Robust Component Analysis via HQ Minimization

Transcription

1 Robust Component Analysis via HQ Minimization Ran He, Wei-shi Zheng and Liang Wang

2 Outline Overview Half-quadratic minimization principal component analysis Robust principal component analysis Robust principal component analysis Low-rank matrix recovery Summary

3 Overview of HQ minimization d n X R β R n d y R st sample n st sample st sample st feature st feature d st feature

4 Overview of HQ minimization T T k k k [g,,g k ] indicates a graph of {x k } T T k k k+ k g x = x, g = [0,..., 0,, 0,...0] g x = x x, g = [0,...,0,,,...0] min Ax y + φ( g x) x k T k Convex Non-convex k = k k + ϕm k e φ( x ) min( x e ) ( e ) k The additive form k = k k + ϕm k p φ( x ) min{ p x ( p )} k The multiplicative form When auxiliary variables e and p are given, the original problem becomes a quadratic problem. [Geman and Reynolds. TPAMI, 99 ][Geman and Yang, TIP, 995]

5 Overview of HQ minimization Half-quadratic minimization min Ax y + λ φ( x i ) x i The multiplicative form λ p t i i i i + ϕ pi ( px ( )) Alternate minimization t i δ( x ) min Ax y + Q ( x, p) x = M The additive form λ p t i i i i + ϕ i (( x p ) ( p )) Alternate minimization t i δ( x ) min Ax y + Q ( x, p) x = A

6 Overview of HQ minimization Huber loss function φ λ H () v v / v λ λ v λ / v > λ δ M H v () v = λ/ v v > The multiplicative form λ λ δ A H () v 0 v = v λsign( v) v > The additive form Soft-thresholding function λ λ λ φh () v = min( p p v ) + λ p

7 Overview of HQ minimization data matrix d n X R orthonormal basis d m U R coefficient matrix m n V R st sample st sample n n d m F i i ( ij jk ki ) i= i= j= k= X UV = X Uv = x u v

8 Outline Overview Half-quadratic minimization Principal component analysis Robust principal component analysis Robust principal component analysis Low-rank matrix recovery Summary

9 Overview of PCA Principal component analysis x i v i u PCA seeks a principal subspace (denoted by the magenta line), such that the orthogonal projection of the data points (red dots) onto this subspace maximizes the variance of the projected points (green dots). An alternative definition of PCA is based on minimizing the sum-of-squares of the projection errors, indicated by the blue lines. [Bishop 006] [C. M. Bishop. Pattern recognition and machine learning. Springer, 006]

10 Overview of PCA Principal component analysis d X = [ x,..., x ] R d U = [ u,..., u ] R n m is a data set of samples is a projection matrix m n = [,..., n ] is the projection coordinates under the projection matrix U. V v v R The mean (or center) of X n m μ = n i x i [C. M. Bishop. Pattern recognition and machine learning. Springer, 006]

11 Overview of PCA Maximum variance formulation of PCA The variance of the projected data n i T T T i μ = ( u x u ) u Su where S is the data covariance matrix S = ( xi μ)( xi μ) T n i [C. M. Bishop. Pattern recognition and machine learning. Springer, 006]

12 Overview of PCA Maximum variance formulation of PCA Lagrange multiplier and the normalization T condition u u = T T + λ u u u Su ( ) By setting the derivative with respect to u equal to zero, we see that this quantity will have a stationary point that Su = λu T u Su = λ [C. M. Bishop. Pattern recognition and machine learning. Springer, 006]

13 Overview of PCA Minimum error formulation of PCA Matrix X UV F vector min x ( μ + Uv ) UV,, μ i i i element m min x ( μ + v u ) UV,, μ ij ij ip pj i j p= Mean square error (MSE) [R. He et al. Neurocomputing, 00]

14 Overview of PCA Graph embedding formulation of PCA T min Tr ( U X ( I W ) X U ) T U U I = where I is the identity matrix, W is a nxn matrix whose elements are all equal to /n. T [S. Yan et al. IEEE TPAMI, 007]

15 Overview of PCA Two problems PCA is sensitive to outliers albeit it is robust to small white noise. Choosing the number of components

16 Outline Overview Half-quadratic minimization Principal component analysis Robust principal component analysis Robust principal component analysis Low-rank matrix recovery Summary

17 Overview of RPCA Self-organizing rules Gibbs distribution energy function A heuristic approach min X A A n min Xi Ai A i = n d min Xij A A i i = = ij L. Xu and A. Yuille. Robust principal component analysis by self-organizing rules based on statistical physics approach, IEEE TNN, 995 A. Baccini et al. A L-norm PCA and a heuristic approach, Ordinal and Symbolic Data Analysis, 996

18 Overview of RPCA 50% occlusion n d m min φ( x u v ) ij jk ki UV, i = j = k = i: each sample j: each dimension k: each basis φ: k: Robust estimator M. Black and A. Jepson. Eigentracking: Robust Matching and Tracking of Articulated Objects Using a View-Based Representation. IJCV, 998

19 Overview of RPCA min θ n i= x i θ p L p M-estimate of location PCA RPCA N. Locantore et al. Robust PCA for functional data, Test, 999

20 Overview of RPCA two objects recovered st recovered st original image random noise 50% occlusion Recognition under Dense corruption A robust hypothesizeand-test paradigm using subsets of image points A selection procedure based on the Minimum Description Length principle A. Leonardis and H. Bischof. Robust recognition using Eigenimages. Computer Vision and Image Understanding, 000

21 Overview of RPCA n d m min φ( x μ u v ) ij j jk ki UV,, μ i = j = k = data PCA RPCA outlier weight i: each sample j: each dimension k: each basis φ: k: M-estimator F. De La Torre and M. Black. A framework for robust subspace learning. IJCV, 003 F. De La Torre and M.J. Black. Robust Parameterized Component Analysis: Theory and Applications to D Facial Appearance Models. CVIU, 003

22 Overview of RPCA min X UV UV, Factorizing a synthetic 30x30 matrix. Left: Missing data (o) and outliers ( ) in synthetic matrix; Middle: Initial weights from least L norm.right: Final weights after weighted L norm minimization. min W ( X UV ) UV F, Huber M-estimator is used to approximate the L norm Q. Ke and T. Kanade, Robust L norm factorization in the presence of outliers and missing data by alternative convex programming, CVPR, 005.

23 Overview of RPCA n T T T φ Xi Xi Xi UU Xi UV, i = min ( ) Huber M-estimator L M-estimator rotational invariant C. Ding et al. R-PCA: rotational invariant l-norm principal component analysis for robust subspace factorization, ICML, 006

24 Overview of RPCA occluded data compatible points selected pixels reconstructed image S. Fidler et al. Combining Reconstructive and Discriminative Subspace Methods for Robust Classification and Regression by Subsampling. IEEE TPAMI, 006.

25 Overview of RPCA original data subspace outliers PCA Robust PCA Robust PCA Robust PCA S. Danijel et al, Weighted and robust learning of subspace representations. Pattern Recognition, 007.

26 Overview of RPCA Segmentation α-pca n = T i i min φ ( UU ( x ) x ) U i Left: the ground truth Right: standard PCA (white) and α-pca (black) a twice-differentiable function J. Iglesias et al. A family of PCA for dealing with outliers. MICCAI, 007.

27 Overview of RPCA n m d min ujkx U i k j = = = ji outlier PCA-L PCA R-PCA i: each sample j: each dimension k: each basis μ = N. Kwak, Principal component analysis based on L-norm maximization, IEEE TPAMI, 008

28 Overview of RPCA min rank ( A ) s. t. A = X ij ij A min A st.. A ij = X ij The functional A * is the nuclear norm of the matrix X, which is the sum of its singular values. Singular value shrinkage operator A * min A F A Y + λ A * J. Cai et al. A singular value thresholding algorithm for matrix completion A. Ganesh et al. Fast algorithms for recovering a corrupted low-rank matrix. CAMSAP, 008.

29 Overview of RPCA Different types of PCA outliers Compute a robust center and covariance matrix in this k-dimensional subspace by applying the reweighted Minimum Covariance Determinant (MCD) estimator M. Hubert et al. Robust PCA for skewed data and its outlier map. Computational Statistics and Data Analysis, 009

30 Overview of RPCA min A + λ E st.. A + E = X A * 0 min A + λ E st.. A + E = X A * Left: video data; Middle: recovered background; Right: detected sparse errors * + λν + min ν A E X - A - E F AE, J. Wright et al. Robust principal component analysis: Exact recovery of corrupted low-rank matrices via convex optimization, 009. Z. Lin et al. Fast convex optimization algorithms for exact recovery of a corrupted lowrank matrix, 009.

31 Overview of RPCA E. J. Candes et al. Robust principal component analysis? Journal of the ACM, 00. E. J. Candes and Y. Plan. Matrix completion with noise. Proceedings of the IEEE, 00, 98(6): A. Ganesh et al. Dense error correction for low-rank matrices via principal component pursuit. the Computing Research Repository, 00. E. J. Candes and T. Tao. The power of convex relaxation: Near-optimal matrix completion. IEEE TIT,

32 Overview of RPCA Renyi quadratic entropy by Parzen window method 4-D data structure Ran He et al. Principal component analysis based on nonparametric maximum entropy. Neurocomputing, 00

33 Overview of RPCA min A * + λφ (E) st.. A + E = X min A * + λφ( X A) A A Φ ( E) = φ( Eij ) φ: k: Robust estimator ij R. He et al. Recovery of corrupted low rank matrices via half-quadratic based nonconvex minimization. IEEE CVPR, 0. Ran He et al. Recovery of Corrupted Low-rank Matrix by Implicit Regularizer. Submitted to IEEE TPAMI, 0.

34 Overview of RPCA - T T H r c d F N min A + αφ (E) s.t. E= C AC -[ λ ] x & λ = FN A * When using L -error terms, these methods often oscillate around the true optimum and only slowly converge towards this optimum due to the non-differentiability of the L -norm at 0. The Huber cost function is not only a more appropriate model for outliers and inliers contaminated by Gaussian noise it also leads to less oscillations and hence faster convergence R. Angst. The Generalized Trace-Norm and its Application to Structurefrom-Motion Problems. ICCV, 0.

35 Overview of RPCA * * = A * Tr ( AA ) ( AA ) W A F t+ arg min ( t ), and t+ ( t+ ( t+ ) * ) = = ϕ( A) = μ F A W A W A A The minimization can be reformulated as a weighted least squares problem with linear constraints; each of the updates for A t+ and W t+ can then be performed explicitly M. Fornasier et al. Low rank matrix recovery via iteratively reweighted least squares minimization. SIAM Journal on Optimization, 0.

36 Overview of RPCA Summary Robust PCAs have been widely used in tracking, matrix complement, robust recognition, and etc since 998. L-norm, Lp-norm and other robust M-estimators have been used to improve the robustness of PCA.

37 Outline Overview Robust principal component analysis PCA and outliers A general framework Robust PCA algorithms Low-rank matrix recovery Summary

38 PCA and outliers Mean square error is sensitive to outliers In robust statistics, outliers are those data points that are significantly different from the original data points. PCA minimizes the sum of squared errors, which is prone to the presence of outliers, because large errors squared dominate the sum.

39 PCA and outliers Outliers mean vector outliers outliers multimodal distributions

40 Robust PCA Outliers mean vector Robust estimation of mean vector [R. He et al. TIP, 0]

41 PCA and outliers data matrix d n X R orthonormal basis d m U R coefficient matrix m n V R n n d m d n m F i i ( ij jk ki ) ( ij jk ki ) i= i= j= k= j= i= k= X UV = X Uv = x u v = x u v matrix vector element

42 PCA and outliers data matrix d n X R X UV F Each entry n d m φ( x u v ) ij jk ki i= j= k= n i= Each sample d j = φ( ( x u v ) ) j ij k jk ki Each feature φ( ( x u v ) ) i ij k jk ki Structure sparsity

43 Outline Overview Robust principle component analysis PCA and outliers A general framework Robust PCA algorithms Low-rank matrix recovery Summary

44 A general framework Each data element The additive form n d m φ( x u v ) ij jk ki i= j= k= n d m xij ujkvki pij + ϕa pij UV,, P i = j = k = min ( ) ( ) The multiplicative form (weighting) n d m pij xij ujkvki + ϕm pij UV,, P i = j = k = min ( ) ( )

45 A general framework Each sample The multiplicative form (weighting) n d m p i xij ujkvki + ϕ pi UV,, P i = j = k = Each feature min ( ) ( ) The multiplicative form (weighting) n j ij k i= d n m p j xij ujkvki + ϕ pj UV,, P j = i = k = min ( ) ( ) φ( ( x u v ) ) jk ki

46 A general framework Each sample The additive form min φ( ( xij ujkvki ) ) UV j k i,,, n = n = j min φ( e ) st.. X = UV + E UV E j i φ() t = ε + t UV,, E n λ φ j j i= min X UV E ( e ) + E

47 A general framework Alternate minimization algorithm Step Calculate HQ auxiliary variable P or E according to HQ minimization functions Step Calculate corrected data X from P or E Step 3 Calculate mean vector μ X μ Solve a standard PCA problem based on,

48 Outline Overview Robust principle component analysis PCA and outliers A general framework Robust PCA algorithms Low-rank matrix recovery Summary

49 Robust PCAs - data entry Outliers each element m min x ( μ + v u ) ij ij ip pj UV, i j p = MSE M-estimator min φ( x ( μ + v u )) ij ij ip pj μ, UV, i j p = where φ(.) is a robust estimator. m

50 Robust PCAs - data entry Outliers each element Examples min φ( x ( μ + v u )) ij ij ip pj μ, UV, i j p = Geman-McClure function [F. De la Torre and M. J. Black 003] Weighted PCA [S. Danijel et al 007] L-norm [N. Kwak 008] Welsch M-estimator [R. He et al. 00] m

51 Robust PCAs - data entry Outliers each entry min φ( x ( μ + v u )) ij ij ip pj μ, UV, i j p = Welsch M-estimator Renyi quadratic entropy Parzen window estimation of distribution From Gaussian distribution to any distribution m

52 Robust PCAs - data entry Algorithm min φ( x ( μ + v u )) ij ij ip pj μ, UV, i j p = The multiplicative form Alternate minimization m m Pij xij μij + vipupj + ϕ Pij P, μ, UV, ij p = min { ( ( )) ( )} m t+ t t t ij = δ ij μ + ij ip pj p= m P ( x ( v u )) min P ( x ( μ + v u )) ij ij ij ip pj μ, UV, ij p =

53 Robust PCAs - data entry Algorithm min φ( x ( μ + v u )) ij ij ip pj μ, UV, i j p = The additive form Huber M-estimator m m xij μij + vipupj Eij + ϕ pij E, μ, UV, ij p = min {( ( ) ) ( )} min { ( x ( μ + v u ) E ) + λ E } ij ij ip pj ij E, μ, UV, ij p = m

54 Robust PCAs - data sample Outliers each sample MSE M-estimator where φ(.) is a robust estimator. Examples min x ( μ + Uv ) UV,, μ μ, UV, i Huber M-estimator [C. Ding et al. 006] Lp M-estimator [J. Iglesias et al. 007] i min φ( x ( μ + Uv )) i Welsch M-estimator [R. He et al. 0] i i i

55 Robust PCAs - data sample Outliers each sample min φ( x ( μ + Uv )) μ, UV, Welsch M-estimator Maximum correntropy criterion i i i

56 Robust PCAs - data sample Algorithm min φ( x ( μ + Uv )) μ, UV, The multiplicative form i i i t+ t t t i = δ i μ + i p ( x ( U v )) μ, Uv, t + i min p ( x ( μ + Uv )) i i i The weight vector p gives outliers some values and removes outlier during iteration.

57 Outline Overview Half-quadratic minimization Robust principal component analysis Low-rank matrix recovery Summary

58 Low-rank matrix recovery Nuclear norm minimization min A s.t. A X A * = min A F A X + μ A * where. * denotes the nuclear norm of a matrix (i.e., the sum of its singular values) and μ is a constant.

59 Low-rank matrix recovery Nuclear norm minimization min A s.t. A X A The additive form * Singular value shrinkage operator The multiplicative form = min A F () v Iteratively reweighted least squares δ A H A X + μ A 0 v = v λsign( v) v > * λ λ * * = A * Tr ( AA ) ( AA ) W A F

60 Low-rank matrix recovery Robust low rank matrix recovery min + λ s.t. + = A * E 0 A E X AE, min + s.t. A * λ E A + E = X AE, Relaxed version min A + E X + μ A + λμ E AE, F Nuclear norm: singular value shrinkage operator L norm: soft shrinkage operator *

61 Low-rank matrix recovery When matrix A is given min A + E X + μ A + λμ E AE, F min E ( X A) + λμ E E Soft shrinkage operator & Huber M-estimator λ F * Matrix A is fixed φh () v = min( p p v ) + λ p λμ φh ij ij E ij { λμ } ( X A ) = min E ( X A) + E [R. He et al. Submitted to IEEE TPAMI, 0] F

62 Low-rank matrix recovery Robust low rank matrix recovery min φ( Aij Xij ) + μ A A Low rank matrix recovery ij A X F + μ A * = Aij Xij + μ A A * ij min min ( ) A min φ( Eij ) + μ A * st.. A + E = X A ij * Mean square error is sensitive to outliers

63 Low-rank matrix recovery M-estimation min φ( Aij Xij ) + μ A A The multiplicative form ij min P ( A X ) A ( P ) AP, ij The additive form ij ij ij + μ * + ϕ ij ij min X A E A ϕ( Eij ) AE, F + μ + * ij * [R. He et al. Submitted to IEEE TPAMI, 0]

64 Low-rank matrix recovery The additive form of M-estimation min X A E A ϕ( Eij ) AE, Huber M-estimator min φ( Aij Xij ) + μ A A ij F + μ + * ij * min AE, F X A E + μ A + μλ E * [R. He et al. Submitted to IEEE TPAMI, 0]

65 Low-rank matrix recovery data matrix d n X R F X A + A * n i= j= Each element d n i= d j = φ( X A ) + A ij ij Each sample φ( ( X A ) ) + A j ij ij Each feature φ( ( A A ) ) + A i ij ij * * * Accelerated proximal gradient approach

66 Summary Robust PCA Mean square error is sensitive to outliers Outliers are data elements, data samples or features Updating mean vector is important to RPCA The two half-quadratic forms of M-estimators

67 Summary Nuclear norm regularized M-estimation min φ( D ), ij Aij + μ x ij * A min φ( Eij ) + μ x * st.. A + E = D AE, M-estimator ij, The additive form The multiplicative form Nuclear norm The additive form The multiplicative form φ( D A ) min( D A p ) ij ij ij ij ij pij ij ij ij ij ij p φ( D A ) min p ( D A ) min A A * = ij F D A + λ A W A F *

68 Summary Half-quadratic minimization Robust PCAs Low rank matrix recovery Convert complex problems to linear least squares problems Two-forms of minimization methods Error correction (the additive form) Error detection (the multiplicative form)

69 Summary Huber and Welsch M-estimators Convex Huber M-estimator It changes from an L metric to L Its dual potential function is absolute function. Nonconvex Welsch M-estimator It changes from an L metric to L and finally to L0 depending upon the distance between samples Correntropy induced metric

70 Some source codes Open Pattern Recognition Project is intended to be an open source platform for sharing algorithms of image processing, computer vision, natural language processing, pattern recognition, machine learning and the related fields. OpenPR is currently supported by the National Laboratory of Pattern Recognition, CASIA.

71 Thank You

72 Some references - robust PCA R.A. Maronna. Robust M-estimators of Multivariate Location and Scatter, Annals of Statistics, 976, 4: N.A. Campbell. Robust Procedures in Multivariate Analysis I: Robust Covariance Estimation. Applied Statistics, 980, 9: G. Li and Z. Chen. Projection-Pursuit Approach to Robust Dispersion Matrices and Principal Components Journal of the American Statistical Association, 985. L. Xu and A. Yuille. Robust principal component analysis by self-organizing rules based on statistical physics approach, IEEE TNN, 995,6: A. Baccini et al. A L-norm PCA and a heuristic approach, Ordinal and Symbolic Data Analysis, 996, : M. Black and A. Jepson. Eigentracking: Robust Matching and Tracking of Articulated Objects Using a View-Based Representation. IJCV, 998, 6(): N. Locantore et al. Robust PCA for functional data, Test, 999, 8 (): 73. C. Croux and G. Haesbroeck. Principal Components Analysis based on Robust Estimators of the Covariance or Correlation Matrix. Biometrika, 000, 87: A. Leonardis and H. Bischof. Robust recognition using eigenimages. Computer Vision and Image Understanding, 000, 78(), G. Boente et al. Influence Functions and Outlier Detection under the Common Principal Components Model: A Robust Approach. Biometrika, 00, 89:

73 Some references - robust PCA M. Hubert et al. A Fast Method for Robust Principal Components with Applications to Chemometrics. Chemometrics and Intelligent Laboratory Systems, 00, 60: 0-. M. Hubert and K. V. Branden. Robust Methods for Partial Least Squares Regression. Journal of Chemometrics, 003. M. Hubert and K. Van Driessen. Fast and Robust Discriminant Analysis. Computational Statistics and Data Analysis, 003. M. Hubert and S. Verboven. A Robust PCR Method for High-Dimensional Regressors. Journal of Chemometrics, 003, 7: F. De La Torre and M. Black. A framework for robust subspace learning. IJCV, 003, 54(-3):7 4. F. De la Torre and M.J. Black. Robust Parameterized Component Analysis: Theory and Applications to D Facial Appearance Models, CVIU, 003, 9: X. Wang and X. Tang. A unified framework for subspace face recognition. IEEE TPAMI, 004, 6(9): 8. R. A. Maronna. Principal components and orthogonal regression based on robust scales. Technometrics, 005, 47: Q. Ke and T. Kanade, Robust L norm factorization in the presence of outliers and missing data by alternative convex programming, in: CVPR, 005.

74 Some references - robust PCA M. Hubert et al. RobPCA a new approach to robust principal component analysis, Technometrics, 005, 47: S. Fidler et al. Combining Reconstructive and Discriminative Subspace Methods for Robust Classification... IEEE TPAMI, 006, 8(3): C. Ding et al. R-pca: rotational invariant l-norm principal component analysis for robust subspace factorization, in: ICML, 006. J. Iglesias et al. A family of PCA for dealing with outliers. in MICCAI, 007. S. Danijel et al, Weighted and robust learning of subspace representations. Pattern Recognition, 007, 40: N. Kwak, Principal component analysis based on L-norm maximization, IEEE TPAMI, 008, 30 (9): M. Hubert et al. Robust PCA for skewed data and its outlier map. Computational Statistics and Data Analysis, 009, 53: Y. Pang et al. L-norm based tensor analysis, IEEE TCSVT, 00, 0 (): Ran He et al. Principal component analysis based on nonparametric maximum entropy. Neurocomputing, 00, 73: R. He et al. Robust principal component analysis based on maximum correntropy criterion. IEEE TIP, 0, 0(6):

75 Some references - low rank matrix J. Cai et al. A singular value thresholding algorithm for matrix completion. preprint, urlhttp://arxiv.org/abs/ , 008. A. Ganesh et al. Fast algorithms for recovering a corrupted low-rank matrix. in International Workshop on CAMSAP, 008. J. Wright et al. Robust principal component analysis: Exact recovery of corrupted lowrank matrices via convex optimization, Journal of the ACM, vol. 4, pp. 44, 009. Z. Lin et al. The augmented Lagrange multiplier method for exact recovery of corrupted low-rank matrices, UIUC Technical Report UILU-ENG-09-5, 009. Z. Lin et al. Fast convex optimization algorithms for exact recovery of a corrupted lowrank matrix. UIUC Technical Report UILU-ENG-09-4, 009. E. J. Candes et al. Robust principal component analysis? Journal of the ACM, 00. E. J. Cand es and Y. Plan, Matrix completion with noise, Proceedings of the IEEE, 00, 98(6): A. Ganesh et al. Dense error correction for low-rank matrices via principal component pursuit, in the Computing Research Repository, 00. G. Liu et al. Robust subspace segmentation by low-rank representation, in ICML, 00.

76 Some references - low rank matrix E. J. Cand es and T. Tao. The power of convex relaxation: Near-optimal matrix completion. IEEE Trans. on Information Theory, 00, 56(5): Y. Mu et al. Accelerated low-rank visual recovery by random projection. in IEEE CVPR, 0. R. He et al. Recovery of corrupted low rank matrices via half-quadratic based nonconvex minimization. in IEEE CVPR, 0. Ran He et al. Recovery of Corrupted Low-rank Matrix by Implicit Regularizer. Submitted to IEEE TPAMI, 0. M. Fornasier et al. Low rank matrix recovery via iteratively reweighted least squares minimization. submitted to SIAM Journal on Optimization, 0. D. Hsu et al. Robust matrix decomposition with sparse corruptions. IEEE Trans. on Information Theory, 0, 57(): D. Gross. Recovering low-rank matrices from few coefficients in any basis. IEEE Trans. on Information Theory, 0, 57(3):