A Walk From Free Probability to Matrix Discrepancy III: Higher Rank Kadison-Singer and Spectrally Thin Trees

2026-09-17

AI Summary

研究解决了高秩Kadison-Singer问题，通过提出一种具有多项式实数运算量的确定性算法，实现了对矩阵不一致性O(√ε log(2r))的控制。

Abstract

Let $A_1,\ldots,A_N$ be positive semidefinite matrices of rank at most $r$, with $\sum_iA_i=I$ and $\|A_i\|\le\varepsilon$. We prove that the original matrices admit signs with discrepancy $O(\sqrt\varepsilon\log(2r))$, independently of their dimension and number which is a significantly stronger result than what was known existentially. We give a deterministic algorithm with polynomial real-arithmetic work, and a separate existence proof requiring no computational assumptions. This extends our companion paper on rank-one Kadison--Singer discrepancy. A concave matrix power interpolates between the trace source, which pays a factor $r$, and the sandwich source, whose density response is harder to control. We prove that source concavity controls this additional response in the same inverse-Sylvester metric as the optimized spectral potential. As an application, a single spanning tree can be chosen simultaneously $O(\varepsilon\log^2(2s))$-spectrally thin for $s$ positive edge weightings of a common graph, provided every edge has leverage at most $\varepsilon$ in every weighting. The reduction preserves one common selection decision per edge. For incidence matrices with at most $t$ ones in every row and column, the diagonal specialization gives a deterministic walk on fractional colorings with discrepancy $O(\sqrt t\log(2t))$. The local-walk mechanism gives both existence and an efficient construction without using the Lovász local lemma. A Lean formalization of our existence proof has been completed and will be released shortly.

Problem

• positive semidefinite matrices
• discrepancy
• Kadison-Singer
• spectrally thin trees
• spanning tree

Innovation

• Matrix Discrepancy
• Kadison-Singer Problem
• Spectrally Thin Trees
• Deterministic Algorithm