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Author: Király, Franz J. × Type: Text × WGL Institute: MFO ×

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Now showing 1 - 5 of 5
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    Algebraic geometric comparison of probability distributions
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2011) Király, Franz J.; von Bünau, Paul; Meinecke, Frank C.; Blythe, Duncan A.J.; Müller, Klaus-Robert
    We propose a novel algebraic framework for treating probability distributions represented by their cumulants such as the mean and covariance matrix. As an example, we consider the unsupervised learning problem of finding the subspace on which several probability distributions agree. Instead of minimizing an objective function involving the estimated cumulants, we show that by treating the cumulants as elements of the polynomial ring we can directly solve the problem, at a lower computational cost and with higher accuracy. Moreover, the algebraic viewpoint on probability distributions allows us to invoke the theory of Algebraic Geometry, which we demonstrate in a compact proof for an identifiability criterion.
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    The algebraic combinatorial approach for low-rank matrix completion
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2013) Király, Franz J.; Theran, Louis; Tomioka, Ryota; Uno, Takeaki
    We propose an algebraic combinatorial framework for the problem of completing partially observed low-rank matrices. We show that the intrinsic properties of the problem, including which entries can be reconstructed, and the degrees of freedom in the reconstruction, do not depend on the values of the observed entries, but only on their position. We associate combinatorial and algebraic objects, differentials and matroids, which are descriptors of the particular reconstruction task, to the set of observed entries, and apply them to obtain reconstruction bounds. We show how similar techniques can be used to obtain reconstruction bounds on general compressed sensing problems with algebraic compression constraints. Using the new theory, we develop several algorithms for low-rank matrix completion, which allow to determine which set of entries can be potentially reconstructed and which not, and how, and we present algorithms which apply algebraic combinatorial methods in order to reconstruct the missing entries.
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    Prediction and quantification of individual athletic performance
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2015) Blythe, Duncan A.J.; Király, Franz J.
    We present a novel, quantitative view on the human athletic performance of individuals. We obtain a predictor for athletic running performances, a parsimonious model, and a training state summary consisting of three numbers, by application of modern validation techniques and recent advances in machine learning to the thepowerof10 database of British athletes’ performances (164,746 individuals, 1,417,432 performances). Our predictor achieves a low average prediction error (out-of-sample), e.g., 3.6 min on elite Marathon performances, and a lower error than the state-of-the-art in performance prediction (30% improvement, RMSE). We are also the first to report on a systematic comparison of predictors for athletic running performance. Our model has three parameters per athlete, and three components which are the same for all athletes. The first component of the model corresponds to a power law with exponent dependent on the athlete which achieves a better goodness-of-fit than known power laws in athletics. Many documented phenomena in quantitative sports science, such as the form of scoring tables, the success of existing prediction methods including Riegel’s formula, the Purdy points scheme, the power law for world records performances and the broken power law for world record speeds may be explained on the basis of our findings in a unified way. We provide strong evidence that the three parameters per athlete are related to physiological and/or behavioural parameters, such as training state, event specialization and age, which allows us to derive novel physiological hypotheses relating to athletic performance. We conjecture on this basis that our findings will be vital in exercise physiology, race planning, the study of aging and training regime design.
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    Obtaining error-minimizing estimates and universal entry-wise error bounds for low-rank matrix completion
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2013) Király, Franz J.; Theran, Louis
    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 rank-one matrices, the new algorithm is fast, admits a highly-parallel implementation, and produces an error minimizing estimate that is qualitatively close to our theoretical and the state-of-the-art Nuclear Norm and OptSpace methods.
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    Algebraic matroids with graph symmetry
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2014) Király, Franz J.; Rosen, Zvi; Theran, Louis
    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, low-rank 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.