We propose and analyze a conservative drifting method for one-step generative modeling. The method replaces the original displacement-based drifting velocity by a kernel density estimator (KDE)-gradient velocity, namely the difference of the kernel-smoothed data score and the kernel-smoothed model score. This velocity is a gradient field, addressing the non-conservatism issue identified for general displacement-based drifting fields. We prove continuous-time finite-particle convergence bounds for the conservative method on $\R^d$: a joint-entropy identity yields bounds for the empirical Stein drift, the smoothed Fisher discrepancy of the KDE, and the squared center velocity. The main finite-particle correction is a reciprocal-KDE self-interaction term, and we give deterministic and high-probability local-occupancy conditions under which this term is controlled. We keep the quadrature constants explicit and track their possible bandwidth dependence: the root residual-velocity rate $N^{-1/(d+4)}$ holds under an additional $h$-uniform quadrature regularity condition, while a more general growth condition yields the optimized root rate $N^{-(2-β)/(2(d+4-β))}$, where $0\le β<2$. We also analyze the non-conservative drifting method with Laplace kernel, corresponding to the original displacement-based velocity proposed in Deng et al., 2026 (arxiv:2602.04770). For this method, a sharp companion kernel decomposes the velocity into a positive scalar preconditioning of a sharp-score mismatch plus a Laplace scale-mismatch residual, producing an analogous finite-particle rate with an unavoidable residual term. Finally, we explain how the continuous-time residual-velocity bounds translate into one-step generation guarantees through the explicit drift size $η$.

Discrete diffusion has become a leading framework for generative modeling in various applications including language, vision, and biology. Existing convergence theory, however, exhibits fundamental limitations. KL-based analyses diverge under singular priors such as the masked distribution, while bounds in total variation (TV) depend on the state space size $S$ and become vacuous for modern language tasks, where vocabularies contain hundreds of thousands of tokens. We develop a unified adjoint-equation-based framework that establishes dimension-free convergence guarantees in any integral probability metric (IPM). To the best of our knowledge, our bounds are the first to be entirely free of $S$ and applicable to both masked and uniform priors. Importantly, our theory relies only on a single standard rate-matrix regularity assumption and is compatible with time-inhomogeneous schedules. Four novel techniques drive our improvements: working in the space of observables via adjoint equations rather than directly with probability measures, a regularity analysis that yields bounds on any IPM, a coupling argument that removes $S$-dependence under uniform transitions, and a score-marginal cancellation technique that removes $S$-dependence under masked transitions. Our framework thus sharply departs from prior analyses and avoids the shortcomings of pathspace-KL and existing TV-based approaches. Beyond convergence bounds, our framework provides a versatile toolkit for further theoretical study of discrete diffusion models.

Modern datasets across many disciplines increasingly consist of time-evolving, potentially infinite-dimensional random objects, such as dynamic functional data, which are naturally modeled in Hilbert spaces. In these settings, characterizing probability measures, for example, through densities, can be ill-defined or technically challenging. Motivated by clustering applications, we propose a Gaussian mixture framework for Hilbert-space-valued data based on kernel mean embeddings and develop efficient optimization algorithms for estimation. We establish theoretical guarantees showing that the proposed algorithm is well defined and that the model yields a dense class of approximations in infinite-dimensional spaces. We evaluate the framework through extensive experiments on diverse structures and data geometries, including $L^2$-functional data and random graphs in Laplacian spaces arising in modern medical applications.

Deep neural networks have been increasingly used in real-world applications, making it critical to ensure their ability to adapt to new, unseen data. In this paper, we study the generalization capability of neural networks trained with (stochastic) gradient flow. We establish a new connection between the loss dynamics of gradient flow and general kernel machines by proposing a new kernel, called loss path kernel. This kernel measures the similarity between two data points by evaluating the agreement between loss gradients along the path determined by the gradient flow. Based on this connection, we derive a new generalization upper bound that applies to general neural network architectures. This new bound is tight and strongly correlated with the true generalization error. We apply our results to guide the design of neural architecture search (NAS) and demonstrate favorable performance compared with state-of-the-art NAS algorithms through numerical experiments.

The NFM framework utilizes the classical idea of multiplicative frailty in survival analysis as a principled way of extending the proportional hazard assumption, at the same time being able to leverage the strong approximation power of neural architectures for handling nonlinear covariate dependence. Two concrete models are derived under the framework that extends neural proportional hazard models and nonparametric hazard regression models. Both models allow efficient training under the likelihood objective. Theoretically, for both proposed models, we establish statistical guarantees of neural function approximation with respect to nonparametric components via characterizing their rate of convergence. Empirically, we provide synthetic experiments that verify our theoretical statements. We also conduct experimental evaluations over 6 benchmark datasets of different scales, showing that the proposed NFM models achieve predictive performance comparable to or sometimes surpassing state-of-the-art survival models.

Given $n$ independent samples from a $d$-dimensional probability distribution, our aim is to generate diffusion-based samples from a distribution obtained by tilting the original, where the degree of tilt is parametrized by $θ\in \mathbb{R}^d$. We define a plug-in estimator and show that it is minimax-optimal. We develop Wasserstein bounds between the distribution of the plug-in estimator and the true distribution as a function of $n$ and $θ$, illustrating regimes where the output and the desired true distribution are close. Further, under some assumptions, we prove the TV-accuracy of running Diffusion on these tilted samples. Our theoretical results are supported by extensive simulations. Applications of our work include finance, weather and climate modelling, and many other domains, where the aim may be to generate samples from a tilted distribution that satisfies practically motivated moment constraints.

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.

Such learning problems are formulated as latent or generative model learning assuming that observations were emerged from the low-dimensional latent states, which includes an intractable posterior inference of latent states for given input data.

An instance of natural occurrences of such constraints is when mixed strategies are used, which correspond to a probability simplex constraint.

FromTheorem 2,thevariational gap is strictly positive as long as the rotation part of the score network remainstobenonzero.