Results 11  20
of
45
Factor Matrix Trace Norm Minimization for LowRank Tensor Completion
"... Most existing lownrank minimization algorithms for tensor completion suffer from high computational cost due to involving multiple singular value decompositions (SVDs) at each iteration. To address this issue, we propose a novel factor matrix trace norm minimization method for tensor completion ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
(Show Context)
Most existing lownrank minimization algorithms for tensor completion suffer from high computational cost due to involving multiple singular value decompositions (SVDs) at each iteration. To address this issue, we propose a novel factor matrix trace norm minimization method for tensor completion problems. Based on the CANDECOMP/PARAFAC (CP) decomposition, we first formulate a factor matrix rank minimization model by deducing the relation between the rank of each factor matrix and the moden rank of a tensor. Then, we introduce a tractable relaxation of our rank function, which leads to a convex combination problem of much smaller scale matrix nuclear norm minimization. Finally, we develop an efficient alternating direction method of multipliers (ADMM) scheme to solve the proposed problem. Experimental results on both synthetic and realworld data validate the effectiveness of our approach. Moreover, our method is significantly faster than the stateoftheart approaches and scales well to handle large datasets. 1
BOUNDING THE EQUIVARIANT BETTI NUMBERS AND COMPUTING THE GENERALIZED EULERPOINCARÉ CHARACTERISTIC OF SYMMETRIC SEMIALGEBRAIC SETS
, 2014
"... Let R be a real closed field. The problem of obtaining tight bounds on the Betti numbers of semialgebraic subsets of R k in terms of the number and degrees of the defining polynomials has been an important problem in real algebraic geometry with the first results due to Oleĭnik and Petrovskiĭ, Tho ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
(Show Context)
Let R be a real closed field. The problem of obtaining tight bounds on the Betti numbers of semialgebraic subsets of R k in terms of the number and degrees of the defining polynomials has been an important problem in real algebraic geometry with the first results due to Oleĭnik and Petrovskiĭ, Thom and Milnor. These bounds are all exponential in the number of variables k. Motivated by several applications in real algebraic geometry, as well as in theoretical computer science, where such bounds have found applications, we consider in this paper the problem of bounding the equivariant Betti numbers of symmetric algebraic and semialgebraic subsets of R k. We obtain several asymptotically tight upper bounds. In particular, we prove that if S ⊂ R k is a semialgebraic subset defined by a finite set of s symmetric polynomials of degree at most d, then the sum of the Skequivariant Betti numbers of S with coefficients in Q is bounded by (skd) O(d). Unlike the classical bounds on the ordinary Betti numbers of real algebraic varieties and semialgebraic sets, the above bound is polynomial in k when the degrees of the defining polynomials are bounded by a constant. As an application we improve the best known bound on the ordinary Betti numbers of the projection of a compact semialgebraic set improving for any fixed degree the best previously known bound for this problem due to Gabrielov, Vorobjov and Zell. As another application of our methods we obtain polynomial time (for fixed degrees) algorithms for computing the generalized EulerPoincaré characteristic of semialgebraic sets defined by symmetric polynomials. This is in contrast to the best complexity of the known algorithms for the same problem in the nonsymmetric situation, which is singly exponential. This singly exponential complexity for the latter problem is unlikely to be improved because of hardness result (#Phardness) coming from discrete complexity theory.
Parallel Algorithms for Constrained Tensor Factorization via Alternating Direction Method of Multipliers
, 2014
"... Abstract—Tensor factorization has proven useful in a wide range of applications, from sensor array processing to communications, speech and audio signal processing, and machine learning. With few recent exceptions, all tensor factorization algorithms were originally developed for centralized, inme ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
Abstract—Tensor factorization has proven useful in a wide range of applications, from sensor array processing to communications, speech and audio signal processing, and machine learning. With few recent exceptions, all tensor factorization algorithms were originally developed for centralized, inmemory computation on a single machine; and the few that break away from this mold do not easily incorporate practically important constraints, such as nonnegativity. A new constrained tensor factorization framework is proposed in this paper, building upon the Alternating Direction Method of Multipliers (ADMoM). It is shown that this simplifies computations, bypassing the need to solve constrained optimization problems in each iteration; and it naturally leads to distributed algorithms suitable for parallel implementation. This opens the door for many emerging big dataenabled applications. The methodology is exemplified using nonnegativity as a baseline constraint, but the proposed framework can incorporate many other types of constraints. Numerical experiments are encouraging, indicating that ADMoMbased nonnegative tensor factorization (NTF) has high potential as an alternative to stateoftheart approaches. Index Terms—Tensor decomposition, PARAFACmodel, parallel algorithms.
L.H.: Distance between subspaces of different dimensions. ArXiv eprints
, 2014
"... ar ..."
(Show Context)
Tensor Spectral Clustering for Partitioning Higherorder Network Structures
"... Spectral graph theorybased methods represent an important class of tools for studying the structure of networks. Spectral methods are based on a firstorder Markov chain derived from a random walk on the graph and thus they cannot take advantage of important higherorder network substructures such ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
Spectral graph theorybased methods represent an important class of tools for studying the structure of networks. Spectral methods are based on a firstorder Markov chain derived from a random walk on the graph and thus they cannot take advantage of important higherorder network substructures such as triangles, cycles, and feedforward loops. Here we propose a Tensor Spectral Clustering (TSC) algorithm that allows for modeling higherorder network structures in a graph partitioning framework. Our TSC algorithm allows the user to specify which higherorder network structures (cycles, feedforward loops, etc.) should be preserved by the network clustering. Higherorder network structures of interest are represented using a tensor, which we then partition by developing a multilinear spectral method. Our framework can be applied to discovering layered flows in networks as well as graph anomaly detection, which we illustrate on synthetic networks. In directed networks, a higherorder structure of particular interest is the directed 3cycle, which captures feedback loops in networks. We demonstrate that our TSC algorithm produces large partitions that cut fewer directed 3cycles than standard spectral clustering algorithms.
A Statistical Model for Tensor PCA
 Neural Information Processing Systems (NIPS
, 2014
"... We consider the Principal Component Analysis problem for large tensors of arbitrary order k under a singlespike (or rankone plus noise) model. On the one hand, we use information theory, and recent results in probability theory, to establish necessary and sufficient conditions under which the prin ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
We consider the Principal Component Analysis problem for large tensors of arbitrary order k under a singlespike (or rankone plus noise) model. On the one hand, we use information theory, and recent results in probability theory, to establish necessary and sufficient conditions under which the principal component can be estimated using unbounded computational resources. It turns out that this is possible as soon as the signaltonoise ratio β becomes larger than C k log k (and in particular β can remain bounded as the problem dimensions increase). On the other hand, we analyze several polynomialtime estimation algorithms, based on tensor unfolding, power iteration and message passing ideas from graphical models. We show that, unless the signaltonoise ratio diverges in the system dimensions, none of these approaches succeeds. This is possibly related to a fundamental limitation of computationally tractable estimators for this problem. We discuss various initializations for tensor power iteration, and show that a tractable initialization based on the spectrum of the matricized tensor outperforms significantly baseline methods, statistically and computationally. Finally, we consider the case in which additional side information is available about the unknown signal. We characterize the amount of side information that allows the iterative algorithms to converge to a good estimate. 1
Online and DifferentiallyPrivate Tensor Decomposition
"... Abstract Tensor decomposition is an important tool for big data analysis. In this paper, we resolve many of the key algorithmic questions regarding robustness, memory efficiency, and differential privacy of tensor decomposition. We propose simple variants of the tensor power method which enjoy thes ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
Abstract Tensor decomposition is an important tool for big data analysis. In this paper, we resolve many of the key algorithmic questions regarding robustness, memory efficiency, and differential privacy of tensor decomposition. We propose simple variants of the tensor power method which enjoy these strong properties. We present the first guarantees for online tensor power method which has a linear memory requirement. Moreover, we present a noise calibrated tensor power method with efficient privacy guarantees. At the heart of all these guarantees lies a careful perturbation analysis derived in this paper which improves up on the existing results significantly.