Abstract not found

High-dimensional modeling is becoming ubiquitous across the sciences and engineering because of advances in sensor technology and storage technology. Computationally-oriented researchers no longer have to avoid what were once intractably large, tensor-structured data sets. The current NSF promotion of “computational thinking” is timely: we need a focused international effort to oversee the transition from matrix-based to tensor-based computational thinking. The successful problem-solving tools provided by the numerical linear algebra community need to be broadened and generalized. However, tensor-based research is not just matrix-based research with additional subscripts. Tensors are data objects in their own right and there is much to learn about their geometry and their connections to statistics and operator theory. This requires full participation of researchers from engineering, the natural sciences, and the information sciences, together with statisticians, mathematicians, numerical analysts, and software/language designers. Representatives from these disciplines participated in the Workshop. We believe that the NSF can help ensure the vitality of “big N” engineering and science by systematically supporting research in tensor-based computation and modeling.

Non-negative matrix factorization (NMF) has become a popular tool for exploratory analysis due to its part based easy interpretable representation. Sparseness is commonly invoked in NMF (SNMF) by regularizing by the l1 − norm both to alleviate the non-uniqueness of the NMF representation as well as promote sparse (i.e. part based) representations. While sparseness can prune excess components thereby potentially also establish the number of components it is an open problem what constitutes the adequate degree of sparseness, i.e. how to tune the pruning. In a hierarchical Bayesian framework SNMF corresponds to imposing an exponential prior while the regularization strength can be expressed in terms of the hyper-parameters of these priors. Thus, within the Bayesian modelling framework Automatic Relevance Determination (ARD) can learn these pruning strengths from data. We demonstrate on three benchmark NMF data how the proposed ARD framework can be used to tune the pruning thereby also estimate the NMF model order. 1.

This paper introduces a novel approach for learning to rank (LETOR) based on the notion of monotone retargeting. It involves minimizing a divergence between all monotonic increasing transformations of the training scores and a parameterized prediction function. The minimization is over the transformations as well as over the parameters. MR is applied to Bregman divergences, a large class of “distance like ” functions that were recently shown to be the unique class that is statistically consistent with the normalized discounted gain (NDCG) criterion [19]. The algorithm uses alternating projection style updates, in which one set of simultaneous projections can be computed independent of the Bregman divergence and the other reduces to parameter estimation of a generalized linear model. This results in an easily implemented, efficiently parallelizable algorithm for the LETOR task that enjoys global optimum guarantees under mild conditions. We present empirical results on benchmark datasets showing that this approach can outperform the state of the art NDCG consistent techniques. 1