Many data can be represented as rankings or permutations, raising the question of developing machine learning models on the symmetric group. When the number of items in the permutations gets large, manipulating permutations can quickly become computationally intractable. I will discuss two computationally efficient embeddings of the symmetric groups in Euclidean spaces leading to fast machine learning algorithms, and illustrate their relevance on biological applications and image classification.[-]

In this talk we consider high-dimensional classification. We discuss first high-dimensional binary classification by sparse logistic regression, propose a model/feature selection procedure based on penalized maximum likelihood with a complexity penalty on the model size and derive the non-asymptotic bounds for the resulting misclassification excess risk. Implementation of any complexity penalty-based criterion, however, requires a combinatorial search over all possible models. To find a model selection procedure computationally feasible for high-dimensional data, we consider logistic Lasso and Slope classifiers and show that they also achieve the optimal rate. We extend further the proposed approach to multiclass classification by sparse multinomial logistic regression.

This is joint work with Vadim Grinshtein and Tomer Levy.[-]

We study the model selection problem in a large class of causal time series models, which includes both the ARMA or AR($\infty$) processes, as well as the GARCH or ARCH($\infty$), APARCH, ARMA-GARCH and many others processes. To tackle this issue, we consider a penalized contrast based on the quasi-likelihood of the model. We provide sufficient conditions for the penalty term to ensure the consistency of the proposed procedure as well as the consistency and the asymptotic normality of the quasi-maximum likelihood estimator of the chosen model. We also propose a tool for diagnosing the goodness-of-fit of the chosen model based on a Portmanteau test. Monte-Carlo experiments and numerical applications on illustrative examples are performed to highlight the obtained asymptotic results. Moreover, using a data-driven choice of the penalty, they show the practical efficiency of this new model selection procedure and Portemanteau test.[-]

A new Central Limit Theorem (CLT) is developed for random variables of the form ξ=z⊤f(z)−divf(z) where z∼N(0,In).
The normal approximation is proved to hold when the squared norm of f(z) dominates the squared Frobenius norm of ∇f(z) in expectation.
Applications of this CLT are given for the asymptotic normality of de-biased estimators in linear regression with correlated design and convex penalty in the regime p/n→γ∈(0,∞). For the estimation of linear functions ⟨a,β⟩ of the unknown coefficient vector β, this analysis leads to asymptotic normality of the de-biased estimate for most normalized directions a0, where "most" is quantified in a precise sense. This asymptotic normality holds for any coercive convex penalty if γ<1 and for any strongly convex penalty if γ≥1. In particular the penalty needs not be separable or permutation invariant.[-]

There is an emerging consensus in the transdiciplinary literature that the ultimate goal of regression analysis is to model the conditional distribution of an outcome, given a set of explanatory variables or covariates. This new approach is called "distributional regression", and marks a clear break from the classical view of regression, which has focused on estimating a conditional mean or quantile only. Isotonic Distributional Regression (IDR) learns conditional distributions that are simultaneously optimal relative to comprehensive classes of relevant loss functions, subject to monotonicity constraints in terms of a partial order on the covariate space. This IDR solution is exactly computable and does not require approximations nor implementation choices, except for the selection of the partial order. Despite being an entirely generic technique, IDR is strongly competitive with state-of-the-art methods in a case study on probabilistic precipitation forecasts from a leading numerical weather prediction model.

Joint work with Alexander Henzi and Johanna F. Ziegel.[-]

The highly influential two-group model in testing a large number of statistical hypotheses assumes that the test statistics are drawn independently from a mixture of a high probability null distribution and a low probability alternative. Optimal control of the marginal false discovery rate (mFDR), in the sense that it provides maximal power (expected true discoveries) subject to mFDR control, is known to be achieved by thresholding the local false discovery rate (locFDR), i.e., the probability of the hypothesis being null given the set of test statistics, with a fixed threshold.
We address the challenge of controlling optimally the popular false discovery rate (FDR) or positive FDR (pFDR) rather than mFDR in the general two-group model, which also allows for dependence between the test statistics. These criteria are less conservative than the mFDR criterion, so they make more rejections in expectation.
We derive their optimal multiple testing (OMT) policies, which turn out to be thresholding the locFDR with a threshold that is a function of the entire set of statistics. We develop an efficient algorithm for finding these policies, and use it for problems with thousands of hypotheses. We illustrate these procedures on gene expression studies. [-]

As a generalization of fill-in free property of a sparse positive definite real symmetric matrix with respect to the Cholesky decomposition, we introduce a notion of (quasi-)Cholesky structure for a real vector space of symmetric matrices. The cone of positive definite symmetric matrices in a vector space with a quasi-Cholesky structure admits explicit calculations and rich analysis similar to the ones for Gaussian selsction model associated to a decomposable graph. In particular, we can apply our method to a decomposable graphical model with a vertex pemutation symmetry.[-]

Many modern applications seek to understand the relationship between an outcome variable of interest and a high-dimensional set of covariates. Often the first question asked is which covariates are important in this relationship, but the immediate next question, which in fact subsumes the first, is \emph{how} important each covariate is in this relationship. In parametric regression this question is answered through confidence intervals on the parameters. But without making substantial assumptions about the relationship between the outcome and the covariates, it is unclear even how to \emph{measure} variable importance, and for most sensible choices even less clear how to provide inference for it under reasonable conditions. In this paper we propose \emph{floodgate}, a novel method to provide asymptotic inference for a scalar measure of variable importance which we argue has universal appeal, while assuming nothing but moment bounds about the relationship between the outcome and the covariates. We take a model-X approach and thus assume the covariate distribution is known, but extend floodgate to the setting that only a \emph{model} for the covariate distribution is known and also quantify its robustness to violations of the modeling assumptions. We demonstrate floodgate's performance through extensive simulations and apply it to data from the UK Biobank to quantify the effects of genetic mutations on traits of interest.[-]

We consider the problem of estimating the mean vector of the multivariate complex normaldistribution with unknown covariance matrix under an invariant loss function when the samplesize is smaller than the dimension of the mean vector. Following the approach of Chételat and Wells (2012, Ann.Statist, p. 3137–3160), we show that a modification of Baranchik-tpye estimatorsbeats the MLE if it satisfies certain conditions. Based on this result, we propose the James-Stein-like shrinkage and its positive-part estimators.[-]