Diagonal and Low-Rank Matrix Decompositions, Correlation Matrices, and Ellipsoid Fitting
By James Saunderson, Venkat Chandrasekaran, Pablo A. Parrilo, Alan S. Willsky. 2012
83 citations recorded by the source corpus.
In this paper we establish links between, and new results for, three problems that are not usually considered together. The first is a matrix decomposition problem that arises in areas such as statistical modeling and signal processing: given a matrix $X$ formed as the sum of an unknown diagonal matrix and an unknown low-rank positive semidefinite matrix, decompose $X$ into these constituents. The second problem we consider is to determine the facial structure of the set of correlation matrices, a convex set also known as the elliptope. This convex body, and particularly its facial structure, plays a role in applications from combinatorial optimization to mathematical finance. The third problem is a basic geometric question: given points $v_1,v_2,\ldots,v_n\in \mathbb{R}^k$ (where $n > k$) determine whether there is a centered ellipsoid passing exactly through all the points. We show that in a precise sense these three problems are equivalent. Furthermore we establish a simple sufficient condition on a su...
Subfields: Computational Mechanics, Computational Theory and Mathematics, Numerical Analysis
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Computational Mechanics
- Dimitris J. Bertsimas
- Rahul Mazumder
- Robert Tibshirani
- Velibor V. Misic
- Trevor J. Hastie
- Peter Radchenko
In Computational Theory and Mathematics
In Numerical Analysis
- Rahul Mazumder
- Dimitris J. Bertsimas
- Martin S. Copenhaver
- Santosh Srinivas Vempala
- Robert Tibshirani
- Blaine Nelson
Citations in the corpus
- Analysis of elliptical copula correlation factor model with Kendall's tau (2013, unknown): “In this paper, the majority of the vectors will belong to R, and the majority of the 4We also observe that, in the noiseless setting, the low-rank and diagonal matrix decomposition problem is treated in [20], which employs a semidefinite program, the minimum trace factor analysis (MTFA), to minimize”
- Certifiably Optimal Low Rank Factor Analysis (2016, methodology): “The estimation Problem (4) is equivalent to Note that Problem (14) has an additional PSD constraint Σ − Θ (cid:23) 0 which does not explicitly appear in Problem (4).”