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Algebraic matroids with graph symmetry
, 2013
"... This paper studies the properties of two kinds of matroids: (a) algebraic matroids and (b) finite and infinite matroids whose ground set have some canonical symmetry, for example row and column symmetry and transposition symmetry. For (a) algebraic matroids, we expose cryptomorphisms making them acc ..."
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Cited by 5 (4 self)
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This paper studies the properties of two kinds of matroids: (a) algebraic matroids and (b) finite and infinite matroids whose ground set have some canonical symmetry, for example row and column symmetry and transposition symmetry. For (a) algebraic matroids, we expose cryptomorphisms making them accessible to techniques from commutative algebra. This allows us to introduce for each circuit in an algebraic matroid an invariant called circuit polynomial, generalizing the minimal polynomial in classical Galois theory, and studying the matroid structure with multivariate methods. For (b) matroids with symmetries we introduce combinatorial invariants capturing structural properties of the rank function and its limit behavior, and obtain proofs which are purely combinatorial and do not assume algebraicity of the matroid; these imply and generalize known results in some specific cases where the matroid is also algebraic. These results are motivated by, and readily applicable to framework rigidity, lowrank matrix completion and determinantal varieties, which lie in the intersection of (a) and (b) where additional results can be derived. We study the corresponding matroids and their associated invariants, and for selected cases, we characterize the matroidal structure and the circuit polynomials completely.
Computing algebraic matroids
, 2014
"... Abstract. An affine variety induces the structure of an algebraic matroid on the set of coordinates of the ambient space. The matroid has two natural decorations: a circuit polynomial attached to each circuit, and the degree of the projection map to each base, called the base degree. Decorated alge ..."
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Abstract. An affine variety induces the structure of an algebraic matroid on the set of coordinates of the ambient space. The matroid has two natural decorations: a circuit polynomial attached to each circuit, and the degree of the projection map to each base, called the base degree. Decorated algebraic matroids can be computed via symbolic computation using Gröbner bases, or through linear algebra in the space of differentials (with decorations calculated using numerical algebraic geometry). Both algorithms are developed here. Failure of the second algorithm occurs on a subvariety called the nonmatroidal or NMlocus. Decorated algebraic matroids have widespread relevance anywhere that coordinates have combinatorial significance. Examples are computed from applied algebra, in algebraic statistics and chemical reaction network theory, as well as more theoretical examples from algebraic geometry and matroid theory. 1.
A Characterization of Deterministic Sampling Patterns for LowRank Matrix Completion
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Matrix Completion with Queries
"... In many applications, e.g., recommender systems and traffic monitoring, the data comes in the form of a matrix that is only partially observed and low rank. A fundamental dataanalysis task for these datasets is matrix completion, where the goal is to accurately infer the entries missing from the ma ..."
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In many applications, e.g., recommender systems and traffic monitoring, the data comes in the form of a matrix that is only partially observed and low rank. A fundamental dataanalysis task for these datasets is matrix completion, where the goal is to accurately infer the entries missing from the matrix. Even when the data satisfies the lowrank assumption, classical matrixcompletion methods may output completions with significant error – in that the reconstructed matrix differs significantly from the true underlying matrix. Often, this is due to the fact that the information contained in the observed entries is insufficient. In this work, we address this problem by proposing an active version of matrix completion, where queries can be made to the true underlying matrix. Subsequently, we design Order&Extend, which is the first algorithm to unify a matrixcompletion approach and a querying strategy into a single algorithm. Order&Extend is able identify and alleviate insufficient information by judiciously querying a small number of additional entries. In an extensive experimental evaluation on realworld datasets, we demonstrate that our algorithm is efficient and is able to accurately reconstruct the true matrix while asking only a small number of queries.
RESEARCH STATEMENT ALGEBRAIC MATROIDS: STRUCTURE AND APPLICATIONS
"... Algebraic matroids are combinatorial objects that can be extracted from geometric problems, describing the independence structure on the coordinates. In this proposal, algebraic matroids are used to analyze applied problems, and their structure is explored. We give historical background to this topi ..."
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Algebraic matroids are combinatorial objects that can be extracted from geometric problems, describing the independence structure on the coordinates. In this proposal, algebraic matroids are used to analyze applied problems, and their structure is explored. We give historical background to this topic, then set forth four projects. 1.
Coherence and Sufficient Sampling Densities for Reconstruction in Compressed Sensing
"... We give a new, very general, formulation of the compressed sensing problem in terms of coordinate projections of an analytic variety, and derive sufficient sampling rates for signal reconstruction. Our bounds are linear in the coherence of the signal space, a geometric parameter independent of the ..."
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We give a new, very general, formulation of the compressed sensing problem in terms of coordinate projections of an analytic variety, and derive sufficient sampling rates for signal reconstruction. Our bounds are linear in the coherence of the signal space, a geometric parameter independent of the specific signal and measurement, and logarithmic in the ambient dimension where the signal is presented. We exemplify our approach by deriving sufficient sampling densities for lowrank matrix completion and distance matrix completion which are independent of the true matrix. 1.
Obtaining ErrorMinimizing Estimates and Universal EntryWise Error Bounds for LowRank Matrix Completion
"... We propose a general framework for reconstructing and denoising single entries of incomplete and noisy entries. We describe: effective algorithms for deciding if and entry can be reconstructed and, if so, for reconstructing and denoising it; and a priori bounds on the error of each entry, individua ..."
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We propose a general framework for reconstructing and denoising single entries of incomplete and noisy entries. We describe: effective algorithms for deciding if and entry can be reconstructed and, if so, for reconstructing and denoising it; and a priori bounds on the error of each entry, individually. In the noiseless case our algorithm is exact. For rankone matrices, the new algorithm is fast, admits a highlyparallel implementation, and produces an error minimizing estimate that is qualitatively close to our theoretical and the stateoftheare Nuclear Norm and OptSpace methods. 1.
AlgebraicCombinatorial Methods for LowRank Matrix Completion with Application to Athletic Performance Prediction
, 2014
"... This paper presents novel algorithms which exploit the intrinsic algebraic and combinatorial structure of the matrix completion task for estimating missing entries in the general low rank setting. For positive data, we achieve results outperforming the state of the art nuclear norm, both in accura ..."
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This paper presents novel algorithms which exploit the intrinsic algebraic and combinatorial structure of the matrix completion task for estimating missing entries in the general low rank setting. For positive data, we achieve results outperforming the state of the art nuclear norm, both in accuracy and computational efficiency, in simulations and in the task of predicting athletic performance from partially observed data. 1