Many types of regularization schemes have been employed in statistical learning, each one motivated by some assumption about the problem domain. In this paper, we present a unified asymptotic analysis of smooth regularizers, which allows us to see how the validity of these assumptions impacts the success of a particular regularizer. In addition, our analysis motivates an algorithm for optimizing regularization parameters, which in turn can be analyzed within our framework. We apply our analysis to several examples, including hybrid generative-discriminative learning and multi-task learning.

Probabilistic topic models have become popular as methods for dimensionality reduction in collections of text documents or images. These models are usually treated as generative models and trained using maximum likelihood or Bayesian methods. In this paper, we discuss an alternative: a discriminative framework in which we assume that supervised side information is present, and in which we wish to take that side information into account in finding a reduced dimensionality representation.Specifically, we present DiscLDA, a discriminative variation on Latent Dirichlet Allocation (LDA) in which a class-dependent linear transformation isintroduced on the topic mixture proportions. This parameter is estimated by maximizing the conditional likelihood. By using the transformed topic mixture proportionsas a new representation of documents, we obtain a supervised dimensionality reduction algorithm that uncovers the latent structure in a document collectionwhile preserving predictive power for the task of classification. We compare the predictive power of the latent structure of DiscLDA with unsupervised LDAon the 20 Newsgroups document classification task and show how our model can identify shared topics across classes as well as class-dependent topics.

Continuously-Adaptive Discretization for Message-Passing (CAD-MP) is a new message-passing algorithm employing adaptive discretization. Most previous message-passing algorithms approximated arbitrary continuous probability distributions using either: a family of continuous distributions such as the exponential family; a particle-set of discrete samples; or a fixed, uniform discretization. In contrast, CAD-MP uses a discretization that is (i) non-uniform, and (ii) adaptive. The non-uniformity allows CAD-MP to localize interesting features (such as sharp peaks) in the marginal belief distributions with time complexity that scales logarithmically with precision, as opposed to uniform discretization which scales at best linearly. We give a principled method for altering the non-uniform discretization according to information-based measures. CAD-MP is shown in experiments on simulated data to estimate marginal beliefs much more precisely than competing approaches for the same computational expense.

We consider the question of determining a real-valued function on the space of permutations of n elements with very limited observations.

Graph clustering methods such as spectral clustering are defined for general weighted graphs. In machine learning, however, data often is not given in form of a graph, but in terms of similarity (or distance) values between points. In this case, first a neighborhood graph is constructed using the similarities between the points and then a graph clustering algorithm is applied to this graph. In this paper we investigate the influence of the construction of the similarity graph on the clustering results. We first study the convergence of graph clustering criteria such as the normalized cut (Ncut) as the sample size tends to infinity. We find that the limit expressions are different for different types of graph, for example the r-neighborhood graph or the k-nearest neighbor graph. In plain words: Ncut on a kNN graph does something systematically different than Ncut on an r-neighborhood graph! This finding shows that graph clustering criteria cannot be studied independently of the kind of graph they are applied to. We also provide examples which show that these differences can be observed for toy and real data already for rather small sample sizes.

We consider the problem of finding the M assignments with maximum probability in a probabilistic graphical model. We show how this problem can be formulated as a linear program (LP) on a particular polytope. We prove that, for tree graphs (and junction trees in general), this polytope has a particularly simple form and differs from the marginal polytope in a single inequality constraint. We use this characterization to provide an approximation scheme for non-tree graphs, by using the set of spanning trees over such graphs. The method we present puts the M-best inference problem in the context of LP relaxations, which have recently received considerable attention and have proven useful in solving difficult inference problems.

For many supervised learning problems, we possess prior knowledge about which features yield similar information about the target variable. In predicting the topic of a document, we might know that two words are synonyms, or when performing image recognition, we know which pixels are adjacent. Such synonymous or neighboring features are near-duplicates and should therefore be expected to have similar weights in a good model. Here we present a framework for regularized learning in settings where one has prior knowledge about which features are expected to have similar and dissimilar weights. This prior knowledge is encoded as a graph whose vertices represent features and whose edges represent similarities and dissimilarities between them. During learning, each feature's weight is penalized by the amount it differs from the average weight of its neighbors. For text classification, regularization using graphs of word co-occurrences outperforms manifold learning and compares favorably to other recently proposed semi-supervised learning methods. For sentiment analysis, feature graphs constructed from declarative human knowledge, as well as from auxiliary task learning, significantly improve prediction accuracy.

We present an approach for learning stochastic geometric models of object categories from single view images. We focus here on models expressible as a spatially contiguous assemblage of blocks. Model topologies are learned across groups of images, and one or more such topologies is linked to an object category (e.g. chairs). Fitting learned topologies to an image can be used to identify the object class, as well as detail its geometry. The latter goes beyond labeling objects, as it provides the geometric structure of particular instances. We learn the models using joint statistical inference over structure parameters, camera parameters, and instance parameters. These produce an image likelihood through a statistical imaging model. We use trans-dimensional sampling to explore topology hypotheses, and alternate between Metropolis-Hastings and stochastic dynamics to explore instance parameters. Experiments on images of furniture objects such as tables and chairs suggest that this is an effective approach for learning models that encode simple representations of category geometry and the statistics thereof, and support inferring both category and geometry on held out single view images.

The problem of ranking arises ubiquitously in almost every aspect of life, and in particular in Machine Learning/Information Retrieval. A statistical model for ranking predicts how humans rank subsets V of some universe U. In this work we define a statistical model for ranking that satisfies certain desirable properties. The model automatically gives rise to a logistic regression based approach to learning how to rank, for which the score and comparison based approaches are dual views. This offers a new generative approach to ranking which can be used for IR.

Speaker comparison, the process of finding the speaker similarity between two speech signals, occupies a central role in a variety of applications---speaker verification, clustering, and identification. Speaker comparison can be placed in a geometric framework by casting the problem as a model comparison process. For a given speech signal, feature vectors are produced and used to adapt a Gaussian mixture model (GMM). Speaker comparison can then be viewed as the process of compensating and finding metrics on the space of adapted models. We propose a framework, inner product discriminant functions (IPDFs), which extends many common techniques for speaker comparison: support vector machines, joint factor analysis, and linear scoring. The framework uses inner products between the parameter vectors of GMM models motivated by several statistical methods. Compensation of nuisances is performed via linear transforms on GMM parameter vectors. Using the IPDF framework, we show that many current techniques are simple variations of each other. We demonstrate, on a 2006 NIST speaker recognition evaluation task, new scoring methods using IPDFs which produce excellent error rates and require significantly less computation than current techniques.