Matrix permanent inequalities for approximating joint assignment matrices in tracking systems

Uhlmann, Jeffrey

doi:10.1016/j.jfranklin.2004.07.003

Cited by 9 publications

(5 citation statements)

References 14 publications

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“…This result suggests several special properties that might be conjectured to hold for some values n > 2 . The most tantalizing relates to the fact that the denominator of the coefficient is the permanent of G. In fact, the overall result represents the joint assignment matrix (JAM) [9] of G, which is the solution 3 to an important combinatorial problem arising in multiple-target tracking and related applications [32,33]. Evaluation of the JAM for a general n × n matrix is believed to be computationally intractable based on the #P-Hard complexity of evaluating the permanent of a matrix [6,9,10,22], so it should not be surprising that JAM-equivalence does not hold for n > 2.…”

Section: The Rga and Its Inversementioning

confidence: 99%

On Radically Expanding the Landscape of Potential Applications for Automated-Proof Methods

Uhlmann

Wang

2021

SN COMPUT. SCI.

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In this paper, we examine the potential of optimization-based computer-assisted proof methods to be applied much more widely than commonly recognized by engineers and computer scientists. More specifically, we contend that there are vast opportunities to derive valuable mathematical results and properties that may be narrow in scope, such as in highly specialized engineering control applications, that are presently overlooked, because they have characteristics atypical of those that are conventionally studied in the areas of pure and applied mathematics. As a concrete example, we demonstrate use of sumof-squares (SOS) optimization for certifying polynomial nonnegativity as a part of a proposed dimension-pinning strategy to prove that the inverse of the relative gain array (RGA) of a d-dimensional positive-definite matrix is doubly stochastic for d ≤ 4 . However, it is not specifically this result and solution method that are of principal interest in this paper but rather how they illustrate the relevance of optimization-based proof techniques to engineering system design more broadly. We believe that our paper is the first to explicitly emphasize the fundamental distinction between methods that can be applied to prove results/properties over a fixed number of dimensions versus those that hold generally. The latter class of problems is the conventional domain of mathematicians, but the former is what we propose to be a fertile and largely unrecognized class of problems that are amenable to automated-proof technologies, e.g., as we demonstrate using our novel dimensionpinning approach.

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Section: The Rga and Its Inversementioning

confidence: 99%

On Radically Expanding the Landscape of Potential Applications for Automated-Proof Methods

Uhlmann

Wang

2021

SN COMPUT. SCI.

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“…Stochastic approximation methods running in polynomial time [e.g. Jerrum et al, 2004, Kuck et al, 2019 and variational bounds [see Uhlmann, 2004, and the references therein] are also available. Given that (17) must be evaluated for nm 2 values of (i, k, l), and accounting for the computation of the matrices Press et al, 2007, Chap.…”

Section: Gaussian Mixture Approachmentioning

confidence: 99%

Scalable Feature Matching Across Large Data Collections

Degras¹

2021

Preprint

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This paper is concerned with matching feature vectors in a one-to-one fashion across large collections of datasets. Formulating this task as a multidimensional assignment problem with decomposable costs (MDADC), we develop extremely fast algorithms with time complexity linear in the number n of datasets and space complexity a small fraction of the data size. These remarkable properties hinge on using the squared Euclidean distance as dissimilarity function, which can reduce n 2 matching problems between pairs of datasets to n problems and enable calculating assignment costs on the fly. To our knowledge, no other method applicable to the MDADC possesses these linear scaling and low-storage properties necessary to large-scale applications. In numerical experiments, the novel algorithms outperform competing methods and show excellent computational and optimization performances. An application of feature matching to a large neuroimaging database is presented. The algorithms of this paper are implemented in the R package matchFeat available at github.com/ddegras/matchFeat.

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“…This result suggests several special properties that might be conjectured to hold for some values n > 2. The most tantalizing relates to the fact that the denominator of the coefficient is the permanent of G. In fact, the overall result represents the joint assignment matrix (JAM) [5] of G, which is the solution 3 to an important combinatorial problem arising in multiple-target tracking and related applications [27,28]. Evaluation of the JAM for a general n×n matrix is believed to be computationally intractable based on the #P-Hard complexity of evaluating the permanent of a matrix [5,3,17,6], so it should not be surprising that JAM-equivalence does not hold for n > 2.…”

Section: The Rga and Its Inversementioning

confidence: 99%

On Radically Expanding the Landscape of Potential Applications for Automated Proof Methods

Uhlmann,

Wang

2019

Preprint

View full text Add to dashboard Cite

In this paper we examine the potential of computer-assisted proof methods to be applied much more broadly than commonly recognized. More specifically, we contend that there are vast opportunities to derive useful mathematical results and properties that are extremely narrow in scope, and of practical relevance only to highly-specialized engineering applications, that are presently overlooked because they have characteristics atypical of those that are conventionally pursued in the areas of pure and applied mathematics. As a concrete example, we demonstrate use of automated methods for certifying polynomial nonnegativity as a part of a dimension-pinning strategy to prove that the inverse of the relative gain array (RGA) of a d-dimensional positive-definite matrix is doublystochastic for d ≤ 4.

show abstract

Matrix permanent inequalities for approximating joint assignment matrices in tracking systems

Cited by 9 publications

References 14 publications

On Radically Expanding the Landscape of Potential Applications for Automated-Proof Methods

On Radically Expanding the Landscape of Potential Applications for Automated-Proof Methods

Scalable Feature Matching Across Large Data Collections

On Radically Expanding the Landscape of Potential Applications for Automated Proof Methods

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