Algorithms for Nonnegative Tensor Factorization

Abstract

Nonnegative Matrix Factorization (NMF) is a decomposition which incorporates nonnegativity constraints in both the weights and the bases of the representation. The nonnegativity constraints in NMF correspond better to the intuitive notion of combining parts in order to create a complete object, since the object is represented using only additions of weighted nonnegative basis images. NMF has proven to be very successful for image analysis, especially for imaged-based object representation, discovery of latent object variables and recognition. A drawback of NMF is that it requires the object tensor (with valence more than one) to be vectorized. This procedure may result in information loss since the local object structure is lost due to vectorization. Recently, in order to remedy this disadvantage of NMF methods, Nonnegative Tensor Factorization (NTF) algorithms that can be applied directly to the tensor representation of object collections, have been introduced. In this chapter, we demonstrate how various algorithms are formulated in order to treat arbitrary valence NTFs and we present the various cost functions that have been used for measuring the quality of the approximation. We discuss the optimization procedures that have been used for deriving the factors of the decomposition. Afterwards, we describe how additional constraints can be incorporated into the cost of the decomposition in order to either enhance the sparsity of the solution or to enhance the discrimination between object classes. The presented NTF schemes are described in a manner that can be easily implemented using, in most cases, only matrix multiplications and publicly available packages for treating tensor representations. Finally, we comment on the various applications of NTF algorithms in visual representation and recognition.