Author

Tarun Kathuria

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Preprint Sep 2026

A Walk From Free Probability to Matrix Discrepancy II: Weaver's Problem and the Kadison-Singer Conjecture

\cite{mss2015} proved Weaver's discrepancy result existentially, resolving the Kadison--Singer conjecture . Finding such signs efficiently for general inputs remained an open algorithmic question. In the real-arithmetic model, we give a deterministic algorithm running in polynomial time with discrepancy at most $35\sqr...

Tarun Kathuria · 2 citations

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 potential measures a soft spectral edge of the evolving discrepancy matrix perturbed by an operator-valued free semicircular element and combines Lehner's variational formula for the free edge with spectral Tsallis regularization, putting the discrepancy and remaining covariance in a single smooth optimization prob...

Tarun Kathuria · 2 citations