This study presents a conditional flow matching framework for solving physics-constrained Bayesian inverse problems. In this setting, samples from the joint distribution of inferred variables and measurements are assumed available, while explicit evaluation of the prior and likelihood densities is not required. We derive a simple and self-contained formulation of both the unconditional and conditional flow matching algorithms, tailored specifically to inverse problems. In the conditional setting, a neural network is trained to learn the velocity field of a probability flow ordinary differential equation that transports samples from a chosen source distribution directly to the posterior distribution conditioned on observed measurements. This black-box formulation accommodates nonlinear, high-dimensional, and potentially non-differentiable forward models without restrictive assumptions on the noise model. We further analyze the behavior of the learned velocity field in the regime of finite training data. Under mild architectural assumptions, we show that overtraining can induce degenerate behavior in the generated conditional distributions, including variance collapse and a phenomenon termed selective memorization, wherein generated samples concentrate around training data points associated with similar observations. A simplified theoretical analysis explains this behavior, and numerical experiments confirm it in practice. We demonstrate that standard early-stopping criteria based on monitoring test loss effectively mitigate such degeneracy. The proposed method is evaluated on several physics-based inverse problems. We investigate the impact of different choices of source distributions, including Gaussian and data-informed priors. Across these examples, conditional flow matching accurately captures complex, multimodal posterior distributions while maintaining computational efficiency.

We study privacy-preserving sparse linear regression in the high-dimensional regime, focusing on the LASSO estimator. We analyze two widely used mechanisms for differential privacy: output perturbation, which injects noise into the estimator, and objective perturbation, which adds a random linear term to the loss function. Using approximate message passing (AMP), we characterize the typical behavior of these estimators under random design and privacy noise. To quantify privacy, we adopt typical-case measures, including the on-average KL divergence, which admits a hypothesis-testing interpretation in terms of distinguishability between neighboring datasets. Our analysis reveals that sparsity plays a central role in shaping the privacy-accuracy trade-off: stronger regularization can improve privacy by stabilizing the estimator against single-point data changes. We further show that the two mechanisms exhibit qualitatively different behaviors. In particular, for objective perturbation, increasing the noise level can have non-monotonic effects, and excessive noise may destabilize the estimator, leading to increased sensitivity to data perturbations. Our results demonstrate that AMP provides a powerful framework for analyzing privacy-accuracy trade-offs in high-dimensional sparse models.

We study learning under regime variation, where the learner, its memory state, and the evaluative conditions may evolve over time. This paper is a foundational and structural contribution: its goal is to define the core learning-theoretic objects required for such settings and to establish their first theorem-supporting consequences. The paper develops a regime-varying framework centered on admissible transport, protected-core preservation, and evaluator-aware learning evolution. It records the immediate closure consequences of admissibility, develops a structural obstruction argument for faithful fixed-ontology reduction in genuinely multi-regime settings, and introduces a protected-stability template together with explicit numerical and symbolic witnesses on controlled subclasses, including convex and deductive settings. It also establishes theorem-layer results on evaluator factorization, morphisms, composition, and partial kernel-level alignment across semantically commensurable layers. A worked two-regime example makes the admissibility certificate, protected evaluative core, and regime-variation cost explicit on a controlled subclass. The symbolic component is deliberately restricted in scope: the paper establishes a first kernel-level compatibility result together with a controlled monotonic deductive witness. The manuscript should therefore be read as introducing a structured learning-theoretic framework for regime-varying learning together with its first theorem-supporting layer, not as a complete quantitative theory of all learning systems.

We study forward and inverse problems for time-dependent probability measures in Bayes--Hilbert spaces. On the forward side, we show that each sufficiently regular Bayes--Hilbert path admits a canonical dynamical realization: a weighted Neumann problem transforms the log-density variation into the unique gradient velocity field of minimum kinetic energy. This construction induces a transport form on Bayes--Hilbert tangent directions, which measures the dynamical cost of realizing prescribed motions, and yields a flow-matching interpretation in which the canonical velocity field is the minimum-energy execution of the prescribed path. On the inverse side, we formulate reconstruction directly on Bayes--Hilbert path space from time-dependent indirect observations. The resulting variational problem combines a data-misfit term with the transport action induced by the forward geometry. In our infinite-dimensional setting, however, this transport geometry alone does not provide sufficient compactness, so we add explicit temporal and spatial regularization to close the theory. The linearized observation operator induces a complementary observability form, which quantifies how strongly tangent directions are seen through the data. Under explicit Sobolev regularity and observability assumptions, we prove existence of minimizers, derive first-variation formulas, establish local stability of the observation map, and deduce recovery of the evolving law, its score, and its canonical velocity field under the strong topologies furnished by the compactness theory.

Abstraction and realization are bilateral processes that are key in deriving intelligence and creativity. In many domains, the two processes are approached through \emph{rules}: high-level principles that reveal invariances within similar yet diverse examples. Under a probabilistic setting for discrete input spaces, we focus on the rule realization problem which generates input sample distributions that follow the given rules. More ambitiously, we go beyond a mechanical realization that takes whatever is given, but instead ask for proactively selecting reasonable rules to realize. This goal is demanding in practice, since the initial rule set may not always be consistent and thus intelligent compromises are needed. We formulate both rule realization and selection as two strongly connected components within a single and symmetric bi-convex problem, and derive an efficient algorithm that works at large scale. Taking music compositional rules as the main example throughout the paper, we demonstrate our model's efficiency in not only music realization (composition) but also music interpretation and understanding (analysis).

We leverage ideas from Stochastic Realization Theory to develop a class of models called B'MOJO to seamlessly combine eidetic and fading memory within an elementary composable module. The overall architecture can be used to implement models that can access short-term eidetic memory "in-context," permanent structural memory "in-weights,"

Many applications seek to recover low-rank approximations of noisy tensor data.