Related papers: Global Optimality in Low-rank Matrix Optimization

While it is well known that finding approximate optima of non-convex functions is computationally intractable, we show that the problem is, in fact, uncomputable in the oracle model. Specifically, we prove that no algorithm with access only…

This paper propose a new frame work for finding global minima which we call optimization by cut. In each iteration, it takes some samples from the feasible region and evaluates the objective function at these points. Based on the…

A central challenge to many fields of science and engineering involves minimizing non-convex error functions over continuous, high dimensional spaces. Gradient descent or quasi-Newton methods are almost ubiquitously used to perform such…

A key question in many low-rank problems throughout optimization, machine learning, and statistics is to characterize the convex hulls of simple low-rank sets and judiciously apply these convex hulls to obtain strong yet computationally…

We propose an algorithmic framework, that employs active subspace techniques, for scalable global optimization of functions with low effective dimension (also referred to as low-rank functions). This proposal replaces the original…

Non-negative matrix factorization (NMF) is a fundamental non-convex optimization problem with numerous applications in Machine Learning (music analysis, document clustering, speech-source separation etc). Despite having received extensive…

Low-rank modeling has a lot of important applications in machine learning, computer vision and social network analysis. While the matrix rank is often approximated by the convex nuclear norm, the use of nonconvex low-rank regularizers has…

Recent work established that rank overparameterization eliminates spurious local minima in nonconvex low-rank matrix recovery under the restricted isometry property (RIP). But this does not fully explain the practical success of…

We consider the global minimization of a particular type of minimum structured optimization problems wherein the variables must belong to some basic set, the feasible domain is described by the intersection of a large number of functional…

When computing bounds, spatial branch-and-bound algorithms often linearly outer approximate convex relaxations for non-convex expressions in order to capitalize on the efficiency and robustness of linear programming solvers. Considering…

We consider a general multi-agent convex optimization problem where the agents are to collectively minimize a global objective function subject to a global inequality constraint, a global equality constraint, and a global constraint set.…

This work demonstrates the utility of gradients for the global optimization of certain differentiable functions with many suboptimal local minima. To this end, a principle for generating search directions from non-local quadratic…

With the extensive application of submodularity, its generalizations are constantly being proposed. However, most of them are tailored for special problems. In this paper, we focus on quasi-submodularity, a universal generalization, which…

We consider using gradient descent to minimize the nonconvex function $f(X)=\phi(XX^{T})$ over an $n\times r$ factor matrix $X$, in which $\phi$ is an underlying smooth convex cost function defined over $n\times n$ matrices. While only a…

We introduce a derivative-free global optimization algorithm that efficiently computes minima for various classes of one-dimensional functions, including non-convex, and non-smooth functions.This algorithm numerically approximates the…

We revisit the landscape of the simple matrix factorization problem. For low-rank matrix factorization, prior work has shown that there exist infinitely many critical points all of which are either global minima or strict saddles. At a…

Motivated by an application in computational biology, we consider low-rank matrix factorization with $\{0,1\}$-constraints on one of the factors and optionally convex constraints on the second one. In addition to the non-convexity shared…

In this paper, we analyze some theoretical properties of the problem of minimizing a quadratic function with a cubic regularization term, arising in many methods for unconstrained and constrained optimization that have been proposed in the…

Many problems in data science can be treated as estimating a low-rank matrix from highly incomplete, sometimes even corrupted, observations. One popular approach is to resort to matrix factorization, where the low-rank matrix factors are…

In this paper we consider convex optimization problems with stochastic composite objective function subject to (possibly) infinite intersection of constraints. The objective function is expressed in terms of expectation operator over a sum…