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17
On model selection consistency of Mestimators with geometrically decomposable penalties
 Advances in Neural Information Processing Systems
, 2013
"... Penalized Mestimators are used in diverse areas of science and engineering to fit highdimensional models with some lowdimensional structure. Often, the penalties are geometrically decomposable, i.e. can be expressed as a sum of support functions over convex sets. We generalize the notion of irre ..."
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Penalized Mestimators are used in diverse areas of science and engineering to fit highdimensional models with some lowdimensional structure. Often, the penalties are geometrically decomposable, i.e. can be expressed as a sum of support functions over convex sets. We generalize the notion of irrepresentable to geometrically decomposable penalties and develop a general framework for establishing consistency and model selection consistency of Mestimators with such penalties. We then use this framework to derive results for some special cases of interest in bioinformatics and statistical learning. 1
Learning Graphs from Signal Observations under Smoothness Prior
, 2014
"... The construction of a meaningful graph plays a crucial role in the success of many graphbased data representations and algorithms, especially in the emerging field of signal processing on graphs. However, a meaningful graph is not always readily available from the data, nor easy to define depending ..."
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The construction of a meaningful graph plays a crucial role in the success of many graphbased data representations and algorithms, especially in the emerging field of signal processing on graphs. However, a meaningful graph is not always readily available from the data, nor easy to define depending on the application domain. In this paper, we address the problem of graph learning, where we are interested in learning graph topologies, namely, the relationships between data entities, that well explain the signal observations. In particular, we want to infer a graph such that the input data forms graph signals with smooth variations on the resulting topology. To this end, we adopt a factor analysis model for the graph signals and impose a Gaussian probabilistic prior on the latent variables that control these graph signals. We show that the Gaussian prior leads to an efficient representation that favors the smoothness property of the graph signals. We then propose an algorithm for learning graphs that enforce such smoothness property for the signal observations by minimizing the variations of the signals on the learned graph. Experiments on both synthetic and real world data demonstrate that the proposed graph learning framework can efficiently infer meaningful graph topologies from only the signal observations.
Robust Multimodal Graph Matching: Sparse Coding Meets Graph Matching Marcelo Fiori
"... Graph matching is a challenging problem with very important applications in a wide range of fields, from image and video analysis to biological and biomedical problems. We propose a robust graph matching algorithm inspired in sparsityrelated techniques. We cast the problem, resembling group or coll ..."
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Graph matching is a challenging problem with very important applications in a wide range of fields, from image and video analysis to biological and biomedical problems. We propose a robust graph matching algorithm inspired in sparsityrelated techniques. We cast the problem, resembling group or collaborative sparsity formulations, as a nonsmooth convex optimization problem that can be efficiently solved using augmented Lagrangian techniques. The method can deal with weighted or unweighted graphs, as well as multimodal data, where different graphs represent different types of data. The proposed approach is also naturally integrated with collaborative graph inference techniques, solving general network inference problems where the observed variables, possibly coming from different modalities, are not in correspondence. The algorithm is tested and compared with stateoftheart graph matching techniques in both synthetic and real graphs. We also present results on multimodal graphs and applications to collaborative inference of brain connectivity from alignmentfree functional magnetic resonance imaging (fMRI) data. The code is publicly available. 1
Exact Estimation of Multiple Directed Acyclic Graphs
"... Probability models based on directed acyclic graphs (DAGs) are widely used to make inferences and predictions concerning interplay in multivariate systems. In many applications, data are collected from related but nonidentical units whose DAGs may differ but are likely to share many features. Stati ..."
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Probability models based on directed acyclic graphs (DAGs) are widely used to make inferences and predictions concerning interplay in multivariate systems. In many applications, data are collected from related but nonidentical units whose DAGs may differ but are likely to share many features. Statistical estimation for multiple related DAGs appears extremely challenging since all graphs must be simultaneously acyclic. Recent work by Oyen and Lane (2013) avoids this problem by making the strong assumption that all units share a common ordering of the variables. In this paper we propose a novel Bayesian formulation for multiple DAGs and, requiring no assumptions on any ordering of the variables, we prove that the maximum a posteriori estimate is characterised as the solution to an integer linear program (ILP). Consequently exact estimation may be achieved using highly optimised techniques for ILP instances, including constraint propagation and cutting plane algorithms. Our framework permits a complex dependency structure on the collection of units, including group and subgroup structure. This dependency structure can itself be efficiently learned from data and a special case of our methodology provides a novel analogue of kmeans clustering for DAGs. Results on simulated data and fMRI data obtained from multiple subjects are presented.
Towards a MultiSubject Analysis of Neural Connectivity
, 2014
"... Directed acyclic graphs (DAGs) and associated probability models are widely used to model neural connectivity and communication channels. In many experiments, data are collected from multiple subjects whose DAGs may differ but are likely to share many features. The first exact algorithm for estimati ..."
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Directed acyclic graphs (DAGs) and associated probability models are widely used to model neural connectivity and communication channels. In many experiments, data are collected from multiple subjects whose DAGs may differ but are likely to share many features. The first exact algorithm for estimation of multiple related DAGs was recently proposed by Oates et al. (2014); in this letter we present examples and discuss implications of the methodology as applied to the analysis of fMRI data from a multisubject experiment. Elicitation of hyperparameters requires care and we illustrate how this may proceed retrospectively based on technical replicate data. In addition to joint learning of subjectspecific DAGs, we simultaneously estimate relationships between the subjects themselves. A special case of the methodology provides a novel analogue of kmeans clustering of subjects based on their DAG structure. It is anticipated that the exact algorithms discussed here will be widely applicable within neuroscience. 1
Graphical Markov models: overview
, 2015
"... AbstractWe describe how graphical Markov models emerged in the last 40 years, based on three essential concepts that had been developed independently more than a century ago. Sequences of joint or single regressions and their regression graphs are singled out as being the subclass that is best suite ..."
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AbstractWe describe how graphical Markov models emerged in the last 40 years, based on three essential concepts that had been developed independently more than a century ago. Sequences of joint or single regressions and their regression graphs are singled out as being the subclass that is best suited for analyzing longitudinal data and for tracing developmental pathways, both in observational and in intervention studies. Interpretations are illustrated using two sets of data. Furthermore, some of the more recent, important results for sequences of regressions are summarized. 1 Some general and historical remarks on the types of model Graphical models aim to describe in concise form the possibly complex interrelations between a set of variables so that key properties can be read directly o ↵ a graph. The central idea is that each variable is represented by a node in a graph. Any pair of nodes may become coupled, that is joined by an edge. Coupled nodes are also said to be adjacent. For many types of graph, a missing edge represents some form of conditional independence between the pair of variables and an edge present can be interpreted as a corresponding conditional dependence. Because the conditioning set may be empty,
Gaussian tree constraints applied to acoustic linguistic functional data
"... Evolutionary models of languages are usually considered to take the form of trees. With the development of socalled tree constraints the plausibility of the tree model assumptions can be addressed by checking whether the moments of observed variables lie within regions consistent with trees. In ou ..."
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Evolutionary models of languages are usually considered to take the form of trees. With the development of socalled tree constraints the plausibility of the tree model assumptions can be addressed by checking whether the moments of observed variables lie within regions consistent with trees. In our linguistic application, the data set comprises acoustic samples (audio recordings) from speakers of five Romance languages or dialects. We wish to assess these functional data for compatibility with a hereditary tree model at the language level. A novel combination of canonical function analysis (CFA) with a separable covariance structure provides a method for generating a representative basis for the data. This resulting basis is formed of components which emphasize language differences whilst maintaining the integrity of the observational languagegroupings. A previously unexploited Gaussian tree constraint is then applied to componentbycomponent projections of the data to investigate adherence to an evolutionary tree. The results indicate that while a tree model is unlikely to be suitable for modeling all aspects of the acoustic linguistic data, certain features of the spoken Romance languages highlighted by the separableCFA basis may indeed be suitably modeled as a tree.
A Junction Tree Framework for Undirected Graphical Model Selection
, 2013
"... An undirected graphical model is a joint probability distribution defined on an undirected graph G ∗, where the vertices in the graph index a collection of random variables and the edges encode conditional independence relationships amongst random variables. The undirected graphical model selection ..."
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An undirected graphical model is a joint probability distribution defined on an undirected graph G ∗, where the vertices in the graph index a collection of random variables and the edges encode conditional independence relationships amongst random variables. The undirected graphical model selection (UGMS) problem is to estimate the graph G ∗ given observations drawn from the undirected graphical model. This paper proposes a framework for decomposing the UGMS problem into multiple subproblems over clusters and subsets of the separators in a junction tree. The junction tree is constructed using a graph that contains a superset of the edges in G ∗. We highlight three main properties of using junction trees for UGMS. First, different regularization parameters or different UGMS algorithms can be used to learn different parts of the graph. This is possible since the subproblems we identify can be solved independently of each other. Second, under certain conditions, a junction tree based UGMS algorithm can produce consistent results with exponentially fewer observations than the usual requirements of existing algorithms. Third, both our theoretical and experimental results show that the junction tree framework does a significantly better job at finding the weakest edges in a graph than existing methods. This property is a consequence of both the first and second properties. Finally, we note that our framework is independent of the choice of the UGMS algorithm and can be used as a wrapper around standard UGMS algorithms for more accurate graph estimation.
Gaussian Approximation of Collective Graphical Models
"... The Collective Graphical Model (CGM) models a population of independent and identically distributed individuals when only collective statistics (i.e., counts of individuals) are observed. Exact inference in CGMs is intractable, and previous work has explored Markov Chain Monte Carlo (MCMC) and M ..."
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The Collective Graphical Model (CGM) models a population of independent and identically distributed individuals when only collective statistics (i.e., counts of individuals) are observed. Exact inference in CGMs is intractable, and previous work has explored Markov Chain Monte Carlo (MCMC) and MAP approximations for learning and inference. This paper studies Gaussian approximations to the CGM. As the population grows large, we show that the CGM distribution converges to a multivariate Gaussian distribution (GCGM) that maintains the conditional independence properties of the original CGM. If the observations are exact marginals of the CGM or marginals that are corrupted by Gaussian noise, inference in the GCGM approximation can be computed efficiently in closed form. If the observations follow a different noise model (e.g., Poisson), then expectation propagation provides efficient and accurate approximate inference. The accuracy and speed of GCGM inference is compared to the MCMC and MAP methods on a simulated bird migration problem. The GCGM matches or exceeds the accuracy of the MAP method while being significantly faster. 1.