Generalized Block Multilinear Factor Analysis: Representing Parts and Wholes with Hierarchical Block Tensors

M. Alex O. Vasilescu
University of California, Los Angeles (UCLA)

By adhering to the dictum, “No causation without manipulation (treatment, intervention)”, cause and effect data analysis represents changes in observed data in terms of changes in the causal factors that generate the data. When causal factors are not amenable for active manipulation in the real world due to current technical limitations or ethical considerations, a counterfactual approach performs an intervention on the model of data formation. In the case of object representation or activity (temporal object) representation, varying object parts is generally unfeasible whether they be spatial or temporal. Multilinear algebra, the algebra of higher order tensors, is a suitable and transparent framework for disentangling the causal factors of data formation. Learning a part-based intrinsic causal factor representation in a multilinear framework requires applying a set of interventions on a part-based multilinear model. This tutorial discusses a unified multilinear model of wholes and parts that generalizes block tensor decomposition. We derive a hierarchical block multilinear factorization, the M-mode Block SVD, that computes a disentangled representation of the causal factors by optimizing simultaneously across the entire object hierarchy. Given computational efficiency considerations, we introduce an incremental bottom-up computational alternative, the Incremental M-mode Block SVD,that employs the lower level abstractions, the part representations,to represent the higher level of abstractions, the parent wholes.This incremental computational approach may also be employed to update the causal model parameters when data becomes available incrementally. The resulting object representation is an interpretable combinatorial choice of wholes’ and parts’ representations that renders object recognition robust to occlusion and reduces training data requirements. Tensor factor analysis is a data agnostic framework and it is well suited for data starved domains.

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