The maximum entropy (ME) model of a knowledge base R consisting of relational probabilistic conditionals can be defined referring to the set of all ground instances of the conditionals. The logic FO-PCL employs the notion of parametric uniformity for avoiding the full grounding of R. We present an implementation of a rule system transforming R into a knowledge base that is parametrically uniform and has the same ME model, simplifying the ME model computation. The implementation provides different execution and evaluation modes, including the generation of all possible solutions.

Electronic discovery is an interesting subproblem of information retrieval in which one identifies documents that are potentially relevant to issues and facts of a legal case from an electronically stored document collection (a corpus). In this paper, we consider representing documents in a topic space using the well-known topic models such as latent Dirichlet allocation and latent semantic indexing, and solving the information retrieval problem via finding document similarities in the topic space rather doing it in the corpus vocabulary space. We also develop an iterative SMART ranking and categorization framework including human-in-the-loop to label a set of seed (training) documents and using them to build a semi-supervised binary document classification model based on Support Vector Machines. To improve this model, we propose a method for choosing seed documents from the whole population via an active learning strategy. We report the results of our experiments on a real dataset in the electronic discovery domain.

K-means is undoubtedly the most widely used partitional clustering algorithm. Unfortunately, this algorithm is highly sensitive to the initial selection of the cluster centers. Numerous initialization methods have been proposed to address this drawback. Many of these methods, however, have superlinear complexity in the number of data points, which makes them impractical for large data sets. On the other hand, linear methods are often random and/or sensitive to the order in which the data points are processed. These methods are generally unreliable in that the quality of their results is unpredictable. In this paper, we propose a linear, deterministic, and order-invariant initialization method based on multidimensional histograms. Experiments on a diverse collection of data sets from the UCI Machine Learning Repository demonstrate the superiority of our method over the well-known maximin method.

Probabilistic reasoning under the so-called principle of maximum entropy is a viable and convenient alternative to Bayesian networks, relieving the user from providing complete (local) probabilistic information and observing rigorous conditional independence assumptions. In this paper, we present a novel approach to performing computational MaxEnt reasoning that makes use of symbolic computations instead of graph-based techniques. Given a probabilistic knowledge base, we encode the MaxEnt optimization problem into a system of polynomial equations, and then apply Gröbner basis theory to find MaxEnt inferences as solutions to the polynomials. We illustrate our approach with an example of a knowledge base that represents findings on fraud detection in enterprises.

We propose a penalized likelihood method to jointly estimate multiple precision matrices for use in quadratic discriminant analysis and model based clustering. A ridge penalty and a ridge fusion penalty are used to introduce shrinkage and promote similarity between precision matrix estimates. Block-wise coordinate descent is used for optimization, and validation likelihood is used for tuning parameter selection. Our method is applied in quadratic discriminant analysis and semi-supervised model based clustering.

Rare data in a large-scale database are called outliers that reveal significant information in the real world. The subspace-based outlier detection is regarded as a feasible approach in very high dimensional space. However, the outliers found in subspaces are only part of the true outliers in high dimensional space, indeed. The outliers hidden in normal-clustered points are sometimes neglected in the projected dimensional subspace. In this paper, we propose a robust subspace method for detecting such inner outliers in a given dataset, which uses two dimensional-projections: detecting outliers in subspaces with local density ratio in the first projected dimensions; finding outliers by comparing neighbor's positions in the second projected dimensions. Each point's weight is calculated by summing up all related values got in the two steps projected dimensions, and then the points scoring the largest weight values are taken as outliers. By taking a series of experiments with the number of dimensions from 10 to 10000, the results show that our proposed method achieves high precision in the case of extremely high dimensional space, and works well in low dimensional space.

We consider the question of learning in general topological vector spaces. By exploiting known (or parametrized) covariance structures, our Main Theorem demonstrates that any continuous linear map corresponds to a certain isomorphism of embedded Hilbert spaces. By inverting this isomorphism and extending continuously, we construct a version of the Ordinary Least Squares estimator in absolute generality. Our Gauss-Markov theorem demonstrates that OLS is a "best linear unbiased estimator", extending the classical result. We construct a stochastic version of the OLS estimator, which is a continuous disintegration exactly for the class of "uncorrelated implies independent" (UII) measures. As a consequence, Gaussian measures always exhibit continuous disintegrations through continuous linear maps, extending a theorem of the first author. Applying this framework to some problems in machine learning, we prove a useful representation theorem for covariance tensors, and show that OLS defines a good kriging predictor for vector-valued arrays on general index spaces. We also construct a support-vector machine classifier in this setting. We hope that our article shines light on some deeper connections between probability theory, statistics and machine learning, and may serve as a point of intersection for these three communities.

Learning to rank is a supervised learning problem where the output space is the space of rankings but the supervision space is the space of relevance scores. We make theoretical contributions to the learning to rank problem both in the online and batch settings. First, we propose a perceptron-like algorithm for learning a ranking function in an online setting. Our algorithm is an extension of the classic perceptron algorithm for the classification problem. Second, in the setting of batch learning, we introduce a sufficient condition for convex ranking surrogates to ensure a generalization bound that is independent of number of objects per query. Our bound holds when linear ranking functions are used: a common practice in many learning to rank algorithms. En route to developing the online algorithm and generalization bound, we propose a novel family of listwise large margin ranking surrogates. Our novel surrogate family is obtained by modifying a well-known pairwise large margin ranking surrogate and is distinct from the listwise large margin surrogates developed using the structured prediction framework. Using the proposed family, we provide a guaranteed upper bound on the cumulative NDCG (or MAP) induced loss under the perceptron-like algorithm. We also show that the novel surrogates satisfy the generalization bound condition.

In this work, we employ quantitative methods from the realm of statistics and machine learning to develop novel methodologies for author attribution and textual analysis. In particular, we develop techniques and software suitable for applications to Classical study, and we illustrate the efficacy of our approach in several interesting open questions in the field. We apply our numerical analysis techniques to questions of authorship attribution in the case of the Greek tragedian Euripides, to instances of intertextuality and influence in the poetry of the Roman statesman Seneca the Younger, and to cases of "interpolated" text with respect to the histories of Livy.

Contrastive divergence (CD) is a promising method of inference in high dimensional distributions with intractable normalizing constants, however, the theoretical foundations justifying its use are somewhat shaky. This document proposes a framework for understanding CD inference, how/when it works, and provides multiple justifications for the CD moment conditions, including framing them as a variational approximation. Algorithms for performing inference are discussed and are applied to social network data using an exponential-family random graph models (ERGM). The framework also provides guidance about how to construct MCMC kernels providing good CD inference, which turn out to be quite different from those used typically to provide fast global mixing.