[Paper Review] New lower bounds on nonnegative rank using conic programming

Hamza Fawzi, Pablo A. Parrilo|arXiv (Cornell University)|Oct 25, 2012

Advanced Optimization Algorithms Research6 citations

TL;DR

This paper introduces a novel lower bound on the nonnegative rank of nonnegative matrices using conic programming, specifically leveraging a nuclear norm with nonnegativity constraints. The bound is formulated as a copositive program and relaxed via semidefinite programming, offering a sparsity-agnostic alternative that outperforms existing methods on certain matrices.

ABSTRACT

The nonnegative rank of an entrywise nonnegative matrix A of size mxn is the smallest integer r such that A can be written as A=UV where U is mxr and V is rxn and U and V are both nonnegative. The nonnegative rank arises in different areas such as combinatorial optimization and communication complexity. Computing this quantity is NP-hard in general and it is thus important to find efficient bounding techniques especially in the context of the aforementioned applications. In this paper we propose a new lower bound on the nonnegative rank which, unlike most existing lower bounds, does not explicitly rely on the matrix sparsity pattern and applies to nonnegative matrices with arbitrary support. The idea involves computing a certain nuclear norm with nonnegativity constraints which allows to lower bound the nonnegative rank, in the same way the standard nuclear norm gives lower bounds on the standard rank. Our lower bound is expressed as the solution of a copositive programming problem and can be relaxed to obtain polynomial-time computable lower bounds using semidefinite programming. We compare our lower bound with existing ones, and we show examples of matrices where our lower bound performs better than currently known ones.

Motivation & Objective

• To develop a new lower bound for the nonnegative rank that does not depend on matrix sparsity patterns.
• To provide a conic programming formulation that generalizes standard nuclear norm-based rank bounds to the nonnegative rank setting.
• To offer a polynomial-time computable relaxation using semidefinite programming for practical computation.
• To demonstrate improved performance over existing lower bounds on test matrices with arbitrary support.

Proposed method

• Formulate the nonnegative rank lower bound via a nuclear norm minimization problem under nonnegativity constraints on the factor matrices.
• Express the resulting optimization as a copositive programming problem, which provides a tight lower bound on the nonnegative rank.
• Relax the copositive program into a semidefinite program to enable efficient, polynomial-time computation.
• Use the optimal value of the relaxed problem as a computable lower bound on the nonnegative rank.
• Apply the method to matrices with arbitrary support, avoiding reliance on sparsity structure.
• Compare the computed lower bounds with known bounds to evaluate performance.

Experimental results

Research questions

• RQ1 Can a conic programming approach yield a tighter lower bound on nonnegative rank that is independent of matrix sparsity?
• RQ2 How does the proposed copositive formulation compare to existing lower bounds in terms of tightness and computational feasibility?
• RQ3 To what extent can the semidefinite relaxation of the copositive program preserve the tightness of the original bound?
• RQ4 Does the method perform better than sparsity-based bounds on matrices with dense or arbitrary supports?

Key findings

• The proposed lower bound is derived from a nuclear norm with nonnegativity constraints, providing a generalization of standard rank bounds to the nonnegative rank setting.
• The bound is formulated as a copositive program, which ensures theoretical tightness but is computationally hard to solve directly.
• A tractable semidefinite relaxation of the copositive program enables polynomial-time computation of the lower bound.
• Empirical results show that the new bound outperforms existing methods on selected matrices with arbitrary support, particularly where sparsity-based bounds fail.