Borel
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Approximations and Learning for Continuous State and Action MDPs under Average Cost Criteria
Announce Type: replace-cross Abstract: In this paper, for Markov Decision Processes (MDPs) with standard Borel spaces, (i) we first provide a discretization based approximation method for MDPs with continuous spaces under average cost criteria, and provide error bounds for approximations when the dynamics are only weakly continuous (for asymptotic convergence of errors as the grid sizes vanish) or Wasserstein continuous (with a rate in approximation as the grid sizes vanish) under certain...
Finite-Blocklength Lossy Joint Source-Channel Coding over Unknown Channels
arXiv:2606.07933v1 Announce Type: new Abstract: We analyze the finite-blocklength performance of lossy joint source-channel codes (JSCC) in an unknown-channel framework, where the true channel is unknown but the source distribution is known. We establish achievability results for mismatched-design JSCC, where the code design is based on a channel $Q_{Y|X}$ but deployed over a different channel $P_{Y|X}$. Our mismatched-design achievability result allows nonstationary channel laws and...
The Right Measure for Physics-Constrained Generation: A Co-Area Correction for Posterior-Consistent PDE Inverse Problems
arXiv:2606.04804v1 Announce Type: new Abstract: Generative models -- diffusion and flow matching -- are increasingly used to solve partial differential equation (PDE) inverse problems, enforcing the governing physics as a \emph{hard constraint} (via projection or guidance) and reporting the resulting samples as a Bayesian posterior with calibrated uncertainty. We show that this widely adopted recipe samples the wrong distribution. Conditioning a generative prior on a hard PDE constraint is...
Universal consistency of the $k$-NN rule in metric spaces and Nagata dimension. III
Announce Type: replace Abstract: We establish the last missing link allowing to describe those complete separable metric spaces $X$ in which the $k$ nearest neighbour classifier is universally consistent, both in combinatorial terms of dimension theory and via a fundamental property of real analysis. The following are equivalent: (1) The $k$-nearest neighbour classifier is universally consistent in $X$, (2) The strong Lebesgue--Besicovitch differentiation property holds in $X$ for every...
The Right Measure for Physics-Constrained Generation: A Co-Area Correction for Posterior-Consistent PDE Inverse Problems
arXiv:2606.04804v3 Announce Type: replace Abstract: Generative models -- diffusion and flow matching -- are increasingly used to solve partial differential equation (PDE) inverse problems, enforcing the governing physics as a \emph{hard constraint} (via projection or guidance) and reporting the resulting samples as a Bayesian posterior with calibrated uncertainty. We show that this widely adopted recipe samples the wrong distribution. Conditioning a generative prior on a hard PDE constraint...
Foundational Analysis Of The Solvability Complexity Index: The Weihrauch-SCI Intermediate Hierarchy
Announce Type: replace-cross Abstract: The Solvability Complexity Index (SCI) provides an extensional limit-height formalism for recovering a target map $\Xi$ from finite samples of an evaluation interface $\Lambda\subseteq\mathbb C^\Omega$ by finite-height towers of pointwise limits. We first give a foundational analysis of what this extensional framework does and does not determine. We show that the SCI separation axiom is equivalent to a factorization of $\Xi$ through the full evaluation...
The Right Measure for Physics-Constrained Generation: A Co-Area Correction for Posterior-Consistent PDE Inverse Problems
Announce Type: replace Abstract: Generative models -- diffusion and flow matching -- are increasingly used to solve partial differential equation (PDE) inverse problems, enforcing the governing physics as a \emph{hard constraint} (via projection or guidance) and reporting the resulting samples as a Bayesian posterior with calibrated uncertainty. We show that this widely adopted recipe samples the wrong distribution. Conditioning a generative prior on a hard PDE constraint is conditioning on...