[论文解读] Self-scaled bounds for atomic cone ranks: applications to nonnegative rank and cp-rank

Hamza Fawzi, Pablo A. Parrilo|DSpace@MIT (Massachusetts Institute of Technology)|Apr 11, 2014

Mathematical Inequalities and Applications被引用 3

一句话总结

本文提出了一种新颖的自缩放原子范数框架，通过使用平方和松弛的半定规划方法，推导出非负秩和cp-秩的更紧下界。该方法得到的下界在缩放下保持不变，优于现有的基于范数的下界，并继承了次可加性及对角缩放不变性等结构特性。

ABSTRACT

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 which improve on previously known bounds. Additionally the bounds we propose can be computed by semidefinite programming. 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 is at least as good as other existing norm-based bounds. We mainly focus our attention on the two important cases of nonnegative rank and cp-rank where our bounds satisfying interesting properties: For the nonnegative rank we show that our lower bound can be interpreted as a non-combinatorial version of the fractional rectangle cover number, while the sum-of-squares relaxation is closely related to the Lovász theta number of the rectangle graph of the matrix. We also prove that the lower bound inherits many of the structural properties satisfied by the nonnegative rank such as invariance under diagonal scaling, subadditivity, etc. We also apply our method to obtain lower bounds on the cp-rank for completely positive matrices. In this case we prove that our lower bound is always greater than or equal the plain rank lower bound, and we show that it has interesting connections with combinatorial lower bounds based on edge-clique cover number.

研究动机与目标

• 开发针对原子秩函数（如非负秩和cp-秩）的更紧致、可扩展的下界。
• 解决计算非负秩有效下界这一挑战，非负秩是优化、通信复杂性与统计学中的关键量。
• 通过自缩放原子范数框架，统一并推广现有的基于范数与组合的下界。
• 确保下界可通过使用平方和松弛的半定规划方法进行计算。
• 为新下界建立理论性质，如对角缩放不变性与次可加性。

提出的方法

• 提出一种自缩放原子范数方法，其中原子根据输入矩阵或向量进行缩放，确保在缩放变换下保持不变。
• 基于原子范数定义一个下界函数 τ₊(A)，该函数在非负矩阵的秩一矩阵的凸组合上实现最小化。
• 使用平方和（SOS）松弛方法，将非凸的原子秩最小化问题转化为可处理的半定规划问题。
• 推导出一个SDP松弛 τ₊^sos(A)，通过块结构的半正定矩阵约束捕捉自缩放下界。
• 利用块对角矩阵的结构，证明该下界在直和运算下的次可加性与可加性。
• 通过图论解释，将SOS松弛与已知的组合下界（如Lovász ϑ-数与分数矩形覆盖数）联系起来。

实验结果

研究问题

• RQ1 自缩放原子范数框架能否为非负秩提供比现有基于范数的方法更紧致且更具不变性的下界？
• RQ2 所提出的下界与组合下界（如分数矩形覆盖数与边团覆盖数）之间有何关系？
• RQ3 新下界在多大程度上继承了次可加性与对角缩放不变性等结构特性？
• RQ4 原子范数的平方和松弛能否被高效计算，并证明其等价于已知的SDP松弛？
• RQ5 在矩阵的矩形图背景下，新下界与Lovász ϑ-数之间存在何种关系？

主要发现

• 所提出的自缩放下界 τ₊(A) 始终不小于其他基于范数的非负秩下界。
• 下界 τ₊(A) 在矩阵 A 的对角缩放下保持不变，即当行与列被正权重缩放时，其值保持不变。
• 平方和松弛 τ₊^sos(A) 被证明等价于矩阵 A 的矩形图的Lovász ϑ-数，从而将其与一个著名的组合下界联系起来。
• 对于非负秩，下界 τ₊(A) 可解释为分数矩形覆盖数的一种非组合变体。
• 下界 τ₊(A) 具有次可加性：τ₊(A ⊕ B) = τ₊(A) + τ₊(B)，且该性质对SOS松弛 τ₊^sos 同样成立。
• 对于cp-秩，下界 τ₊(A) 始终大于等于普通秩，并与矩阵支持图的边团覆盖数存在关联。