Directed interpretable discovery in tensors with sparse projection

Abstract

Tensors or multiway arrays are useful constructs capable of representing complex graphs, spatio-temporal data and multi-source data. Analyzing such complex data requires enforcing simplicity (such as sparsity) so as to give meaningful insights. Sparse tensor decomposition typically requires adding a penalty term in addition to the reconstruction error term to the optimization problem which provides a number of challenges. Not only is tuning the weights on the terms time consuming, but the resulting sparse decomposition typically has a nonuniform distribution of sparsity. Such decomposition is undesirable in some datasets (such as fMRI scans) where we want each factor to have similar sparsity for easier interpretation. In this paper, we propose an alternative method that can directly obtain a sparse, nonnegative and interpretable decomposition without the need for tuning. We allow the user to specify the exact amount of sparsity and where the sparsity should be. This is achieved by augmenting the alternating nonnegative least squares (ANLS) algorithm with a projection step rather than imposing a penalty term on the objective function. We demonstrate our works usefulness in finding interpretable features for real world problems in fMRI scan data and face image analysis without the need to pre-process the data.