Matrix estimation by Universal Singular Value Thresholding

Chatterjee, Sourav

doi:10.1214/14-aos1272

Cited by 377 publications

(554 citation statements)

References 100 publications

(138 reference statements)

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“…This independence assumption is similar to assumptions typically made for low-rank matrix and tensor completion (e.g., [56,12,42]). The typical assumption is that entries of the matrix (or tensor) are sampled with uniform probability, which is equivalent to assuming the x u i are jointly independent [33].…”

Section: Defining the Risk Gapmentioning

confidence: 83%

“…Matrix completion is well-studied [20,63,51,9,10,38,1,54,30,12,23], but tensor completion is still an open problem. Tensor rank is NP-hard to compute [27] and has poor continuity properties [16].…”

Section: Tensor Completionmentioning

confidence: 99%

“…Incoherence conditions are common in the matrix and tensor completion literature [51,9,10,38,1,54,30,12,23]. One interpretation of A5 is it forces G jq to represent the difference in risk (flattened to just two variables) between keeping x j , x q coupled versus decoupled in the ideal partition Γ * .…”

Section: Defining the Risk Gapmentioning

confidence: 99%

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Low-Rank Approximation and Completion of Positive Tensors

Aswani¹

2016

SIAM J. Matrix Anal. & Appl.

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Abstract. Unlike the matrix case, computing low-rank approximations of tensors is NP-hard and numerically ill-posed in general. Even the best rank-1 approximation of a tensor is NP-hard. In this paper, we use convex optimization to develop polynomial-time algorithms for low-rank approximation and completion of positive tensors. Our approach is to use algebraic topology to define a new (numerically well-posed) decomposition for positive tensors, which we show is equivalent to the standard tensor decomposition in important cases. Though computing this decomposition is a nonconvex optimization problem, we prove it can be exactly reformulated as a convex optimization problem. This allows us to construct polynomial-time randomized algorithms for computing this decomposition and for solving low-rank tensor approximation problems. Among the consequences is that best rank-1 approximations of positive tensors can be computed in polynomial time. Our framework is next extended to the tensor completion problem, where noisy entries of a tensor are observed and then used to estimate missing entries. We provide a polynomial-time algorithm that for specific cases requires a polynomial (in tensor order) number of measurements, in contrast to existing approaches that require an exponential number of measurements. These algorithms are extended to exploit sparsity in the tensor to reduce the number of measurements needed. We conclude by providing a novel interpretation of statistical regression problems with categorical variables as tensor completion problems, and numerical examples with synthetic data and data from a bioengineered metabolic network show the improved performance of our approach on this problem.

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Section: Defining the Risk Gapmentioning

confidence: 83%

Section: Tensor Completionmentioning

confidence: 99%

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Low-Rank Approximation and Completion of Positive Tensors

Aswani¹

2016

SIAM J. Matrix Anal. & Appl.

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“…What matrix corresponds to our data table? Here, we give a simple proposal for how to construct such a matrix, motivated by [KMO10,JNS13,Cha14]. Our key insight is that the SVD is the solution to our problem when the entries in the table have mean zero and variance one (and all the loss functions are quadratic).…”

Section: Svdmentioning

confidence: 99%

Generalized Low Rank Models

Udell

Horn

Zadeh

et al. 2016

FNT in Machine Learning

196

129

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Principal components analysis (PCA) is a well-known technique for approximating a tabular data set by a low rank matrix. Here, we extend the idea of PCA to handle arbitrary data sets consisting of numerical, Boolean, categorical, ordinal, and other data types. This framework encompasses many well known techniques in data analysis, such as nonnegative matrix factorization, matrix completion, sparse and robust PCA, k-means, k-SVD, and maximum margin matrix factorization. The method handles heterogeneous data sets, and leads to coherent schemes for compressing, denoising, and imputing missing entries across all data types simultaneously. It also admits a number of interesting interpretations of the low rank factors, which allow clustering of examples or of features. We propose several parallel algorithms for fitting generalized low rank models, and describe implementations and numerical results.

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“…The regular image segmentation algorithms such as thresholding [7], edge [8] and color clustering [9] fail to extract full features of the mudras. This is due to occlusions of fingers during capture and colouring used for fingers during performances.…”

Section: Introductionmentioning

confidence: 99%