In this paper, we establish some theoretical connections between Sum-Product Networks (SPNs) and Bayesian Networks (BNs). We prove that every SPN can be converted into a BN in linear time and space in terms of the network size. The key insight is to use Algebraic Decision Diagrams (ADDs) to compactly represent the local conditional probability distributions at each node in the resulting BN by exploiting context-specific independence (CSI). The generated BN has a simple directed bipartite graphical structure. We show that by applying the Variable Elimination algorithm (VE) to the generated BN with ADD representations, we can recover the original SPN where the SPN can be viewed as a history record or caching of the VE inference process. To help state the proof clearly, we introduce the notion of normal SPN and present a theoretical analysis of the consistency and decomposability properties. We conclude the paper with some discussion of the implications of the proof and establish a connection between the depth of an SPN and a lower bound of the tree-width of its corresponding BN.
Since their introduction by , SPNs have generated a lot of interest as a tractable class of models for probabilistic inference in machine learning. Discriminative learning techniques for SPNs have been proposed and applied to image classification @cite_14 . Later, automatic structure learning algorithms were developed to build tree-structured SPNs directly from data @cite_25 @cite_6 @cite_12 @cite_9 . SPNs have also been applied to various fields and have generated promising results, including activity modeling @cite_16 , speech modeling @cite_3 and language modeling @cite_5 . Theoretical work investigating the influence of the depth of SPNs on expressiveness exists @cite_26 , but is quite limited. As discussed later, our results reinforce previous theoretical results about the depth of SPNs and provide further insights about the structure of SPNs by examining the structure of equivalent BNs.
@cite_14 @cite_26 @cite_9 @cite_6 @cite_3 @cite_5 @cite_16 @cite_25 @cite_12
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This paper argues that the judicial use of formal language theory and grammatical inference are invaluable tools in understanding how deep neural networks can and cannot represent and learn long-term dependencies in temporal sequences. Learning experiments were conducted with two types of Recurrent Neural Networks (RNNs) on six formal languages drawn from the Strictly Local (SL) and Strictly Piecewise (SP) classes. The networks were Simple RNNs (s-RNNs) and Long Short-Term Memory RNNs (LSTMs) of varying sizes. The SL and SP classes are among the simplest in a mathematically well-understood hierarchy of subregular classes. They encode local and long-term dependencies, respectively. The grammatical inference algorithm Regular Positive and Negative Inference (RPNI) provided a baseline. According to earlier research, the LSTM architecture should be capable of learning long-term dependencies and should outperform s-RNNs. The results of these experiments challenge this narrative. First, the LSTMs' performance was generally worse in the SP experiments than in the SL ones. Second, the s-RNNs out-performed the LSTMs on the most complex SP experiment and performed comparably to them on the others. The need for feasible inference in Probabilistic Graphical Models PGMs has lead to tractable models like Sum-Product Networks SPNs. Their highly expressive power and their ability to provide exact and tractable inference make them very attractive for several real world applications, from computer vision to NLP. Recently, great attention around SPNs has focused on structure learning, leading to different algorithms being able to learn both the network and its parameters from data. Here, we enhance one of the best structure learner, LearnSPN, aiming to improve both the structural quality of the learned networks and their achieved likelihoods. Our algorithmic variations are able to learn simpler, deeper and more robust networks. These results have been obtained by exploiting some insights in the building process done by LearnSPN, by hybridizing the network adopting tree-structured models as leaves, and by blending bagging estimations into mixture creation. We prove our claims by empirically evaluating the learned SPNs on several benchmark datasets against other competitive SPN and PGM structure learners. Recurrent neural nets (RNN) and convolutional neural nets (CNN) are widely used on NLP tasks to capture the long-term and local dependencies, respectively. Attention mechanisms have recently attracted enormous interest due to their highly parallelizable computation, significantly less training time, and flexibility in modeling dependencies. We propose a novel attention mechanism in which the attention between elements from input sequence(s) is directional and multi-dimensional (i.e., feature-wise). A light-weight neural net, "Directional Self-Attention Network (DiSAN)", is then proposed to learn sentence embedding, based solely on the proposed attention without any RNN CNN structure. DiSAN is only composed of a directional self-attention with temporal order encoded, followed by a multi-dimensional attention that compresses the sequence into a vector representation. Despite its simple form, DiSAN outperforms complicated RNN models on both prediction quality and time efficiency. It achieves the best test accuracy among all sentence encoding methods and improves the most recent best result by 1.02 on the Stanford Natural Language Inference (SNLI) dataset, and shows state-of-the-art test accuracy on the Stanford Sentiment Treebank (SST), Multi-Genre natural language inference (MultiNLI), Sentences Involving Compositional Knowledge (SICK), Customer Review, MPQA, TREC question-type classification and Subjectivity (SUBJ) datasets. The key limiting factor in graphical model inference and learning is the complexity of the partition function. We thus ask the question: what are general conditions under which the partition function is tractable? The answer leads to a new kind of deep architecture, which we call sum-product networks (SPNs). SPNs are directed acyclic graphs with variables as leaves, sums and products as internal nodes, and weighted edges. We show that if an SPN is complete and consistent it represents the partition function and all marginals of some graphical model, and give semantics to its nodes. Essentially all tractable graphical models can be cast as SPNs, but SPNs are also strictly more general. We then propose learning algorithms for SPNs, based on backpropagation and EM. Experiments show that inference and learning with SPNs can be both faster and more accurate than with standard deep networks. For example, SPNs perform image completion better than state-of-the-art deep networks for this task. SPNs also have intriguing potential connections to the architecture of the cortex.