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In the realm of machine learning theory, to prevent unnatural coding schemes between teacher and learner, No-Clash Teaching Dimension was introduced as provably optimal complexity measure for collusion-free teaching. However, whether…

We bound the number of nearly orthogonal vectors with fixed VC-dimension over $\setpm^n$. Our bounds are of interest in machine learning and empirical process theory and improve previous bounds by Haussler. The bounds are based on a simple…

Generalization in deep learning has been the topic of much recent theoretical and empirical research. Here we introduce desiderata for techniques that predict generalization errors for deep learning models in supervised learning. Such…

We are interested in a class of groups, quasi-Frobenius groups (with involutions), whose internal structure generalizes that of the classical groups GA1(C), PGL 2(C) and SO3(R) : a subgroup and its conjugates, of finite index in their…

High-dimensional group inference is an essential part of statistical methods for analysing complex data sets, including hierarchical testing, tests of interaction, detection of heterogeneous treatment effects and inference for local…

In this dissertation, I derive a new method to estimate the Vapnik-Chervonenkis Dimension (VCD) for the class of linear functions. This method is inspired by the technique developed by Vapnik et al. Vapnik et al. (1994). My contribution…

The study of strategic or adversarial manipulation of testing data to fool a classifier has attracted much recent attention. Most previous works have focused on two extreme situations where any testing data point either is completely…

The goal of this paper is to show that fundamental concepts in higher-order Fourier analysis can be nauturally extended to the non-commutative setting. We generalize Gowers norms to arbitrary compact non-commutative groups. On the…

We present a new general-purpose algorithm for learning classes of $[0,1]$-valued functions in a generalization of the prediction model, and prove a general upper bound on the expected absolute error of this algorithm in terms of a…

Let ${\rm cs}(G)$ denote the set of conjugacy class sizes of a group $G$, and let ${\rm cs}^*(G)={\rm cs}(G)\setminus\{1\}$ be the sizes of non-central classes. We prove three results. We classify all finite groups $G$ with ${\rm…

Deep neural networks often contain far more parameters than training examples, yet they still manage to generalize well in practice. Classical complexity measures such as VC-dimension or PAC-Bayes bounds usually become vacuous in this…

The Gordon--Rodriguez-Villegas theorem says that, in a finite group, the number of solutions to a system of coefficient-free equations is divisible by the order of the group if the rank of the matrix composed of the exponent sums of $j$-th…

In this paper, generalized metrics mean metrics taking values in general linearly ordered Abelian groups. Using the Hahn fields, we first prove that for every generalized metric space, if the set of the Archimedean equivalence classes of…

In repeated Measure Designs with multiple groups, the primary purpose is to compare different groups in various aspects. For several reasons, the number of measurements and therefore the dimension of the observation vectors can depend on…

This monograph deals with adaptive supervised classification, using tools borrowed from statistical mechanics and information theory, stemming from the PACBayesian approach pioneered by David McAllester and applied to a conception of…

This paper is an expanded and updated version of the preprint arXiv:math/0406499. It includes a more detailed description of the basics of the theory of Cherednik and Hecke algebras of varieties started in arXiv:math/0406499, as well as a…

In this paper we construct a model for the classifying space, BVCG, of a crystallographic group G of rank n relative to the family VC of virtually-cyclic subgroups of G. The model is used to show that there exists no other model for the…

We study wide Bayesian neural networks focusing on the rare but statistically dominant fluctuations that govern posterior concentration, beyond Gaussian-process limits. Large-deviation theory provides explicit variational objectives-rate…

A seminal result in learning theory characterizes the PAC learnability of binary classes through the Vapnik-Chervonenkis dimension. Extending this characterization to the general multiclass setting has been open since the pioneering works…

We develop a fast and accurate grouped penalized credible region approach for variable selection and prediction in Bayesian high-dimensional linear regression. Most existing Bayesian methods either are subject to high computational costs…