Tensor Decompositions for Very Large Scale Problems

Modern applications such as neuroscience, text mining, and large-scale social networks generate massive amounts of data with multiple aspects and high dimensionality. Tensors (i.e., multi-way arrays) provide a natural representation for such massive data. Consequently, tensor decompositions and factorizations are emerging as novel and promising tools for exploratory analysis of multidimensional data in diverse disciplines including social networks analysis. Tensor decompositions, especially DEDICOM, PARAFAC and TUCKER models are important tools for exploration of social networks by capturing multi-linear and multi-aspects structures in massive higher-order data-sets. However, most of the existing algorithms for tensor decompositions and factorizations are not suitable for large-scale problems since usually memory overflows occur during the tensor decomposition process [1], [2]. In this paper we propose novel algorithms for 3D nonnegative DEDICOM and TUCKER models which are suitable for very large scale problems. In fact, our algorithms allow us to work with much larger dense tensors that were too big to handle before. In our approach, we first decompose three-way data into Tucker-3 or Tucker-2, and next we retrieve the desired DEDICOM matrices from the Tucker factors and core tensor. For very large-scale problems we developed novel divide and conquer procedure which splits a data tensor to sub-tensors and next performs TUCKER-3 for the each sub-tensor in parallel way, and next reassemble the results to estimate desired factor (component) matrices of the DEDICOM model. We illustrate validity and high performance of the proposed algorithms by two illustrative examples.