Selected work

Representative Papers

Stochastic Optimization Under Power-Law Spectra: Tight Bounds and Shuffling Analysis

Sep 28, 2026

This study addresses the absence of a unified theoretical foundation for stochastic gradient descent (SGD) convergence bounds and data shuffling strategies in high-dimensional machine learning. The proposed approach extends power-law spectral theory to stochastic optimization, demonstrating that the spectral exponent governs SGD dynamics. By integrating high-dimensional statistical learning theory with precise modeling of Gaussian data, tight convergence bounds featuring exact constants are derived. This work represents the first effort to unify abstract spectral theory with practical training sampling mechanisms, rigorously establishing the superiority of single shuffle over alternative sampling schemes. Ultimately, this research elucidates the intrinsic mechanism by which data geometry drives optimization speed, bridges the gap between theory and practice, and provides a comprehensive theoretical framework for stochastic training.

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Fast Evaluation of Polynomials with Rational Preprocessing

Sep 05, 2026

Horner's rule evaluates a monic degree-$n$ polynomial using $n-1$ multiplications. We show that with rational preprocessing of the coefficients, any such polynomial can be evaluated using only $\lfloor n/2 \rfloor + 1$ multiplications over fields of characteristic zero or of characteristic $p>n$. This resolves the multiplication side of a conjecture of Rabin and Winograd (Comm. Pure Appl. Math 1972), who achieved $n/2 + 2\lceil\log_2 n\rceil$ multiplications and conjectured the logarithmic overhead was necessary. We show that this multiplication count can't be beaten in general, proving that three multiplications do not suffice for degree~$6$. This strengthens the lower bound of Pan (STOC 1978), who proved a tight bound for general, complex preprocessing. In characteristic~2, for every $n>1$ and every finite field of size at least $2n$, we prove that an $n$-multiplication chain cannot parametrize all value vectors at $2n$ distinct evaluation points, even with arbitrary preprocessing. We give $\lfloor n/2 \rfloor + 1$ multiplication schedules over characteristic~2, each with an explicit inverse, for every odd degree $n\le 25$ and conjecture that this is possible for all $n$. We also give an injective polynomial construction for universal hashing that uses $N$ multiplications to hash $2N$ values with a single random key. This improves the best previous construction by Daniel J. Bernstein (cryp.to).

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DRAMPyML: A Formal Description of DRAM Protocols with Timed Petri Nets

Feb 11, 2026

This work addresses the growing complexity of JEDEC DRAM protocols, which renders traditional finite-state machines inadequate for accurately capturing concurrent multi-bank operations and intricate timing constraints, thereby hindering comprehension and verification. To overcome these limitations, the paper introduces timed Petri nets—a formalism well-suited for modeling concurrency and real-time behavior—into DRAM protocol specification for the first time. Leveraging Python, the authors develop an executable formal model that precisely encodes the protocol’s parallelism and timing semantics. This model not only substantially enhances the clarity and efficiency of protocol understanding but also enables rigorous correctness verification of DRAM controllers (RTL), DRAM logic, and memory simulators, while supporting evaluation of key performance metrics—capabilities unattainable with conventional state-machine-based approaches.

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A Complete Decomposition of Stochastic Differential Equations

Jan 12, 2026

This work proposes a novel structured decomposition for stochastic differential equations (SDEs) with prescribed time-varying marginal distributions. By integrating probability flow analysis, SDE theory, and matrix decomposition techniques, the authors uniquely decompose any such SDE into three distinct components: a scalar potential field governing the evolution of the marginal distribution, a symmetric positive semi-definite diffusion matrix field, and a skew-symmetric matrix field. This tripartite decomposition provides the first complete characterization of the intrinsic geometric and probabilistic structure of SDEs under given marginal constraints. The decomposition is not only mathematically unique but also establishes a rigorous theoretical foundation for modeling, analyzing, and controlling complex stochastic systems.

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Lattice Random Walk Discretisations of Stochastic Differential Equations

Aug 28, 2025

Existing numerical solvers for stochastic differential equations (SDEs) suffer from computationally expensive drift/diffusion evaluations, reliance on Gaussian sampling, sensitivity to quantization errors, and instability with non-Lipschitz drift terms. To address these challenges, this paper proposes a lattice-based random walk discretization method. It replaces continuous drift and diffusion dynamics with 1–2-bit stochastic operations—using binary or ternary increment sampling—thereby eliminating floating-point arithmetic and Gaussian sampling entirely and enabling native compatibility with bitstream probabilistic computing architectures. Theoretically, the method achieves first-order weak convergence, exhibits robustness to quantization errors, and remains stable under non-Lipschitz drift conditions. Empirical evaluation demonstrates its effectiveness on canonical SDEs and state-of-the-art diffusion models, while delivering substantial hardware efficiency gains and computational speedup.

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Recent publications

Latest Papers

Stochastic Optimization Under Power-Law Spectra: Tight Bounds and Shuffling Analysis

Sep 28, 2026

This study addresses the absence of a unified theoretical foundation for stochastic gradient descent (SGD) convergence bounds and data shuffling strategies in high-dimensional machine learning. The proposed approach extends power-law spectral theory to stochastic optimization, demonstrating that the spectral exponent governs SGD dynamics. By integrating high-dimensional statistical learning theory with precise modeling of Gaussian data, tight convergence bounds featuring exact constants are derived. This work represents the first effort to unify abstract spectral theory with practical training sampling mechanisms, rigorously establishing the superiority of single shuffle over alternative sampling schemes. Ultimately, this research elucidates the intrinsic mechanism by which data geometry drives optimization speed, bridges the gap between theory and practice, and provides a comprehensive theoretical framework for stochastic training.

0 citationsRead paper

Fast Evaluation of Polynomials with Rational Preprocessing

Sep 05, 2026

Horner's rule evaluates a monic degree-$n$ polynomial using $n-1$ multiplications. We show that with rational preprocessing of the coefficients, any such polynomial can be evaluated using only $\lfloor n/2 \rfloor + 1$ multiplications over fields of characteristic zero or of characteristic $p>n$. This resolves the multiplication side of a conjecture of Rabin and Winograd (Comm. Pure Appl. Math 1972), who achieved $n/2 + 2\lceil\log_2 n\rceil$ multiplications and conjectured the logarithmic overhead was necessary. We show that this multiplication count can't be beaten in general, proving that three multiplications do not suffice for degree~$6$. This strengthens the lower bound of Pan (STOC 1978), who proved a tight bound for general, complex preprocessing. In characteristic~2, for every $n>1$ and every finite field of size at least $2n$, we prove that an $n$-multiplication chain cannot parametrize all value vectors at $2n$ distinct evaluation points, even with arbitrary preprocessing. We give $\lfloor n/2 \rfloor + 1$ multiplication schedules over characteristic~2, each with an explicit inverse, for every odd degree $n\le 25$ and conjecture that this is possible for all $n$. We also give an injective polynomial construction for universal hashing that uses $N$ multiplications to hash $2N$ values with a single random key. This improves the best previous construction by Daniel J. Bernstein (cryp.to).

0 citationsRead paper

DRAMPyML: A Formal Description of DRAM Protocols with Timed Petri Nets

Feb 11, 2026

This work addresses the growing complexity of JEDEC DRAM protocols, which renders traditional finite-state machines inadequate for accurately capturing concurrent multi-bank operations and intricate timing constraints, thereby hindering comprehension and verification. To overcome these limitations, the paper introduces timed Petri nets—a formalism well-suited for modeling concurrency and real-time behavior—into DRAM protocol specification for the first time. Leveraging Python, the authors develop an executable formal model that precisely encodes the protocol’s parallelism and timing semantics. This model not only substantially enhances the clarity and efficiency of protocol understanding but also enables rigorous correctness verification of DRAM controllers (RTL), DRAM logic, and memory simulators, while supporting evaluation of key performance metrics—capabilities unattainable with conventional state-machine-based approaches.

0 citationsRead paper

A Complete Decomposition of Stochastic Differential Equations

Jan 12, 2026

This work proposes a novel structured decomposition for stochastic differential equations (SDEs) with prescribed time-varying marginal distributions. By integrating probability flow analysis, SDE theory, and matrix decomposition techniques, the authors uniquely decompose any such SDE into three distinct components: a scalar potential field governing the evolution of the marginal distribution, a symmetric positive semi-definite diffusion matrix field, and a skew-symmetric matrix field. This tripartite decomposition provides the first complete characterization of the intrinsic geometric and probabilistic structure of SDEs under given marginal constraints. The decomposition is not only mathematically unique but also establishes a rigorous theoretical foundation for modeling, analyzing, and controlling complex stochastic systems.

0 citationsRead paper

Lattice Random Walk Discretisations of Stochastic Differential Equations

Aug 28, 2025

Existing numerical solvers for stochastic differential equations (SDEs) suffer from computationally expensive drift/diffusion evaluations, reliance on Gaussian sampling, sensitivity to quantization errors, and instability with non-Lipschitz drift terms. To address these challenges, this paper proposes a lattice-based random walk discretization method. It replaces continuous drift and diffusion dynamics with 1–2-bit stochastic operations—using binary or ternary increment sampling—thereby eliminating floating-point arithmetic and Gaussian sampling entirely and enabling native compatibility with bitstream probabilistic computing architectures. Theoretically, the method achieves first-order weak convergence, exhibits robustness to quantization errors, and remains stable under non-Lipschitz drift conditions. Empirical evaluation demonstrates its effectiveness on canonical SDEs and state-of-the-art diffusion models, while delivering substantial hardware efficiency gains and computational speedup.

0 citationsRead paper