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This work investigates the geometry of a nonconvex reformulation of minimizing a general convex loss function $f(X)$ regularized by the matrix nuclear norm $|X|_*$. Nuclear-norm regularized matrix inverse problems are at the heart of many applications in machine learning, signal processing, and control. The statistical performance of nuclear norm regularization has been studied extensively in literature using convex analysis techniques. Despite its optimal performance, the resulting optimization has high computational complexity when solved using standard or even tailored fast convex solvers. To develop faster and more scalable algorithms, we follow the proposal of Burer-Monteiro to factor the matrix variable $X$ into the product of two smaller rectangular matrices $X=UV^T$ and also replace the nuclear norm $|X|_*$ with $(|U|_F^2+|V|_F^2)/2$. In spite of the nonconvexity of the factored formulation, we prove that when the convex loss function $f(X)$ is $(2r,4r)$-restricted well-conditioned, each critical point of the factored problem either corresponds to the optimal solution $X^star$ of the original convex optimization or is a strict saddle point where the Hessian matrix has a strictly negative eigenvalue. Such a geometric structure of the factored formulation allows many local search algorithms to converge to the global optimum with random initializations.
The process of rank aggregation is intimately intertwined with the structure of skew-symmetric matrices. We apply recent advances in the theory and algorithms of matrix completion to skew-symmetric matrices. This combination of ideas produces a new m
We give a formal and complete characterization of the explicit regularizer induced by dropout in deep linear networks with squared loss. We show that (a) the explicit regularizer is composed of an $ell_2$-path regularizer and other terms that are als
In this paper, we investigate tensor recovery problems within the tensor singular value decomposition (t-SVD) framework. We propose the partial sum of the tubal nuclear norm (PSTNN) of a tensor. The PSTNN is a surrogate of the tensor tubal multi-rank
We describe novel subgradient methods for a broad class of matrix optimization problems involving nuclear norm regularization. Unlike existing approaches, our method executes very cheap iterations by combining low-rank stochastic subgradients with ef
This work investigates the use of mixed-norm regularization for sensor selection in Event-Related Potential (ERP) based Brain-Computer Interfaces (BCI). The classification problem is cast as a discriminative optimization framework where sensor select