CausalX: Causal Explanations and Block Multilinear Factor Analysis

TL;DR

The paper addresses disentangling data-formation causal factors in visual data when real-world manipulation is limited by proposing a hierarchical multilinear tensor framework. It introduces a unified model of wholes and parts and derives the M-mode Block SVD to compute disentangled causal representations across the object hierarchy. An Incremental M-mode Block SVD is proposed for bottom-up, online updates using lower-level part representations to inform higher-level abstractions. The approach yields interpretable, combinatorial intrinsic causal factors that improve occlusion robustness and reduce data requirements, advancing trustworthy AI in vision by enabling causal explanations and manipulable representations.

Abstract

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. When causal factors are not amenable for active manipulation in the real world due to current technological 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 and/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 representations in a multilinear framework requires applying a set of interventions on a part-based multilinear model. We propose a unified multilinear model of wholes and parts. 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 intrinsic causal factor representations related to an object's recursive hierarchy of wholes and parts that renders object recognition robust to occlusion and reduces training data requirements.

Figures (2)

• Figure 1: Data tensor, ${\mathbf{\mathcal{D}}}$, expressed in terms of a hierarchical data tensor, ${\mathbf{\mathcal{D}}}_{\hbox{\tiny${\mathbf{\mathcal{H}}}$}}$, a mathematical instantiation of a tree data structure where ${\mathbf{\mathcal{D}}} = {\mathbf{\mathcal{D}}}_{\hbox{\tiny${\mathbf{\mathcal{H}}}$}} \times_{\hbox{\tiny1}} {\mathbf{I}}_{\hbox{\tiny1x}} \dots \times_{\hbox{\tinyc}} {\mathbf{I}}_{\hbox{\tinycx}} \dots {\mathbf{I}}_{\hbox{\tinyCx}}$, versus a bag of independent parts/sub-parts, or a data tensor with a reparameterized measurement mode in terms of regions and sub-regions. An object hierarchy may be based on adaptive quad/triangle based subdivision of various depths vasilescu92, or a set of perceptual parts of arbitrary shape, size and location. Images of non-articulated objects are best expressed with hierarchical data tensors that have a partially compositional form, where all the parts share the same extrinsic causal factor representations, Fig. \ref{['fig:Block-Tucker-Base-Case']}b. Images of objects with articulated parts are best expressed in terms of hierarchical data tensors that are fully compositional in the causal factors, Fig. \ref{['fig:Block-Tucker-Base-Case']}c. Images of non-articulated objects may also be represented by a fully compositional hierarchical data tensor, as depicted by the TensorTrinity example above.
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