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Voleon Group

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Selected work

Representative Papers

High-Dimensional Expanders, the Sparsest Cut Problem, and Steurer's Conjecture

May 29, 2026

This work refutes Steurer’s conjecture that any family of unit vectors with low pairwise correlations must contain a constant-separated subset of linear size. By constructing a counterexample based on sparse high-dimensional expanders, the authors demonstrate that even when the average correlation among vectors is polynomially small, no linear-sized approximately separated subset may exist. The proof hinges on three key technical contributions: the design of high-dimensional expanders, a refined analysis of vector correlations, and an estimate of the $L_2$ mixing time for reweighted random walks. As a byproduct, this construction yields the first example of a vertex expander on which every reweighted simple random walk exhibits an $L_2$ mixing time of at least $\log^{5/4 - o(1)} n$, thereby revealing a fundamental limitation in the relationship between correlation and separability.

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Optimal $e^{(γ+o(1))n}$-Approximation of the Permanent of Positive Semidefinite Matrices

May 20, 2026

This work establishes the optimal approximation ratio for computing the permanent of Hermitian positive semidefinite matrices in deterministic polynomial time. By formulating a concave optimization problem based on the row vectors of a matrix factorization, analyzing the Wick integral formula via entropy methods, and leveraging complex matrix decomposition techniques, the authors derive tight upper and lower bounds on the permanent. Their main contribution is the first complete characterization of the optimal approximation ratio at exponential precision, proving that $\operatorname{per}(A)$ satisfies $e^{-\gamma n} \widehat P(A) \le \operatorname{per}(A) \le \widehat P(A)$. This yields an approximation ratio of $e^{(\gamma+o(1))n}$, which exactly matches the known computational hardness lower bound, thereby closing a longstanding gap in the theoretical understanding of this problem.

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Uncertainty quantification for Markov chains with application to temporal difference learning

Feb 19, 2025

This paper addresses the statistical inference challenge for Markov chain-driven reinforcement learning (e.g., temporal difference learning) under data dependence. It establishes, for the first time, concentration inequalities and Berry–Esseen-type bounds on distributional convergence rates for high-dimensional vector- and matrix-valued functions of Markov chains. Methodologically, it integrates stochastic process theory, spectral analysis, and high-dimensional probability tools to derive high-probability consistency guarantees that match the asymptotic variance, achieving a Gaussian approximation rate of $O(T^{-1/4}log T)$ in convex distance. The key contributions are: (1) the first non-asymptotic, high-probability convergence guarantee for TD estimators under non-i.i.d. data—sharper than prior results; and (2) a rigorous theoretical foundation for uncertainty quantification and statistical inference in reinforcement learning.

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Latest Papers

High-Dimensional Expanders, the Sparsest Cut Problem, and Steurer's Conjecture

May 29, 2026

This work refutes Steurer’s conjecture that any family of unit vectors with low pairwise correlations must contain a constant-separated subset of linear size. By constructing a counterexample based on sparse high-dimensional expanders, the authors demonstrate that even when the average correlation among vectors is polynomially small, no linear-sized approximately separated subset may exist. The proof hinges on three key technical contributions: the design of high-dimensional expanders, a refined analysis of vector correlations, and an estimate of the $L_2$ mixing time for reweighted random walks. As a byproduct, this construction yields the first example of a vertex expander on which every reweighted simple random walk exhibits an $L_2$ mixing time of at least $\log^{5/4 - o(1)} n$, thereby revealing a fundamental limitation in the relationship between correlation and separability.

0 citationsRead paper

Optimal $e^{(γ+o(1))n}$-Approximation of the Permanent of Positive Semidefinite Matrices

May 20, 2026

This work establishes the optimal approximation ratio for computing the permanent of Hermitian positive semidefinite matrices in deterministic polynomial time. By formulating a concave optimization problem based on the row vectors of a matrix factorization, analyzing the Wick integral formula via entropy methods, and leveraging complex matrix decomposition techniques, the authors derive tight upper and lower bounds on the permanent. Their main contribution is the first complete characterization of the optimal approximation ratio at exponential precision, proving that $\operatorname{per}(A)$ satisfies $e^{-\gamma n} \widehat P(A) \le \operatorname{per}(A) \le \widehat P(A)$. This yields an approximation ratio of $e^{(\gamma+o(1))n}$, which exactly matches the known computational hardness lower bound, thereby closing a longstanding gap in the theoretical understanding of this problem.

0 citationsRead paper

Uncertainty quantification for Markov chains with application to temporal difference learning

Feb 19, 2025

This paper addresses the statistical inference challenge for Markov chain-driven reinforcement learning (e.g., temporal difference learning) under data dependence. It establishes, for the first time, concentration inequalities and Berry–Esseen-type bounds on distributional convergence rates for high-dimensional vector- and matrix-valued functions of Markov chains. Methodologically, it integrates stochastic process theory, spectral analysis, and high-dimensional probability tools to derive high-probability consistency guarantees that match the asymptotic variance, achieving a Gaussian approximation rate of $O(T^{-1/4}log T)$ in convex distance. The key contributions are: (1) the first non-asymptotic, high-probability convergence guarantee for TD estimators under non-i.i.d. data—sharper than prior results; and (2) a rigorous theoretical foundation for uncertainty quantification and statistical inference in reinforcement learning.

0 citationsRead paper