cvpr cvpr2013 cvpr2013-405 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Bo Wang, Zhuowen Tu
Abstract: With the increasing availability of high dimensional data and demand in sophisticated data analysis algorithms, manifold learning becomes a critical technique to perform dimensionality reduction, unraveling the intrinsic data structure. The real-world data however often come with noises and outliers; seldom, all the data live in a single linear subspace. Inspired by the recent advances in sparse subspace learning and diffusion-based approaches, we propose a new manifold denoising algorithm in which data neighborhoods are adaptively inferred via sparse subspace reconstruction; we then derive a new formulation to perform denoising to the original data. Experiments carried out on both toy and real applications demonstrate the effectiveness of our method; it is insensitive to parameter tuning and we show significant improvement over the competing algorithms.
Reference: text
sentIndex sentText sentNum sentScore
1 com l Abstract With the increasing availability of high dimensional data and demand in sophisticated data analysis algorithms, manifold learning becomes a critical technique to perform dimensionality reduction, unraveling the intrinsic data structure. [sent-5, score-0.348]
2 For simple data such as well-aligned faces, PCA techniques are shown to be well applicable; for data of high complexity, a single linear subspace then may not be sufficient to capture the intrinsic data structure. [sent-12, score-0.376]
3 mixture of PCA [22]) and product of linear subspaces (generalized principal component analysis [24]) are proposed. [sent-15, score-0.147]
4 In more general cases of manifold learning, nonlinearality is often assumed to preserve the lower dimension embedding [18, 21, 16, 3]. [sent-16, score-0.245]
5 Both the ISOMAP [21] and LLE [16] make an assumption of local Euclidean space and try to preserve similar neighborhood structures in the embedded spaces. [sent-17, score-0.091]
6 The spectral manifold learning approaches [3] compute the Eigen-maps for the graph Laplacian to directly perform non-linear dimensionality reduction. [sent-18, score-0.38]
7 From a different angle, diffusion maps [8] defines diffusion distances between data samples; an input similarity measurement is improved through a diffusion process. [sent-19, score-0.279]
8 Diffusion-based methods [8, 14] essentially perform denoising on the similarity metric on which spectral manifold learning methods can be applied. [sent-20, score-0.699]
9 The idea is to explicitly construct a new embedding space with a corresponding metric which is more faithful to the manifold structure and hence induces a better distance/similarity measure. [sent-21, score-0.245]
10 All the manifold learning methods above depend on the construction of a good neighborhood graph, either explicitly [21, 16] or implicitly [3, 8, 25, 26]. [sent-24, score-0.344]
11 Essentially, the neighborhood or k-nearest neighborhood definition forms a sparse sample assumption. [sent-26, score-0.2]
12 Recent seminal work in compressive sensing [4] demonstrates the significance of having the sparse assumption in reconstruction, if the data is indeed sparse. [sent-27, score-0.074]
13 The idea of local sparse data reconstruction [27] has achieved some significant improvement in codebook-based image classification task. [sent-28, score-0.128]
14 Moreover, as discussed before, data manifolds are often composed of multiple subspaces. [sent-29, score-0.098]
15 The sparse subspace clustering algorithm (SSC) [9] provides a tractable way of learning subspaces with the in-class sparse self-reconstruction assumption. [sent-30, score-0.709]
16 Here, inspired by sparse subspace learning and diffusion-based approaches, we propose a new manifold denoising algorithm in which data neighborhoods are adaptively inferred via sparse subspace reconstruction; we then 444666668 derive a new formulation to perform denoising to the original data. [sent-31, score-1.899]
17 In the experiments, we show significant improvements over the existing manifold denoising algorithms. [sent-32, score-0.592]
18 Next, we review related work in manifold denoising and then give the detailed formulation of our algorithm. [sent-34, score-0.592]
19 Related Work Although manifold learning is an important topic in machine learning, manifold denoising has received relatively less attention. [sent-36, score-0.873]
20 Unlike the traditional manifold learning and dimension reduction algorithms, the goal of a denoising algorithm is to obtain a cleaner output in the same dimensionality as the input data. [sent-37, score-0.702]
21 The outputs of a denoising algorithm can be further used to facilitate tasks such as feature selection, clustering, and classification. [sent-38, score-0.379]
22 The simplest denoising technique is PCA which adopts a strategy of projection and backward mapping. [sent-39, score-0.347]
23 The idea behind them is that noises in the samples follows a normal distribution and averaging local manifolds can reduce the noise degree. [sent-43, score-0.305]
24 These methods obtain denoised outputs by reversing a diffusion process of which graph Laplacian is the generator. [sent-48, score-0.434]
25 However, the problem associated with them is that graph Laplacian only captures global gradient distribution of the manifolds and it fails to reduce the local noise inherited in sub-manifolds which are important to pattern discovery such as clustering and classification. [sent-49, score-0.255]
26 Lastly, both categories face the same challenge to choose the number of neighbors when either inferring the local manifolds or choosing local reliable clusters. [sent-50, score-0.202]
27 In this paper, we try to overcame these problems using two strategies: 1) We use a recent subspace learning algorithm to capture the subspace structures which indicate a good potential clusters and also a reliable similarity measure. [sent-51, score-0.791]
28 2) We design a local sparse regularization term which shares several common algebraic properties with graph Laplacian but with the difference in emphasizing the local subspace distributions. [sent-52, score-0.45]
29 , xn} where each xi ∈ Rm and m indicaast Ses =the { input data dim}e wnsihoerne. [sent-60, score-0.087]
30 The objective of manifold denoising in this paper ∈is R therefore to compute a new denoised data matrix X? [sent-65, score-0.968]
31 ∈ Rm×n to facilitate clustering and classification in furtX? [sent-66, score-0.112]
32 Each data sample xi here is associated with a label indicator vector yi ∈ {0, 1}C such that yi (k) = 1if xi belongs to the k-th subspace, }and otherwise yi (k) = 0. [sent-70, score-0.22]
33 ; ynT]T ∈ {0, 1}n×C is used to represent a clustering scheme. [sent-74, score-0.08]
34 , hxen }ve, ratnidce tshe V edges Ep are weights adtean soatmedby an n n similarity mnda ttrhiex Wdg ews Eith a Wij indicating ttehed similarity b netw siemeinl xi yan mda xj . [sent-79, score-0.212]
35 W Ni represents a set of xi’s neighbors in graph G, excluding xi and Ki = |Ni |. [sent-80, score-0.176]
36 Background: Sparse Subspace Clustering Sparse subspace clustering(SSC) [9] builds clustering on sparse learning with applications demonstrated in motion segmentation [9]. [sent-83, score-0.534]
37 The motivation of SSC is simple in that each data sample xi can be expressed as a? [sent-84, score-0.087]
38 linear combination of all the data within the cluster, xi = ? [sent-85, score-0.087]
39 =eairn a sparse coefficient matrix A ∈ Rn×n such that aij ? [sent-90, score-0.128]
40 The sparse matrix A has two important usages: 1) For each datum xi, its neighbors Ni can be easily inferred by the nonzero ele,m itesn ntse ginh bthoer si -Nth column of A. [sent-103, score-0.284]
41 Therefore, we don’t need to specify the number of nearest neighbors for each datum as most of other methods do. [sent-104, score-0.098]
42 In addition, we can theoretically guarantee that all chosen neighbors of xi belong to the same subspace with xi. [sent-105, score-0.516]
43 1 Motivations A straightforward idea behind a good denoising algorithm is that, for each subspace, the new denoised data still reserves the intrinsic allocation of subspaces and also can be easily reconstructed by simple PCA algorithm. [sent-111, score-0.912]
44 Hence, we proposed an optimization framework that aims to minimize both subspace coherence error and reconstruction er- ror. [sent-112, score-0.588]
45 An good denoised output should satisfy two metrics: 1) Points belonging to the same subspaces should be still in the same subspaces; 2) They are close to what they were. [sent-117, score-0.423]
46 The first metric can be achieved by minimizing the subspace coherence error while the second is satisfied by minimizing the reconstruction error. [sent-118, score-0.56]
47 We can see that, those noisy points are moved towards the center of the subspace they belong to and at the same time, we reserve the positions of most of the points. [sent-121, score-0.344]
48 2 Subspace Coherence Error Instead of using partition matrix Y directly, we adopted a Scaled Partition Matrix, G, such that G= Y(YTY)−21 = [g1, g2, . [sent-127, score-0.095]
49 gik = Yik/√nk, w ≤he kre nk is) itsh eth esiz ke- ohf c tohlek-th subspace, can be regarded as the confidence that xi is assigned to the k-th subspace. [sent-134, score-0.168]
50 is The principle of local denoising is that the subspace assignments in the neighborhood of each patient should be as smooth as possible. [sent-136, score-0.793]
51 Specifically, it assumes that the cluster indicator value at xi should be well inferred by the local neighbors Ni. [sent-137, score-0.287]
52 So given a similarity graph G = (V, E), we extract a subgraph Gi = (Vi, Ei) such that Vi = Ni ∪ xi aenxtdr aEcit =a sEub(Vgir)ap. [sent-138, score-0.223]
53 hT hGe similarity m) sautrcixh athsasotc Viat=ed Nwith∪ t hxe subgraph G Ei( iVs Wi = W(Ei). [sent-139, score-0.104]
54 Using the label diffusion algorithm h[ 2G9], we can Wrec(Eonstruct a virtual label indicator vector pk such that pk = (1 − α)(I − αLi)−1g(kVi), 1 ≤ k ≤ C, (2) where α is a constant (0 < α < 1) and g(kVi) is the scaled subspace indicator vector on the subgraph (VVi. [sent-140, score-0.763]
55 Li represents tshueb snpoarcmea ilnizdeicda ttroarn vseictitoonr mona ttrhiex fuobrg trhaep subgraph Wi, i. [sent-141, score-0.112]
56 g ik = pk (Ki + 1) is the estimated cluster assignm+en 1t ainnddi gc? [sent-148, score-0.148]
57 1−βiβ(i(Kli)+1)0 ifot xhjeriwsi tshee l-th element in Ni (5) This sparse matrix B represents a linear relationship that g? [sent-158, score-0.128]
58 1 = Trace(GT(I Denote Z = (I − − B)T(I B)T(I − − B)G) (6) B), then our subspace as- signment er Zesul =ts can b Be )obt(Iain −ed B by solving rt sheub following optimization problems: G∈mRnin×C Trace(GTZG) s. [sent-187, score-0.344]
59 444676880 The idea behind the denoising principle is that the denoised features should be coherent to the subspace assignments. [sent-192, score-1.042]
60 So for each new feature f, we define the coherence error CE as follows: ? [sent-193, score-0.162]
61 Nj From (8), we can see that the coherence error (CE) evaluates two aspects : 1) The extent of match between the feature and the local structure of the image manifold; 2)The fitness between the feature and the subspace assignments. [sent-200, score-0.546]
62 The smaller CE is, the more relevant to the subspace structures the feature is. [sent-201, score-0.372]
63 Therefore, one objective can be defined as the coherence error for the whole denoised data X? [sent-220, score-0.484]
64 3 Subspace Reconstruction Error Another direction of denosing is to make sure the new data points are not far away from the initial data manifolds which are inferred by traditional data reconstruction tech- × nique PCA. [sent-236, score-0.21]
65 We look at each data point xi and its neighbors Ni, and assume them to be random samples from a linear subspace, approximating a local manifold around xi (To be exact, it can be guaranteed that neighbours inferred by sparse subspace clustering(1) share the same subspace [9]). [sent-237, score-1.329]
66 Ni, we can approximate it with PCA: Ri = UiUiTXi(I − llT/(1 + ki)) + XillT/(1 + ki) (11) where Ui ∈ Rm×di denote the di principal components from PCA ∈and R ki = |Ni |. [sent-239, score-0.218]
67 The number of the principal component di can b=e e s|Ntim|a. [sent-240, score-0.076]
68 Wf teh eest pimrinacteip dail by setting the variance of the chosen di principal components account for at least 90% the whole variance. [sent-242, score-0.076]
69 xal reconstructed data for the j-th neighbor for xi in (11). [sent-255, score-0.087]
70 , Sn] be a 1 n block matrix where each block Si ∈ Rn×(1+]ki b)e corresponds ctok mthea neighborhood indicator m∈at Rrix. [sent-260, score-0.163]
71 Therefore we can express the reconstruction error in matrix form: LRE = ? [sent-262, score-0.154]
72 Hence the overall error can be expressed as a combination of subspace coherence error and subspace reconstruction error: L(λ) = LCE + λLRE = Trace(ZX? [sent-275, score-0.95]
73 ur algorithm depends on the neighborhood graphs, so our algorithm is potential to be generalized into an iterative way. [sent-286, score-0.09]
74 ∗ (λ) in (15), the sparse subspace matrix can be refined. [sent-288, score-0.472]
75 Analysis In this section, we provide some preliminary analysis of our method and its relationship with some existing denoising algorithms. [sent-295, score-0.347]
76 (1); Step2: Compute subspace coherence error by constructing Z from eqn. [sent-308, score-0.506]
77 Another line of denoising algorithms tends to use Laplacian-Bertrami operator on smooth manifolds [13, 11]. [sent-311, score-0.484]
78 They often use Normalized Laplacian L+ = I D−21 WD−21 where D is the − degree dm Laatrpixla coiaf nth Le similarity matrix W. [sent-313, score-0.093]
79 The key difference from ours is that, Laplacian regularization only considers the global structure while our algorithm focus on local subspace structure. [sent-314, score-0.344]
80 As pointed by [11], pure Laplacian denoising tends to be over smooth and local structure is more helpful to reduce the effect of noise for applications such as clustering and classification afterwards. [sent-315, score-0.511]
81 We can see that, our SSD can successfully remove the added noise and therefore facilitate the applications afterwards. [sent-329, score-0.077]
82 We can see that, when noise is small, one round of SSD is enough to clean the data, while iterative SSD can obtain cleaner data with large noise. [sent-343, score-0.138]
83 And we compare our method to a few existing manifold denoising algorithms : 1) Generalized Blurring Mean Shift (GBMS) [5]; 2) Manifold Blurring Mean Shift (MBMS) [28]; 3) Manifold Denoising (MD) [13]; 4) Locally Linear Denoising (LLD) [11]. [sent-348, score-0.592]
84 Firstly, we perform clustering on three benchmark face recognition datasets (ORL, Yale, Yale B). [sent-368, score-0.127]
85 We use Normalized Mutual Information (NMI) [19] to evaluate the performance of clustering results. [sent-369, score-0.08]
86 We also show some illustrative of the denoised images from YaleB in Fig. [sent-452, score-0.371]
87 It is noticeable that SSD is capable to adjust the face orientations and illuminations to make the face image clearer and discriminative. [sent-454, score-0.094]
88 The upper panel is the initial faces while the lower panel is the denoised output. [sent-457, score-0.526]
89 Instead, our algorithm can infer the cluster distribution since we aim to minimize the subspace coherence error. [sent-471, score-0.527]
90 However, our method reserves the basic topology of the subspace distribution due to the objective of minimizing the reconstruction error. [sent-473, score-0.479]
91 With denoising algorithms, the classification accuracy can be improved. [sent-503, score-0.347]
92 In this case, other denoising algorithms failed to improve the data features. [sent-511, score-0.347]
93 In addition, we show several examples of denoised images compared with the original ones in Fig. [sent-513, score-0.322]
94 The most obvious change is that the denoised ones look smoother and easier to identify. [sent-515, score-0.353]
95 Conclusion In this paper, we have introduced a general manifold learning algorithm which takes the advantages of subspace 4We downloaded the data from http://www. [sent-518, score-0.625]
96 The upper panel is the initial faces while the lower panel is the denoised output. [sent-528, score-0.526]
97 It is observed that SSD reserves the distinctive style of each digit or letter while smoothing a few minor outliers, e. [sent-529, score-0.119]
98 Our algorithm requires nearly no parameter tuning across different datasets and we show significant improvement over the competing methods on various applications such as clustering and classification. [sent-533, score-0.08]
99 The lower panel are the results using SVM with linear kernels. [sent-594, score-0.086]
100 The baseline refers the results on raw features without any denoising algorithm. [sent-595, score-0.347]
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