Floriana Esposito

Alejandro Molina, Antonio Vergari, Nicola Di Mauro et al.

While all kinds of mixed data -from personal data, over panel and scientific data, to public and commercial data- are collected and stored, building probabilistic graphical models for these hybrid domains becomes more difficult. Users spend significant amounts of time in identifying the parametric form of the random variables (Gaussian, Poisson, Logit, etc.) involved and learning the mixed models. To make this difficult task easier, we propose the first trainable probabilistic deep architecture for hybrid domains that features tractable queries. It is based on Sum-Product Networks (SPNs) with piecewise polynomial leave distributions together with novel nonparametric decomposition and conditioning steps using the Hirschfeld-Gebelein-Rényi Maximum Correlation Coefficient. This relieves the user from deciding a-priori the parametric form of the random variables but is still expressive enough to effectively approximate any continuous distribution and permits efficient learning and inference. Our empirical evidence shows that the architecture, called Mixed SPNs, can indeed capture complex distributions across a wide range of hybrid domains.

Antonio Vergari, Nicola Di Mauro, Floriana Esposito

Sum-Product Networks (SPNs) are recently introduced deep tractable probabilistic models by which several kinds of inference queries can be answered exactly and in a tractable time. Up to now, they have been largely used as black box density estimators, assessed only by comparing their likelihood scores only. In this paper we explore and exploit the inner representations learned by SPNs. We do this with a threefold aim: first we want to get a better understanding of the inner workings of SPNs; secondly, we seek additional ways to evaluate one SPN model and compare it against other probabilistic models, providing diagnostic tools to practitioners; lastly, we want to empirically evaluate how good and meaningful the extracted representations are, as in a classic Representation Learning framework. In order to do so we revise their interpretation as deep neural networks and we propose to exploit several visualization techniques on their node activations and network outputs under different types of inference queries. To investigate these models as feature extractors, we plug some SPNs, learned in a greedy unsupervised fashion on image datasets, in supervised classification learning tasks. We extract several embedding types from node activations by filtering nodes by their type, by their associated feature abstraction level and by their scope. In a thorough empirical comparison we prove them to be competitive against those generated from popular feature extractors as Restricted Boltzmann Machines. Finally, we investigate embeddings generated from random probabilistic marginal queries as means to compare other tractable probabilistic models on a common ground, extending our experiments to Mixtures of Trees.

Antonio Vergari, Nicola Di Mauro, Floriana Esposito

Probabilistic models learned as density estimators can be exploited in representation learning beside being toolboxes used to answer inference queries only. However, how to extract useful representations highly depends on the particular model involved. We argue that tractable inference, i.e. inference that can be computed in polynomial time, can enable general schemes to extract features from black box models. We plan to investigate how Tractable Probabilistic Models (TPMs) can be exploited to generate embeddings by random query evaluations. We devise two experimental designs to assess and compare different TPMs as feature extractors in an unsupervised representation learning framework. We show some experimental results on standard image datasets by applying such a method to Sum-Product Networks and Mixture of Trees as tractable models generating embeddings.

Nicola Di Mauro, Floriana Esposito

Dealing with structured data needs the use of expressive representation formalisms that, however, puts the problem to deal with the computational complexity of the machine learning process. Furthermore, real world domains require tools able to manage their typical uncertainty. Many statistical relational learning approaches try to deal with these problems by combining the construction of relevant relational features with a probabilistic tool. When the combination is static (static propositionalization), the constructed features are considered as boolean features and used offline as input to a statistical learner; while, when the combination is dynamic (dynamic propositionalization), the feature construction and probabilistic tool are combined into a single process. In this paper we propose a selective propositionalization method that search the optimal set of relational features to be used by a probabilistic learner in order to minimize a loss function. The new propositionalization approach has been combined with the random subspace ensemble method. Experiments on real-world datasets shows the validity of the proposed method.

Claudio Taranto, Nicola Di Mauro, Floriana Esposito

The probabilistic graphs framework models the uncertainty inherent in real-world domains by means of probabilistic edges whose value quantifies the likelihood of the edge existence or the strength of the link it represents. The goal of this paper is to provide a learning method to compute the most likely relationship between two nodes in a framework based on probabilistic graphs. In particular, given a probabilistic graph we adopted the language-constraint reachability method to compute the probability of possible interconnections that may exists between two nodes. Each of these connections may be viewed as feature, or a factor, between the two nodes and the corresponding probability as its weight. Each observed link is considered as a positive instance for its corresponding link label. Given the training set of observed links a L2-regularized Logistic Regression has been adopted to learn a model able to predict unobserved link labels. The experiments on a real world collaborative filtering problem proved that the proposed approach achieves better results than that obtained adopting classical methods.