Low-rank matrix recovery is a fundamental problem in machine learning with numerous applications. In practice, the problem can be solved by convex optimization namely nuclear norm minimization, or by non-convex optimization as it is well-known that for low-rank matrix problems like matrix sensing and matrix completion, all local optima of the natural non-convex objectives are also globally optimal under certain ideal assumptions.
In this paper, we study new approaches for matrix sensing in a semi-random model where an adversary can add any number of arbitrary sensing matrices. More precisely, the problem is to recover a low-rank matrix $X^\star$ from linear measurements $b_i = \langle A_i, X^\star \rangle$, where an unknown subset of the sensing matrices satisfies the Restricted Isometry Property (RIP) and the rest of the $A_i$'s are chosen adversarially.
It is known that in the semi-random model, existing non-convex objectives can have bad local optima. To fix this, we present a descent-style algorithm that provably recovers the ground-truth matrix $X^\star$. For the closely-related problem of semi-random matrix completion, prior work [CG18] showed that all bad local optima can be eliminated by reweighting the input data. However, the analogous approach for matrix sensing requires reweighting a set of matrices to satisfy RIP, which is a condition that is NP-hard to check. Instead, we build on the framework proposed in [KLL$^+$23] for semi-random sparse linear regression, where the algorithm in each iteration reweights the input based on the current solution, and then takes a weighted gradient step that is guaranteed to work well locally. Our analysis crucially exploits the connection between sparsity in vector problems and low-rankness in matrix problems, which may have other applications in obtaining robust algorithms for sparse and low-rank problems.
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Recovery analysis of damped spectrally sparse signals and its relation to MUSIC
Abstract One of the classical approaches for estimating the frequencies and damping factors in a spectrally sparse signal is the MUltiple SIgnal Classification (MUSIC) algorithm, which exploits the low-rank structure of an autocorrelation matrix. Low-rank matrices have also received considerable attention recently in the context of optimization algorithms with partial observations, and nuclear norm minimization (NNM) has been widely used as a popular heuristic of rank minimization for low-rank matrix recovery problems. On the other hand, it has been shown that NNM can be viewed as a special case of atomic norm minimization (ANM), which has achieved great success in solving line spectrum estimation problems. However, as far as we know, the general ANM (not NNM) considered in many existing works can only handle frequency estimation in undamped sinusoids. In this work, we aim to fill this gap and deal with damped spectrally sparse signal recovery problems. In particular, inspired by the dual analysis used in ANM, we offer a novel optimization-based perspective on the classical MUSIC algorithm and propose an algorithm for spectral estimation that involves searching for the peaks of the dual polynomial corresponding to a certain NNM problem, and we show that this algorithm is in fact equivalent to MUSIC itself. Building on this connection, we also extend the classical MUSIC algorithm to the missing data case. We provide exact recovery guarantees for our proposed algorithms and quantify how the sample complexity depends on the true spectral parameters. In particular, we provide a parameter-specific recovery bound for low-rank matrix recovery of jointly sparse signals rather than use certain incoherence properties as in existing literature. Simulation results also indicate that the proposed algorithms significantly outperform some relevant existing methods (e.g., ANM) in frequency estimation of damped exponentials.
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- PAR ID:
- 10253367
- Date Published:
- Journal Name:
- Information and Inference: A Journal of the IMA
- ISSN:
- 2049-8772
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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The low-rank matrix recovery is an important machine learning research topic with various scientific applications. Most existing low-rank matrix recovery methods relax the rank minimization problem via the trace norm minimization. However, such a relaxation makes the solution seriously deviate from the original one. Meanwhile, most matrix recovery methods minimize the squared prediction errors on the observed entries, which is sensitive to outliers. In this paper, we propose a new robust matrix recovery model to address the above two challenges. The joint capped trace norm and capped $\ell_1$-norm are used to tightly approximate the rank minimization and enhance the robustness to outliers. The evaluation experiments are performed on both synthetic data and real world applications in collaborative filtering and social network link prediction. All empirical results show our new method outperforms the existing matrix recovery methods.
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Abstract Low-rank matrix models have been universally useful for numerous applications, from classical system identification to more modern matrix completion in signal processing and statistics. The nuclear norm has been employed as a convex surrogate of the low-rankness since it induces a low-rank solution to inverse problems. While the nuclear norm for low rankness has an excellent analogy with the $\ell _1$ norm for sparsity through the singular value decomposition, other matrix norms also induce low-rankness. Particularly as one interprets a matrix as a linear operator between Banach spaces, various tensor product norms generalize the role of the nuclear norm. We provide a tensor-norm-constrained estimator for the recovery of approximately low-rank matrices from local measurements corrupted with noise. A tensor-norm regularizer is designed to adapt to the local structure. We derive statistical analysis of the estimator over matrix completion and decentralized sketching by applying Maurey’s empirical method to tensor products of Banach spaces. The estimator provides a near-optimal error bound in a minimax sense and admits a polynomial-time algorithm for these applications.
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imaiai/iaaa024
