Inference on high-order graphical models has become increasingly important in recent years. We consider energies with simple 'sparse' high-order potentials. Previous work in this area uses either specialized message-passing or transforms each high-order potential to the pairwise case. We take a fundamentally different approach, transforming the entire original problem into a comparatively small instance of a submodular vertex-cover problem. These vertex-cover instances can then be attacked by standard pairwise methods, where they run much faster (4--15 times) and are often more effective than on the original problem. We evaluate our approach on synthetic data, and we show that our algorithm can be useful in a fast hierarchical clustering and model estimation framework.

Given $\alpha,\epsilon$, we study the time complexity required to improperly learn a halfspace with misclassification error rate of at most $(1+\alpha)\,L^*_\gamma + \epsilon$, where $L^*_\gamma$ is the optimal $\gamma$-margin error rate. For $\alpha = 1/\gamma$, polynomial time and sample complexity is achievable using the hinge-loss. For $\alpha = 0$, \cite{ShalevShSr11} showed that $\poly(1/\gamma)$ time is impossible, while learning is possible in time $\exp(\tilde{O}(1/\gamma))$. An immediate question, which this paper tackles, is what is achievable if $\alpha \in (0,1/\gamma)$. We derive positive results interpolating between the polynomial time for $\alpha = 1/\gamma$ and the exponential time for $\alpha=0$. In particular, we show that there are cases in which $\alpha = o(1/\gamma)$ but the problem is still solvable in polynomial time. Our results naturally extend to the adversarial online learning model and to the PAC learning with malicious noise model.

The empirical success of the belief propagation approximate inference algorithm has inspired numerous theoretical and algorithmic advances. Yet, for continuous non-Gaussian domains performing belief propagation remains a challenging task: recent innovations such as nonparametric or kernel belief propagation, while useful, come with a substantial computational cost and offer little theoretical guarantees, even for tree structured models. In this work we present Nonparanormal BP for performing efficient inference on distributions parameterized by a Gaussian copulas network and any univariate marginals. For tree structured networks, our approach is guaranteed to be exact for this powerful class of non-Gaussian models. Importantly, the method is as efficient as standard Gaussian BP, and its convergence properties do not depend on the complexity of the univariate marginals, even when a nonparametric representation is used.

The ability to predict action content from neural signals in real time before the action occurshas been long sought in the neuroscientific study of decision-making, agency and volition. Online real-time (ORT) prediction is important for understanding therelation between neural correlates of decision-making and conscious, voluntary action as well as for brain-machine interfaces. Here, epilepsy patients, implanted with intracranial depth microelectrodes or subdural grid electrodes for clinical purposes, participated in a "matching-pennies" game against an opponent. In each trial, subjects were given a 5 s countdown, after which they had to raise their left or right hand immediately as the "go" signal appeared on a computer screen. They won a fixed amount of money if they raised a different hand than their opponent and lost that amount otherwise.

We advocate the use of a new distribution family--the transelliptical--for robust inference of high dimensional graphical models. The transelliptical family is an extension of the nonparanormal family proposed by Liu et al. (2009). Just as the nonparanormal extends the normal by transforming the variables using univariate functions, the transelliptical extends the elliptical family in the same way. We propose a nonparametric rank-based regularization estimator which achieves the parametric rates of convergence for both graph recovery and parameter estimation. Sucha result suggests that the extra robustness and flexibility obtained by the semiparametric transelliptical modeling incurs almost no efficiency loss. We also discuss the relationship between this work with the transelliptical component analysis proposed by Han and Liu (2012).

Graphical models are a very useful tool to describe and understand natural phenomena, from gene expression to climate change and social interactions. The topological structure of these graphs/networks is a fundamental part of the analysis, and in many cases the main goal of the study. However, little work has been done on incorporating prior topological knowledge onto the estimation of the underlying graphical models from sample data. In this work we propose extensions to the basic joint regression model for network estimation, which explicitly incorporate graph-topological constraints into the corresponding optimization approach. The first proposed extension includes an eigenvector centrality constraint, thereby promoting this important prior topological property. The second developed extension promotes the formation of certain motifs, triangle-shaped ones in particular, which are known to exist for example in genetic regulatory networks. The presentation of the underlying formulations, which serve as examples of the introduction of topological constraints in network estimation, is complemented with examples in diverse datasets demonstrating the importance of incorporating such critical prior knowledge.

We formulate clustering aggregation as a special instance of Maximum-Weight Independent Set (MWIS) problem. For a given dataset, an attributed graph is constructed from the union of the input clusterings generated by different underlying clustering algorithms with different parameters. The vertices, which represent the distinct clusters, are weighted by an internal index measuring both cohesion and separation. The edges connect the vertices whose corresponding clusters overlap. Intuitively, an optimal aggregated clustering can be obtained by selecting an optimal subset of non-overlapping clusters partitioning the dataset together. We formalize this intuition as the MWIS problem on the attributed graph, i.e., finding the heaviest subset of mutually non-adjacent vertices. This MWIS problem exhibits a special structure. Since the clusters of each input clustering form a partition of the dataset, the vertices corresponding to each clustering form a maximal independent set (MIS) in the attributed graph. We propose a variant of simulated annealing method that takes advantage of this special structure. Our algorithm starts from each MIS, which is close to a distinct local optimum of the MWIS problem, and utilizes a local search heuristic to explore its neighborhood in order to find the MWIS. Extensive experiments on many challenging datasets show that: 1. our approach to clustering aggregation automatically decides the optimal number of clusters; 2. it does not require any parameter tuning for the underlying clustering algorithms; 3. it can combine the advantages of different underlying clustering algorithms to achieve superior performance; 4. it is robust against moderate or even bad input clusterings.

W e propose a multiresolution Gaussian process to capture lo ng-range, non-Markovian dependencies while allowing for abrupt changes a nd non-stationarity. The multiresolution GP hierarchically couples a collectio n of smooth GPs, each defined over an element of a random nested partition. Long-ra nge dependencies are captured by the top-level GP while the partition poi nts define the abrupt changes. Due to the inherent conjugacy of the GPs, one can analytically marginalize the GPs and compute the marginal likelihood of the observations given the partition tree. This property allows for efficient inference of the partition itself, for which we employ graph-theoretic techniques. W e apply the multiresolution GP to the analysis of magnetoencephalography (MEG) recordings of brain activity.

The usability of Brain Computer Interfaces (BCI) based on the P300 speller is severely hindered by the need for long training times and many repetitions of the same stimulus. In this contribution we introduce a set of unsupervised hierarchical probabilistic models that tackle both problems simultaneously by incorporating prior knowledge from two sources: information from other training subjects (through transfer learning) and information about the words being spelled (through language models). We show, that due to this prior knowledge, the performance of the unsupervised models parallels and in some cases even surpasses that of supervised models, while eliminating the tedious training session.

Identifying patterns from the neuroimaging recordings of brain activity related to the unobservable psychological or mental state of an individual can be treated as a unsupervised pattern recognition problem. The main challenges, however, for such an analysis of fMRI data are: a) defining a physiologically meaningful feature-space for representing the spatial patterns across time; b) dealing with the high-dimensionality of the data; and c) robustness to the various artifacts and confounds in the fMRI time-series. In this paper, we present a network-aware feature-space to represent the states of a general network, that enables comparing and clustering such states in a manner that is a) meaningful in terms of the network connectivity structure; b)computationally efficient; c) low-dimensional; and d) relatively robust to structured and random noise artifacts. This feature-space is obtained from a spherical relaxation of the transportation distance metric which measures the cost of transporting ``mass'' over the network to transform one function into another. Through theoretical and empirical assessments, we demonstrate the accuracy and efficiency of the approximation, especially for large problems. While the application presented here is for identifying distinct brain activity patterns from fMRI, this feature-space can be applied to the problem of identifying recurring patterns and detecting outliers in measurements on many different types of networks, including sensor, control and social networks.