Self-scaled bounds for atomic cone ranks: applications to nonnegative rank and cp-rank

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• 作者：Hamza Fawzi ; Pablo A. Parrilo
• 刊名：Mathematical Programming
• 出版年：2016
• 出版时间：July 2016
• 年：2016
• 卷：158
• 期：1-2
• 页码：417-465
• 全文大小：1,111 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Calculus of Variations and Optimal Control
  Mathematics of Computing
  Numerical Analysis
  Combinatorics
  Mathematical and Computational Physics
  Mathematical Methods in Physics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1436-4646
• 卷排序：158

文摘

The nonnegative rank of a matrix A is the smallest integer r such that A can be written as the sum of r rank-one nonnegative matrices. The nonnegative rank has received a lot of attention recently due to its application in optimization, probability and communication complexity. In this paper we study a class of atomic rank functions defined on a convex cone which generalize several notions of “positive” ranks such as nonnegative rank or cp-rank (for completely positive matrices). The main contribution of the paper is a new method to obtain lower bounds for such ranks. Additionally the bounds we propose can be computed by semidefinite programming using sum-of-squares relaxations. The idea of the lower bound relies on an atomic norm approach where the atoms are self-scaled according to the vector (or matrix, in the case of nonnegative rank) of interest. This results in a lower bound that is invariant under scaling and that enjoys other interesting structural properties. For the case of the nonnegative rank we show that our bound has an appealing connection with existing combinatorial bounds and other norm-based bounds. For example we show that our lower bound is a non-combinatorial version of the fractional rectangle cover number, while the sum-of-squares relaxation is closely related to the Lovász \(\bar{\vartheta }\) number of the rectangle graph of the matrix. We also prove that the lower bound is always greater than or equal to the hyperplane separation bound (and other similar “norm-based” bounds). We also discuss the case of the tensor nonnegative rank as well as the cp-rank, and compare our bound with existing results.Mathematics Subject Classification90C2515A2315B48