Confidence sets play a fundamental role in statistical inference. In this paper, we consider confidence intervals for high dimensional linear regression with random design. We first establish the convergence rates of the minimax expected length for confidence intervals in the oracle setting where the sparsity parameter is given. The focus is then on the problem of adaptation to sparsity for the construction of confidence intervals. Ideally, an adaptive confidence interval should have its length automatically adjusted to the sparsity of…

The energy of data is the value of a real function of distances between data in metric spaces. The name energy derives from Newton's gravitational potential energy which is also a function of distances between physical objects. One of the advantages of working with energy functions (energy statistics) is that even if the observations/data are complex objects, like functions or graphs, we can use their real valued distances for inference. Other advantages will be illustrated and discussed in the talk.…

Learning the governing equations for time-varying measurement data is of great interest across different scientific fields. When such data is moreover highly corrupted, for example, due to the recording mechanism failing over unknown intervals of time, recovering the governing equations becomes quite challenging. In this work, we show that if the data exhibits chaotic behavior, it is possible to recover the underlying governing nonlinear differential equations even if a large percentage of the data is corrupted by outliers, by solving…

Latent variable models have become a key tool for the modern statistician, letting us express complex assumptions about the hidden structures that underlie our data. Latent variable models have been successfully applied in numerous fields. The central computational problem in latent variable modeling is posterior inference, the problem of approximating the conditional distribution of the latent variables given the observations. Posterior inference is central to both exploratory tasks and predictive tasks. Approximate posterior inference algorithms have revolutionized Bayesian statistics, revealing…

I will discuss trend filtering, a newly proposed tool of Steidl et al. (2006), Kim et al. (2009) for nonparametric regression. The trend filtering estimate is defined as the minimizer of a penalized least squares criterion, in which the penalty term sums the absolute kth order discrete derivatives over the input points. I will give an overview of some interesting connections between these estimates and adaptive spline estimation, and also of the provable statistical superiority of trend filtering to other…

Particle filters provide Monte Carlo approximations of intractable quantities, such as likelihood evaluations in state-space models. In many cases, the interest does not lie in the values of the estimates themselves, but in the comparison of these values for various parameters. For instance, we might want to compare the likelihood at two parameter values. Such a comparison is facilitated by introducing positive correlations between the estimators, which is a standard variance reduction technique. In the context of particle filters, this…

In this talk, I will discuss the prediction properties of techniques commonly used to scale up kernel methods and Gaussian processes. In particular, I will focus on data dependent and independent sub-sampling methods, namely Nystrom and random features, and study their generalization properties within a statistical learning theory framework. On the one hand I will show that these methods can achieve optimal learning errors while being computational efficient. On the other hand, I will show that subsampling can be seen…