2,005 research outputs found

    On the Volume of Isolated Singularities

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    We give an equivalent definition of the local volume of an isolated singularity Vol_{BdFF}(X,0) given in [BdFF12] in the Q-Gorenstein case and we generalize it to the non-Q-Gorenstein case. We prove that there is a positive lower bound depending only on the dimension for the non-zero local volume of an isolated singularity if X is Gorenstein. We also give a non-Q-Gorenstein example with Vol_{BdFF}(X,0)=0, which does not allow a boundary \Delta such that the pair (X,\Delta) is log canonical.Comment: 12 pages. Final version. To appear in Compos. Mat

    Stochastic Primal-Dual Coordinate Method for Regularized Empirical Risk Minimization

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    We consider a generic convex optimization problem associated with regularized empirical risk minimization of linear predictors. The problem structure allows us to reformulate it as a convex-concave saddle point problem. We propose a stochastic primal-dual coordinate (SPDC) method, which alternates between maximizing over a randomly chosen dual variable and minimizing over the primal variable. An extrapolation step on the primal variable is performed to obtain accelerated convergence rate. We also develop a mini-batch version of the SPDC method which facilitates parallel computing, and an extension with weighted sampling probabilities on the dual variables, which has a better complexity than uniform sampling on unnormalized data. Both theoretically and empirically, we show that the SPDC method has comparable or better performance than several state-of-the-art optimization methods

    Weakly Supervised Audio Source Separation via Spectrum Energy Preserved Wasserstein Learning

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    Separating audio mixtures into individual instrument tracks has been a long standing challenging task. We introduce a novel weakly supervised audio source separation approach based on deep adversarial learning. Specifically, our loss function adopts the Wasserstein distance which directly measures the distribution distance between the separated sources and the real sources for each individual source. Moreover, a global regularization term is added to fulfill the spectrum energy preservation property regardless separation. Unlike state-of-the-art weakly supervised models which often involve deliberately devised constraints or careful model selection, our approach need little prior model specification on the data, and can be straightforwardly learned in an end-to-end fashion. We show that the proposed method performs competitively on public benchmark against state-of-the-art weakly supervised methods
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