turkmath.org

Matrix decompositions are widely used for developing models and expressing algorithms in signal processing, machine learning, data mining and related fields. This view, where the central object is a matrix has proven to be quite fruitful as it enables complex algorithms to be implemented using simple but powerful primitives, supported by a wide availability of tools for numerical computation. Yet, there are many situations when a matrix based description may become insufficient, or at best obscures the data model or the simplicity of an algorithm. In this talk, we will argue that multiway arrays with several indices, that we call tentatively as tensors, provide a natural framework for developing useful models for modern datasets as well as efficient algorithms for data processing. We express a tensor factorization models using a graph formalism reminiscent to probabilistic graphical models, reminiscent to tensor networks. The setting provides a structured and efficient approach that enables very easy development of application specific custom models, as well as algorithms for the so called coupled (collective) factorizations where an arbitrary set of tensors are factorized simultaneously with shared factors. Extensions to full Bayesian inference for model selection, via variational approximations or Monte Carlo methods are also feasible. We will also mention parallel and distributed inference algorithms and privacy preserving approaches to highlight more recent research directions. We will illustrate the approach in various applications.