# Mathematical & Statistical Methods

### Using multiple samples to learn mixture models

In the mixture models problem it is assumed that there are $K$ distributions $\theta_{1},\ldots,\theta_{K}$ and one gets to observe a sample from a mixture of these distributions with unknown coefficients. The goal is to associate instances with their generating distributions, or to identify the parameters of the hidden distributions. In this work we make the assumption that we have access to several samples drawn from the same $K$ underlying distributions, but with different mixing weights. As with topic modeling, having multiple samples is often a reasonable assumption. Instead of pooling the data into one sample, we prove that it is possible to use the differences between the samples to better recover the underlying structure.

### Large Margin Learning of Upstream Scene Understanding Models

Upstream supervised topic models have been widely used for complicated scene understanding. However, existing maximum likelihood estimation (MLE) schemes can make the prediction model learning independent of latent topic discovery and result in an imbalanced prediction rule for scene classification. This paper presents a joint max-margin and max-likelihood learning method for upstream scene understanding models, in which latent topic discovery and prediction model estimation are closely coupled and well-balanced. The optimization problem is efficiently solved with a variational EM procedure, which iteratively solves an online loss-augmented SVM. We demonstrate the advantages of the large-margin approach on both an 8-category sports dataset and the 67-class MIT indoor scene dataset for scene categorization.

### Exact learning curves for Gaussian process regression on large random graphs

We study learning curves for Gaussian process regression which characterise performance in terms of the Bayes error averaged over datasets of a given size. Whilst learning curves are in general very difficult to calculate we show that for discrete input domains, where similarity between input points is characterised in terms of a graph, accurate predictions can be obtained. These should in fact become exact for large graphs drawn from a broad range of random graph ensembles with arbitrary degree distributions where each input (node) is connected only to a finite number of others. The method is based on translating the appropriate belief propagation equations to the graph ensemble. We demonstrate the accuracy of the predictions for Poisson (Erdos-Renyi) and regular random graphs, and discuss when and why previous approximations to the learning curve fail.

### Kernels and learning curves for Gaussian process regression on random graphs

We investigate how well Gaussian process regression can learn functions defined on graphs, using large regular random graphs as a paradigmatic example. Random-walk based kernels are shown to have some surprising properties: within the standard approximation of a locally tree-like graph structure, the kernel does not become constant, i.e.neighbouring function values do not become fully correlated, when the lengthscale $\sigma$ of the kernel is made large. Instead the kernel attains a non-trivial limiting form, which we calculate. The fully correlated limit is reached only once loops become relevant, and we estimate where the crossover to this regime occurs. Our main subject are learning curves of Bayes error versus training set size.

### A mixture model for the evolution of gene expression in non-homogeneous datasets

We address the challenge of assessing conservation of gene expression in complex, non-homogeneous datasets. Recent studies have demonstrated the success of probabilistic models in studying the evolution of gene expression in simple eukaryotic organisms such as yeast, for which measurements are typically scalar and independent. Models capable of studying expression evolution in much more complex organisms such as vertebrates are particularly important given the medical and scientific interest in species such as human and mouse. We present a statistical model that makes a number of significant extensions to previous models to enable characterization of changes in expression among highly complex organisms. We demonstrate the efficacy of our method on a microarray dataset containing diverse tissues from multiple vertebrate species.

### Optimal Web-Scale Tiering as a Flow Problem

We present a fast online solver for large scale maximum-flow problems as they occur in portfolio optimization, inventory management, computer vision, and logistics. Our algorithm solves an integer linear program in an online fashion. It exploits total unimodularity of the constraint matrix and a Lagrangian relaxation to solve the problem as a convex online game. The algorithm generates approximate solutions of max-flow problems by performing stochastic gradient descent on a set of flows. We apply the algorithm to optimize tier arrangement of over 80 Million web pages on a layered set of caches to serve an incoming query stream optimally.

### Cross Species Expression Analysis using a Dirichlet Process Mixture Model with Latent Matchings

Recent studies compare gene expression data across species to identify core and species specific genes in biological systems. To perform such comparisons researchers need to match genes across species. This is a challenging task since the correct matches (orthologs) are not known for most genes. Previous work in this area used deterministic matchings or reduced multidimensional expression data to binary representation. Here we develop a new method that can utilize soft matches (given as priors) to infer both, unique and similar expression patterns across species and a matching for the genes in both species.

### A Smoothed Approximate Linear Program

We present a novel linear program for the approximation of the dynamic programming cost-to-go function in high-dimensional stochastic control problems. LP approaches to approximate DP naturally restrict attention to approximations that are lower bounds to the optimal cost-to-go function. Our program -- the smoothed approximate linear program -- relaxes this restriction in an appropriate fashion while remaining computationally tractable. Doing so appears to have several advantages: First, we demonstrate superior bounds on the quality of approximation to the optimal cost-to-go function afforded by our approach. Second, experiments with our approach on a challenging problem (the game of Tetris) show that the approach outperforms the existing LP approach (which has previously been shown to be competitive with several ADP algorithms) by an order of magnitude.

### A Stochastic Gradient Method with an Exponential Convergence _Rate for Finite Training Sets

We propose a new stochastic gradient method for optimizing the sum of a finite set of smooth functions, where the sum is strongly convex. While standard stochastic gradient methods converge at sublinear rates for this problem, the proposed method incorporates a memory of previous gradient values in order to achieve a linear convergence rate. In a machine learning context, numerical experiments indicate that the new algorithm can dramatically outperform standard algorithms, both in terms of optimizing the training error and reducing the test error quickly. Papers published at the Neural Information Processing Systems Conference.

### Scalable Training of Mixture Models via Coresets

How can we train a statistical mixture model on a massive data set? In this paper, we show how to construct coresets for mixtures of Gaussians and natural generalizations. A coreset is a weighted subset of the data, which guarantees that models fitting the coreset will also provide a good fit for the original data set. We show that, perhaps surprisingly, Gaussian mixtures admit coresets of size independent of the size of the data set. More precisely, we prove that a weighted set of $O(dk 3/\eps 2)$ data points suffices for computing a $(1 \eps)$-approximation for the optimal model on the original $n$ data points.