Neural probabilistic language models (NPLMs) have been shown to be competitive with and occasionally superior to the widely-used n-gram language models. The main drawback of NPLMs is their extremely long training and testing times. Morin and Bengio have proposed a hierarchical language model built around a binary tree of words that was two orders of magnitude faster than the non-hierarchical language model it was based on. However, it performed considerably worse than its non-hierarchical counterpart in spite of using a word tree created using expert knowledge. We introduce a fast hierarchical language model along with a simple feature-based algorithm for automatic construction of word trees from the data. We then show that the resulting models can outperform non-hierarchical models and achieve state-of-the-art performance.

We consider the problem of binary classification where the classifier may abstain instead of classifying each observation. The Bayes decision rule for this setup, known as Chow's rule, is defined by two thresholds on posterior probabilities. From simple desiderata, namely the consistency and the sparsity of the classifier, we derive the double hinge loss function that focuses on estimating conditional probabilities only in the vicinity of the threshold points of the optimal decision rule. We show that, for suitable kernel machines, our approach is universally consistent. We cast the problem of minimizing the double hinge loss as a quadratic program akin to the standard SVM optimization problem and propose an active set method to solve it efficiently. We finally provide preliminary experimental results illustrating the interest of our constructive approach to devising loss functions.

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.

Fast retrieval methods are increasingly critical for many large-scale analysis tasks, and there have been several recent methods that attempt to learn hash functions for fast and accurate nearest neighbor searches. In this paper, we develop an algorithm for learning hash functions based on explicitly minimizing the reconstruction error between the original distances and the Hamming distances of the corresponding binary embeddings. We develop a scalable coordinate-descent algorithm for our proposed hashing objective that is able to efficiently learn hash functions in a variety of settings. Unlike existing methods such as semantic hashing and spectral hashing, our method is easily kernelized and does not require restrictive assumptions about the underlying distribution of the data. We present results over several domains to demonstrate that our method outperforms existing state-of-the-art techniques.

We propose a novel non-parametric adaptive anomaly detection algorithm for high dimensional data based on score functions derived from nearest neighbor graphs on n-point nominal data. Anomalies are declared whenever the score of a test sample falls below q, which is supposed to be the desired false alarm level. The resulting anomaly detector is shown to be asymptotically optimal in that it is uniformly most powerful for the specified false alarm level, q, for the case when the anomaly density is a mixture of the nominal and a known density. Our algorithm is computationally efficient, being linear in dimension and quadratic in data size. It does not require choosing complicated tuning parameters or function approximation classes and it can adapt to local structure such as local change in dimensionality. We demonstrate the algorithm on both artificial and real data sets in high dimensional feature spaces.

Pattern classification approaches to analyzing functional neuroimaging data have become increasingly popular [12] [3] [4].

Many machine learning algorithms require the summation of Gaussian kernel functions, an expensive operation if implemented straightforwardly. Several methods have been proposed to reduce the computational complexity of evaluating such sums, including tree and analysis based methods. These achieve varying speedups depending on the bandwidth, dimension, and prescribed error, making the choice between methods difficult for machine learning tasks. We provide an algorithm that combines tree methods with the Improved Fast Gauss Transform (IFGT). As originally proposed the IFGT suffers from two problems: (1) the Taylor series expansion does not perform well for very low bandwidths, and (2) parameter selection is not trivial and can drastically affect performance and ease of use. We address the first problem by employing a tree data structure, resulting in four evaluation methods whose performance varies based on the distribution of sources and targets and input parameters such as desired accuracy and bandwidth. To solve the second problem, we present an online tuning approach that results in a black box method that automatically chooses the evaluation method and its parameters to yield the best performance for the input data, desired accuracy, and bandwidth. In addition, the new IFGT parameter selection approach allows for tighter error bounds. Our approach chooses the fastest method at negligible additional cost, and has superior performance in comparisons with previous approaches.

We prove certain theoretical properties of a graph-regularized transductive learning objective that is based on minimizing a Kullback-Leibler divergence based loss. These include showing that the iterative alternating minimization procedure used to minimize the objective converges to the correct solution and deriving a test for convergence. We also propose a graph node ordering algorithm that is cache cognizant and leads to a linear speedup in parallel computations. This ensures that the algorithm scales to large data sets. By making use of empirical evaluation on the TIMIT and Switchboard I corpora, we show this approach is able to out-perform other state-of-the-art SSL approaches. In one instance, we solve a problem on a 120 million node graph.

Principal component analysis is a fundamental operation in computational data analysis, with myriad applications ranging from web search to bioinformatics to computer vision and image analysis. However, its performance and applicability in real scenarios are limited by a lack of robustness to outlying or corrupted observations. Thispaper considers the idealized "robust principal component analysis" problem of recovering a low rank matrix A from corrupted observations D A E. Here, the corrupted entries E are unknown and the errors can be arbitrarily large (modeling grossly corrupted observations common in visual and bioinformatic data), but are assumed to be sparse. We prove that most matrices A can be efficiently and exactly recovered from most error sign-and-support patterns bysolving a simple convex program, for which we give a fast and provably convergent algorithm. Our result holds even when the rank of A grows nearly proportionally (up to a logarithmic factor) to the dimensionality of the observation spaceand the number of errors E grows in proportion to the total number of entries in the matrix. A byproduct of our analysis is the first proportional growth results for the related problem of completing a low-rank matrix from a small fraction ofits entries. Simulations and real-data examples corroborate the theoretical results, and suggest potential applications in computer vision.

We describe a primal-dual framework for the design and analysis of online strongly convex optimization algorithms. Our framework yields the tightest known logarithmic regret bounds for Follow-The-Leader and for the gradient descent algorithm proposed in HazanKaKaAg06. We then show that one can interpolate between these two extreme cases. In particular, we derive a new algorithm that shares the computational simplicity of gradient descent but achieves lower regret in many practical situations. Finally, we further extend our framework for generalized strongly convex functions.