Results 11  20
of
74
Universal lowrank matrix recovery from Pauli measurements
"... We study the problem of reconstructing an unknown matrix M of rank r and dimension d using O(rd poly log d) Pauli measurements. This has applications in quantum state tomography, and is a noncommutative analogue of a wellknown problem in compressed sensing: recovering a sparse vector from a few of ..."
Abstract

Cited by 20 (0 self)
 Add to MetaCart
(Show Context)
We study the problem of reconstructing an unknown matrix M of rank r and dimension d using O(rd poly log d) Pauli measurements. This has applications in quantum state tomography, and is a noncommutative analogue of a wellknown problem in compressed sensing: recovering a sparse vector from a few of its Fourier coefficients. We show that almost all sets of O(rd log 6 d) Pauli measurements satisfy the rankr restricted isometry property (RIP). This implies that M can be recovered from a fixed (“universal”) set of Pauli measurements, using nuclearnorm minimization (e.g., the matrix Lasso), with nearlyoptimal bounds on the error. A similar result holds for any class of measurements that use an orthonormal operator basis whose elements have small operator norm. Our proof uses Dudley’s inequality for Gaussian processes, together with bounds on covering numbers obtained via entropy duality. 1
Uniqueness of lowrank matrix completion by rigidity theory
, 2009
"... Abstract. The problem of completing a lowrank matrix from a subset of its entries is often encountered in the analysis of incomplete data sets exhibiting an underlying factor model with applications in collaborative filtering, computer vision and control. Most recent work had been focused on constr ..."
Abstract

Cited by 20 (1 self)
 Add to MetaCart
(Show Context)
Abstract. The problem of completing a lowrank matrix from a subset of its entries is often encountered in the analysis of incomplete data sets exhibiting an underlying factor model with applications in collaborative filtering, computer vision and control. Most recent work had been focused on constructing efficient algorithms for exact or approximate recovery of the missing matrix entries and proving lower bounds for the number of known entries that guarantee a successful recovery with high probability. A related problem from both the mathematical and algorithmic point of view is the distance geometry problem of realizing points in a Euclidean space from a given subset of their pairwise distances. Rigidity theory answers basic questions regarding the uniqueness of the realization satisfying a given partial set of distances. We observe that basic ideas and tools of rigidity theory can be adapted to determine uniqueness of lowrank matrix completion, where inner products play the role that distances play in rigidity theory. This observation leads to an efficient randomized algorithm for testing both local and global unique completion. Crucial to our analysis is a new matrix, which we call the completion matrix, that serves as the analogue of the rigidity matrix. Key words. Low rank matrices, missing values, rigidity theory, rigid graphs, iterative methods.
Subsampling Algorithms for Semidefinite Programming
, 2009
"... We derive a stochastic gradient algorithm for semidefinite optimization using randomization techniques. The algorithm uses subsampling to reduce the computational cost of each iteration and the subsampling ratio explicitly controls the algorithm’s granularity, i.e. the tradeoff between cost per iter ..."
Abstract

Cited by 17 (0 self)
 Add to MetaCart
We derive a stochastic gradient algorithm for semidefinite optimization using randomization techniques. The algorithm uses subsampling to reduce the computational cost of each iteration and the subsampling ratio explicitly controls the algorithm’s granularity, i.e. the tradeoff between cost per iteration and total number of iterations. Furthermore, the total computational cost is directly proportional to the complexity (i.e. rank) of the solution. We study numerical performance on some largescale problems arising in statistical learning.
Learning Social Infectivity in Sparse Lowrank Networks Using Multidimensional Hawkes Processes
"... How will the behaviors of individuals in a social network be influenced by their neighbors, the authorities and the communities in a quantitative way? Such critical and valuable knowledge is unfortunately not readily accessible and we tend to only observe its manifestation in the form of recurrent ..."
Abstract

Cited by 16 (7 self)
 Add to MetaCart
(Show Context)
How will the behaviors of individuals in a social network be influenced by their neighbors, the authorities and the communities in a quantitative way? Such critical and valuable knowledge is unfortunately not readily accessible and we tend to only observe its manifestation in the form of recurrent and timestamped events occurring at the individuals involved in the social network. It is an important yet challenging problem to infer the underlying network of social inference based on the temporal patterns of those historical events that we can observe. In this paper, we propose a convex optimization approach to discover the hidden network of social influence by modeling the recurrent events at different individuals as multidimensional Hawkes processes, emphasizing the mutualexcitation nature of the dynamics of event occurrence. Furthermore, our estimation procedure, using nuclear and!1 norm regularization simultaneously on the parameters, is able to take into account the prior knowledge of the presence of neighbor interaction, authority influence, and community coordination in the social network. To efficiently solve the resulting optimization problem, we also design an algorithm ADM4 which combines techniques of alternating direction method of multipliers and majorization minimization. We experimented with both synthetic and real world data sets, and showed that the proposed method can discover the hidden network more accurately and produce a better predictive model than several baselines.
PACBayesian Analysis of Coclustering and Beyond
"... We derive PACBayesian generalization bounds for supervised and unsupervised learning models based on clustering, such as coclustering, matrix trifactorization, graphical models, graph clustering, and pairwise clustering. 1 We begin with the analysis of coclustering, which is a widely used approa ..."
Abstract

Cited by 15 (7 self)
 Add to MetaCart
(Show Context)
We derive PACBayesian generalization bounds for supervised and unsupervised learning models based on clustering, such as coclustering, matrix trifactorization, graphical models, graph clustering, and pairwise clustering. 1 We begin with the analysis of coclustering, which is a widely used approach to the analysis of data matrices. We distinguish among two tasks in matrix data analysis: discriminative prediction of the missing entries in data matrices and estimation of the joint probability distribution of row and column variables in cooccurrence matrices. We derive PACBayesian generalization bounds for the expected outofsample performance of coclusteringbased solutions for these two tasks. The analysis yields regularization terms that were absent in the previous formulations of coclustering. The bounds suggest that the expected performance of coclustering is governed by a tradeoff between its empirical performance and the mutual information preserved by the cluster variables on row and column IDs. We derive an iterative projection algorithm for finding a local optimum of this tradeoff for discriminative prediction tasks. This algorithm achieved stateoftheart performance in the MovieLens collaborative filtering task. Our coclustering model can also be seen as matrix trifactorization and the results provide generalization bounds, regularization
Structured lowrank approximation with missing data
 SIAM J. Matrix Anal. Appl
, 2013
"... We consider lowrank approximation of affinely structured matrices with missing elements. The method proposed is based on reformulation of the problem as inner and outer optimization. The inner minimization is a singular linear leastnorm problem and admits an analytic solution. The outer problem i ..."
Abstract

Cited by 14 (10 self)
 Add to MetaCart
(Show Context)
We consider lowrank approximation of affinely structured matrices with missing elements. The method proposed is based on reformulation of the problem as inner and outer optimization. The inner minimization is a singular linear leastnorm problem and admits an analytic solution. The outer problem is a nonlinear least squares problem and is solved by local optimization methods: minimization subject to quadratic equality constraints and unconstrained minimization with regularized cost function. The method is generalized to weighted lowrank approximation with missing values and is illustrated on approximate lowrank matrix completion, system identification, and datadriven simulation problems. An extended version of the paper is a literate program, implementing the method and reproducing the presented results.
Accelerated lowrank visual recovery by random projection
 in: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR
"... Exact recovery from contaminated visual data plays an important role in various tasks. By assuming the observed data matrix as the addition of a lowrank matrix and a sparse matrix, theoretic guarantee exists under mild conditions for exact data recovery. Practically matrix nuclear norm is adopted ..."
Abstract

Cited by 13 (0 self)
 Add to MetaCart
(Show Context)
Exact recovery from contaminated visual data plays an important role in various tasks. By assuming the observed data matrix as the addition of a lowrank matrix and a sparse matrix, theoretic guarantee exists under mild conditions for exact data recovery. Practically matrix nuclear norm is adopted as a convex surrogate of the nonconvex matrix rank function to encourage lowrank property and serves as the major component of recentlyproposed Robust Principal Component Analysis (RPCA). Recent endeavors have focused on enhancing the scalability of RPCA to largescale datasets, especially mitigating the computational burden of frequent largescale Singular Value Decomposition (SVD) inherent with the nuclear norm optimization. In our proposed scheme, the nuclear norm of an auxiliary matrix is minimized instead, which is related to the original lowrank matrix by random projection. By design, the modified optimization entails SVD on matrices of much smaller scale, as compared to the original optimization problem. Theoretic analysis well justifies the proposed scheme, along with greatly reduced optimization complexity. Both qualitative and quantitative studies are provided on various computer vision benchmarks to validate its effectiveness, including facial shadow removal, surveillance background modeling and largescale image tag transduction. It is also highlighted that the proposed solution can serve as a general principal to accelerate many other nuclear norm oriented problems in numerous tasks. 1.
CONVERGENCE OF FIXEDPOINT CONTINUATION ALGORITHMS FOR MATRIX RANK MINIMIZATION
, 2009
"... Abstract. The matrix rank minimization problem has applications in many fields such as system identification, optimal control, lowdimensional embedding, etc. As this problem is NPhard in general, its convex relaxation, the nuclear norm minimization problem, is often solved instead. Recently, Ma, G ..."
Abstract

Cited by 12 (0 self)
 Add to MetaCart
(Show Context)
Abstract. The matrix rank minimization problem has applications in many fields such as system identification, optimal control, lowdimensional embedding, etc. As this problem is NPhard in general, its convex relaxation, the nuclear norm minimization problem, is often solved instead. Recently, Ma, Goldfarb and Chen proposed a fixedpoint continuation algorithm for solving the nuclear norm minimization problem [33]. By incorporating an approximate singular value decomposition technique in this algorithm, the solution to the matrix rank minimization problem is usually obtained. In this paper, we study the convergence/recoverability properties of the fixedpoint continuation algorithm and its variants for matrix rank minimization. Heuristics for determining the rank of the matrix when its true rank is not known are also proposed. Some of these algorithms are closely related to greedy algorithms in compressed sensing. Numerical results for these algorithms for solving affinely constrained matrix rank minimization problems are reported.
Clustering using maxnorm constrained optimization. ICML
, 2012
"... We suggest using the maxnorm as a convex surrogate constraint for clustering. We show how this yields a better exact cluster recovery guarantee than previously suggested nuclearnorm relaxation, and study the effectiveness of our method, and other related convex relaxations, compared to other clust ..."
Abstract

Cited by 10 (3 self)
 Add to MetaCart
We suggest using the maxnorm as a convex surrogate constraint for clustering. We show how this yields a better exact cluster recovery guarantee than previously suggested nuclearnorm relaxation, and study the effectiveness of our method, and other related convex relaxations, compared to other clustering approaches. 1
Beyond Brain Blobs: Machine Learning Classifiers as Instruments for Analyzing Functional Magnetic Resonance Imaging Data
, 1998
"... Vector Decomposition MachineEsta tese é dedicada aos meus pais Paula e José, avós Clementina e Sidónio e à minha irmã Mariana, por terem sempre confiado em mim de todas as formas possiveis, mesmo The thesis put forth in this dissertation is that machine learning classifiers can be used as instrument ..."
Abstract

Cited by 10 (1 self)
 Add to MetaCart
(Show Context)
Vector Decomposition MachineEsta tese é dedicada aos meus pais Paula e José, avós Clementina e Sidónio e à minha irmã Mariana, por terem sempre confiado em mim de todas as formas possiveis, mesmo The thesis put forth in this dissertation is that machine learning classifiers can be used as instruments for decoding variables of interest from functional magnetic resonance imaging (fMRI) data. There are two main goals in decoding: • Showing that the variable of interest can be predicted from the data in a statistically reliable manner (i.e. there’s enough information present). • Shedding light on how the data encode the information needed to predict, taking into account what the classifier used can learn and any criteria by which the data are filtered (e.g. how voxels and time points used are chosen). Chapter 2 considers the issues that arise when using traditional linear classifiers and several different voxel selection techniques to strive towards these