The paper investigates how Markov chains converge within a geometry defined by Hilbertian embeddings of positive definite kernels, also known as Maximum Mean Discrepancy (MMD). By assuming a contraction in this kernel-induced geometry and a standard spectral gap, the authors derive non-asymptotic, high‑probability bounds on the empirical occupation measure’s convergence to the invariant measure. These bounds separate the influence of the spectral gap, which sets the effective sample size, from the kernel contraction, which appears only as a linear correction. The results are applied to one‑step prediction of dynamical systems from a single trajectory, yielding fast learning rates for a nonlinear least‑squares estimator that illustrate the interaction between kernel‑induced mixing and classical source and capacity conditions.
arXiv:2607. 22399v1 Announce Type: cross Abstract: We consider the problem of learning from a single finite trajectory of an ergodic stochastic dynamical system.
arXiv:2607. 24235v1 Announce Type: cross Abstract: Over the past 20 years, kernel discrepancies have been leveraged as a highly powerful tool for quantifying the disagreement of distributions, with numerous successful applications in two-sample, goodness-of-fit, and independence testing, among others.
arXiv:2607. 10592v1 Announce Type: new Abstract: Many geometric statistics and manifold learning pipelines routinely produce observations -- such as tangent vectors or local frames -- whose natural home is a varying family of fibers attached to different points of a base manifold, rather than a single shared vector space.
arXiv:2606. 02232v1 Announce Type: new Abstract: Learning a Markov transition model is not merely conditional density estimation: the learned object must be a valid transition kernel before it is iterated in downstream dynamics.
We develop a comprehensive theory for regularized M-estimation in reproducing kernel Hilbert spaces. Under mild conditions on the loss we establish existence and measurability of the estimator, covering a wide range of convex and non-convex losses, including bounded robust losses.
arXiv:2504.18184v5 Announce Type: replace Abstract: We consider a class of statistical inverse problems involving the estimation of a regression operator from a Polish space to a separable Hilbert sp...
arXiv:2402.04691v5 Announce Type: replace-cross Abstract: This study investigates the use of stochastic gradient descent (SGD) to learn operators between general Hilbert spaces. We study weak and str...
The monograph explores the relationships between Gaussian processes and reproducing kernel Hilbert spaces (RKHS), two widely used approaches that rely on positive definite kernels. It examines how these frameworks connect and are equivalent across key topics such as regression, interpolation, numerical integration, distributional discrepancies, statistical dependence, and Gaussian process sample path properties. By establishing a unifying perspective based on the equivalence between the Gaussian Hilbert space and the RKHS, the work aims to bridge methods developed independently by the machine learning, statistics, and numerical analysis communities.
The paper tackles learning structured linear dynamical systems when only a few observations are available at each time step. It proposes an estimator that minimizes a bias‑corrected, potentially non‑convex objective and derives non‑asymptotic error bounds that depend on the local complexity of the constraint set, the trajectory length, and the sub‑sampling probability. The authors also prove convergence of projected gradient descent and demonstrate the theory for three specific constraint sets, showing that accurate recovery of the transition matrix is achievable with much shorter trajectories and very low observation rates.
The paper presents a fully data-driven method for optimal control of nonlinear control‑affine systems modeled as stochastic diffusions. It embeds state probability densities into a reproducing kernel Hilbert space to learn Markov transition operators, which are then used in convex operator‑theoretic Hamilton‑Jacobi‑Bellman recursions that scale linearly with state dimensionality. Numerical experiments show the approach can solve diverse nonlinear control tasks, such as depth regulation of an autonomous underwater vehicle.
arXiv:2606. 26316v1 Announce Type: new Abstract: We study first-order methods for smooth objectives satisfying the Polyak-\L{}ojasiewicz (PL) condition when gradient samples are generated by an exogenous Markov chain.
arXiv:2610.09731v1 Announce Type: cross Abstract: Kernel methods are among the most powerful tools in machine learning and statistics, with a large number of successful applications. Their immense su...