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We analyze a class of estimators based on convex relaxation for solving high-dimensional matrix decomposition problems. The observations are noisy realizations of a linear transformation $\mathfrak{X}$ of the sum of an approximately) low…

Machine Learning · Statistics 2012-08-09 Alekh Agarwal , Sahand N. Negahban , Martin J. Wainwright

This paper considers the problem of recovery of a low-rank matrix in the situation when most of its entries are not observed and a fraction of observed entries are corrupted. The observations are noisy realizations of the sum of a low rank…

Statistics Theory · Mathematics 2016-07-05 Olga Klopp , Karim Lounici , Alexandre B. Tsybakov

In this letter, we consider the problem of detecting a high dimensional signal based on compressed measurements with physical layer secrecy guarantees. We assume that the network operates in the presence of an eavesdropper who intends to…

Information Theory · Computer Science 2015-06-02 Bhavya Kailkhura , Sijia Liu , Thakshila Wimalajeewa , Pramod K. Varshney

We describe several algorithms for matrix completion and matrix approximation when only some of its entries are known. The approximation constraint can be any whose approximated solution is known for the full matrix. For low rank…

Numerical Analysis · Mathematics 2014-07-01 Gil Shabat , Yaniv Shmueli , Amir Averbuch

We consider the problem of noisy 1-bit matrix completion under an exact rank constraint on the true underlying matrix $M^*$. Instead of observing a subset of the noisy continuous-valued entries of a matrix $M^*$, we observe a subset of…

Machine Learning · Statistics 2015-02-25 Sonia Bhaskar , Adel Javanmard

Dense subgraph discovery is an important primitive in graph mining, which has a wide variety of applications in diverse domains. In the densest subgraph problem, given an undirected graph $G=(V,E)$ with an edge-weight vector $w=(w_e)_{e\in…

Social and Information Networks · Computer Science 2021-10-27 Atsushi Miyauchi , Akiko Takeda

We consider a problem of considerable practical interest: the recovery of a data matrix from a sampling of its entries. Suppose that we observe m entries selected uniformly at random from a matrix M. Can we complete the matrix and recover…

Information Theory · Computer Science 2008-05-30 Emmanuel J. Candes , Benjamin Recht

Given a dynamic network, where edges appear and disappear over time, we are interested in finding sets of edges that have similar temporal behavior and form a dense subgraph. Formally, we define the problem as the enumeration of the maximal…

Social and Information Networks · Computer Science 2021-03-02 Giulia Preti , Polina Rozenshtein , Aristides Gionis , Yannis Velegrakis

This paper is concerned with the problem of recovering an unknown matrix from a small fraction of its entries. This is known as the matrix completion problem, and comes up in a great number of applications, including the famous Netflix…

Information Theory · Computer Science 2009-03-10 Emmanuel J. Candes , Terence Tao

This paper deals with the problem of robust matrix completion -- retrieving a low-rank matrix and a sparse matrix from the compressed counterpart of their superposition. Though seemingly not an unresolved issue, we point out that the…

Information Theory · Computer Science 2024-10-10 Yinjian Wang

We consider the problem of finding a dense submatrix of a matrix with i.i.d. Gaussian entries, where density is measured by average value. This problem arose from practical applications in biology and social sciences…

Probability · Mathematics 2025-07-28 Shankar Bhamidi , David Gamarnik , Shuyang Gong

In distributed networks, it is often useful for the nodes to be aware of dense subgraphs, e.g., such a dense subgraph could reveal dense subtructures in otherwise sparse graphs (e.g. the World Wide Web or social networks); these might…

Distributed, Parallel, and Cluster Computing · Computer Science 2012-08-08 Atish Das Sarma , Ashwin Lall , Danupon Nanongkai , Amitabh Trehan

The problem of finding the densest subgraph in a given graph has several applications in graph mining, particularly in areas like social network analysis, protein and gene analyses etc. Depending on the application, finding dense subgraphs…

Data Structures and Algorithms · Computer Science 2019-11-07 Naga V. C. Gudapati , Enrico Malaguti , Michele Monaci

In this work, we consider constrained stochastic optimization problems under hidden convexity, i.e., those that admit a convex reformulation via non-linear (but invertible) map $c(\cdot)$. A number of non-convex problems ranging from…

Optimization and Control · Mathematics 2024-11-12 Ilyas Fatkhullin , Niao He , Yifan Hu

In this paper, we introduce the Maximum Matrix Contraction problem, where we aim to contract as much as possible a binary matrix in order to maximize its density. We study the complexity and the polynomial approximability of the problem.…

Computational Complexity · Computer Science 2023-06-05 Dimitri Watel , Pierre-Louis Poirion

Recent years have seen a flurry of activities in designing provably efficient nonconvex procedures for solving statistical estimation problems. Due to the highly nonconvex nature of the empirical loss, state-of-the-art procedures often…

Machine Learning · Computer Science 2020-06-09 Cong Ma , Kaizheng Wang , Yuejie Chi , Yuxin Chen

A useful approach to the mathematical analysis of large-scale biological networks is based upon their decompositions into monotone dynamical systems. This paper deals with two computational problems associated to finding decompositions…

Molecular Networks · Quantitative Biology 2007-05-23 Bhaskar DasGupta , German Andres Enciso , Eduardo Sontag , Yi Zhang

We give the first fully dynamic algorithm which maintains a $(1-\epsilon)$-approximate densest subgraph in worst-case time $\text{poly}(\log n, \epsilon^{-1})$ per update. Dense subgraph discovery is an important primitive for many…

Data Structures and Algorithms · Computer Science 2020-03-12 Saurabh Sawlani , Junxing Wang

We study the problem of recovering an incomplete $m\times n$ matrix of rank $r$ with columns arriving online over time. This is known as the problem of life-long matrix completion, and is widely applied to recommendation system, computer…

Machine Learning · Computer Science 2016-12-04 Maria-Florina Balcan , Hongyang Zhang

Finding the densest subgraph (DS) from a graph is a fundamental problem in graph databases. The DS obtained, which reveals closely related entities, has been found to be useful in various application domains such as e-commerce, social…

Databases · Computer Science 2025-04-16 Yi Yang , Chenhao Ma , Reynold Cheng , Laks V. S. Lakshmanan , Xiaolin Han