arXiv research

Sharp condition found for Burer-Monteiro method to work for MaxCut-type SDPs.

Paper shows Burer-Monteiro method can solve SDPs in polynomial time under smoothed analysis.

This paper studies clustering for possibly high dimensional data (e.g. images, time series, gene expression data, and many other settings), and rephrase it as low rank matrix estimation in the PAC-Bayesian framework. Our approach leverages the well known Burer-Monteiro factorisation strategy from large scale optimisati…

Improved guarantees for nonconvex matrix factorization with rank overparameterization.

We study the projected gradient descent method on low-rank matrix problems with a strongly convex objective. We use the Burer-Monteiro factorization approach to implicitly enforce low-rankness; such factorization introduces non-convexity in the objective. We focus on constraint sets that include both positive semi-defi…

Paper develops efficient AltMin algorithm for SRPCP robust matrix recovery.

A new algorithm solves semidefinite programs using Langevin diffusion.

We address the rectangular matrix completion problem by lifting the unknown matrix to a positive semidefinite matrix in higher dimension, and optimizing a nonconvex objective over the semidefinite factor using a simple gradient descent scheme. With $O( μr^2 κ^2 n \max(μ, \log n))$ random observations of a $n_1 \times n…

We consider the non-square matrix sensing problem, under restricted isometry property (RIP) assumptions. We focus on the non-convex formulation, where any rank-$r$ matrix $X \in \mathbb{R}^{m \times n}$ is represented as $UV^\top$, where $U \in \mathbb{R}^{m \times r}$ and $V \in \mathbb{R}^{n \times r}$. In this paper…

Semidefinite programming (SDP) with diagonal constraints arise in many optimization problems, such as Max-Cut, community detection and group synchronization. Although SDPs can be solved to arbitrary precision in polynomial time, generic convex solvers do not scale well with the dimension of the problem. In order to add…

This work is concerned with the non-negative rank-1 robust principal component analysis (RPCA), where the goal is to recover the dominant non-negative principal components of a data matrix precisely, where a number of measurements could be grossly corrupted with sparse and arbitrary large noise. Most of the known techn…

Semidefinite programs (SDP) are important in learning and combinatorial optimization with numerous applications. In pursuit of low-rank solutions and low complexity algorithms, we consider the Burer--Monteiro factorization approach for solving SDPs. We show that all approximate local optima are global optima for the pe…

Low-rank factorization is a standard way to make structured optimization problems in machine learning more tractable by replacing matrix variables with compact factors. For positive semidefinite (PSD) variables, the symmetric Burer--Monteiro factorization (sBMF) writes $Z=XX^\top$ with a single low-rank factor $X$. A r…

New algorithm improves clustering accuracy without sacrificing scalability.

Gradient descent with preconditioning finds global optima in overparameterized nonconvex factorization.

This paper studies noisy low-rank matrix completion: given partial and noisy entries of a large low-rank matrix, the goal is to estimate the underlying matrix faithfully and efficiently. Arguably one of the most popular paradigms to tackle this problem is convex relaxation, which achieves remarkable efficacy in practic…

Geometric technique determines exactness of SDP robustness certificate.

Maximum A posteriori Probability (MAP) inference in graphical models amounts to solving a graph-structured combinatorial optimization problem. Popular inference algorithms such as belief propagation (BP) and generalized belief propagation (GBP) are intimately related to linear programming (LP) relaxation within the She…

Paper develops methods for non-quadratic loss low-rank matrix recovery.

Efficiently recovers low-tubal-rank tensors from few measurements.

APGD algorithm reconstructs point set from partial distance measurements.

Paper tackles robust matrix completion with heavy-tailed noise.

Simplifies solving noisy SDPs for low rank matrix recovery problems.

Consider an unknown smooth function $f: [0,1]^d \rightarrow \mathbb{R}$, and say we are given $n$ noisy mod 1 samples of $f$, i.e., $y_i = (f(x_i) + η_i)\mod 1$, for $x_i \in [0,1]^d$, where $η_i$ denotes the noise. Given the samples $(x_i,y_i)_{i=1}^{n}$, our goal is to recover smooth, robust estimates of the clean sa…