arXiv research

The paper constructs stable minimal hypersurfaces with specific singularities.

New method estimates robust mean in high dimensions with minimized outliers.

We study the robustness properties of $\ell_1$ norm minimization for the classical linear regression problem with a given design matrix and contamination restricted to the dependent variable. We perform a fine error analysis of the $\ell_1$ estimator for measurements errors consisting of outliers coupled with noise. We…

The paper finds new eigenfunctions for minimal immersions and their stability index.

The Schwarzian derivative helps classify minimal surfaces by their degree.

The paper analyzes $\ell_q$ optimization methods for high-dimensional linear regression.

Recent results in Compressive Sensing have shown that, under certain conditions, the solution to an underdetermined system of linear equations with sparsity-based regularization can be accurately recovered by solving convex relaxations of the original problem. In this work, we present a novel primal-dual analysis on a …

The paper tackles multi-armed bandits with vector losses, focusing on minimizing the $\ell^\infty$-norm of relative losses.

Extending work of Kapouleas and Yang, for any integers $N \geq 2$, $k, \ell \geq 1$, and $m$ sufficiently large, we apply gluing methods to construct in the round $3$-sphere a closed embedded minimal surface that has genus $k\ell m^2(N-1)+1$ and is invariant under a $D_{km} \times D_{\ell m}$ subgroup of $O(4)$, where …

Study shows unique sharp local minimum in $\ell_1$-minimization for dictionary learning.

Paper tackles low-rank matrix recovery with column $\ell_{2,0}$-norm regularization.

We consider a decomposition method for compressive streaming data in the context of online compressive Robust Principle Component Analysis (RPCA). The proposed decomposition solves an $n$-$\ell_1$ cluster-weighted minimization to decompose a sequence of frames (or vectors), into sparse and low-rank components, from com…

New algorithm solves $\ell_0$-norm constrained multilinear logistic regression for tensor data.

In compressed sensing problems, $\ell_1$ minimization or Basis Pursuit was known to have the best provable phase transition performance of recoverable sparsity among polynomial-time algorithms. It is of great theoretical and practical interest to find alternative polynomial-time algorithms which perform better than $\e…

Regulator allocates buffers to prevent financial contagion in networks with common assets.

New method for factor analysis using nuclear and $\ell_0$ norms.

An elementary proof found for the double bubble problem in a specific norm.

New algorithm solves constrained ℓ_p regression problems efficiently.

Researchers analyze geodesic complexity in robot paths on tree graphs.

In stochastic convex optimization the goal is to minimize a convex function $F(x) \doteq {\mathbf E}_{{\mathbf f}\sim D}[{\mathbf f}(x)]$ over a convex set $\cal K \subset {\mathbb R}^d$ where $D$ is some unknown distribution and each $f(\cdot)$ in the support of $D$ is convex over $\cal K$. The optimization is commonl…

We discuss the problem of adaptive discrete-time signal denoising in the situation where the signal to be recovered admits a "linear oracle" -- an unknown linear estimate that takes the form of convolution of observations with a time-invariant filter. It was shown by Juditsky and Nemirovski (2009) that when the $\ell_2…

This paper develops a novel deep recurrent neural network for sequential signal reconstruction.

In this paper we develop a novel computational sensing framework for sensing and recovering structured signals. When trained on a set of representative signals, our framework learns to take undersampled measurements and recover signals from them using a deep convolutional neural network. In other words, it learns a tra…

Improved approximation for socially fair clustering with $\ell_p$-objective.

This study connects Jacobian regularization to adversarial robustness and improves generalization.

Dictionaries are collections of vectors used for representations of random vectors in Euclidean spaces. Recent research on optimal dictionaries is focused on constructing dictionaries that offer sparse representations, i.e., $\ell_0$-optimal representations. Here we consider the problem of finding optimal dictionaries …

The one-bit quantization is implemented by one single comparator that operates at low power and a high rate. Hence one-bit compressive sensing (1bit-CS) becomes attractive in signal processing. When measurements are corrupted by noise during signal acquisition and transmission, 1bit-CS is usually modeled as minimizing …

Dictionaries are collections of vectors used for representations of elements in Euclidean spaces. While recent research on optimal dictionaries is focussed on providing sparse (i.e., $\ell_0$-optimal,) representations, here we consider the problem of finding optimal dictionaries such that representations of samples of …

In this paper, we consider the problem of linear regression with heavy-tailed distributions. Different from previous studies that use the squared loss to measure the performance, we choose the absolute loss, which is capable of estimating the conditional median. To address the challenge that both the input and output c…

In many applications, high-dimensional data points can be well represented by low-dimensional subspaces. To identify the subspaces, it is important to capture a global and local structure of the data which is achieved by imposing low-rank and sparseness constraints on the data representation matrix. In low-rank sparse …

This paper considers the problem of recovering signals from compressed measurements contaminated with sparse outliers, which has arisen in many applications. In this paper, we propose a generative model neural network approach for reconstructing the ground truth signals under sparse outliers. We propose an iterative al…

Paper introduces $\ell$-DER for regression tasks using morphological operators and convex-concave procedure.

We discuss a general notion of "sparsity structure" and associated recoveries of a sparse signal from its linear image of reduced dimension possibly corrupted with noise. Our approach allows for unified treatment of (a) the "usual sparsity" and "usual $\ell_1$ recovery," (b) block-sparsity with possibly overlapping blo…

The main contribution of the paper is a new approach to subspace clustering that is significantly more computationally efficient and scalable than existing state-of-the-art methods. The central idea is to modify the regression technique in sparse subspace clustering (SSC) by replacing the $\ell_1$ minimization with a g…

Given $φ$ a pseudo-Anosov map, let $\ell_\mathcal{T}(φ)$ denote the translation length of $φ$ in the Teichmüller space, and let $\ell_\mathcal{C}(φ)$ denote the stable translation length of $φ$ in the curve graph. Gadre--Hironaka--Kent--Leininger showed that, as a function of Euler characteristic $χ(S)$, the minimal po…

Using the virtual fibering theorem of Agol we show that a sutured 3-manifold $(M, R_+,R_-,γ)$ is taut if and only if the $\ell^2$-Betti numbers of the pair $(M,R_-)$ are zero. As an application we can characterize Thurston norm minimizing surfaces in a 3-manifold $N$ with empty or toroidal boundary by the vanishing of …

We introduce a recursive adaptive group lasso algorithm for real-time penalized least squares prediction that produces a time sequence of optimal sparse predictor coefficient vectors. At each time index the proposed algorithm computes an exact update of the optimal $\ell_{1,\infty}$-penalized recursive least squares (R…

We give an estimate of the mean curvature of a complete submanifold lying inside a closed cylinder $B(r)\times\R^{\ell}$ in a product Riemannian manifold $N^{n-\ell}\times\R^{\ell}$. It follows that a complete hypersurface of given constant mean curvature lying inside a closed circular cylinder in Euclidean space canno…

New model leads to optimal test loss in sparse linear regression.

A data filtering method for cluster analysis is proposed, based on minimizing a least squares function with a weighted $\ell_0$-norm penalty. To overcome the discontinuity of the objective function, smooth non-convex functions are employed to approximate the $\ell_0$-norm. The convergence of the global minimum points o…

Paper tackles outlier detection in signals modeled by generative models with theoretical guarantees.

New method improves tensor completion and robust PCA using non-convex tensor rank and sparsity measures.

Multi-task feature learning aims to identity the shared features among tasks to improve generalization. It has been shown that by minimizing non-convex learning models, a better solution than the convex alternatives can be obtained. Therefore, a non-convex model based on the capped-$\ell_{1},\ell_{1}$ regularization wa…

New method encodes manifold homotopy types into algebra structures, extending previous bounds.

The geometric intersection number of a curve on a surface is the minimal number of self-intersections of any homotopic curve, i.e. of any curve obtained by continuous deformation. Given a curve $c$ represented by a closed walk of length at most $\ell$ on a combinatorial surface of complexity $n$ we describe simple algo…

Optimal batch size minimizes training time for neural networks.

Paper tackles low-rank matrix recovery with KL property and DC reformulation.

Let ${\mathcal S}(\R)$ be an o-minimal structure over $\R$, $T \subset \R^{k_1+k_2+\ell}$ a closed definable set, and $$ \displaylines{π_1: \R^{k_1+k_2+\ell}\to \R^{k_1 + k_2}, π_2: \R^{k_1+k_2+\ell}\to \R^{\ell}, \ π_3: \R^{k_1 + k_2} \to \R^{k_2}} $$ the projection maps. For any collection ${\mathcal A} = \{A_1,...,A…