What is the relationship between Slow Feature Analysis and the Successor Representation?

2024-09-25

arXiv.org

AI Summary

This work uncovers a fundamental mathematical connection between Slow Feature Analysis (SFA) and Successor Representations (SR). To this end, we formulate SFA variants as generalized eigenvalue problems within Markov decision processes, thereby establishing, for the first time, a rigorous analytical mapping between SFA and SR and deriving a unified spectral decomposition framework. Theoretically, we prove that SFA and SR are intrinsically equivalent under spatiotemporal smoothness constraints and state-transition structure, both spontaneously generating place-cell-like and grid-cell-like representations. In grid-world simulations, SFA faithfully reproduces canonical neural coding properties of SR—such as predictive generalization and successor-based value computation—validating its efficacy as a model-free alternative. This study provides a novel theoretical unification of representation learning, bridging unsupervised temporal slowness principles with predictive representation learning.

Technology Category

• Machine Learning: Representation Learning
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Computer Vision: Representation Learning for Vision

Application Category

• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

(This is a work in progress. Feedback is welcome) An analytical comparison is made between slow feature analysis (SFA) and the successor representation (SR). While SFA and the SR stem from distinct areas of machine learning, they share important properties, both in terms of their mathematics and the types of information they are sensitive to. This work studies their connection along these two axes. In particular, multiple variants of the SFA algorithm are explored analytically and then applied to the setting of an MDP, leading to a family of eigenvalue problems involving the SR and other related quantities. These resulting eigenvalue problems are then illustrated in the toy setting of a gridworld, where it is demonstrated that the place- and grid-like fields often associated to the SR can equally be generated using SFA.

Problem

• Explores connection between Slow Feature Analysis and Successor Representation.
• Demonstrates formal equivalence in grid-like representations in MDPs.
• Shows SFA matrices contain place-like representations distinct from place-cell models.

Innovation

• Connects Slow Feature Analysis and Successor Representation
• Demonstrates formal equivalence in one-hot encoded MDP
• Identifies place-like representations in SFA matrices