Author

Zhao Song

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Preprint Sep 2026

Matrix Spencer: Eight Standard Deviations Suffice and an Almost-Linear Time Algorithm for Dense Input

The Matrix Spencer conjecture asserts that for all symmetric matrices $A_1,\ldots,A_n\in\mathbb{R}^{n\times n}$ with $\|A_i\|\le1$ there are signs $\varepsilon_1,\ldots,\varepsilon_n\in\{-1,1\}$ with $\|\sum_{i=1}^n\varepsilon_iA_i\|=O(\sqrt n)$. We prove it: a signing of discrepancy below $8\sqrt n$ always exists. We...

Zhao Song, Li-Cheng Zhang · 0 citations

Preprint Aug 2026

Hadamard Flattening and Gaussian Pooling Sketch for Least Squares with Coordinate-wise Guarantee

A new fast, dense randomized transform is introduced, which combines a randomized Hadamard flattening, a random permutation, and balanced, disjoint Gaussian pooling to achieve a truly nearly-linear-in-d row count.

Zhao Song, Li-Cheng Zhang · 0 citations