A scaling characterization of nc-rank via unbounded gradient flow

2025/02/08 by Hirai, Hiroshi · 2 citations
#53C35 #90C25 #Differential Geometry (math.DG) #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2502.05412

Abstract

Given a tuple of n × n complex matrices \cal A = (A₁,A₂,…, A_m), the linear symbolic matrix A = A₁x₁ + A₂x₂ + ⋯ + A_m x_m is nonsingular in the noncommutative sense if and only if the completely positive operators T_{\cal A} (X) = ∑_i=1^m A_i X A_i^† and T_{\cal A}^*(X) = ∑_i=1^m A_i^† X A_i can be scaled to be doubly stochastic: For every ε> 0 there are g,h ∈ GL(n,ℂ) such that ‖T_g^†\cal Ah(I)- I‖ < ε, ‖ T^*_g^†\cal Ah(I) - I‖ < ε. In this paper, we show a refinement: The noncommutative corank of A is equal to one-half of the minimum residual ‖T_g^†\cal Ah(I) - I‖₁ + ‖T^*_g^†\cal Ah(I) - I‖₁ over all possible scalings g^†\cal Ah, where ‖⋅ ‖₁ is the trace norm. To show this, we interpret the residuals as gradients of a convex function on symmetric space GL(n,ℂ)/U_n, and establish a general duality relation of the minimum gradient-norm of a lower-unbounded convex function f on GL(n,ℂ)/U_n with an invariant Finsler metric, by utilizing the unbounded gradient flow of f at infinity.

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• Generalized gradient flows in Hadamard manifolds and convex optimization on entanglement polytopes
• Discrete-time gradient flows for unbounded convex functions on Gromov hyperbolic spaces

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