Related papers: Hardness of High-Dimensional Linear Classification

Let $\textbf{A}$ be a symmetric convex quadratic form on $\mathbb{R}^{Nn}$ and $\Omega\Subset \mathbb{R}^n$ a bounded convex domain. We consider the problem of existence of solutions $u: \Omega \subset \mathbb{R}^n \longrightarrow…

We study the complexity of geometric problems on spaces of low fractal dimension. It was recently shown by [Sidiropoulos & Sridhar, SoCG 2017] that several problems admit improved solutions when the input is a pointset in Euclidean space…

Delineation of curvilinear structures is an important problem in Computer Vision with multiple practical applications. With the advent of Deep Learning, many current approaches on automatic delineation have focused on finding more powerful…

We develop a linear-algebraic framework for dimensional analysis in systems with constraints, particularly when variables are numerous or related by implicit relations so that direct elimination is impractical. By expressing both…

We prove deviation inequalities for sums of high-dimensional random matrices and operators with dependence and {\rc heavy tails}. Estimation of high-dimensional matrices is a concern for numerous modern applications. However, most results…

Spanners for low dimensional spaces (e.g. Euclidean space of constant dimension, or doubling metrics) are well understood. This lies in contrast to the situation in high dimensional spaces, where except for the work of Har-Peled, Indyk and…

Decision making and learning in the presence of uncertainty has attracted significant attention in view of the increasing need to achieve robust and reliable operations. In the case where uncertainty stems from the presence of adversarial…

In Part I we construct the upper bound, in the spirit of $\Gamma$- $\limsup$, achieved by multidimensional profiles, for some general classes of singular perturbation problems, with or without the prescribed differential constraint, taking…

We provide new algorithms and conditional hardness for the problem of estimating effective resistances in $n$-node $m$-edge undirected, expander graphs. We provide an $\widetilde{O}(m\epsilon^{-1})$-time algorithm that produces with high…

High dimensional classification has been highlighted for last two decades and much research has been conducted in order to circumvent challenges encountered in high dimensions. While existing methods have focused mainly on developing…

Neural ordinary differential equations (NODE) have garnered significant attention for their design of continuous-depth neural networks and the ability to learn data/feature dynamics. However, for high-dimensional systems, estimating…

Evaluating the statistical dimension is a common tool to determine the asymptotic phase transition in compressed sensing problems with Gaussian ensemble. Unfortunately, the exact evaluation of the statistical dimension is very difficult and…

We study the problem of agnostically learning halfspaces which is defined by a fixed but unknown distribution $\mathcal{D}$ on $\mathbb{Q}^n\times \{\pm 1\}$. We define $\mathrm{Err}_{\mathrm{HALF}}(\mathcal{D})$ as the least error of a…

In semidefinite programming a proposed optimal solution may be quite poor in spite of having sufficiently small residual in the optimality conditions. This issue may be framed in terms of the discrepancy between forward error (the…

We present two effective methods for solving high-dimensional partial differential equations (PDE) based on randomized neural networks. Motivated by the universal approximation property of this type of networks, both methods extend the…

A number of non-standard finite element methods have been proposed in recent years, each of which derives from a specific class of PDE-constrained norm minimization problems. The most notable examples are $\mathcal{L}\mathcal{L}^*$ methods.…

Supervised dimensionality reduction has emerged as an important theme in the last decade. Despite the plethora of models and formulations, there is a lack of a simple model which aims to project the set of patterns into a space defined by…

We develop a new tool, namely polynomial and linear algebraic methods, for studying systems of word equations. We illustrate its usefulness by giving essentially simpler proofs of several hard problems. At the same time we prove extensions…

Temporal difference (TD) learning algorithms with neural network function parameterization have well-established empirical success in many practical large-scale reinforcement learning tasks. However, theoretical understanding of these…

We prove 3SUM-hardness (no strongly subquadratic-time algorithm, assuming the 3SUM conjecture) of several problems related to finding Abelian square and additive square factors in a string. In particular, we conclude conditional optimality…