4,497 research outputs found
Extreme statistics of non-intersecting Brownian paths
We consider finite collections of non-intersecting Brownian paths on the
line and on the half-line with both absorbing and reflecting boundary
conditions (corresponding to Brownian excursions and reflected Brownian
motions) and compute in each case the joint distribution of the maximal height
of the top path and the location at which this maximum is attained. The
resulting formulas are analogous to the ones obtained in [MFQR13] for the joint
distribution of and , where is the
Airy process, and we use them to show that in the three cases the joint
distribution converges, as , to the joint distribution of
and . In the case of non-intersecting Brownian
bridges on the line, we also establish small deviation inequalities for the
argmax which match the tail behavior of . Our proofs are based on
the method introduced in [CQR13,BCR15] for obtaining formulas for the
probability that the top line of these line ensembles stays below a given
curve, which are given in terms of the Fredholm determinant of certain
"path-integral" kernels.Comment: Minor corrections, improved exposition. To appear in Electron. J.
Proba
Fast Parallel Randomized Algorithm for Nonnegative Matrix Factorization with KL Divergence for Large Sparse Datasets
Nonnegative Matrix Factorization (NMF) with Kullback-Leibler Divergence
(NMF-KL) is one of the most significant NMF problems and equivalent to
Probabilistic Latent Semantic Indexing (PLSI), which has been successfully
applied in many applications. For sparse count data, a Poisson distribution and
KL divergence provide sparse models and sparse representation, which describe
the random variation better than a normal distribution and Frobenius norm.
Specially, sparse models provide more concise understanding of the appearance
of attributes over latent components, while sparse representation provides
concise interpretability of the contribution of latent components over
instances. However, minimizing NMF with KL divergence is much more difficult
than minimizing NMF with Frobenius norm; and sparse models, sparse
representation and fast algorithms for large sparse datasets are still
challenges for NMF with KL divergence. In this paper, we propose a fast
parallel randomized coordinate descent algorithm having fast convergence for
large sparse datasets to archive sparse models and sparse representation. The
proposed algorithm's experimental results overperform the current studies' ones
in this problem
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