[ { "id": "authors:739bj-bet81", "collection": "authors", "collection_id": "739bj-bet81", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20121004-133456681", "type": "article", "title": "Diagonal and Low-Rank Matrix Decompositions, Correlation Matrices, and Ellipsoid Fitting", "author": [ { "family_name": "Saunderson", "given_name": "J.", "orcid": "0000-0002-5456-0180", "clpid": "Saunderson-James" }, { "family_name": "Chandrasekaran", "given_name": "V.", "clpid": "Chandrasekaran-V" }, { "family_name": "Parrilo", "given_name": "P. A.", "orcid": "0000-0003-1132-8477", "clpid": "Parrilo-P-A" }, { "family_name": "Willsky", "given_name": "A. S.", "clpid": "Willsky-A-S" } ], "abstract": "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 v1, v2, \u2026 , vn \u2208 R^k (where n > k) determine whether there is a centered ellipsoid passing exactly through all of the points. \n\n\nWe show that in a precise sense these three problems are equivalent. Furthermore we establish a simple sufficient condition on a subspace U that ensures any positive semidefinite matrix L with column space U can be recovered from D+L for any diagonal matrix D using a convex optimization-based heuristic known as minimum trace factor analysis. This result leads to a new understanding of the structure of rank-deficient correlation matrices and a simple condition on a set of points that ensures there is a centered ellipsoid passing through them.", "doi": "10.1137/120872516", "issn": "0895-4798", "publisher": "Society for Industrial and Applied Mathematics", "publication": "SIAM Journal on Matrix Analysis and Applications", "publication_date": "2012-12-19", "series_number": "4", "volume": "33", "issue": "4", "pages": "1395-1416" }, { "id": "authors:w276e-jcy72", "collection": "authors", "collection_id": "w276e-jcy72", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20121004-152325296", "type": "article", "title": "The Convex Geometry of Linear Inverse Problems", "author": [ { "family_name": "Chandrasekaran", "given_name": "Venkat", "clpid": "Chandrasekaran-V" }, { "family_name": "Recht", "given_name": "Benjamin", "clpid": "Recht-B" }, { "family_name": "Parrilo", "given_name": "Pablo A.", "orcid": "0000-0003-1132-8477", "clpid": "Parrilo-P-A" }, { "family_name": "Willsky", "given_name": "Alan S.", "clpid": "Willsky-A-S" } ], "abstract": "In applications throughout science and engineering one is often faced with the challenge of solving an ill-posed inverse problem, where the number of available measurements is smaller than the dimension of the model to be estimated. However in many practical situations of interest, models are constrained structurally so that they only have a few degrees of freedom relative to their ambient dimension. This paper provides a general framework to convert notions of simplicity into convex penalty functions, resulting in convex optimization solutions to linear, underdetermined inverse problems. The class of simple models considered includes those formed as the sum of a few atoms from some (possibly infinite) elementary atomic set; examples include well-studied cases from many technical fields such as sparse vectors (signal processing, statistics) and low-rank matrices (control, statistics), as well as several others including sums of a few permutation matrices (ranked elections, multiobject tracking), low-rank tensors (computer vision, neuroscience), orthogonal matrices (machine learning), and atomic measures (system identification). The convex programming formulation is based on minimizing the norm induced by the convex hull of the atomic set; this norm is referred to as the atomic norm. The facial structure of the atomic norm ball carries a number of favorable properties that are useful for recovering simple models, and an analysis of the underlying convex geometry provides sharp estimates of the number of generic measurements required for exact and robust recovery of models from partial information. These estimates are based on computing the Gaussian widths of tangent cones to the atomic norm ball. When the atomic set has algebraic structure the resulting optimization problems can be solved or approximated via semidefinite programming. The quality of these approximations affects the number of measurements required for recovery, and this tradeoff is characterized via some examples. Thus this work extends the catalog of simple models (beyond sparse vectors and low-rank matrices) that can be recovered from limited linear information via tractable convex programming.", "doi": "10.1007/s10208-012-9135-7", "issn": "1615-3375", "publisher": "Springer", "publication": "Foundations of Computational Mathematics", "publication_date": "2012-12", "series_number": "6", "volume": "12", "issue": "6", "pages": "805-849" }, { "id": "authors:w826x-t7e72", "collection": "authors", "collection_id": "w826x-t7e72", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130110-100237034", "type": "article", "title": "Convex Graph Invariants", "author": [ { "family_name": "Chandrasekaran", "given_name": "Venkat", "clpid": "Chandrasekaran-V" }, { "family_name": "Parrilo", "given_name": "Pablo A.", "orcid": "0000-0003-1132-8477", "clpid": "Parrilo-P-A" }, { "family_name": "Willsky", "given_name": "Alan S.", "clpid": "Willsky-A-S" } ], "abstract": "The structural properties of graphs are usually characterized in terms of invariants, which\nare functions of graphs that do not depend on the labeling of the nodes. In this paper\nwe study convex graph invariants, which are graph invariants that are convex functions\nof the adjacency matrix of a graph. Some examples include functions of a graph such as\nthe maximum degree, the MAXCUT value (and its semidefinite relaxation), and spectral\ninvariants such as the sum of the k largest eigenvalues. Such functions can be used to\nconstruct convex sets that impose various structural constraints on graphs and thus provide\na unified framework for solving a number of interesting graph problems via convex\noptimization. We give a representation of all convex graph invariants in terms of certain\nelementary invariants, and we describe methods to compute or approximate convex graph\ninvariants tractably. We discuss the interesting subclass of spectral invariants, and also\ncompare convex and nonconvex invariants. Finally, we use convex graph invariants to provide\nefficient convex programming solutions to graph problems such as the deconvolution of\nthe composition of two graphs into the individual components, hypothesis testing between\ngraph families, and the generation of graphs with certain desired structural properties.", "doi": "10.1137/100816900", "issn": "0036-1445", "publisher": "Society for Industrial and Applied Mathematics", "publication": "SIAM Review", "publication_date": "2012-08-07", "series_number": "3", "volume": "54", "issue": "3", "pages": "513-541" }, { "id": "authors:gqqee-ncb30", "collection": "authors", "collection_id": "gqqee-ncb30", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130205-141004858", "type": "article", "title": "Rejoinder: Latent variable graphical model selection via convex optimization", "author": [ { "family_name": "Chandrasekaran", "given_name": "Venkat", "clpid": "Chandrasekaran-V" }, { "family_name": "Parrilo", "given_name": "Pablo A.", "orcid": "0000-0003-1132-8477", "clpid": "Parrilo-P-A" }, { "family_name": "Willsky", "given_name": "Alan S.", "clpid": "Willsky-A-S" } ], "abstract": "We thank all the discussants for their careful reading of our paper, and for their insightful critiques. We would also like to thank the editors for organizing this discussion. Our paper contributes to the area of high-dimensional statistics which has received much attention over the past several years across the statistics, machine learning and signal processing communities. In this rejoinder we clarify and comment on some of the points raised in the discussions. Finally, we also remark on some interesting challenges that lie ahead in latent variable modeling.", "doi": "10.1214/12-AOS1020", "issn": "0090-5364", "publisher": "Institute of Mathematical Statistics", "publication": "Annals of Statistics", "publication_date": "2012-08", "series_number": "4", "volume": "40", "issue": "4", "pages": "2005-2013" }, { "id": "authors:a8hmk-wjd18", "collection": "authors", "collection_id": "a8hmk-wjd18", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130207-085454891", "type": "article", "title": "Latent Variable Graphical Model Selection via Convex Optimization", "author": [ { "family_name": "Chandrasekaran", "given_name": "Venkat", "clpid": "Chandrasekaran-V" }, { "family_name": "Parrilo", "given_name": "Pablo A.", "orcid": "0000-0003-1132-8477", "clpid": "Parrilo-P-A" }, { "family_name": "Willsky", "given_name": "Alan S.", "clpid": "Willsky-A-S" } ], "abstract": "Suppose we have samples of a subset of a collection of random variables. No additional information is provided about the number of latent variables, nor of the relationship between the latent and observed variables. Is it possible to discover the number of hidden components, and to learn a statistical model over the entire collection of variables? We address this question in the setting in which the latent and observed variables are jointly Gaussian, with the conditional statistics of the observed variables conditioned on the latent variables being specified by a graphical model. As a first step we give natural conditions under which such latent-variable Gaussian graphical models are identifiable given marginal statistics of only the observed variables. Essentially these conditions require that the conditional graphical model among the observed variables is sparse, while the effect of the latent variables is \"spread out\" over most of the observed variables. Next we propose a tractable convex program based on regularized maximum-likelihood for model selection in this latent-variable setting; the regularizer uses both the \u2113_1 norm and the nuclear norm. Our modeling framework can be viewed as a combination of dimensionality reduction (to identify latent variables) and graphical modeling (to capture remaining statistical structure not attributable to the latent variables), and it consistently estimates both the number of hidden components and the conditional graphical model structure among the observed variables. These results are applicable in the high-dimensional setting in which the number of latent/observed variables grows with the number of samples of the observed variables. The geometric properties of the algebraic varieties of sparse matrices and of low-rank matrices play an important role in our analysis.", "doi": "10.1214/11-AOS949", "issn": "0090-5364", "publisher": "Institute of Mathematical Statistics", "publication": "Annals of Statistics", "publication_date": "2012-08", "series_number": "4", "volume": "40", "issue": "4", "pages": "1935-1967" }, { "id": "authors:0t84c-nmc65", "collection": "authors", "collection_id": "0t84c-nmc65", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20121008-095909823", "type": "article", "title": "Rank-Sparsity Incoherence for Matrix Decomposition", "author": [ { "family_name": "Chandrasekaran", "given_name": "Venkat", "clpid": "Chandrasekaran-V" }, { "family_name": "Sanghavi", "given_name": "Sujay", "orcid": "0000-0003-0754-9154", "clpid": "Sanghavi-S" }, { "family_name": "Parrilo", "given_name": "Pablo A.", "orcid": "0000-0003-1132-8477", "clpid": "Parrilo-P-A" }, { "family_name": "Willsky", "given_name": "Alan S.", "clpid": "Willsky-A-S" } ], "abstract": "Suppose we are given a matrix that is formed by adding an unknown sparse matrix to an unknown low-rank matrix. Our goal is to decompose the given matrix into its sparse and low-rank components. Such a problem arises in a number of applications in model and system identification and is intractable to solve in general. In this paper we consider a convex optimization formulation to splitting the specified matrix into its components by minimizing a linear combination of the \u2113_1 norm and the nuclear norm of the components. We develop a notion of rank-sparsity incoherence, expressed as an uncertainty principle between the sparsity pattern of a matrix and its row and column spaces, and we use it to characterize both fundamental identifiability as well as (deterministic) sufficient conditions for exact recovery. Our analysis is geometric in nature with the tangent spaces to the algebraic varieties of sparse and low-rank matrices playing a prominent role. When the sparse and low-rank matrices are drawn from certain natural random ensembles, we show that the sufficient conditions for exact recovery are satisfied with high probability. We conclude with simulation results on synthetic matrix decomposition problems.", "doi": "10.1137/090761793", "issn": "1052-6234", "publisher": "Society for Industrial and Applied Mathematics", "publication": "SIAM Journal of Optimization", "publication_date": "2011-06-30", "series_number": "2", "volume": "21", "issue": "2", "pages": "572-596" }, { "id": "authors:zq2bs-k7j84", "collection": "authors", "collection_id": "zq2bs-k7j84", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20110602-133842599", "type": "article", "title": "Symmetry Analysis of Reversible Markov Chains", "author": [ { "family_name": "Boyd", "given_name": "Stephen", "clpid": "Boyd-S" }, { "family_name": "Diaconis", "given_name": "Persi", "clpid": "Diaconis-P" }, { "family_name": "Parrilo", "given_name": "Pablo", "orcid": "0000-0003-1132-8477", "clpid": "Parrilo-P-A" }, { "family_name": "Xiao", "given_name": "Lin", "clpid": "Xiao-Lin" } ], "abstract": "We show how to use subgroups of the symmetry group of a reversible Markov chain to give useful bounds on eigenvalues and their multiplicity. We supplement classical representation theoretic tools involving a group commuting with a self-adjoint operator with criteria for an eigenvector to descend to an orbit graph. As examples, we show that the Metropolis construction can dominate a max-degree construction by an arbitrary amount and that, in turn, the fastest mixing Markov chain can dominate the Metropolis construction by an arbitrary amount.", "doi": "10.1080/15427951.2005.10129100", "issn": "1944-9488", "publisher": "Taylor & Francis", "publication": "Internet Mathematics", "publication_date": "2008", "series_number": "1", "volume": "2", "issue": "1", "pages": "31-71" }, { "id": "authors:601jw-f4f86", "collection": "authors", "collection_id": "601jw-f4f86", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:DOHpra05", "type": "article", "title": "Detecting multipartite entanglement", "author": [ { "family_name": "Doherty", "given_name": "Andrew C.", "clpid": "Doherty-A-C" }, { "family_name": "Spedalieri", "given_name": "Federico M.", "clpid": "Spedalieri-F-M" }, { "family_name": "Parrilo", "given_name": "Pablo A.", "orcid": "0000-0003-1132-8477", "clpid": "Parrilo-P-A" } ], "abstract": "We discuss the problem of determining whether the state of several quantum mechanical subsystems is entangled. As in previous work on two subsystems we introduce a procedure for checking separability that is based on finding state extensions with appropriate properties and may be implemented as a semidefinite program. The main result of this work is to show that there is a series of tests of this kind such that if a multiparty state is entangled this will eventually be detected by one of the tests. The procedure also provides a means of constructing entanglement witnesses that could in principle be measured in order to demonstrate that the state is entangled.", "doi": "10.1103/PhysRevA.71.032333", "issn": "1050-2947", "publisher": "Physical Review A", "publication": "Physical Review A", "publication_date": "2005-03-01", "series_number": "3", "volume": "71", "issue": "3", "pages": "Art. No. 032333" }, { "id": "authors:5vny8-wby74", "collection": "authors", "collection_id": "5vny8-wby74", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:PRAieeetac04", "type": "article", "title": "Nonlinear control synthesis by convex optimization", "author": [ { "family_name": "Prajna", "given_name": "Stephen", "clpid": "Prajna-S" }, { "family_name": "Parrilo", "given_name": "Pablo A.", "orcid": "0000-0003-1132-8477", "clpid": "Parrilo-P-A" }, { "family_name": "Rantzer", "given_name": "Anders", "clpid": "Rantzer-A" } ], "abstract": "A stability criterion for nonlinear systems, recently derived by the third author, can be viewed as a dual to Lyapunov's second theorem. The criterion is stated in terms of a function which can be interpreted as the stationary density of a substance that is generated all over the state-space and flows along the system trajectories toward the equilibrium. The new criterion has a remarkable convexity property, which in this note is used for controller synthesis via convex optimization. Recent numerical methods for verification of positivity of multivariate polynomials based on sum of squares decompositions are used.", "doi": "10.1109/TAC.2003.823000", "issn": "0018-9286", "publisher": "IEEE", "publication": "IEEE Transactions on Automatic Control", "publication_date": "2004-02-01", "series_number": "2", "volume": "49", "issue": "2", "pages": "310-314" }, { "id": "authors:hvjax-bwp85", "collection": "authors", "collection_id": "hvjax-bwp85", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:DOHpra04", "type": "article", "title": "Complete family of separability criteria", "author": [ { "family_name": "Doherty", "given_name": "Andrew C.", "clpid": "Doherty-A-C" }, { "family_name": "Parrilo", "given_name": "Pablo A.", "orcid": "0000-0003-1132-8477", "clpid": "Parrilo-P-A" }, { "family_name": "Spedalieri", "given_name": "Federico M.", "clpid": "Spedalieri-F-M" } ], "abstract": "We introduce a family of separability criteria that are based on the existence of extensions of a bipartite quantum state rho to a larger number of parties satisfying certain symmetry properties. It can be easily shown that all separable states have the required extensions, so the nonexistence of such an extension for a particular state implies that the state is entangled. One of the main advantages of this approach is that searching for the extension can be cast as a convex optimization problem known as a semidefinite program. Whenever an extension does not exist, the dual optimization constructs an explicit entanglement witness for the particular state. These separability tests can be ordered in a hierarchical structure whose first step corresponds to the well-known positive partial transpose (Peres-Horodecki) criterion, and each test in the hierarchy is at least as powerful as the preceding one. This hierarchy is complete, in the sense that any entangled state is guaranteed to fail a test at some finite point in the hierarchy, thus showing it is entangled. The entanglement witnesses corresponding to each step of the hierarchy have well-defined and very interesting algebraic properties that, in turn, allow for a characterization of the interior of the set of positive maps. Coupled with some recent results on the computational complexity of the separability problem, which has been shown to be NP hard, this hierarchy of tests gives a complete and also computationally and theoretically appealing characterization of mixed bipartite entangled states.", "doi": "10.1103/PhysRevA.69.022308", "issn": "1050-2947", "publisher": "Physical Review A", "publication": "Physical Review A", "publication_date": "2004-02-01", "series_number": "2", "volume": "69", "issue": "2", "pages": "Art. No. 022308" }, { "id": "authors:v84v3-3sg67", "collection": "authors", "collection_id": "v84v3-3sg67", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200324-151107160", "type": "article", "title": "Semidefinite programming relaxations for semialgebraic problems", "author": [ { "family_name": "Parrilo", "given_name": "Pablo A.", "orcid": "0000-0003-1132-8477", "clpid": "Parrilo-P-A" } ], "abstract": "A hierarchy of convex relaxations for semialgebraic problems is introduced. For questions reducible to a finite number of polynomial equalities and inequalities, it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility. The main tools employed are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials, and some results from real algebraic geometry. The techniques provide a constructive approach for finding bounded degree solutions to the Positivstellensatz, and are illustrated with examples from diverse application fields.", "doi": "10.1007/s10107-003-0387-5", "issn": "0025-5610", "publisher": "Springer", "publication": "Mathematical Programming", "publication_date": "2003-05", "series_number": "2", "volume": "96", "issue": "2", "pages": "293-320" }, { "id": "authors:kcfkx-9fq18", "collection": "authors", "collection_id": "kcfkx-9fq18", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:DOHprl02", "type": "article", "title": "Distinguishing Separable and Entangled States", "author": [ { "family_name": "Doherty", "given_name": "A. C.", "clpid": "Doherty-A-C" }, { "family_name": "Parrilo", "given_name": "Pablo A.", "orcid": "0000-0003-1132-8477", "clpid": "Parrilo-P-A" }, { "family_name": "Spedalieri", "given_name": "Federico M.", "clpid": "Spedalieri-F-M" } ], "abstract": "We show how to design families of operational criteria that distinguish entangled from separable quantum states. The simplest of these tests corresponds to the well-known Peres-Horodecki positive partial transpose (PPT) criterion, and the more complicated tests are strictly stronger. The new criteria are tractable due to powerful computational and theoretical methods for the class of convex optimization problems known as semidefinite programs. We successfully applied the results to many low-dimensional states from the literature where the PPT test fails. As a by-product of the criteria, we provide an explicit construction of the corresponding entanglement witnesses.", "doi": "10.1103/PhysRevLett.88.187904", "issn": "0031-9007", "publisher": "American Physical Society", "publication": "Physical Review Letters", "publication_date": "2002-05-06", "series_number": "18", "volume": "88", "issue": "18", "pages": "Art. No. 187904" } ]