Computing the spark: mixed-integer programming for the (vector) matroid girth problem

Tillmann, Andreas M.

doi:10.1007/s10589-019-00114-9

Cited by 10 publications

(18 citation statements)

References 67 publications

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“…In matroid terminology, a full spark frame corresponds to the uniform matroid of the correct parameters. Calculating the spark of a matrix is in general NP Hard , although there are special cases which can be solved using exact (mixed‐)integer programming models and linear programming heuristics . In what follows, we will calculate the spark of an infinite class of frames and for an infinite subclass of these frames, we determine all of the subsets of

spark Φ

vectors which are linearly dependent.…”

Section: Gabor–steiner Equiangular Tight Framesmentioning

confidence: 99%

Optimal arrangements of classical and quantum states with limited purity

Bodmann

King

2019

J. London Math. Soc.

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We consider sets of trace‐normalized non‐negative operators in Hilbert–Schmidt balls that maximize their mutual Hilbert–Schmidt distance; these are optimal arrangements in the sets of purity‐limited classical or quantum states on a finite‐dimensional Hilbert space. Classical states are understood to be represented by diagonal matrices, with the diagonal entries forming a probability vector. We also introduce the concept of spectrahedron arrangements which provides a unified framework for classical and quantum arrangements and the flexibility to define new types of optimal packings. Continuing a prior work, we combine combinatorial structures and line packings associated with frames to arrive at optimal arrangements of higher rank quantum states. One new construction that is presented involves generating an optimal arrangement we call a Gabor–Steiner equiangular tight frame as the orbit of a projective representation of the Weyl–Heisenberg group over any finite abelian group. The minimal sets of linearly dependent vectors, the so‐called binder, of the Gabor–Steiner equiangular tight frames are then characterized; under certain conditions, these form combinatorial block designs and in one case generate a new class of block designs. The projections onto the span of minimal linearly dependent sets in the Gabor–Steiner equiangular tight frame are then used to generate further optimal spectrahedron arrangements.

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spark Φ

vectors which are linearly dependent.…”

Section: Gabor–steiner Equiangular Tight Framesmentioning

confidence: 99%

Optimal arrangements of classical and quantum states with limited purity

Bodmann

King

2019

J. London Math. Soc.

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“…, and generalized variants (with other norms or sparsity-inducing penalty functions) of some such problems [99], as well as related problems such as matroid (co-)girth and (co-)spark [211,318,312,308], MinULR/MaxFS [6,7], and matrix sparsification [249,306,178].…”

Section: Exact Models and Solution Methodsmentioning

confidence: 99%

“…In light of the earlier discussion, this is polynomially equivalent to spark(D), where D ∈ R (n−m)×n is such that A is a basis for its nullspace (cf. [308,Lemma 3.1]). In particular, a solution v to min{ A v 0 : v = 0} can be retrieved from a solution x to spark(D) as the unique solution to A v = x, i.e., v = AA −1 Ax (recall that A ∈ R n×m with full column-rank m < n).…”

Section: Miscellaneous Related Problems and Extensionsmentioning

confidence: 99%

“…Thus, every value M > 0 works in a big-M cardinality modeling approach. Spark computation is discussed in detail in [308]; in particular, a formulation with M = 1 was employed and-utilizing additional auxiliary binary variables to model the nontriviality constraint x = 0-the resulting MIP was given as…”

Section: Cardinality Minimizationmentioning

confidence: 99%

“…A fundamental question from the signal processing perspective is that of uniqueness of the recovery problem solution. In particular, for the basic reconstruction problem 0 -min(Ax = b), uniqueness can be characterized by means of a matrix parameter called the spark (see [176,127,308]), which is defined as the smallest number of linearly dependent columns, i.e., ( 0 -min(Ax = 0, x = 0)) spark(A) = min{ x 0 : Ax = 0, x = 0}.…”

mentioning

confidence: 99%

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Cardinality Minimization, Constraints, and Regularization: A Survey

Tillmann¹,

Bienstock²,

Lodi³

et al. 2021

Preprint

Self Cite

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We survey optimization problems that involve the cardinality of variable vectors in constraints or the objective function. We provide a unified viewpoint on the general problem classes and models, and give concrete examples from diverse application fields such as signal and image processing, portfolio selection, or machine learning. The paper discusses general-purpose modeling techniques and broadly applicable as well as problem-specific exact and heuristic solution approaches. While our perspective is that of mathematical optimization, a main goal of this work is to reach out to and build bridges between the different communities in which cardinality optimization problems are frequently encountered. In particular, we highlight that modern mixed-integer programming, which is often regarded as impractical due to commonly unsatisfactory behavior of black-box solvers applied to generic problem formulations, can in fact produce provably high-quality or even optimal solutions for cardinality optimization problems, even in large-scale real-world settings. Achieving such performance typically draws on the merits of problem-specific knowledge that may stem from different fields of application and, e.g., shed light on structural properties of a model or its solutions, or lead to the development of efficient heuristics; we also provide some illustrative examples.

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Rough set approximations based on a matroidal structure over three sets

Wang

Mao²,

Liu³

et al. 2022

Appl Intell

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Pawlak’s classical model of rough set approximations provides an efficient tool for extracting information exactly by employing available knowledge (i.e., known knowledge) in an information system, since many problems in rough set theory are NP-hard and their solution process is therefore greedy and approximate. Many extensions of Pawlak’s classical model have been proposed in recent years. Most of them are considered over one or two sets, that is, one- or two-dimensional space or one- or two-dimensional data. Aided by relation-based rough set models, a few of these extensions are considered over three sets. However, the real world is in three-dimensional space. Therefore, it is necessary to solve these problems with other models, such as covering rough set models. For this purpose, we propose the TP-matroid—a matroidal structure over three sets. Employing the family of feasible sets of a TP-matroid as the available knowledge, a pair of rough set approximations—lower and upper approximations—is provided. In addition, for an information system defined over three sets, assisted by formal concept analysis, we establish a pair of rough set approximations. Furthermore, two TP-matroids are established based on the above pair of rough set approximations. The integration between the two pairs of rough set approximations presented here is discussed. The results show that for an information system in three-dimensional space, the rough set approximations provided here can effectively explore unknown knowledge by using available knowledge based on the family of feasible sets of a TP-matroid.

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Computing the spark: mixed-integer programming for the (vector) matroid girth problem

Cited by 10 publications

References 67 publications

Optimal arrangements of classical and quantum states with limited purity

Optimal arrangements of classical and quantum states with limited purity

Cardinality Minimization, Constraints, and Regularization: A Survey

Rough set approximations based on a matroidal structure over three sets

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