Computer Vision News - June 2016

What is a tensor ? Tensors are generalization of scalars and vectors ; a scalar is a tensor of order zero. Vectors and matrices are first- and second-order tensors, respectively. Arrays with three or more dimensions are called higher-order tensors. 12 Computer Vision News Tool Tensorlab: Matlab package for tensor computations Example of a 3D Tensor: 165 face images 320x243 pixels each What is TensorLab and what it is good for? TensorLab is a relatively new and updated Matlab toolbox for tensor computations and optimization . It provides algorithms for (coupled) tensor decompositions of dense, sparse, incomplete and structured tensors with the possibility of imposing structure on the factors, as well as visualization methods. TensorLab requires MATLAB 7.9 (R2009b) or higher and can be easily downloaded and installed from this link . Formally, a Tensor decomposition (also known as Canonical Polyadic Decomposition or CPD ) is a low rank approximation of a tensor with a sum of rank-one tensors (i.e. vectors). The CPD model was independently suggested by Harshman and by Carroll and Chang in 1970 (Carroll & Chang, 1970; Harshman, 1970). In contrast to the one score and one loading in PCA, a component in CPD contains one score and several loading vectors. These components are then used to interpret the underlying structure of the data. Such decompositions given by the CPD model may be regarded as generalizations of Principal Component Analysis (PCA) and Singular Value Decomposition (SVD). Algorithms for modeling Tensor decompositions are iterative and are commonly based on Alternating Least Squares (ALS). Where N is the number of modes in the model, R is the number of components (factors) in the model, and E is the residual error term. The reader interested in more rigorous details is invited to refer to this link . In our demonstration below, R and N will be set to 3 and the residual error term E will be ignored. Thus, the decomposition can be illustrated as follows: Mathematically, for a given tensor ∈ ℛ 1 , 2 … the CPD model is defined as: