Learning the Multilinear Structure of Visual Data

Statistical decomposition methods are of paramount importance in discovering the modes of variations of visual data. Probably the most prominent linear decomposition method is the Principal Component Analysis (PCA), which discovers a single mode of variation in the data. However, in practice, visual data exhibit several modes of variations. For instance, the appearance of faces varies in identity, expression, pose etc. To extract these modes of variations from visual data, several supervised methods, such as the TensorFaces, that rely on multilinear (tensor) decomposition (e.g., Higher Order SVD) have been developed. The main drawbacks of such methods is that they require both labels regarding the modes of variations and the same number of samples under all modes of variations (e.g., the same face under different expressions, poses etc.). Therefore, their applicability is limited to well-organised data, usually captured in well-controlled conditions. In this paper, we propose the first general multilinear method, to the best of our knowledge, that discovers the multilinear structure of visual data in unsupervised setting. That is, without the presence of labels. We demonstrate the applicability of the proposed method in two applications, namely Shape from Shading (SfS) and expression transfer.