Stable low-rank matrix recovery via null space properties

The problem of recovering a matrix of low rank from an incomplete and possibly noisy set of linear measurements arises in a number of areas. In order to derive rigorous recovery results, the measurement map is usually modeled probabilistically. We derive sufficient conditions on the minimal amount of measurements ensuring recovery via convex optimization. We establish our results via certain properties of the null space of the measurement map. In the setting where the measurements are realized as Frobenius inner products with independent standard Gaussian random matrices we show that $10 r (n_1 + n_2)$ measurements are enough to uniformly and stably recover an $n_1 \times n_2$ matrix of rank at most $r$. We then significantly generalize this result by only requiring independent mean-zero, variance one entries with four finite moments at the cost of replacing $10$ by some universal constant. We also study the case of recovering Hermitian rank-$r$ matrices from measurement matrices proportional to rank-one projectors. For $m \geq C r n$ rank-one projective measurements onto independent standard Gaussian vectors, we show that nuclear norm minimization uniformly and stably reconstructs Hermitian rank-$r$ matrices with high probability. Next, we partially de-randomize this by establishing an analogous statement for projectors onto independent elements of a complex projective 4-designs at the cost of a slightly higher sampling rate $m \geq C rn \log n$. Moreover, if the Hermitian matrix to be recovered is known to be positive semidefinite, then we show that the nuclear norm minimization approach may be replaced by minimizing the $\ell_2$-norm of the residual subject to the positive semidefinite constraint. Then no estimate of the noise level is required a priori. We discuss applications in quantum physics and the phase retrieval problem.Comment: 26 page

Non-convex Optimization for Machine Learning

A vast majority of machine learning algorithms train their models and perform inference by solving optimization problems. In order to capture the learning and prediction problems accurately, structural constraints such as sparsity or low rank are frequently imposed or else the objective itself is designed to be a non-convex function. This is especially true of algorithms that operate in high-dimensional spaces or that train non-linear models such as tensor models and deep networks. The freedom to express the learning problem as a non-convex optimization problem gives immense modeling power to the algorithm designer, but often such problems are NP-hard to solve. A popular workaround to this has been to relax non-convex problems to convex ones and use traditional methods to solve the (convex) relaxed optimization problems. However this approach may be lossy and nevertheless presents significant challenges for large scale optimization. On the other hand, direct approaches to non-convex optimization have met with resounding success in several domains and remain the methods of choice for the practitioner, as they frequently outperform relaxation-based techniques - popular heuristics include projected gradient descent and alternating minimization. However, these are often poorly understood in terms of their convergence and other properties. This monograph presents a selection of recent advances that bridge a long-standing gap in our understanding of these heuristics. The monograph will lead the reader through several widely used non-convex optimization techniques, as well as applications thereof. The goal of this monograph is to both, introduce the rich literature in this area, as well as equip the reader with the tools and techniques needed to analyze these simple procedures for non-convex problems.Comment: The official publication is available from now publishers via http://dx.doi.org/10.1561/220000005

Blind Demixing for Low-Latency Communication

In the next generation wireless networks, lowlatency communication is critical to support emerging diversified applications, e.g., Tactile Internet and Virtual Reality. In this paper, a novel blind demixing approach is developed to reduce the channel signaling overhead, thereby supporting low-latency communication. Specifically, we develop a low-rank approach to recover the original information only based on a single observed vector without any channel estimation. Unfortunately, this problem turns out to be a highly intractable non-convex optimization problem due to the multiple non-convex rankone constraints. To address the unique challenges, the quotient manifold geometry of product of complex asymmetric rankone matrices is exploited by equivalently reformulating original complex asymmetric matrices to the Hermitian positive semidefinite matrices. We further generalize the geometric concepts of the complex product manifolds via element-wise extension of the geometric concepts of the individual manifolds. A scalable Riemannian trust-region algorithm is then developed to solve the blind demixing problem efficiently with fast convergence rates and low iteration cost. Numerical results will demonstrate the algorithmic advantages and admirable performance of the proposed algorithm compared with the state-of-art methods.Comment: 14 pages, accepted by IEEE Transaction on Wireless Communicatio

Informed Non-convex Robust Principal Component Analysis with Features

We revisit the problem of robust principal component analysis with features acting as prior side information. To this aim, a novel, elegant, non-convex optimization approach is proposed to decompose a given observation matrix into a low-rank core and the corresponding sparse residual. Rigorous theoretical analysis of the proposed algorithm results in exact recovery guarantees with low computational complexity. Aptly designed synthetic experiments demonstrate that our method is the first to wholly harness the power of non-convexity over convexity in terms of both recoverability and speed. That is, the proposed non-convex approach is more accurate and faster compared to the best available algorithms for the problem under study. Two real-world applications, namely image classification and face denoising further exemplify the practical superiority of the proposed method