We address the consistency of a kernel ridge regression estimate of the conditional mean embedding (CME), which is an embedding of the conditional distribution of Y given X into a target reproducing kernel Hilbert space HY . The CME allows us to take conditional expectations of target RKHS functions, and has been employed in nonparametric causal and Bayesian inference. We address the misspecified setting, where the target CME is in the space of Hilbert-Schmidt operators acting from an input interpolation space between HX and L2, to HY . This space of operators is shown to be isomorphic to a newly defined vector-valued interpolation space. Using this isomorphism, we derive a novel and adaptive statistical learning rate for the empirical CME estimator under the misspecified setting. Our analysis reveals that our rates match the optimal O(logn/n) rates without assuming HY to be finite dimensional. We further establish a lower bound on the learning rate, which shows that the obtained upper bound is optimal.

We study a class of dynamical systems modelled as Markov chains that admit an invariant distribution via the corresponding transfer, or Koopman, operator. While data-driven algorithms to reconstruct such operators are well known, their relationship with statistical learning is largely unexplored. We formalize a framework to learn the Koopman operator from finite data trajectories of the dynamical system. We consider the restriction of this operator to a reproducing kernel Hilbert space and introduce a notion of risk, from which different estimators naturally arise. We link the risk with the estimation of the spectral decomposition of the Koopman operator. These observations motivate a reduced-rank operator regression (RRR) estimator. We derive learning bounds for the proposed estimator, holding both in i.i.d. and non i.i.d.

We are interested in a framework of online learning with kernels for lowdimensional, but large-scale and potentially adversarial datasets. We study the computational and theoretical performance of online variations of kernel Ridge regression.

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

In a general framework, we study how the choice of loss and the assumed smoothness of the underlying density together determine the minimax rate.

We introduce a unified framework for learning the spatio-temporal dynamics of vector valued functions by combining operator valued reproducing kernel Hilbert spaces (OV-RKHS) with kernel based Koopman operator methods. The approach enables nonparametric and data driven estimation of complex time evolving vector fields while preserving both spatial and temporal structure. We establish representer theorems for time dependent OV-RKHS interpolation, derive Sobolev type approximation bounds for smooth vector fields, and provide spectral convergence guarantees for kernel Koopman operator approximations. This framework supports efficient reduced order modeling and long term prediction of high dimensional nonlinear systems, offering theoretically grounded tools for forecasting, control, and uncertainty quantification in spatio-temporal machine learning.

Distributed learning has attracted significant attention recently with the increasing popularity of distributed systems and distributed data.

In this paper, we study an online regularized learning algorithm in a reproducing kernel Hilbert spaces (RKHS) based on a class of dependent processes. We choose such a process where the degree of dependence is measured by mixing coefficients. As a representative example, we analyze a strictly stationary Markov chain, where the dependence structure is characterized by the \(ϕ\)- and \(β\)-mixing coefficients. Under these assumptions, we derive probabilistic upper bounds as well as convergence rates for both the exponential and polynomial decay of the mixing coefficients.