Maximum A Posteriori Inference in Sum-Product Networks

Sum-product networks (SPNs) are a class of probabilistic graphical models that allow tractable marginal inference. However, the maximum a posteriori (MAP) inference in SPNs is NP-hard. We investigate MAP inference in SPNs from both theoretical and algorithmic perspectives. For the theoretical part, we reduce general MAP inference to its special case without evidence and hidden variables; we also show that it is NP-hard to approximate the MAP problem to $2^{n^\epsilon}$ for fixed $0 \leq \epsilon < 1$, where $n$ is the input size. For the algorithmic part, we first present an exact MAP solver that runs reasonably fast and could handle SPNs with up to 1k variables and 150k arcs in our experiments. We then present a new approximate MAP solver with a good balance between speed and accuracy, and our comprehensive experiments on real-world datasets show that it has better overall performance than existing approximate solvers

Detection for 5G-NOMA: An Online Adaptive Machine Learning Approach

Non-orthogonal multiple access (NOMA) has emerged as a promising radio access technique for enabling the performance enhancements promised by the fifth-generation (5G) networks in terms of connectivity, low latency, and high spectrum efficiency. In the NOMA uplink, successive interference cancellation (SIC) based detection with device clustering has been suggested. In the case of multiple receive antennas, SIC can be combined with the minimum mean-squared error (MMSE) beamforming. However, there exists a tradeoff between the NOMA cluster size and the incurred SIC error. Larger clusters lead to larger errors but they are desirable from the spectrum efficiency and connectivity point of view. We propose a novel online learning based detection for the NOMA uplink. In particular, we design an online adaptive filter in the sum space of linear and Gaussian reproducing kernel Hilbert spaces (RKHSs). Such a sum space design is robust against variations of a dynamic wireless network that can deteriorate the performance of a purely nonlinear adaptive filter. We demonstrate by simulations that the proposed method outperforms the MMSE-SIC based detection for large cluster sizes.Comment: Accepted at ICC 201

Training Input-Output Recurrent Neural Networks through Spectral Methods

We consider the problem of training input-output recurrent neural networks (RNN) for sequence labeling tasks. We propose a novel spectral approach for learning the network parameters. It is based on decomposition of the cross-moment tensor between the output and a non-linear transformation of the input, based on score functions. We guarantee consistent learning with polynomial sample and computational complexity under transparent conditions such as non-degeneracy of model parameters, polynomial activations for the neurons, and a Markovian evolution of the input sequence. We also extend our results to Bidirectional RNN which uses both previous and future information to output the label at each time point, and is employed in many NLP tasks such as POS tagging

Visualizing and Understanding Sum-Product Networks

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.Comment: Machine Learning Journal paper (First Online), 24 page