arXiv subjects

Alexander W. Hsu

Publications and source records attributed to Alexander W. Hsu.

5 recordsLinked to original sources

arXiv2026

Approximating Measures on Function Spaces: Transport and Truncation

Measures on function spaces arise throughout Bayesian inverse problems and generative modeling, often with low-dimensional structure relative to a tractable reference measure. We introduce the class $\mathcal{P}_ψ(μ)$ of measures that differ from a reference measure $μ$ only through a finite-dimensional map $ψ$ while preserving the reference conditionals on its fibers. Class members are determined by their $d$-dimensional pushforwards under $ψ$ and admit convenient block-triangular transport map representations. Draws are taken from this $d$-dimensional distribution and then completed to function space through sampling of the reference conditionals. For Gaussian references, these transport maps are finite rank perturbations of the identity. In contrast, optimal transport maps do not preserve this low-dimensional structure. For covariance perturbations that are trace class in the Cameron-Martin geometry of the reference, we show the optimal map is a trace class perturbation of the identity. Even finite rank perturbations yield corrections whose rank, governed by an invariant subspace, is typically infinite. We develop approximation theory for $\mathcal{P}_ψ(μ)$ and error analysis for fitting within it, splitting the total error into an irreducible class error and a finite-dimensional marginal term set by the dimension of $ψ$ rather than the ambient discretization. We present numerical results including inference from low-dimensional nonlinear observation maps, deconvolution under a jump process prior, and state estimation for Navier-Stokes flows.

math.PR↗

arXiv2026

A joint optimization approach to identifying sparse dynamics using least squares kernel collocation

We develop an all-at-once modeling framework for learning systems of ordinary differential equations (ODE) from scarce, partial, and noisy observations of the states. The proposed methodology amounts to a combination of sparse recovery strategies for the ODE over a function library combined with techniques from reproducing kernel Hilbert space (RKHS) theory for estimating the state and discretizing the ODE. Our numerical experiments reveal that the proposed strategy leads to significant gains in terms of accuracy, sample efficiency, and robustness to noise, both in terms of learning the equation and estimating the unknown states. This work demonstrates capabilities well beyond existing and widely used algorithms while extending the modeling flexibility of other recent developments in equation discovery.

stat.ME↗

arXiv2025

Sharp bounds on the failure of the hot spots conjecture

The hot spots ratio of a domain $Ω\subset \mathbb{R}^d$ measures the degree of failure of Rauch's hot spots conjecture on that domain. We identify the largest possible value of this ratio over all connected Lipschitz domains $Ω\subset \mathbb{R}^d$, for any dimension $d$. As $d\to \infty$, we show that this maximal ratio converges to $\sqrt{e}$, which asymptotically matches the previous best known upper bound by Mariano, Panzo and Wang. For $d\ge 2$, we show that sets extremizing the hot spots ratio do not exist, and extremizing sequences must converge to a ball at a quantitative rate. We then give a sharp bound on the measure of the set for which the first Neumann eigenfunction exceeds its maximal boundary value. From this we deduce that the hot spots conjecture is asymptotically true "in measure'' as $d\to \infty$.

math.SP↗

arXiv2024

Corrected Correlation Estimates for Meta-Analysis

Meta-analysis allows rigorous aggregation of estimates and uncertainty across multiple studies. When a given study reports multiple estimates, such as log odds ratios (ORs) or log relative risks (RRs) across exposure groups, accounting for within-study correlations improves accuracy and efficiency of meta-analytic results. Canonical approaches of Greenland-Longnecker and Hamling estimate pseudo cases and non-cases for exposure groups to obtain within-study correlations. However, currently available implementations for both methods fail on simple examples. We review both GL and Hamling methods through the lens of optimization. For ORs, we provide modifications of each approach that ensure convergence for any feasible inputs. For GL, this is achieved through a new connection to entropic minimization. For Hamling, a modification leads to a provably solvable equivalent set of equations given a specific initialization. For each, we provide implementations a guaranteed to work for any feasible input. For RRs, we show the new GL approach is always guaranteed to succeed, but any Hamling approach may fail: we give counter-examples where no solutions exist. We derive a sufficient condition on reported RRs that guarantees success when reported variances are all equal.

stat.ME↗

arXiv2024

Conditional Optimal Transport on Function Spaces

We present a systematic study of conditional triangular transport maps in function spaces from the perspective of optimal transportation and with a view towards amortized Bayesian inference. More specifically, we develop a theory of constrained optimal transport problems that describe block-triangular Monge maps that characterize conditional measures along with their Kantorovich relaxations. This generalizes the theory of optimal triangular transport to separable infinite-dimensional function spaces with general cost functions. We further tailor our results to the case of Bayesian inference problems and obtain regularity estimates on the conditioning maps from the prior to the posterior. Finally, we present numerical experiments that demonstrate the computational applicability of our theoretical results for amortized and likelihood-free inference of functional parameters.

math.OC↗