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Visual objects are composed of a recursive hierarchy of perceptual wholes andparts, whose properties, such as shape, reflectance, and color, constitute ahierarchy of intrinsic causal factors of object appearance. However, objectappearance is the compositional consequence of both an object's intrinsic andextrinsic causal factors, where the extrinsic causal factors are related toillumination, and imaging conditions. Therefore, this paper proposes a unifiedtensor model of wholes and parts, and introduces a compositional hierarchicaltensor factorization that disentangles the hierarchical causal structure ofobject image formation, and subsumes multilinear block tensor decomposition asa special case. The resulting object representation is an interpretablecombinatorial choice of wholes' and parts' representations that renders objectrecognition robust to occlusion and reduces training data requirements. Wedemonstrate ourapproach in the context of face recognition by training on anextremely reduced dataset of synthetic images, and report encouragingfaceverification results on two datasets - the Freiburg dataset, andthe LabeledFace in the Wild (LFW) dataset consisting of real world images, thus,substantiating the suitability of our approach for data starved domains.

Compositional Hierarchical Tensor Factorization: Representing Hierarchical Intrinsic and Extrinsic Causal Factors