A Tensor Approximation Approach to
Hongcheng Wang and Narendra Ahuja
Dimensionality reduction has recently been extensively studied for computer vision applications.
We present a novel multilinear algebra based approach to reduced dimensionality representation of
multidimensional data, such as image ensembles, video sequences and volume data. Before reducing
the dimensionality we do not convert it into a vector as is done by traditional dimensionality reduction
techniques like PCA. Our approach works directly on the multidimensional form of the data (matrix
in 2D and tensor in higher dimensions) to yield what we call a Datum-as-Is representation. This
helps exploit spatio-temporal redundancies with less information loss than image-as-vector methods.
An efficient rank-R tensor approximation algorithm is presented to approximate higher-order tensors.
We show that rank-R tensor approximation using Datum-as-Is representation generalizes many existing
approaches that use image-as-matrix representation, such as generalized low rank approximation of
matrices (GLRAM) , rank-one decomposition of matrices (RODM)  and rank-one decomposition
of tensors (RODT) . Our approach yields the most compact data representation among all known
image-as-matrix methods. In addition, we propose another rank-R tensor approximation algorithm based
on slice projection of third-order tensors, which needs fewer iterations for convergence for the important
special case of 2D image ensembles, e.g., video. We evaluated the performance of our approach vs. other