Author

Tarun Kathuria

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Preprint Sep 2026

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

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...

Tarun Kathuria · 0 citations

Preprint Sep 2026

A Walk From Free Probability to Matrix Discrepancy I: Matrix Spencer

The Matrix Spencer conjecture asks whether any $n$ real symmetric matrices A_1,...,A_n \in \mathbb{R}^{m \times m} of operator norm at most one admit a signing $x\in\{-1,1\}^n$ such that the operator norm of the signed sum is at most O(\sqrt{n \log(2m/n)}) We give a randomized algorithm establishing this bound with pol...

Tarun Kathuria · 2 citations