Face recognition and identification are very important applications in machine learning. Due to the increasing amount of available data, traditional approaches based on matricization and matrix PCA methods can be difficult to implement. Moreover, the tensorial approaches are a natural choice, due to the mere structure of the databases, for example in the case of color images. Nevertheless, even though various authors proposed factorization strategies for tensors, the size of the considered tensors can pose some serious issues. Indeed, the most demanding part of the computational effort in recognition or identification problems resides in the training process. When only a few features are needed to construct the projection space, there is no need to compute a SVD on the whole data. Two versions of the tensor Golub–Kahan algorithm are considered in this manuscript, as an alternative to the classical use of the tensor SVD which is based on truncated strategies. In this paper, we consider the Tensor Tubal Golub–Kahan Principal Component Analysis method which purpose it to extract the main features of images using the tensor singular value decomposition (SVD) based on the tensor cosine product that uses the discrete cosine transform. This approach is applied for classification and face recognition and numerical tests show its effectiveness.

In this work we revisited some properties of Sylvester Hadamard matrices of order 2^k. Based onlyon the existence of a base from which any Sylvester Hadamard matrix can be constructed, we prove that theirrows (columns) are closed under addition and that the numbers of sign interchanges along any row (column)give all integers 0, 1, . . . , 2^k − 1 in some order. These two properties can be used to check if an Hadamardmatrix is equivalent to a Sylvester Hadamard matrix.