# Matrix estimation by Universal Singular Value Thresholding

@article{Chatterjee2015MatrixEB, title={Matrix estimation by Universal Singular Value Thresholding}, author={Sourav Chatterjee}, journal={Annals of Statistics}, year={2015}, volume={43}, pages={177-214} }

Consider the problem of estimating the entries of a large matrix, when the observed entries are noisy versions of a small random fraction of the original entries. This problem has received widespread attention in recent times, especially after the pioneering works of Emmanuel Cand\`{e}s and collaborators. This paper introduces a simple estimation procedure, called Universal Singular Value Thresholding (USVT), that works for any matrix that has "a little bit of structure." Surprisingly, this…

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## References

SHOWING 1-10 OF 118 REFERENCES

Estimation of high-dimensional low-rank matrices

- Mathematics, Computer Science
- 2010

This work investigates penalized least squares estimators with a Schatten-p quasi-norm penalty term and derives bounds for the kth entropy numbers of the quasi-convex Schatten class embeddings S M p → S M 2 , p < 1, which are of independent interest.

Nuclear norm penalization and optimal rates for noisy low rank matrix completion

- Computer Science, Mathematics
- 2010

A new nuclear norm penalized estimator of A_0 is proposed and a general sharp oracle inequality for this estimator is established for arbitrary values of $n,m_1,m-2$ under the condition of isometry in expectation to find the best trace regression model approximating the data.

The Power of Convex Relaxation: Near-Optimal Matrix Completion

- Computer ScienceIEEE Transactions on Information Theory
- 2010

This paper shows that, under certain incoherence assumptions on the singular vectors of the matrix, recovery is possible by solving a convenient convex program as soon as the number of entries is on the order of the information theoretic limit (up to logarithmic factors).

A Singular Value Thresholding Algorithm for Matrix Completion

- Computer ScienceSIAM J. Optim.
- 2010

This paper develops a simple first-order and easy-to-implement algorithm that is extremely efficient at addressing problems in which the optimal solution has low rank, and develops a framework in which one can understand these algorithms in terms of well-known Lagrange multiplier algorithms.

Spectral Regularization Algorithms for Learning Large Incomplete Matrices

- Computer ScienceJ. Mach. Learn. Res.
- 2010

Using the nuclear norm as a regularizer, the algorithm Soft-Impute iteratively replaces the missing elements with those obtained from a soft-thresholded SVD in a sequence of regularized low-rank solutions for large-scale matrix completion problems.

OptShrink: An Algorithm for Improved Low-Rank Signal Matrix Denoising by Optimal, Data-Driven Singular Value Shrinkage

- Computer ScienceIEEE Transactions on Information Theory
- 2014

This analysis brings into sharp focus the shrinkage-and-thresholding form of the optimal weights, the nonconvex nature of the associated shrinkage function (on the singular values), and explains why matrix regularization via singular value thresholding with convex penalty functions will always be suboptimal.

Estimation of (near) low-rank matrices with noise and high-dimensional scaling

- Computer ScienceICML
- 2010

Simulations show excellent agreement with the high-dimensional scaling of the error predicted by the theory, and illustrate their consequences for a number of specific learning models, including low-rank multivariate or multi-task regression, system identification in vector autoregressive processes, and recovery of low- rank matrices from random projections.

1-Bit Matrix Completion

- Computer ScienceArXiv
- 2012

A theory of matrix completion for the extreme case of noisy 1-bit observations is developed and it is shown that the maximum likelihood estimate under a suitable constraint returns an accurate estimate of M when ||M||_{\infty} <= \alpha, and rank(M) <= r.

Matrix Completion from Noisy Entries

- Computer ScienceJ. Mach. Learn. Res.
- 2009

This work studies a low complexity algorithm, introduced in [1], based on a combination of spectral techniques and manifold optimization, that is called here OPTSPACE, and proves performance guarantees that are order-optimal in a number of circumstances.

Matrix Completion With Noise

- Computer ScienceProceedings of the IEEE
- 2010

This paper surveys the novel literature on matrix completion and introduces novel results showing that matrix completion is provably accurate even when the few observed entries are corrupted with a small amount of noise, and shows that, in practice, nuclear-norm minimization accurately fills in the many missing entries of large low-rank matrices from just a few noisy samples.