Feasibility Preservation under Monotone Retrieval Truncation

Retrieval-based systems approximate access to a corpus by exposing only a truncated subset of available evidence. Even when relevant information exists in the corpus, truncation can prevent compatible evidence from co-occurring, leading to failures that are not captured by relevance-based evaluation. This paper studies retrieval from a structural perspective, modeling query answering as a feasibility problem under truncation. We formalize retrieval as a sequence of candidate evidence sets and characterize conditions under which feasibility in the limit implies feasibility at finite retrieval depth. We show that monotone truncation suffices to guarantee finite witnessability for individual queries. For classes of queries, we identify finite generation of witness certificates as the additional condition required to obtain a uniform retrieval bound, and we show that this condition is necessary. We further exhibit sharp counterexamples demonstrating failure under non-monotone truncation, non-finitely-generated query classes, and purely slotwise coverage. Together, these results isolate feasibility preservation as a correctness criterion for retrieval independent of relevance scoring or optimization, and clarify structural limitations inherent to truncation-based retrieval.

Thermodynamic Characterizations of Singular Bayesian Models: Specific Heat, Susceptibility, and Entropy Flow in Posterior Geometry

Singular learning theory (SLT) \citep{watanabe2009algebraic,watanabe2018mathematical} provides a rigorous asymptotic framework for Bayesian models with non-identifiable parameterizations, yet the statistical meaning of its second-order invariant, the \emph{singular fluctuation}, has remained unclear. In this work, we show that singular fluctuation admits a precise and natural interpretation as a \emph{specific heat}: the second derivative of the Bayesian free energy with respect to temperature. Equivalently, it measures the posterior variance of the log-likelihood observable under the tempered Gibbs posterior. We further introduce a collection of related thermodynamic quantities, including entropy flow, prior susceptibility, and cross-susceptibility, that together provide a detailed geometric diagnosis of singular posterior structure. Through extensive numerical experiments spanning discrete symmetries, boundary singularities, continuous gauge freedoms, and piecewise (ReLU) models, we demonstrate that these thermodynamic signatures cleanly distinguish singularity types, exhibit stable finite-sample behavior, and reveal phase-transition--like phenomena as temperature varies. We also show empirically that the widely used WAIC estimator \citep{watanabe2010asymptotic, watanabe2013widely} is exactly twice the thermodynamic specific heat at unit temperature, clarifying its robustness in singular models.Our results establish a concrete bridge between singular learning theory and statistical mechanics, providing both theoretical insight and practical diagnostics for modern Bayesian models.

Statistical Guarantees for Transformation Based Models with Applications to Implicit Variational Inference

Transformation-based methods have been an attractive approach in non-parametric inference for problems such as unconditional and conditional density estimation due to their unique hierarchical structure that models the data as flexible transformation of a set of common latent variables. More recently, transformation-based models have been used in variational inference (VI) to construct flexible implicit families of variational distributions. However, their use in both non-parametric inference and variational inference lacks theoretical justification. We provide theoretical justification for the use of non-linear latent variable models (NL-LVMs) in non-parametric inference by showing that the support of the transformation induced prior in the space of densities is sufficiently large in the $L_1$ sense. We also show that, when a Gaussian process (GP) prior is placed on the transformation function, the posterior concentrates at the optimal rate up to a logarithmic factor. Adopting the flexibility demonstrated in the non-parametric setting, we use the NL-LVM to construct an implicit family of variational distributions, deemed GP-IVI. We delineate sufficient conditions under which GP-IVI achieves optimal risk bounds and approximates the true posterior in the sense of the Kullback-Leibler divergence. To the best of our knowledge, this is the first work on providing theoretical guarantees for implicit variational inference.