Results 1  10
of
33
Semidefinite characterization and computation of zerodimensional real radical ideals
, 2007
"... real radical ideals ..."
Polar polytopes and recovery of sparse representations
, 2005
"... Suppose we have a signal y which we wish to represent using a linear combination of a number of basis atoms ai, y = � i xiai = Ax. The problem of finding the minimum ℓ0 norm representation for y is a hard problem. The Basis Pursuit (BP) approach proposes to find the minimum ℓ1 norm representation in ..."
Abstract

Cited by 22 (6 self)
 Add to MetaCart
(Show Context)
Suppose we have a signal y which we wish to represent using a linear combination of a number of basis atoms ai, y = � i xiai = Ax. The problem of finding the minimum ℓ0 norm representation for y is a hard problem. The Basis Pursuit (BP) approach proposes to find the minimum ℓ1 norm representation instead, which corresponds to a linear program (LP) that can be solved using modern LP techniques, and several recent authors have given conditions for the BP (minimum ℓ1 norm) and sparse (minimum ℓ0 solutions) representations to be identical. In this paper, we explore this sparse representation problem using the geometry of convex polytopes, as recently introduced into the field by Donoho. By considering the dual LP we find that the socalled polar polytope P ∗ of the centrallysymmetric polytope P whose vertices are the atom pairs ±ai is particularly helpful in providing us with geometrical insight into optimality conditions given by Fuchs and Tropp for nonunitnorm atom sets. In exploring this geometry we are able to tighten some of these earlier results, showing for example that the Fuchs condition is both necessary and sufficient for ℓ1uniqueoptimality, and that there are situations where Orthogonal Matching Pursuit (OMP) can eventually find all ℓ1uniqueoptimal solutions with m nonzeros even if ERC fails for m, if allowed to run for more than m steps.
Tag Completion for Image Retrieval
"... Abstract—Many social image search engines are based on keyword/tag matching. This is because tag based image retrieval (TBIR) is not only efficient but also effective. The performance of TBIR is highly dependent on the availability and quality of manual tags. Recent studies have shown that manual ta ..."
Abstract

Cited by 18 (1 self)
 Add to MetaCart
Abstract—Many social image search engines are based on keyword/tag matching. This is because tag based image retrieval (TBIR) is not only efficient but also effective. The performance of TBIR is highly dependent on the availability and quality of manual tags. Recent studies have shown that manual tags are often unreliable and inconsistent. In addition, since many users tend to choose general and ambiguous tags in order to minimize their efforts in choosing appropriate words, tags that are specific to the visual content of images tend to be missing or noisy, leading to a limited performance of TBIR. To address this challenge, we study the problem of tag completion where the goal is to automatically fill in the missing tags as well as correct noisy tags for given images. We represent the imagetag relation by a tag matrix, and search for the optimal tag matrix consistent with both the observed tags and the visual similarity. We propose a new algorithm for solving this optimization problem. Extensive empirical studies show that the proposed algorithm is significantly more effective than the stateoftheart algorithms. Our studies also verify that the proposed algorithm is computationally efficient and scales well to large databases. Index Terms—tag completion, matrix completion, tagbased image retrieval, image annotation, image retrieval, metric learning. 1
Linear Algebra Algorithms as Dynamical Systems
, 2008
"... Any logical procedure that is used to reason or infer either deductively or inductively so as to draw conclusions or make decisions can be called, in a broad sense, a realization process. A realization process usually assumes the recursive form that one state gets developed into another state by fol ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
Any logical procedure that is used to reason or infer either deductively or inductively so as to draw conclusions or make decisions can be called, in a broad sense, a realization process. A realization process usually assumes the recursive form that one state gets developed into another state by following a certain specific rule. Such an action is qualified as what is generally known as a dynamical system. In mathematics, especially for existence questions, a realization process often appears in the form of an iterative procedure or a differential equation. For years researchers have taken great effort to describe, analyze, and modify realization processes for various applications. The thrust in this exposition is to exploit the notion of dynamical systems as a special realization process for problems arising from the field of linear algebra. Several differential equations whose solutions evolve in some submanifolds of matrices are cast in fairly general frameworks of which special cases have been found to afford unified and fundamental insights into the structure and behavior of existing discrete methods and, now and then, suggest new and
Hingeloss Markov random fields and probabilistic soft logic
, 2015
"... A fundamental challenge in developing highimpact machine learning technologies is balancing the ability to model rich, structured domains with the ability to scale to big data. Many important problem areas are both richly structured and large scale, from social and biological networks, to knowledge ..."
Abstract

Cited by 6 (4 self)
 Add to MetaCart
A fundamental challenge in developing highimpact machine learning technologies is balancing the ability to model rich, structured domains with the ability to scale to big data. Many important problem areas are both richly structured and large scale, from social and biological networks, to knowledge graphs and the Web, to images, video, and natural language. In this paper, we introduce two new formalisms for modeling structured data, distinguished from previous approaches by their ability to both capture rich structure and scale to big data. The first, hingeloss Markov random fields (HLMRFs), is a new kind of probabilistic graphical model that generalizes different approaches to convex inference. We unite three approaches from the randomized algorithms, probabilistic graphical models, and fuzzy logic communities, showing that all three lead to the same inference objective. We then derive HLMRFs by generalizing this unified objective. The second new formalism, probabilistic soft logic (PSL), is a probabilistic programming language that makes HLMRFs easy to define using a syntax based on firstorder logic. We next introduce an algorithm for inferring mostprobable variable assignments (MAP inference) that is much more scalable than generalpurpose convex optimization software, because it uses message passing to take advantage of sparse dependency structures. We then show how to learn the parameters of HLMRFs. The learned HLMRFs are as accurate as analogous discrete models, but much more scalable. Together, these algorithms enable HLMRFs and PSL to model rich, structured data at scales not previously possible.
ON WARM STARTS FOR INTERIOR METHODS
, 2005
"... An appealing feature of interior methods for linear programming is that the number of iterations required to solve a problem tends to be relatively insensitive to the choice of initial point. This feature has the drawback that it is difficult to design interior methods that efficiently utilize infor ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
(Show Context)
An appealing feature of interior methods for linear programming is that the number of iterations required to solve a problem tends to be relatively insensitive to the choice of initial point. This feature has the drawback that it is difficult to design interior methods that efficiently utilize information from an optimal solution to a “nearby ” problem. We discuss this feature in the context of general nonlinear programming and specialize to linear programming. We demonstrate that warm start for a particular nonlinear programming problem, given a nearoptimal solution for a “nearby ” problem, is closely related to an SQP method applied to an equalityconstrained problem. These results are further refined for the case of linear programming.
Numerical experiments with universal barrier functions for cones of chebyshev systems
 Computational Optimization And Applications (To Appear
, 1983
"... Abstract. Based on previous explicit computations of universal barrier functions, we describe numerical experiments for solving certain classes of convex optimization problems. The comparison is given of the performance of the classical affinescaling algorithm with the similar algorithm built upon ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
Abstract. Based on previous explicit computations of universal barrier functions, we describe numerical experiments for solving certain classes of convex optimization problems. The comparison is given of the performance of the classical affinescaling algorithm with the similar algorithm built upon the universal barrier function. Key words. Affinescaling algorithm, approximations of universal barrier functions AMS subject classifications. 90C51,90C25,90C24 1. Introduction. By
Shakedown Analysis Combined With the Problem of Heat Conduction
"... This paper deals with the computation of shakedown loads of engineering structures subjected to varying loads. In particular, we focus on thermal loading and the resulting heat conduction problem in combination with shakedown analysis. The analysis is based on the lower bound shakedown theorem by M ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
This paper deals with the computation of shakedown loads of engineering structures subjected to varying loads. In particular, we focus on thermal loading and the resulting heat conduction problem in combination with shakedown analysis. The analysis is based on the lower bound shakedown theorem by Melan. The calculation is carried out by use of an interiorpoint algorithm. Emphasis is placed on the presentation of theoretical derivations, whereas numerical aspects are out of scope. The methodology is illustrated by application to a simplified model of a tube sheet in heat exchangers.