relation to global coordination [8], although by a different construction of the problem, we avoid hill-climbing a spiky posterior and instead develop a closed-form solution. 2 Estimating locally linear scale and intrinsic dimensionality . We begin with matrix of sample points Y = [y1 , · · · , yN ], yn ∈ RD populating a Ddimensional sample space, and a conjecture that these points are samples from a manifold M of intrinsic dimensionality d < D. We seek a mapping onto a vector space . G(Y) → X = [x1 , · · · , xN ], xn ∈ Rd and 1-to-1 reverse mapping G−1 (X) → Y such that local relations between nearby points are preserved (this will be formalized below). The map G should be non-catastrophic, that is, without folds: Parallel lines on the manifold in RD should map to continuous smooth non-intersecting curves in Rd . This guarantees that linear operations on X such as interpolation will have reasonable analogues on Y. Smoothness means that at some scale r the mapping from a neighborhood on M to Rd is effectively linear. Consider a ball of radius r centered on a data point and containing n(r) data points. The count n(r) grows as rd , but only at the locally linear scale; the grow rate is inﬂated by isotropic noise at smaller scales and by embedding curvature at larger scales. . To estimate r, we look at how the r-ball grows as points are added to it, tracking c(r) = d d log n(r) log r. At noise scales, c(r) ≈ 1/D < 1/d, because noise has distributed points in all directions with equal probability. At the scale at which curvature becomes signiﬁ cant, c(r) < 1/d, because the manifold is no longer perpendicular to the surface of the ball, so the ball does not have to grow as fast to accommodate new points. At the locally linear scale, the process peaks at c(r) = 1/d, because points are distributed only in the directions of the manifold’s local tangent space. The maximum of c(r) therefore gives an estimate of both the scale and the local dimensionality of the manifold (see ﬁ gure 1), provided that the ball hasn’t expanded to a manifold boundary— boundaries have lower dimension than Scale behavior of a 1D manifold in 2-space Point−count growth process on a 2D manifold in 3−space 1 10 radial growth process 1D hypothesis 2D hypothesis 3D hypothesis radius (log scale) samples noise scale locally linear scale curvature scale 0 10 2 1 10 2 10 #points (log scale) 3 10 Figure 1: Point growth processes. L EFT: At the locally linear scale, the number of points in an r-ball grows as rd ; at noise and curvature scales it grows faster. R IGHT: Using the point-count growth process to ﬁ the intrinsic dimensionality of a 2D manifold nonlinearly nd embedded in 3-space (see ﬁ gure 2). Lines of slope 1/3 , 1/2 , and 1 are ﬁ tted to sections of the log r/ log nr curve. For neighborhoods of radius r ≈ 1 with roughly n ≈ 10 points, the slope peaks at 1/2 indicating a dimensionality of d = 2. Below that, the data appears 3 D because it is dominated by noise (except for n ≤ D points); above, the data appears >2 D because of manifold curvature. As the r-ball expands to cover the entire data-set the dimensionality appears to drop to 1 as the process begins to track the 1D edges of the 2D sheet. the manifold. For low-dimensional manifolds such as sheets, the boundary submanifolds (edges and corners) are very small relative to the full manifold, so the boundary effect is typically limited to a small rise in c(r) as r approaches the scale of the entire data set. In practice, our code simply expands an r-ball at every point and looks for the ﬁ peak in rst c(r), averaged over many nearby r-balls. One can estimate d and r globally or per-point. 3 Charting the data In the charting step we ﬁ a soft partitioning of the data into locally linear low-dimensional nd neighborhoods, as a prelude to computing the connection that gives the global lowdimensional embedding. To minimize information loss in the connection, we require that the data points project into a subspace associated with each neighborhood with (1) minimal loss of local variance and (2) maximal agreement of the projections of nearby points into nearby neighborhoods. Criterion (1) is served by maximizing the likelihood function of a Gaussian mixture model (GMM) density ﬁ tted to the data: . p(yi |µ, Σ) = ∑ j p(yi |µ j , Σ j ) p j = ∑ j N (yi ; µ j , Σ j ) p j . (1) Each gaussian component deﬁ a local neighborhood centered around µ j with axes denes ﬁ ned by the eigenvectors of Σ j . The amount of data variance along each axis is indicated by the eigenvalues of Σ j ; if the data manifold is locally linear in the vicinity of the µ j , all but the d dominant eigenvalues will be near-zero, implying that the associated eigenvectors constitute the optimal variance-preserving local coordinate system. To some degree likelihood maximization will naturally realize this property: It requires that the GMM components shrink in volume to ﬁ the data as tightly as possible, which is best achieved by t positioning the components so that they “ pancake” onto locally ﬂat collections of datapoints. However, this state of affairs is easily violated by degenerate (zero-variance) GMM components or components ﬁ tted to overly small enough locales where the data density off the manifold is comparable to density on the manifold (e.g., at the noise scale). Consequently a prior is needed. Criterion (2) implies that neighboring partitions should have dominant axes that span similar subspaces, since disagreement (large subspace angles) would lead to inconsistent projections of a point and therefore uncertainty about its location in a low-dimensional coordinate space. The principal insight is that criterion (2) is exactly the cost of coding the location of a point in one neighborhood when it is generated by another neighborhood— the cross-entropy between the gaussian models deﬁ ning the two neighborhoods: D(N1 N2 ) = = dy N (y; µ1 ,Σ1 ) log N (y; µ1 ,Σ1 ) N (y; µ2 ,Σ2 ) (log |Σ−1 Σ2 | + trace(Σ−1 Σ1 ) + (µ2 −µ1 ) Σ−1 (µ2 −µ1 ) − D)/2. (2) 1 2 2 Roughly speaking, the terms in (2) measure differences in size, orientation, and position, respectively, of two coordinate frames located at the means µ1 , µ2 with axes speciﬁ by ed the eigenvectors of Σ1 , Σ2 . All three terms decline to zero as the overlap between the two frames is maximized. To maximize consistency between adjacent neighborhoods, we form . the prior p(µ, Σ) = exp[− ∑i= j mi (µ j )D(Ni N j )], where mi (µ j ) is a measure of co-locality. Unlike global coordination [8], we are not asking that the dominant axes in neighboring charts are aligned— only that they span nearly the same subspace. This is a much easier objective to satisfy, and it contains a useful special case where the posterior p(µ, Σ|Y) ∝ ∑i p(yi |µ, Σ)p(µ, Σ) is unimodal and can be maximized in closed form: Let us associate a gaussian neighborhood with each data-point, setting µi = yi ; take all neighborhoods to be a priori equally probable, setting pi = 1/N; and let the co-locality measure be determined from some local kernel. For example, in this paper we use mi (µ j ) ∝ N (µ j ; µi , σ2 ), with the scale parameter σ specifying the expected size of a neighborhood on the manifold in sample space. A reasonable choice is σ = r/2, so that 2erf(2) > 99.5% of the density of mi (µ j ) is contained in the area around yi where the manifold is expected to be locally linear. With uniform pi and µi , mi (µ j ) and ﬁ xed, the MAP estimates of the GMM covariances are Σi = ∑ mi (µ j ) (y j − µi )(y j − µi ) + (µ j − µi )(µ j − µi ) + Σ j j ∑ mi (µ j ) (3) . j Note that each covariance Σi is dependent on all other Σ j . The MAP estimators for all covariances can be arranged into a set of fully constrained linear equations and solved exactly for their mutually optimal values. This key step brings nonlocal information about the manifold’s shape into the local description of each neighborhood, ensuring that adjoining neighborhoods have similar covariances and small angles between their respective subspaces. Even if a local subset of data points are dense in a direction perpendicular to the manifold, the prior encourages the local chart to orient parallel to the manifold as part of a globally optimal solution, protecting against a pathology noted in [8]. Equation (3) is easily adapted to give a reduced number of charts and/or charts centered on local centroids. 4 Connecting the charts We now build a connection for set of charts speciﬁ as an arbitrary nondegenerate GMM. A ed GMM gives a soft partitioning of the dataset into neighborhoods of mean µk and covariance Σk . The optimal variance-preserving low-dimensional coordinate system for each neighborhood derives from its weighted principal component analysis, which is exactly speciﬁ ed by the eigenvectors of its covariance matrix: Eigendecompose Vk Λk Vk ← Σk with eigen. values in descending order on the diagonal of Λk and let Wk = [Id , 0]Vk be the operator . th projecting points into the k local chart, such that local chart coordinate uki = Wk (yi − µk ) . and Uk = [uk1 , · · · , ukN ] holds the local coordinates of all points. Our goal is to sew together all charts into a globally consistent low-dimensional coordinate system. For each chart there will be a low-dimensional afﬁ transform Gk ∈ R(d+1)×d ne that projects Uk into the global coordinate space. Summing over all charts, the weighted average of the projections of point yi into the low-dimensional vector space is W j (y − µ j ) 1 . x|y = ∑ G j j p j|y (y) . xi |yi = ∑ G j ⇒ u ji 1 j p j|y (yi ), (4) where pk|y (y) ∝ pk N (y; µk , Σk ), ∑k pk|y (y) = 1 is the probability that chart k generates point y. As pointed out in [8], if a point has nonzero probabilities in two charts, then there should be afﬁ transforms of those two charts that map the point to the same place in a ne global coordinate space. We set this up as a weighted least-squares problem: . G = [G1 , · · · , GK ] = arg min uki 1 ∑ pk|y (yi )p j|y (yi ) Gk Gk ,G j i −Gj u ji 1 2 . (5) F Equation (5) generates a homogeneous set of equations that determines a solution up to an afﬁ transform of G. There are two solution methods. First, let us temporarily anchor one ne neighborhood at the origin to ﬁ this indeterminacy. This adds the constraint G1 = [I, 0] . x . To solve, deﬁ indicator matrix Fk = [0, · · · , 0, I, 0, · · · , 0] with the identity mane . trix occupying the kth block, such that Gk = GFk . Let the diagonal of Pk = diag([pk|y (y1 ), · · · , pk|y (yN )]) record the per-point posteriors of chart k. The squared error of the connection is then a sum of of all patch-to-anchor and patch-to-patch inconsistencies: . E =∑ (GUk − k U1 0 2 )Pk P1 F + ∑ (GU j − GUk )P j Pk j=k 2 F ; . Uk = Fk Uk 1 . (6) Setting dE /dG = 0 and solving to minimize convex E gives −1 G = ∑ Uk P2 k k ∑ j=k P2 j Uk − ∑ ∑ Uk P2 P2 k 1 Uk P2 P2 U j k j k j=k U1 0 . (7) We now remove the dependence on a reference neighborhood G1 by rewriting equation 5, G = arg min ∑ j=k (GU j − GUk )P j Pk G 2 F = GQ 2 F = trace(GQQ G ) , (8) . where Q = ∑ j=k U j − Uk P j Pk . If we require that GG = I to prevent degenerate solutions, then equation (8) is solved (up to rotation in coordinate space) by setting G to the eigenvectors associated with the smallest eigenvalues of QQ . The eigenvectors can be computed efﬁ ciently without explicitly forming QQ ; other numerical efﬁ ciencies obtain by zeroing any vanishingly small probabilities in each Pk , yielding a sparse eigenproblem. A more interesting strategy is to numerically condition the problem by calculating the trailing eigenvectors of QQ + 1. It can be shown that this maximizes the posterior 2 p(G|Q) ∝ p(Q|G)p(G) ∝ e− GQ F e− G1 , where the prior p(G) favors a mapping G whose unit-norm rows are also zero-mean. This maximizes variance in each row of G and thereby spreads the projected points broadly and evenly over coordinate space. The solutions for MAP charts (equation (5)) and connection (equation (8)) can be applied to any well-ﬁ tted mixture of gaussians/factors1 /PCAs density model; thus large eigenproblems can be avoided by connecting just a small number of charts that cover the data. 1 We thank reviewers for calling our attention to Teh & Roweis ([11]— in this volume), which shows how to connect a set of given local dimensionality reducers in a generalized eigenvalue problem that is related to equation (8). LLE, n=5 charting (projection onto coordinate space) charting best Isomap LLE, n=6 LLE, n=7 LLE, n=8 random subset of local charts XYZ view LLE, n=9 LLE, n=10 XZ view data (linked) embedding, XY view XY view original data reconstruction (back−projected coordinate grid) best LLE (regularized) Figure 2: The twisted curl problem. L EFT: Comparison of charting, I SO M AP, & LLE. 400 points are randomly sampled from the manifold with noise. Charting is the only method that recovers the original space without catastrophes (folding), albeit with some shear. R IGHT: The manifold is regularly sampled (with noise) to illustrate the forward and backward projections. Samples are shown linked into lines to help visualize the manifold structure. Coordinate axes of a random selection of charts are shown as bold lines. Connecting subsets of charts such as this will also give good mappings. The upper right quadrant shows various LLE results. At bottom we show the charting solution and the reconstructed (back-projected) manifold, which smooths out the noise. Once the connection is solved, equation (4) gives the forward projection of any point y down into coordinate space. There are several numerically distinct candidates for the backprojection: posterior mean, mode, or exact inverse. In general, there may not be a unique posterior mode and the exact inverse is not solvable in closed form (this is also true of [8]). Note that chart-wise projection deﬁ a complementary density in coordinate space nes px|k (x) = N (x; Gk 0 1 , Gk [Id , 0]Λk [Id , 0] 0 0 0 Gk ). (9) Let p(y|x, k), used to map x into subspace k on the surface of the manifold, be a Dirac delta function whose mean is a linear function of x. Then the posterior mean back-projection is obtained by integrating out uncertainty over which chart generates x: y|x = ∑ pk|x (x) k µk + Wk Gk I 0 + x − Gk 0 1 , (10) where (·)+ denotes pseudo-inverse. In general, a back-projecting map should not reconstruct the original points. Instead, equation (10) generates a surface that passes through the weighted average of the µi of all the neighborhoods in which yi has nonzero probability, much like a principal curve passes through the center of each local group of points. 5 Experiments Synthetic examples: 400 2 D points were randomly sampled from a 2 D square and embedded in 3 D via a curl and twist, then contaminated with gaussian noise. Even if noiselessly sampled, this manifold cannot be “ unrolled” without distortion. In addition, the outer curl is sampled much less densely than the inner curl. With an order of magnitude fewer points, higher noise levels, no possibility of an isometric mapping, and uneven sampling, this is arguably a much more challenging problem than the “ swiss roll” and “ s-curve” problems featured in [12, 9, 8, 1]. Figure 2LEFT contrasts the (unique) output of charting and the best outputs obtained from I SO M AP and LLE (considering all neighborhood sizes between 2 and 20 points). I SO M AP and LLE show catastrophic folding; we had to change LLE’s b. data, yz view c. local charts d. 2D embedding e. 1D embedding 1D ordinate a. data, xy view true manifold arc length Figure 3: Untying a trefoil knot ( ) by charting. 900 noisy samples from a 3 D-embedded 1 D manifold are shown as connected dots in front (a) and side (b) views. A subset of charts is shown in (c). Solving for the 2 D connection gives the “ unknot” in (d). After removing some points to cut the knot, charting gives a 1 D embedding which we plot against true manifold arc length in (e); monotonicity (modulo noise) indicates correctness. Three principal degrees of freedom recovered from raw jittered images pose scale expression images synthesized via backprojection of straight lines in coordinate space Figure 4: Modeling the manifold of facial images from raw video. Each row contains images synthesized by back-projecting an axis-parallel straight line in coordinate space onto the manifold in image space. Blurry images correspond to points on the manifold whose neighborhoods contain few if any nearby data points. regularization in order to coax out nondegenerate (>1 D) solutions. Although charting is not designed for isometry, after afﬁ transform the forward-projected points disagree with ne the original points with an RMS error of only 1.0429, lower than the best LLE (3.1423) or best I SO M AP (1.1424, not shown). Figure 2RIGHT shows the same problem where points are sampled regularly from a grid, with noise added before and after embedding. Figure 3 shows a similar treatment of a 1 D line that was threaded into a 3 D trefoil knot, contaminated with gaussian noise, and then “ untied” via charting. Video: We obtained a 1965-frame video sequence (courtesy S. Roweis and B. Frey) of 20 × 28-pixel images in which B.F. strikes a variety of poses and expressions. The video is heavily contaminated with synthetic camera jitters. We used raw images, though image processing could have removed this and other uninteresting sources of variation. We took a 500-frame subsequence and left-right mirrored it to obtain 1000 points in 20 × 28 = 560D image space. The point-growth process peaked just above d = 3 dimensions. We solved for 25 charts, each centered on a random point, and a 3D connection. The recovered degrees of freedom— recognizable as pose, scale, and expression— are visualized in ﬁ gure 4. original data stereographic map to 3D fishbowl charting Figure 5: Flattening a ﬁ shbowl. From the left: Original 2000×2D points; their stereographic mapping to a 3D ﬁ shbowl; its 2D embedding recovered using 500 charts; and the stereographic map. Fewer charts lead to isometric mappings that fold the bowl (not shown). Conformality: Some manifolds can be ﬂattened conformally (preserving local angles) but not isometrically. Figure 5 shows that if the data is ﬁ nely charted, the connection behaves more conformally than isometrically. This problem was suggested by J. Tenenbaum. 6 Discussion Charting breaks kernel-based NLDR into two subproblems: (1) Finding a set of datacovering locally linear neighborhoods (“ charts” ) such that adjoining neighborhoods span maximally similar subspaces, and (2) computing a minimal-distortion merger (“ connection” ) of all charts. The solution to (1) is optimal w.r.t. the estimated scale of local linearity r; the solution to (2) is optimal w.r.t. the solution to (1) and the desired dimensionality d. Both problems have Bayesian settings. By ofﬂoading the nonlinearity onto the kernels, we obtain least-squares problems and closed form solutions. This scheme is also attractive because large eigenproblems can be avoided by using a reduced set of charts. The dependence on r, like trusted-set methods, is a potential source of solution instability. In practice the point-growth estimate seems fairly robust to data perturbations (to be expected if the data density changes slowly over a manifold of integral Hausdorff dimension), while the use of a soft neighborhood partitioning appears to make charting solutions reasonably stable to variations in r. Eigenvalue stability analyses may prove useful here. Ultimately, we would prefer to integrate r out. In contrast, use of d appears to be a virtue: Unlike other eigenvector-based methods, the best d-dimensional embedding is not merely a linear projection of the best d + 1-dimensional embedding; a unique distortion is found for each value of d that maximizes the information content of its embedding. Why does charting performs well on datasets where the signal-to-noise ratio confounds recent state-of-the-art methods? Two reasons may be adduced: (1) Nonlocal information is used to construct both the system of local charts and their global connection. (2) The mapping only preserves the component of local point-to-point distances that project onto the manifold; relationships perpendicular to the manifold are discarded. Thus charting uses global shape information to suppress noise in the constraints that determine the mapping. Acknowledgments Thanks to J. Buhmann, S. Makar, S. Roweis, J. Tenenbaum, and anonymous reviewers for insightful comments and suggested “ challenge” problems. References [1] M. Balasubramanian and E. L. Schwartz. The IsoMap algorithm and topological stability. Science, 295(5552):7, January 2002. [2] C. Bregler and S. Omohundro. Nonlinear image interpolation using manifold learning. In NIPS–7, 1995. [3] D. DeMers and G. Cottrell. Nonlinear dimensionality reduction. In NIPS–5, 1993. [4] J. Gomes and A. Mojsilovic. A variational approach to recovering a manifold from sample points. In ECCV, 2002. [5] T. Hastie and W. Stuetzle. Principal curves. J. Am. Statistical Assoc, 84(406):502–516, 1989. [6] G. Hinton, P. Dayan, and M. Revow. Modeling the manifolds of handwritten digits. IEEE Trans. Neural Networks, 8, 1997. [7] N. Kambhatla and T. Leen. Dimensionality reduction by local principal component analysis. Neural Computation, 9, 1997. [8] S. Roweis, L. Saul, and G. Hinton. Global coordination of linear models. In NIPS–13, 2002. [9] S. T. Roweis and L. K. Saul. Nonlinear dimensionality reduction by locally linear embedding. Science, 290:2323–2326, December 22 2000. [10] A. Smola, S. Mika, B. Schölkopf, and R. Williamson. Regularized principal manifolds. Machine Learning, 1999. [11] Y. W. Teh and S. T. Roweis. Automatic alignment of hidden representations. In NIPS–15, 2003. [12] J. B. Tenenbaum, V. de Silva, and J. C. Langford. A global geometric framework for nonlinear dimensionality reduction. Science, 290:2319–2323, December 22 2000.

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