Weighted empirical measures on compact manifolds appear in importance sampling, particle approximations, posterior summaries, quadrature, and representation learning. Ordinary effective sample size and related weight summaries ignore the geometry of the support. We introduce heat-kernel entropy profiles to measure nonuniformity after intrinsic diffusion at a range of scales. For order-two Rényi entropy, pairwise heat-kernel overlaps give an exact profile and a geometric effective sample size. This effective sample size discounts nearby or duplicate particles. It approaches ordinary effective sample size as overlaps between distinct particles vanish. On compact boundaryless manifolds, we establish profile monotonicity, gESS scale limits, deterministic-weight consistency, and a bounded-ratio result for self-normalized importance sampling. On spheres, the unlogged profile decomposes into spherical-harmonic energies. The first terms are squared mean-resultant and traceless-second-moment energies, which give vMF- and Bingham-type scalar summaries. Experiments identify antipodal, girdle, multimodal, and duplicate-particle structures that weight-only and first-moment summaries miss.
In high-dimensional Ising model estimation, target sample sizes are often limited, and effectively using auxiliary binary datasets of unknown relevance remains challenging. To address this, we propose Trans-Ising, a transfer learning method that combines a loss-based source scree…
Modern LLM deployments use a number of implementation choices and inference optimizations (e.g., batching, custom kernels, and quantization) on top of fixed weights, so two engines serving "the same model" can produce meaningfully different distributions. We study the problem of…
Longitudinal studies often collect data at sparse, irregular, and unequally spaced time points. Such heterogeneity is often driven by subject-specific covariates, yet existing methods have been restricted to a scalar endpoint value, completely neglecting the underlying response t…
Many real-world processes can be represented as compositions of functions along a directed acyclic graph (DAG). In causal modelling, these correspond to the underlying mechanisms; in engineering, to multiple fidelity levels; and in gene-regulatory networks, to transcription facto…
Many problems in science and engineering are difficult to model accurately, either due to unknown physical mechanisms, poorly quantified measurement uncertainty, or prohibitive computational costs of high-fidelity simulations. These challenges limit the applicability of classical…
Integrating information across related tasks can improve estimation and prediction in transfer, multi-task, and federated learning, but contamination and heterogeneity make robust borrowing challenging. We study a contaminated multi-task empirical risk minimization (ERM) framewor…