arXiv research

Detecting a planted submatrix in random matrices with non-asymptotic methods.

The principal submatrix localization problem deals with recovering a $K\times K$ principal submatrix of elevated mean $μ$ in a large $n\times n$ symmetric matrix subject to additive standard Gaussian noise. This problem serves as a prototypical example for community detection, in which the community corresponds to the …

We consider two closely related problems: planted clustering and submatrix localization. The planted clustering problem assumes that a random graph is generated based on some underlying clusters of the nodes; the task is to recover these clusters given the graph. The submatrix localization problem concerns locating hid…

Paper develops new method for detecting latent structure in large symmetric data matrices.

The interplay between computational efficiency and statistical accuracy in high-dimensional inference has drawn increasing attention in the literature. In this paper, we study computational and statistical boundaries for submatrix localization. Given one observation of (one or multiple non-overlapping) signal submatrix…

Spectral algorithms solve optimal community detection and related problems.

A problem of paramount importance in both pure (Restricted Invertibility problem) and applied mathematics (Feature extraction) is the one of selecting a submatrix of a given matrix, such that this submatrix has its smallest singular value above a specified level. Such problems can be addressed using perturbation analys…

The paper solves a maximum entropy sampling problem with efficient algorithms and performance guarantees.

BIND removes background noise from binary matrices, improving detection accuracy and fairness.

This paper considers probabilistic estimation of a low-rank matrix from non-linear element-wise measurements of its elements. We derive the corresponding approximate message passing (AMP) algorithm and its state evolution. Relying on non-rigorous but standard assumptions motivated by statistical physics, we characteriz…

The computation of the sparse principal component of a matrix is equivalent to the identification of its principal submatrix with the largest maximum eigenvalue. Finding this optimal submatrix is what renders the problem ${\mathcal{NP}}$-hard. In this work, we prove that, if the matrix is positive semidefinite and its …

New findings on computational limits for estimating hidden structures.

Many high dimensional sparse learning problems are formulated as nonconvex optimization. A popular approach to solve these nonconvex optimization problems is through convex relaxations such as linear and semidefinite programming. In this paper, we study the statistical limits of convex relaxations. Particularly, we con…

High throughput biomedical measurements normally capture multiple overlaid biologically relevant signals and often also signals representing different types of technical artefacts like e.g. batch effects. Signal identification and decomposition are accordingly main objectives in statistical biomedical modeling and data…

New method detects inconsistencies in AHP matrices using triadic preference reversals.

New method explains computational barriers in high-dimensional statistical models.

A filter detects and removes noisy labels in semi-supervised learning.

We consider the following general hidden hubs model: an $n \times n$ random matrix $A$ with a subset $S$ of $k$ special rows (hubs): entries in rows outside $S$ are generated from the probability distribution $p_0 \sim N(0,σ_0^2)$; for each row in $S$, some $k$ of its entries are generated from $p_1 \sim N(0,σ_1^2)$, $…

Paper extends FOFC algorithm to work with mixed data types.

Predictive State Representations (PSRs) are powerful techniques for modelling dynamical systems, which represent a state as a vector of predictions about future observable events (tests). In PSRs, one of the fundamental problems is the learning of the PSR model of the underlying system. Recently, spectral methods have …

Given a large data matrix $A\in\mathbb{R}^{n\times n}$, we consider the problem of determining whether its entries are i.i.d. with some known marginal distribution $A_{ij}\sim P_0$, or instead $A$ contains a principal submatrix $A_{{\sf Q},{\sf Q}}$ whose entries have marginal distribution $A_{ij}\sim P_1\neq P_0$. As …

We show that for the problem of testing if a matrix $A \in F^{n \times n}$ has rank at most $d$, or requires changing an $ε$-fraction of entries to have rank at most $d$, there is a non-adaptive query algorithm making $\widetilde{O}(d^2/ε)$ queries. Our algorithm works for any field $F$. This improves upon the previous…

We consider the problem of estimating a low-rank matrix from a noisy observed matrix. Previous work has shown that the optimal method depends crucially on the choice of loss function. In this paper, we use a family of weighted loss functions, which arise naturally for problems such as submatrix denoising, denoising wit…

New algorithms detect and estimate rank-one signals with prior directional information.

Efficiently transforms Gaussian data to simulate various target distributions.

Study on overlaps of singular vectors in Gaussian matrix submatrices.

Improved matrix completion for non-uniformly sampled data.

New insights link diverse statistical problems via secret leakage planted clique.

Study analyzes perturbations in singular subspaces under random noise.

Most existing word embedding methods can be categorized into Neural Embedding Models and Matrix Factorization (MF)-based methods. However some models are opaque to probabilistic interpretation, and MF-based methods, typically solved using Singular Value Decomposition (SVD), may incur loss of corpus information. In addi…

Researchers link knot Floer homology, Burau representation, and quantum gl(1|1).

We present a solution to scale spectral algorithms for learning sequence functions. We are interested in the case where these functions are sparse (that is, for most sequences they return 0). Spectral algorithms reduce the learning problem to the task of computing an SVD decomposition over a special type of matrix call…

Ensemble methods that average over a collection of independent predictors that are each limited to a subsampling of both the examples and features of the training data command a significant presence in machine learning, such as the ever-popular random forest, yet the nature of the subsampling effect, particularly of th…

We study dual volume sampling, a method for selecting k columns from an n x m short and wide matrix (n <= k <= m) such that the probability of selection is proportional to the volume spanned by the rows of the induced submatrix. This method was proposed by Avron and Boutsidis (2013), who showed it to be a promising met…

In many situations it is desirable to identify clusters that differ with respect to only a subset of features. Such clusters may represent homogeneous subgroups of patients with a disease, such as cancer or chronic pain. We define a bicluster to be a submatrix U of a larger data matrix X such that the features and obse…

The exact nonnegative matrix factorization (exact NMF) problem is the following: given an $m$-by-$n$ nonnegative matrix $X$ and a factorization rank $r$, find, if possible, an $m$-by-$r$ nonnegative matrix $W$ and an $r$-by-$n$ nonnegative matrix $H$ such that $X = WH$. In this paper, we propose two heuristics for exac…

We consider the densest $k$-subgraph problem, which seeks to identify the $k$-node subgraph of a given input graph with maximum number of edges. This problem is well-known to be NP-hard, by reduction to the maximum clique problem. We propose a new convex relaxation for the densest $k$-subgraph problem, based on a nucle…

Boolean matrix has been used to represent digital information in many fields, including bank transaction, crime records, natural language processing, protein-protein interaction, etc. Boolean matrix factorization (BMF) aims to find an approximation of a binary matrix as the Boolean product of two low rank Boolean matri…

We study the complexity of sampling from a distribution over all index subsets of the set $\{1,...,n\}$ with the probability of a subset $S$ proportional to the determinant of the submatrix $\mathbf{L}_S$ of some $n\times n$ p.s.d. matrix $\mathbf{L}$, where $\mathbf{L}_S$ corresponds to the entries of $\mathbf{L}$ ind…

We study a semidefinite programming (SDP) relaxation of the maximum likelihood estimation for exactly recovering a hidden community of cardinality $K$ from an $n \times n$ symmetric data matrix $A$, where for distinct indices $i,j$, $A_{ij} \sim P$ if $i, j$ are both in the community and $A_{ij} \sim Q$ otherwise, for …

We study the problem of recovering a hidden community of cardinality $K$ from an $n \times n$ symmetric data matrix $A$, where for distinct indices $i,j$, $A_{ij} \sim P$ if $i, j$ both belong to the community and $A_{ij} \sim Q$ otherwise, for two known probability distributions $P$ and $Q$ depending on $n$. If $P={\r…

New link detection results using closures of 3-braids.

Deep learning improves anomaly detection across various fields.

Graph energy helps detect communities in networks better than traditional methods.

Monitoring gas turbine combustors health, in particular, early detecting abnormal behaviors and incipient faults, is critical in ensuring gas turbines operating efficiently and in preventing costly unplanned maintenance. One popular means of detecting combustor abnormalities is through continuously monitoring exhaust g…

Develops slope detection for 3-manifolds with torus boundaries.

2DSig-Detect detects adversarial perturbations in images.

Develops a method to detect changes in linear systems with temporal correlations.