Author

Lek-Heng Lim

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Preprint Oct 2026

Optimal Monte Carlo integration on matrix manifolds

This article develops optimal order (up to logarithmic factors) weighted cubature rules for Monte Carlo numerical integration in Sobolev reproducing kernel Hilbert spaces over common matrix manifolds. Specifically, we consider matrix models of the Stiefel, Grassmann, flag, and Lagrangian Grassmann manifolds each over e...

Aleksei G. Sorokin, Yi-Han Guo, Lek-Heng Lim · 0 citations

Preprint Sep 2026

Pierce-Birkhoff conjecture is false

We provide a counterexample to the Pierce-Birkhoff conjecture: a continuous function that is piecewise quadratic on a finite collection of semialgebraic sets that partition $\mathbb{R}^n$ but that cannot be expressed as a finite lattice combination of polynomials. The counterexample was found with the assistance of our...

Ze-Hua Lai, Lek-Heng Lim, Jun-Yu Ren · 0 citations

#machine learning Jun 2026

Low-dimensional topology of deep neural networks

The results suggest that low-dimensional topology can be a useful tool to guide designs of AI architectures, and generalize the results from $d = 3$ to arbitrary $d>3$.

Jun Ren, Lek-Heng Lim · arXiv.org · 0 citations