We investigate the accuracy of the two most common estimators for the maximum expected value of a general set of random variables: a generalization of the maximum sample average, and cross validation. No unbiased estimator exists and we show that it is non-trivial to select a good estimator without knowledge about the distributions of the random variables. We investigate and bound the bias and variance of the aforementioned estimators and prove consistency. The variance of cross validation can be significantly reduced, but not without risking a large bias. The bias and variance of different variants of cross validation are shown to be very problem-dependent, and a wrong choice can lead to very inaccurate estimates.

Imaging devices such as telescopes and microscopes collect incoming light using lenses or mirrors of finite size. This finite size imposes a finite aperture on the light that reaches the optical system, leading to effects of diffraction. In particular, diffraction ensures that the image of a point can never be a point. For instance, an imaging system using a lens with an F -number f/D (where f is the focal length, and D is the diameter of the circular aperture) has an impulse response function (Airy disk) whose radius is 1.22λf/D on the sensor, where λ is the wave length of the light (for simplicity, assumed to be monochromatic). Another way to express the same insight uses the transfer function. For a lens focused at infinity, the transfer function is constant within a circle of radius ν 1/(2λf/D), and zero outside [23, p. 136]. This means, in a nutshell, that if we try to image a sinusoidal pattern with spatial frequency larger than ν, diffraction will annihilate that pattern. Likewise, if we decompose a general object into spatial frequencies by Fourier analysis, all components larger than ν will vanish. This article has been accepted for publication at the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Portland, 2013.

In recent years, the problem of detecting communities in networks has received a large amount of attention, with important applications in the social and biological sciences, among others (Fortunato, 2010). The vast majority of this expansive literature focuses on developing realistic models of (random) networks (Albert and Barabási, 2002; Barabási and Albert, 1999), on designing methods for extracting communities from such networks (Girvan and Newman, 2002; Newman, 2006; Reichardt and Bornholdt, 2006) and on fitting models to network data (Bickel et al., 2011). The underlying model is that of graph G (E,V), where E is the set of edges and V is the set of nodes. For example, in a social network, a node would represent an individual and an edge between two nodes would symbolize a friendship or kinship of some sort shared by these two individuals. In the literature just mentioned, almost all the methodology has concentrated on devising graph partitioning methods, with the end goal of clustering the nodes in V into groups with strong inner-connectivity and weak inter-connectivity (Bickel and Chen, 2009; Lancichinetti and Fortunato, 2009; Newman and Girvan, 2004).

Motivated by social balance theory, we develop a theory of link classification in signed networks using the correlation clustering index as measure of label regularity. We derive learning bounds in terms of correlation clustering within three fundamental transductive learning settings: online, batch and active. Our main algorithmic contribution is in the active setting, where we introduce a new family of efficient link classifiers based on covering the input graph with small circuits. These are the first active algorithms for link classification with mistake bounds that hold for arbitrary signed networks.

We present very efficient active learning algorithms for link classification in signed networks. Our algorithms are motivated by a stochastic model in which edge labels are obtained through perturbations of a initial sign assignment consistent with a two-clustering of the nodes. We provide a theoretical analysis within this model, showing that we can achieve an optimal (to whithin a constant factor) number of mistakes on any graph G = (V,E) such that |E| = \Omega(|V|^{3/2}) by querying O(|V|^{3/2}) edge labels. More generally, we show an algorithm that achieves optimality to within a factor of O(k) by querying at most order of |V| + (|V|/k)^{3/2} edge labels. The running time of this algorithm is at most of order |E| + |V|\log|V|.

Topic models are probabilistic models for discovering topical themes in collections of documents. In real world applications, these models provide us with the means of organizing what would otherwise be unstructured collections. They can help us cluster a huge collection into different topics or find a subset of the collection that resembles the topical theme found in an article at hand. The first wave of topic models developed were able to discover the prevailing topics in a big collection of documents spanning a period of time. It was later realized that these time-invariant models were not capable of modeling 1) the time varying number of topics they discover and 2) the time changing structure of these topics. Few models were developed to address this two deficiencies. The online-hierarchical Dirichlet process models the documents with a time varying number of topics. It varies the structure of the topics over time as well. However, it relies on document order, not timestamps to evolve the model over time. The continuous-time dynamic topic model evolves topic structure in continuous-time. However, it uses a fixed number of topics over time. In this dissertation, I present a model, the continuous-time infinite dynamic topic model, that combines the advantages of these two models 1) the online-hierarchical Dirichlet process, and 2) the continuous-time dynamic topic model. More specifically, the model I present is a probabilistic topic model that does the following: 1) it changes the number of topics over continuous time, and 2) it changes the topic structure over continuous-time. I compared the model I developed with the two other models with different setting values. The results obtained were favorable to my model and showed the need for having a model that has a continuous-time varying number of topics and topic structure.

The task of clustering a set of objects based on multiple sources of data arises in several modern applications. We propose an integrative statistical model that permits a separate clustering of the objects for each data source. These separate clusterings adhere loosely to an overall consensus clustering, and hence they are not independent. We describe a computationally scalable Bayesian framework for simultaneous estimation of both the consensus clustering and the source-specific clusterings. We demonstrate that this flexible approach is more robust than joint clustering of all data sources, and is more powerful than clustering each data source separately. This work is motivated by the integrated analysis of heterogeneous biomedical data, and we present an application to subtype identification of breast cancer tumor samples using publicly available data from The Cancer Genome Atlas. Several fields of research now analyze multi-source data (also called multimodal data), in which multiple heterogeneous datasets describe a common set of objects.

Multilabel classification is a relatively recent subfield of machine learning. Unlike to the classical approach, where instances are labeled with only one category, in multilabel classification, an arbitrary number of categories is chosen to label an instance. Due to the problem complexity (the solution is one among an exponential number of alternatives), a very common solution (the binary method) is frequently used, learning a binary classifier for every category, and combining them all afterwards. The assumption taken in this solution is not realistic, and in this work we give examples where the decisions for all the labels are not taken independently, and thus, a supervised approach should learn those existing relationships among categories to make a better classification. Therefore, we show here a generic methodology that can improve the results obtained by a set of independent probabilistic binary classifiers, by using a combination procedure with a classifier trained on the co-occurrences of the labels. We show an exhaustive experimentation in three different standard corpora of labeled documents (Reuters-21578, Ohsumed-23 and RCV1), which present noticeable improvements in all of them, when using our methodology, in three probabilistic base classifiers.

We describe algorithms for learning Bayesian networks from a combination of user knowledge and statistical data. The algorithms have two components: a scoring metric and a search procedure. The scoring metric takes a network structure, statistical data, and a user's prior knowledge, and returns a score proportional to the posterior probability of the network structure given the data. The search procedure generates networks for evaluation by the scoring metric. Previous work has concentrated on metrics for domains containing only discrete variables, under the assumption that data represents a multinomial sample. In this paper, we extend this work, developing scoring metrics for domains containing all continuous variables or a mixture of discrete and continuous variables, under the assumption that continuous data is sampled from a multivariate normal distribution. Our work extends traditional statistical approaches for identifying vanishing regression coefficients in that we identify two important assumptions, called event equivalence and parameter modularity, that when combined allow the construction of prior distributions for multivariate normal parameters from a single prior Bayesian network specified by a user.

Integration is affected by the curse of dimensionality and quickly becomes intractable as the dimensionality of the problem grows. We propose a randomized algorithm that, with high probability, gives a constant-factor approximation of a general discrete integral defined over an exponentially large set. This algorithm relies on solving only a small number of instances of a discrete combinatorial optimization problem subject to randomly generated parity constraints used as a hash function. As an application, we demonstrate that with a small number of MAP queries we can efficiently approximate the partition function of discrete graphical models, which can in turn be used, for instance, for marginal computation or model selection.