We present a new framework for semi-supervised learning with sparse eigenfunction bases of kernel matrices. It turns out that when the \emph{cluster assumption} holds, that is, when the high density regions are sufficiently separated by low density valleys, each high density area corresponds to a unique representative eigenvector. Linear combination of such eigenvectors (or, more precisely, of their Nystrom extensions) provide good candidates for good classification functions. By first choosing an appropriate basis of these eigenvectors from unlabeled data and then using labeled data with Lasso to select a classifier in the span of these eigenvectors, we obtain a classifier, which has a very sparse representation in this basis. Importantly, the sparsity appears naturally from the cluster assumption. Experimental results on a number of real-world data-sets show that our method is competitive with the state of the art semi-supervised learning algorithms and outperforms the natural base-line algorithm (Lasso in the Kernel PCA basis).

The learning of appropriate distance metrics is a critical problem in classification. In this work, we propose a boosting-based technique, termed BoostMetric, for learning a Mahalanobis distance metric. One of the primary difficulties in learning such a metric is to ensure that the Mahalanobis matrix remains positive semidefinite. Semidefinite programming is sometimes used to enforce this constraint, but does not scale well. BoostMetric is instead based on a key observation that any positive semidefinite matrix can be decomposed into a linear positive combination of trace-one rank-one matrices. BoostMetric thus uses rank-one positive semidefinite matrices as weak learners within an efficient and scalable boosting-based learning process. The resulting method is easy to implement, does not require tuning, and can accommodate various types of constraints. Experiments on various datasets show that the proposed algorithm compares favorably to those state-of-the-art methods in terms of classification accuracy and running time.

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.

We show how to model documents as bags of words using family of two-layer, undirected graphical models. Each member of the family has the same number of binary hidden units but a different number of ``softmax visible units. All of the softmax units in all of the models in the family share the same weights to the binary hidden units. We describe efficient inference and learning procedures for such a family. Each member of the family models the probability distribution of documents of a specific length as a product of topic-specific distributions rather than as a mixture and this gives much better generalization than Latent Dirichlet Allocation for modeling the log probabilities of held-out documents. The low-dimensional topic vectors learned by the undirected family are also much better than LDA topic vectors for retrieving documents that are similar to a query document. The learned topics are more general than those found by LDA because precision is achieved by intersecting many general topics rather than by selecting a single precise topic to generate each word.

In this paper, we investigate how, given an image, similar images sharing the same global description can help with unsupervised scene segmentation. In contrast to recent work in semantic alignment of scenes, we allow an input image to be explained by partial matches of similar scenes. This allows for a better explanation of the input scenes. We perform MRF-based segmentation that optimizes over matches, while respecting boundary information. The recovered segments are then used to re-query a large database of images to retrieve better matches for the target regions. We show improved performance in detecting the principal occluding and contact boundaries for the scene over previous methods on data gathered from the LabelMe database.

Hypergraph clustering refers to the process of extracting maximally coherent groups from a set of objects using high-order (rather than pairwise) similarities. Traditional approaches to this problem are based on the idea of partitioning the input data into a user-defined number of classes, thereby obtaining the clusters as a by-product of the partitioning process. In this paper, we provide a radically different perspective to the problem. In contrast to the classical approach, we attempt to provide a meaningful formalization of the very notion of a cluster and we show that game theory offers an attractive and unexplored perspective that serves well our purpose. Specifically, we show that the hypergraph clustering problem can be naturally cast into a non-cooperative multi-player ``clustering game, whereby the notion of a cluster is equivalent to a classical game-theoretic equilibrium concept. From the computational viewpoint, we show that the problem of finding the equilibria of our clustering game is equivalent to locally optimizing a polynomial function over the standard simplex, and we provide a discrete-time dynamics to perform this optimization. Experiments are presented which show the superiority of our approach over state-of-the-art hypergraph clustering techniques.

This paper uses information-theoretic techniques to determine minimax rates for estimating nonparametric sparse additive regression models under high-dimensional scaling. We assume an additive decomposition of the form $f^*(X_1, \ldots, X_p) = \sum_{j \in S} h_j(X_j)$, where each component function $h_j$ lies in some Hilbert Space $\Hilb$ and $S \subset \{1, \ldots, \pdim \}$ is an unknown subset with cardinality $\s = |S$. Given $\numobs$ i.i.d. observations of $f^*(X)$ corrupted with white Gaussian noise where the covariate vectors $(X_1, X_2, X_3,...,X_{\pdim})$ are drawn with i.i.d. components from some distribution $\mP$, we determine tight lower bounds on the minimax rate for estimating the regression function with respect to squared $\LTP$ error. The main result shows that the minimax rates are $\max{\big(\frac{\s \log \pdim / \s}{n}, \LowerRateSq \big)}$. The first term reflects the difficulty of performing \emph{subset selection} and is independent of the Hilbert space $\Hilb$; the second term $\LowerRateSq$ is an \emph{\s-dimensional estimation} term, depending only on the low dimension $\s$ but not the ambient dimension $\pdim$, that captures the difficulty of estimating a sum of $\s$ univariate functions in the Hilbert space $\Hilb$. As a special case, if $\Hilb$ corresponds to the $\m$-th order Sobolev space $\SobM$ of functions that are $m$-times differentiable, the $\s$-dimensional estimation term takes the form $\LowerRateSq \asymp \s \; n^{-2\m/(2\m+1)}$. The minimax rates are compared with rates achieved by an $\ell_1$-penalty based approach, it can be shown that a certain $\ell_1$-based approach achieves the minimax optimal rate.

The longstanding problem of efficient nearest-neighbor (NN) search has ubiquitous applicationsranging from astrophysics to MP3 fingerprinting to bioinformatics to movie recommendations. As the dimensionality of the dataset increases, exact NNsearch becomes computationally prohibitive;(1) distance-approximate NN search can provide large speedups but risks losing the meaning of NN search present in the ranks (ordering) of the distances. This paper presents a simple, practical algorithm allowing the user to, for the first time, directly control the true accuracy of NN search (in terms of ranks) while still achieving the large speedups over exact NN. Experiments on high-dimensional datasets show that our algorithm often achieves faster and more accurate results than the best-known distance-approximate method, with much more stable behavior.

Many transductive inference algorithms assume that distributions over training and test estimates should be related, e.g. by providing a large margin of separation on both sets. We use this idea to design a transduction algorithm which can be used without modification for classification, regression, and structured estimation. At its heart we exploit the fact that for a good learner the distributions over the outputs on training and test sets should match. This is a classical two-sample problem which can be solved efficiently in its most general form by using distance measures in Hilbert Space. It turns out that a number of existing heuristics can be viewed as special cases of our approach.

We develop a convex relaxation of maximum a posteriori estimation of a mixture of regression models. Although our relaxation involves a semidefinite matrix variable, we reformulate the problem to eliminate the need for general semidefinite programming. In particular, we provide two reformulations that admit fast algorithms. The first is a max-min spectral reformulation exploiting quasi-Newton descent. The second is a min-min reformulation consisting of fast alternating steps of closed-form updates. We evaluate the methods against Expectation-Maximization in a real problem of motion segmentation from video data.