Related papers: Indexability, concentration, and VC theory

Deep learning methods minimise the empirical risk using loss functions such as the cross entropy loss. When minimising the empirical risk, the generalisation of the learnt function still depends on the performance on the training data, the…

We introduce a symmetrization technique that allows us to translate a problem of controlling the deviation of some functionals on a product space from their mean into a problem of controlling the deviation between two independent copies of…

This paper gives a review of concentration inequalities which are widely employed in non-asymptotical analyses of mathematical statistics in a wide range of settings, from distribution-free to distribution-dependent, from sub-Gaussian to…

Testing for the equality of two high-dimensional distributions is a challenging problem, and this becomes even more challenging when the sample size is small. Over the last few decades, several graph-based two-sample tests have been…

Popular clustering algorithms based on usual distance functions (e.g., Euclidean distance) often suffer in high dimension, low sample size (HDLSS) situations, where concentration of pairwise distances has adverse effects on their…

For any family of measurable sets in a probability space, we show that either (i) the family has infinite Vapnik-Chervonenkis (VC) dimension or (ii) for every epsilon > 0 there is a finite partition pi such the pi-boundary of each set has…

The VC-dimension, introduced by Vapnik and Chervonenkis in 1968 in the context of learning theory, has in recent years provided a rich source of problems in combinatorial geometry. Given $E\subseteq \mathbb{F}_q^d$ or $E\subseteq…

We present new concentration of measure inequalities for Markov chains, generalising results for chains that are contracting in Wasserstein distance. These are particularly suited to establishing the cut-off phenomenon for suitable chains.…

This work adresses the question of density of piecewise constant (resp. rigid) functions in the space of vector valued functions with bounded variation (resp. deformation) with respect to the strict convergence. Such an approximation…

The characteristics of data like distribution and heterogeneity, become more complex and counterintuitive as dimensionality increases. This phenomenon is known as curse of dimensionality, where common patterns and relationships (e.g.,…

We develop a new high-dimensional statistical learning model which can take advantage of structured correlation in data even in the presence of randomness. We completely characterize learnability in this model in terms of…

Concentration inequalities, a major tool in probability theory, quantify how much a random variable deviates from a certain quantity. This paper proposes a systematic convex optimization approach to studying and generating concentration…

Convolutional networks are at the center of best-in-class computer vision applications for a wide assortment of undertakings. Since 2014, a profound amount of work began to make better convolutional architectures, yielding generous…

For various classes of Lipschitz functions we provide dimension free concentration inequalities for infinitely divisible random vectors with independent components and finite exponential moments.

We study various generalizations of concentration of measure on the unit sphere, in particular by means of log-Sobolev inequalities. First, we show Sudakov-type concentration results and local semicircular laws for weighted random matrices.…

We present the main results of a numerical study of weak lensing cluster counting. We examine the scaling with cosmology of the projected-density-peak mass function. Our main conclusion is that the projected-peak and the three-dimensional…

We consider the problem of determining which classes of functions can be tested more efficiently than they can be learned, in the distribution-free sample-based model that corresponds to the standard PAC learning setting. Our main result…

We establish concentration inequalities for random dynamical systems (RDSs), assuming that the observables of interest are separately Lipschitz. Under a weak average contraction condition, we obtain deviation bounds for several random…

We derive an objective function that can be optimized to give an estimator of the Vapnik- Chervonenkis dimension for model selection in regression problems. We verify our estimator is consistent. Then, we verify it performs well compared to…

A theoretical framework is presented for a (copula-based) notion of dissimilarity between continuous random vectors and its main properties are studied. The proposed dissimilarity assigns the smallest value to a pair of random vectors that…