On Uniform Weighted Deep Polynomial approximation

It is a classical result in rational approximation theory that certain non-smooth or singular functions, such as $|x|$ and $x^{1/p}$, can be efficiently approximated using rational functions with root-exponential convergence in terms of degrees of freedom \cite{Sta, GN}. In contrast, polynomial approximations admit only algebraic convergence by Jackson's theorem \cite{Lub2}. Recent work shows that composite polynomial architectures can recover exponential approximation rates even without smoothness \cite{KY}. In this work, we introduce and analyze a class of weighted deep polynomial approximants tailored for functions with asymmetric behavior-growing unbounded on one side and decaying on the other. By multiplying a learnable deep polynomial with a one-sided weight, we capture both local non-smoothness and global growth. We show numerically that this framework outperforms Taylor, Chebyshev, and standard deep polynomial approximants, even when all use the same number of parameters. To optimize these approximants in practice, we propose a stable graph-based parameterization strategy building on \cite{Jar}.

BOLT: Block-Orthonormal Lanczos for Trace estimation of matrix functions

Efficient matrix trace estimation is essential for scalable computation of log-determinants, matrix norms, and distributional divergences. In many large-scale applications, the matrices involved are too large to store or access in full, making even a single matrix-vector (mat-vec) product infeasible. Instead, one often has access only to small subblocks of the matrix or localized matrix-vector products on restricted index sets. Hutch++ achieves optimal convergence rate but relies on randomized SVD and assumes full mat-vec access, making it difficult to apply in these constrained settings. We propose the Block-Orthonormal Stochastic Lanczos Quadrature (BOLT), which matches Hutch++ accuracy with a simpler implementation based on orthonormal block probes and Lanczos iterations. BOLT builds on the Stochastic Lanczos Quadrature (SLQ) framework, which combines random probing with Krylov subspace methods to efficiently approximate traces of matrix functions, and performs better than Hutch++ in near flat-spectrum regimes. To address memory limitations and partial access constraints, we introduce Subblock SLQ, a variant of BOLT that operates only on small principal submatrices. As a result, this framework yields a proxy KL divergence estimator and an efficient method for computing the Wasserstein-2 distance between Gaussians - both compatible with low-memory and partial-access regimes. We provide theoretical guarantees and demonstrate strong empirical performance across a range of high-dimensional settings.