We analyse the matrix factorization problem. Given a noisy measurement of a product of two matrices, the problem is to estimate back the original matrices. It arises in many applications such as dictionary learning, blind matrix calibration, sparse principal component analysis, blind source separation, low rank matrix completion, robust principal component analysis or factor analysis. It is also important in machine learning: unsupervised representation learning can often be studied through matrix factorization. We use the tools of statistical mechanics - the cavity and replica methods - to analyze the achievability and computational tractability of the inference problems in the setting of Bayes-optimal inference, which amounts to assuming that the two matrices have random independent elements generated from some known distribution, and this information is available to the inference algorithm. In this setting, we compute the minimal mean-squared-error achievable in principle in any computational time, and the error that can be achieved by an efficient approximate message passing algorithm. The computation is based on the asymptotic state-evolution analysis of the algorithm. The performance that our analysis predicts, both in terms of the achieved mean-squared-error, and in terms of sample complexity, is extremely promising and motivating for a further development of the algorithm.
This paper analyzes consumer choices over lunchtime restaurants using data from a sample of several thousand anonymous mobile phone users in the San Francisco Bay Area. The data is used to identify users' approximate typical morning location, as well as their choices of lunchtime restaurants. We build a model where restaurants have latent characteristics (whose distribution may depend on restaurant observables, such as star ratings, food category, and price range), each user has preferences for these latent characteristics, and these preferences are heterogeneous across users. Similarly, each item has latent characteristics that describe users' willingness to travel to the restaurant, and each user has individual-specific preferences for those latent characteristics. Thus, both users' willingness to travel and their base utility for each restaurant vary across user-restaurant pairs. We use a Bayesian approach to estimation. To make the estimation computationally feasible, we rely on variational inference to approximate the posterior distribution, as well as stochastic gradient descent as a computational approach. Our model performs better than more standard competing models such as multinomial logit and nested logit models, in part due to the personalization of the estimates. We analyze how consumers re-allocate their demand after a restaurant closes to nearby restaurants versus more distant restaurants with similar characteristics, and we compare our predictions to actual outcomes. Finally, we show how the model can be used to analyze counterfactual questions such as what type of restaurant would attract the most consumers in a given location.
Leslie Grate and Mark Herbster and Richard Hughey and David Haussler Baskin (;enter for Computer Engineering and Computer and Information Sciences University of California Santa Cruz, CA 95064 Keywords: RNA secondary structure, Gibbs sampler, Expectation Maximization, stochastic contextfree grammars, hidden Markov models, tP NA, snRNA, 16S rRNA, linguistic methods Abstract A new method of discovering the common secondary structure of a family of homologous RNA sequences using Gibbs sampling and stochastic context-free grammars is proposed. These parameters describe a statistical model of the family. After the Gibbs sampling has produced a crude statistical model for the family, this model is translated into a stochastic context-free grammar, which is then refined by an Expectation Maximization (EM) procedure produce a more complete model. A prototype implementation of the method is tested on tRNA, pieces of 16S rRNA and on U5 snRNA with good results. I. Saira Mian and Harry Noller Sinsheimer Laboratories University of California Santa Cruz, CA 95064 Introduction Tools for analyzing RNA are becoming increasingly important as in vitro evolution and selection techniques produce greater numbers of synthesized RNA families to supplement those related by phylogeny. Two principal methods have been established for predicting RNA secondary structure base pairings. The second technique employs thermodynamics to compare the free energy changes predicted for formation of possible s,'covdary structure and relies on finding the structure with the lowest free energy (Tinoco Jr., Uhlenbeck, & Levine 1971: Turner, Sugimoto, & Freier 1988; *This work was supported in part by NSF grants C,I)A-9115268 and IR1-9123692, and NIIt gratnt (.;M17129. When several related sequences are available that all share a common secondary structure, combinations of different approaches have been used to obtain improved results (Waterman 1989; Le & Zuker 1991; Han& Kim 1993; Chiu & Kolodziejczak 1991; Sankoff 1985; Winker et al. 1990; Lapedes 1992; Klinger & Brutlag 1993; Gutell et aL 1992). Recent efforts have applied Stochastic Context-Free Grammars (SCFGs) to the problems of statistical modeling, multiple alignment, discrimination and prediction of the secondary structure of RNA families (Sakakibara el al. 1994; 1993; Eddy & Durbin 1994; Searls 1993).
When applying machine learning to real world classification problems two complications that often arise are imbalanced classes (one class occurs much more often than the other (Kubat, Holte, & Matwin 1998; Ezawa, Singh, & Norton 1996; Fawcett & Provost 1996)) and asymmetric misclassification costs (the cost of misclassifying an example from one class is much larger than the cost of misclassifying an example from the other class (Domingos 1999; Pazzani et al. 1997)).
An essential step in understanding the function of sensory nervous systems isto characterize as accurately as possible the stimulus-response function (SRF) of the neurons that relay and process sensory information. Oneincreasingly common experimental approach is to present a rapidly varying complex stimulus to the animal while recording the responses ofone or more neurons, and then to directly estimate a functional transformation of the input that accounts for the neuronal firing. The estimation techniques usually employed, such as Wiener filtering or other correlation-based estimation of the Wiener or Volterra kernels, are equivalent to maximum likelihood estimation in a Gaussian-output-noise regression model. We explore the use of Bayesian evidence-optimization techniques to condition these estimates. We show that by learning hyperparameters thatcontrol the smoothness and sparsity of the transfer function it is possible to improve dramatically the quality of SRF estimates, as measured by their success in predicting responses to novel input.