Scalable Boolean Tensor Factorizations using Random Walks

Tensors are becoming increasingly common in data mining, and consequently, tensor factorizations are becoming more and more important tools for data miners. When the data is binary, it is natural to ask if we can factorize it into binary factors while simultaneously making sure that the reconstructed tensor is still binary. Such factorizations, called Boolean tensor factorizations, can provide improved interpretability and find Boolean structure that is hard to express using normal factorizations. Unfortunately the algorithms for computing Boolean tensor factorizations do not usually scale well. In this paper we present a novel algorithm for finding Boolean CP and Tucker decompositions of large and sparse binary tensors. In our experimental evaluation we show that our algorithm can handle large tensors and accurately reconstructs the latent Boolean structure

Clustering Boolean Tensors

Tensor factorizations are computationally hard problems, and in particular, are often significantly harder than their matrix counterparts. In case of Boolean tensor factorizations -- where the input tensor and all the factors are required to be binary and we use Boolean algebra -- much of that hardness comes from the possibility of overlapping components. Yet, in many applications we are perfectly happy to partition at least one of the modes. In this paper we investigate what consequences does this partitioning have on the computational complexity of the Boolean tensor factorizations and present a new algorithm for the resulting clustering problem. This algorithm can alternatively be seen as a particularly regularized clustering algorithm that can handle extremely high-dimensional observations. We analyse our algorithms with the goal of maximizing the similarity and argue that this is more meaningful than minimizing the dissimilarity. As a by-product we obtain a PTAS and an efficient 0.828-approximation algorithm for rank-1 binary factorizations. Our algorithm for Boolean tensor clustering achieves high scalability, high similarity, and good generalization to unseen data with both synthetic and real-world data sets

Retrospective Higher-Order Markov Processes for User Trails

Users form information trails as they browse the web, checkin with a geolocation, rate items, or consume media. A common problem is to predict what a user might do next for the purposes of guidance, recommendation, or prefetching. First-order and higher-order Markov chains have been widely used methods to study such sequences of data. First-order Markov chains are easy to estimate, but lack accuracy when history matters. Higher-order Markov chains, in contrast, have too many parameters and suffer from overfitting the training data. Fitting these parameters with regularization and smoothing only offers mild improvements. In this paper we propose the retrospective higher-order Markov process (RHOMP) as a low-parameter model for such sequences. This model is a special case of a higher-order Markov chain where the transitions depend retrospectively on a single history state instead of an arbitrary combination of history states. There are two immediate computational advantages: the number of parameters is linear in the order of the Markov chain and the model can be fit to large state spaces. Furthermore, by providing a specific structure to the higher-order chain, RHOMPs improve the model accuracy by efficiently utilizing history states without risks of overfitting the data. We demonstrate how to estimate a RHOMP from data and we demonstrate the effectiveness of our method on various real application datasets spanning geolocation data, review sequences, and business locations. The RHOMP model uniformly outperforms higher-order Markov chains, Kneser-Ney regularization, and tensor factorizations in terms of prediction accuracy