从自由概率到矩阵差异性的漫步 III：高阶 Kadison-Singer 与谱薄树
A Walk From Free Probability to Matrix Discrepancy III: Higher Rank Kadison-Singer and Spectrally Thin Trees
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中文总结 AI 辅助
本文证明高阶半正定矩阵存在与维度无关的O(√ε log(2r))符号差异性，给出确定性算法，并应用于同时谱薄生成树与分数染色，无需Lovász局部引理。
中文摘要 AI 辅助
设 $A_1,\ldots,A_N$ 为秩至多 $r$ 的半正定矩阵，满足 $\sum_iA_i=I$ 且 $\\|A_i\\|\le\varepsilon$。我们证明这些原始矩阵存在符号选择，使得差异性为 $O(\sqrt\varepsilon\log(2r))$，该结果与矩阵的维度和数量无关，这比已知的存在性结果显著更强。我们给出一个具有多项式实数运算复杂度的确定性算法，以及一个不需要任何计算假设的独立存在性证明。这扩展了我们关于秩一 Kadison-Singer 差异性的姊妹论文。一个凹矩阵幂在迹源（其代价因子为 $r$）与夹心源（其密度响应更难控制）之间进行插值。我们证明源凹性在相同的逆 Sylvester 度量下控制这一额外响应，如同优化的谱势。作为应用，对于共同图上的 $s$ 个正边权，若每条边在每个权中的杠杆至多为 $\varepsilon$，则可同时选择一棵生成树，使其对每个权都是 $O(\varepsilon\log^2(2s))$-谱薄的。该归约保留了每条边的一个共同选择决策。对于每行每列至多有 $t$ 个 1 的关联矩阵，对角化特化给出了一个关于分数染色的确定性漫步，其差异性为 $O(\sqrt t\log(2t))$。局部漫步机制同时给出了存在性和高效构造，而无需使用 Lovász 局部引理。我们存在性证明的 Lean 形式化已完成，并将很快发布。
英文摘要
Let $A_1,\ldots,A_N$ be positive semidefinite matrices of rank at most $r$, with $\sum_i A_i=I$ and $\norm{A_i}\le\varepsilon$. We prove that one sign can be assigned to each original matrix with discrepancy $O(\sqrt{\varepsilon\log(2r)})$, independently of their dimension and number, which is known to be optimal upto constants. We give a separate existence proof and a deterministic algorithm with polynomial work in a real-arithmetic model with semidefinite-value and exact spectral primitives. Separate Lean formalizations verify the existence proof and the algorithm in this arithmetic model. The proof extends the variational approach to discrepancy developed in the companion papers, motivated by operator-valued free interpolation, Lehner's formula, and spectral Tsallis regularization. A concave matrix power interpolates between a trace source and a sandwich source. Its concavity controls the response of the optimizing density. The walk maintains independent source reserves and a second matrix recording reserve expenditure. This matrix certifies contraction of the unfinished input mass between epochs; within an epoch, preparation and a negative-curvature step control the spectral potential while the coefficients advance toward signs. As applications, one spanning tree can be chosen simultaneously $O(\varepsilon\log(2s))$-spectrally thin for $s$ positive weightings of a common graph whose edge leverages are at most $\varepsilon$. The diagonal specialization gives discrepancy $O(\sqrt{L\log(2k)})$ for matrices with row sums at most $L$ and at most $k$ nonzero entries per column, through a walk of fractional colorings.
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- Google（谷歌）
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