Related papers: Sampling discretization error for function classes
We present a general technique for the analysis of first-order methods. The technique relies on the construction of a duality gap for an appropriate approximation of the objective function, where the function approximation improves as the…
This article conducts a large dimensional study of a simple yet quite versatile classification model, encompassing at once multi-task and semi-supervised learning, and taking into account uncertain labeling. Using tools from random matrix…
Diffusion models have had a profound impact on many application areas, including those where data are intrinsically infinite-dimensional, such as images or time series. The standard approach is first to discretize and then to apply…
Deep neural networks are widely used prediction algorithms whose performance often improves as the number of weights increases, leading to over-parametrization. We consider a two-layered neural network whose first layer is frozen while the…
Strong approximation errors of both finite element semi-discretization and spatio-temporal full discretization are analyzed for the stochastic Allen-Cahn equation driven by additive noise in space dimension $d \leq 3$. The full…
Density modelling is the task of learning an unknown probability density function from samples, and is one of the central problems of unsupervised machine learning. In this work, we show that there exists a density modelling problem for…
This article introduces a general statistical modeling principle called "Density Sharpening" and applies it to the analysis of discrete count data. The underlying foundation is based on a new theory of nonparametric approximation and…
Metric learning seeks perceptual embeddings where visually similar instances are close and dissimilar instances are apart, but learned representations can be sub-optimal when the distribution of intra-class samples is diverse and distinct…
In this chapter, we discuss recent work on learning sparse approximations to high-dimensional functions on data, where the target functions may be scalar-, vector- or even Hilbert space-valued. Our main objective is to study how the…
One-shot learning is proposed to make a pretrained classifier workable on a new dataset based on one labeled samples from each pattern. However, few of researchers consider whether the dataset itself supports one-shot learning. In this…
We derive optimal rates of convergence in the supremum norm for estimating the H\"older-smooth mean function of a stochastic process which is repeatedly and discretely observed with additional errors at fixed, multivariate, synchronous…
We study numerical integration of functions depending on an infinite number of variables. We provide lower error bounds for general deterministic linear algorithms and provide matching upper error bounds with the help of suitable multilevel…
We consider the problem of classification of functional data into two groups by linear classifiers based on one-dimensional projections of functions. We reformulate the task to find the best classifier as an optimization problem and solve…
This paper is concerned with the problem of sampling and interpolation involving derivatives in shift-invariant spaces and the error analysis of the derivative sampling expansions for fundamentally large classes of functions. A new type of…
In this paper we consider the following question: can we optimize objective functions from the training data we use to learn them? We formalize this question through a novel framework we call optimization from samples (OPS). In OPS, we are…
We perform extensive numerical simulations of different versions of the sandpile model. We find that previous claims about universality classes are unfounded, since the method previously employed to analyze the data suffered a systematic…
We study two numerical approximations of solutions of nonlocal diffusion evolution problems which are inspired in algorithms for computing the bilateral denoising filtering of an image, and which are based on functional rearrangements and…
Deep neural networks, when optimized with sufficient data, provide accurate representations of high-dimensional functions; in contrast, function approximation techniques that have predominated in scientific computing do not scale well with…
In this work we study loss functions for learning and evaluating probability distributions over large discrete domains. Unlike classification or regression where a wide variety of loss functions are used, in the distribution learning and…
Functional linear discriminant analysis offers a simple yet efficient method for classification, with the possibility of achieving a perfect classification. Several methods are proposed in the literature that mostly address the…