Mathematics

In the regression model Y=b(X)+σ(X)ε , where X has a density f , this paper deals with an oracle inequality for an estimator of bf , involving a kernel in the sense of Lerasle et al. (2016), selected via the PCO method. In addition to the bandwidth selection for kernel-based estimators already studied in Lacour, Massart and Rivoirard (2017) and Comte and Marie (2020), the dimension selection for anisotropic projection estimators of f and bf is covered.

Global sensitivity analysis is the main quantitative technique for identifying the most influential input variables in a numerical simulation model. In particular when the inputs are independent, Sobol' sensitivity indices attribute a portion of the output of interest variance to each input and all possible interactions in the model, thanks to a functional ANOVA decomposition. On the other hand, moment-independent sensitivity indices focus on the impact of input variables on the whole output distribution instead of the variance only, thus providing complementary insight on the inputs / output relationship. Unfortunately they do not enjoy the nice decomposition property of Sobol' indices and are consequently harder to analyze. In this paper, we introduce two moment-independent indices based on kernel-embeddings of probability distributions and show that the RKHS framework used for their definition makes it possible to exhibit a kernel-based ANOVA decomposition. This is the first time such a desirable property is proved for sensitivity indices apart from Sobol' ones. When the inputs are dependent, we also use these new sensitivity indices as building blocks to design kernel-embedding Shapley effects which generalize the traditional variance-based ones used in sensitivity analysis. Several estimation procedures are discussed and illustrated on test cases with various output types such as categorical variables and probability distributions. All these examples show their potential for enhancing traditional sensitivity analysis with a kernel point of view.

We provide an Information-Geometric formulation of Classical Mechanics on the Riemannian manifold of probability distributions, which is an affine manifold endowed with a dually-flat connection. In a non-parametric formalism, we consider the full set of positive probability functions on a finite sample space, and we provide a specific expression for the tangent and cotangent spaces over the statistical manifold, in terms of a Hilbert bundle structure that we call the Statistical Bundle. In this setting, we compute velocities and accelerations of a one-dimensional statistical model using the canonical dual pair of parallel transports and define a coherent formalism for Lagrangian and Hamiltonian mechanics on the bundle. Finally, in a series of examples, we show how our formalism provides a consistent framework for accelerated natural gradient dynamics on the probability simplex, paving the way for direct applications in optimization, game theory and neural networks.

Simultaneous, post-hoc inference is desirable in large-scale hypotheses testing as it allows for exploration of data while deciding on criteria for proclaiming discoveries. It was recently proved that all admissible post-hoc inference methods for the number of true discoveries must be based on closed testing. In this paper we investigate tractable and efficient closed testing with local tests of different properties, such as monotonicty, symmetry and separability, meaning that the test thresholds a monotonic or symmetric function or a function of sums of test scores for the individual hypotheses. This class includes well-known global null tests by Fisher, Stouffer and Ruschendorf, as well as newly proposed ones based on harmonic means and Cauchy combinations. Under monotonicity, we propose a new linear time statistic ("coma") that quantifies the cost of multiplicity adjustments. If the tests are also symmetric and separable, we develop several fast (mostly linear-time) algorithms for post-hoc inference, making closed testing tractable. Paired with recent advances in global null tests based on generalized means, our work immediately instantiates a series of simultaneous inference methods that can handle many complex dependence structures and signal compositions. We provide guidance on choosing from these methods via theoretical investigation of the conservativeness and sensitivity for different local tests, as well as simulations that find analogous behavior for local tests and full closed testing. One result of independent interest is the following: if P 1 ,?? P d are p -values from a multivariate Gaussian with arbitrary covariance, then their arithmetic average P satisfies Pr(P?�t)?�t for t??1 2d .

In applications such as rank aggregation, mixture models for permutations are frequently used when the population exhibits heterogeneity. In this work, we study the widely used Mallows mixture model. In the high-dimensional setting, we propose a polynomial-time algorithm that learns a Mallows mixture of permutations on n elements with the optimal sample complexity that is proportional to logn , improving upon previous results that scale polynomially with n . In the high-noise regime, we characterize the optimal dependency of the sample complexity on the noise parameter. Both objectives are accomplished by first studying demixing permutations under a noiseless query model using groups of pairwise comparisons, which can be viewed as moments of the mixing distribution, and then extending these results to the noisy Mallows model by simulating the noiseless oracle.

We consider stochastic systems of interacting particles or agents, with dynamics determined by an interaction kernel which only depends on pairwise distances. We study the problem of inferring this interaction kernel from observations of the positions of the particles, in either continuous or discrete time, along multiple independent trajectories. We introduce a nonparametric inference approach to this inverse problem, based on a regularized maximum likelihood estimator constrained to suitable hypothesis spaces adaptive to data. We show that a coercivity condition enables us to control the condition number of this problem and prove the consistency of our estimator, and that in fact it converges at a near-optimal learning rate, equal to the min-max rate of 1 -dimensional non-parametric regression. In particular, this rate is independent of the dimension of the state space, which is typically very high. We also analyze the discretization errors in the case of discrete-time observations, showing that it is of order 1/2 in terms of the time gaps between observations. This term, when large, dominates the sampling error and the approximation error, preventing convergence of the estimator. Finally, we exhibit an efficient parallel algorithm to construct the estimator from data, and we demonstrate the effectiveness of our algorithm with numerical tests on prototype systems including stochastic opinion dynamics and a Lennard-Jones model.

This paper analyzes a new regularized learning scheme for high dimensional partially linear support vector machine. The proposed approach consists of an empirical risk and the Lasso-type penalty for linear part, as well as the standard functional norm for nonlinear part. Here the linear kernel is used for model interpretation and feature selection, while the nonlinear kernel is adopted to enhance algorithmic flexibility. In this paper, we develop a new technical analysis on the weighted empirical process, and establish the sharp learning rates for the semi-parametric estimator under the regularized conditions. Specially, our derived learning rates for semi-parametric SVM depend on not only the sample size and the functional complexity, but also the sparsity and the margin parameters.

Combining information both within and across trajectories, we propose a simple estimator for the local regularity of the trajectories of a stochastic process. Independent trajectories are measured with errors at randomly sampled time points. Non-asymptotic bounds for the concentration of the estimator are derived. Given the estimate of the local regularity, we build a nearly optimal local polynomial smoother from the curves from a new, possibly very large sample of noisy trajectories. We derive non-asymptotic pointwise risk bounds uniformly over the new set of curves. Our estimates perform well in simulations. Real data sets illustrate the effectiveness of the new approaches.

Tree tensor networks, or tree-based tensor formats, are prominent model classes for the approximation of high-dimensional functions in computational and data science. They correspond to sum-product neural networks with a sparse connectivity associated with a dimension tree and widths given by a tuple of tensor ranks. The approximation power of these models has been proved to be (near to) optimal for classical smoothness classes. However, in an empirical risk minimization framework with a limited number of observations, the dimension tree and ranks should be selected carefully to balance estimation and approximation errors. We propose and analyze a complexity-based model selection method for tree tensor networks in an empirical risk minimization framework and we analyze its performance over a wide range of smoothness classes. Given a family of model classes associated with different trees, ranks, tensor product feature spaces and sparsity patterns for sparse tensor networks, a model is selected (à la Barron, Birgé, Massart) by minimizing a penalized empirical risk, with a penalty depending on the complexity of the model class and derived from estimates of the metric entropy of tree tensor networks. This choice of penalty yields a risk bound for the selected predictor. In a least-squares setting, after deriving fast rates of convergence of the risk, we show that our strategy is (near to) minimax adaptive to a wide range of smoothness classes including Sobolev or Besov spaces (with isotropic, anisotropic or mixed dominating smoothness) and analytic functions. We discuss the role of sparsity of the tensor network for obtaining optimal performance in several regimes. In practice, the amplitude of the penalty is calibrated with a slope heuristics method. Numerical experiments in a least-squares regression setting illustrate the performance of the strategy.

We study the parameter estimation problem of Vasicek Model driven by sub-fractional Brownian processes from discrete observations, and let {S_t^H,t>=0} denote a sub-fractional Brownian motion whose Hurst parameter 1/2<H<1 . The studies are as follows: firstly, two unknown parameters in the model are estimated by the least squares method. Secondly, the strong consistency and the asymptotic distribution of the estimators are studied respectively. Finally, our estimators are validated by numerical simulation.