Related papers: Curse of Dimensionality in Pivot-based Indexes
Many datasets exhibit a well-defined structure that can be exploited to design faster search tools, but it is not always clear when such acceleration is possible. Here, we introduce a framework for similarity search based on characterizing…
We explore linear and non-linear dimensionality reduction techniques for statistical inference of parameters in cosmology. Given the importance of compressing the increasingly complex data vectors used in cosmology, we address questions…
This thesis deals with the nonparametric estimation of density f of the regression error term E of the model Y=m(X)+E, assuming its independence with the covariate X. The difficulty linked to this study is the fact that the regression error…
Recently, it has been proposed in the literature to employ deep neural networks (DNNs) together with stochastic gradient descent methods to approximate solutions of PDEs. There are also a few results in the literature which prove that DNNs…
Dense retrieval, which encodes queries and documents into a single dense vector, has become the dominant neural retrieval approach due to its simplicity and compatibility with fast approximate nearest neighbor algorithms. As the tasks dense…
As datasets grow it becomes infeasible to process them completely with a desired model. For giant datasets, we frame the order in which computation is performed as a decision problem. The order is designed so that partial computations are…
We study beyond worst-case dimensionality reduction for $s$-sparse vectors. Our work is divided into two parts, each focusing on a different facet of beyond worst-case analysis: We first consider average-case guarantees. A folklore upper…
In this paper we develop a new machinery to study the capacity of artificial neural networks (ANNs) to approximate high-dimensional functions without suffering from the curse of dimensionality. Specifically, we introduce a concept which we…
Despite advances in representation learning, high-dimensional classification remains challenging in low-sample-size regimes, where the dominant signal may vary across applications and labeled data are often limited. We propose a…
We consider the regression problem of estimating functions on $\mathbb{R}^D$ but supported on a $d$-dimensional manifold $ \mathcal{M} \subset \mathbb{R}^D $ with $ d \ll D $. Drawing ideas from multi-resolution analysis and nonlinear…
Backward stochastic differential equations (BSDEs) belong nowadays to the most frequently studied equations in stochastic analysis and computational stochastics. BSDEs in applications are often nonlinear and high-dimensional. In nearly all…
In this paper we propose an improved approximation scheme for the Vector Bin Packing problem (VBP), based on the combination of (near-)optimal solution of the Linear Programming (LP) relaxation and a greedy (modified first-fit) heuristic.…
We study testing of local properties in one-dimensional and multi-dimensional arrays. A property of $d$-dimensional arrays $f:[n]^d \to \Sigma$ is $k$-local if it can be defined by a family of $k \times \ldots \times k$ forbidden…
We refine and generalize what is known about coresets for classification problems via the sensitivity sampling framework. Such coresets seek the smallest possible subsets of input data, so one can optimize a loss function on the coreset and…
When the nonconvex problem is complicated by stochasticity, the sample complexity of stochastic first-order methods may depend linearly on the problem dimension, which is undesirable for large-scale problems. In this work, we propose…
It has been widely realized that Monte Carlo methods (approximation via a sample ensemble) may fail in large scale systems. This work offers some theoretical insight into this phenomenon in the context of the particle filter. We demonstrate…
Max-Cut is a fundamental problem that has been studied extensively in various settings. We design an algorithm for Euclidean Max-Cut, where the input is a set of points in $\mathbb{R}^d$, in the model of dynamic geometric streams, where the…
Existing theories on deep nonparametric regression have shown that when the input data lie on a low-dimensional manifold, deep neural networks can adapt to the intrinsic data structures. In real world applications, such an assumption of…
We present a novel view of nonlinear manifold learning using derivative-free optimization techniques. Specifically, we propose an extension of the classical multi-dimensional scaling (MDS) method, where instead of performing gradient…
Parabolic partial differential equations (PDEs) are widely used in the mathematical modeling of natural phenomena and man made complex systems. In particular, parabolic PDEs are a fundamental tool to determine fair prices of financial…