Related papers: Model-free Nonconvex Matrix Completion: Local Mini…
This work proposes a novel convex-non-convex formulation of the image segmentation and the image completion problems. The proposed approach is based on the minimization of a functional involving two distinct regularization terms: one…
Constrained quasiconvex optimization problems appear in many fields, such as economics, engineering, and management science. In particular, fractional programming, which models ratio indicators such as the profit/cost ratio as fractional…
The low-rank matrix optimization with affine set (rank-MOA) is to minimize a continuously differentiable function over a low-rank set intersecting with an affine set. Under some suitable assumptions, the intersection rule of the Fr\'{e}chet…
The matrix rank minimization problem has applications in many fields such as system identification, optimal control, low-dimensional embedding, etc. As this problem is NP-hard in general, its convex relaxation, the nuclear norm minimization…
We study the Riemannian optimization methods on the embedded manifold of low rank matrices for the problem of matrix completion, which is about recovering a low rank matrix from its partial entries. Assume $m$ entries of an $n\times n$ rank…
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…
We propose an efficient matrix rank reduction method for non-negative matrices, whose time complexity is quadratic in the number of rows or columns of a matrix. Our key insight is to formulate rank reduction as a mean-field approximation by…
This paper describes a suite of algorithms for constructing low-rank approximations of an input matrix from a random linear image of the matrix, called a sketch. These methods can preserve structural properties of the input matrix, such as…
Low-rank representation~(LRR) has been a significant method for segmenting data that are generated from a union of subspaces. It is, however, known that solving the LRR program is challenging in terms of time complexity and memory…
Kernel approximation using randomized feature maps has recently gained a lot of interest. In this work, we identify that previous approaches for polynomial kernel approximation create maps that are rank deficient, and therefore do not…
Affine matrix rank minimization problem is a fundamental problem with a lot of important applications in many fields. It is well known that this problem is combinatorial and NP-hard in general. In this paper, a continuous promoting low rank…
The Nystrom method is an efficient technique used to speed up large-scale learning applications by generating low-rank approximations. Crucial to the performance of this technique is the assumption that a matrix can be well approximated by…
Low-rank matrix recovery has found many applications in science and engineering such as machine learning, signal processing, collaborative filtering, system identification, and Euclidean embedding. But the low-rank matrix recovery problem…
Non-convex optimization is a critical tool in advancing machine learning, especially for complex models like deep neural networks and support vector machines. Despite challenges such as multiple local minima and saddle points, non-convex…
In this paper we study the problem of recovering a low-rank matrix from a number of random linear measurements that are corrupted by outliers taking arbitrary values. We consider a nonsmooth nonconvex formulation of the problem, in which we…
Rank estimation is a classical model order selection problem that arises in a variety of important statistical signal and array processing systems, yet is addressed relatively infrequently in the extant literature. Here we present sample…
This paper considers general rank-constrained optimization problems that minimize a general objective function $f(X)$ over the set of rectangular $n\times m$ matrices that have rank at most $r$. To tackle the rank constraint and also to…
We consider optimization problems in which the goal is find a $k$-dimensional subspace of $\mathbb{R}^n$, $k<<n$, which minimizes a convex and smooth loss. Such problems generalize the fundamental task of principal component analysis (PCA)…
We study the basic problem of robust subspace recovery. That is, we assume a data set that some of its points are sampled around a fixed subspace and the rest of them are spread in the whole ambient space, and we aim to recover the fixed…
We present a unified framework for low-rank matrix estimation with nonconvex penalties. We first prove that the proposed estimator attains a faster statistical rate than the traditional low-rank matrix estimator with nuclear norm penalty.…