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Analogical Reasoning Within a Conceptual Hyperspace

Howard Goldowsky, Vasanth Sarathy

Abstract

We propose an approach to analogical inference that marries the neuro-symbolic computational power of complex-sampled hyperdimensional computing (HDC) with Conceptual Spaces Theory (CST), a promising theory of semantic meaning. CST sketches, at an abstract level, approaches to analogical inference that go beyond the standard predicate-based structure mapping theories. But it does not describe how such an approach can be operationalized. We propose a concrete HDC-based architecture that computes several types of analogy classified by CST. We present preliminary proof-of-concept experimental results within a toy domain and describe how it can perform category-based and property-based analogical reasoning.

Analogical Reasoning Within a Conceptual Hyperspace

Abstract

We propose an approach to analogical inference that marries the neuro-symbolic computational power of complex-sampled hyperdimensional computing (HDC) with Conceptual Spaces Theory (CST), a promising theory of semantic meaning. CST sketches, at an abstract level, approaches to analogical inference that go beyond the standard predicate-based structure mapping theories. But it does not describe how such an approach can be operationalized. We propose a concrete HDC-based architecture that computes several types of analogy classified by CST. We present preliminary proof-of-concept experimental results within a toy domain and describe how it can perform category-based and property-based analogical reasoning.

Paper Structure

This paper contains 16 sections, 13 equations, 4 figures, 4 algorithms.

Table of Contents

  1. Introduction and Motivation
  2. Background
  3. Conceptual Spaces
  4. Hyperdimensional Computing (HDC)
  5. Proposed Approach: Conceptual Hyperspaces
  6. Problem Definition
  7. Analogical Mapping Algorithm
  8. The Conceptual Hyperspace
  9. Encoding Concepts in Hypervectors
  10. Analogical Mapping Algorithm for Category-based analogies
  11. Solving a Property-based Analogy
  12. Discussion
  13. The Importance of HDC as a Modeling Tool
  14. The Origin of Property Dimensions
  15. Exploration of Kernel Functions
  16. ...and 1 more sections

Figures (4)

  • Figure 1: Color domain showing the location of prototype points for PURPLE, BLUE, ORANGE, and YELLOW.
  • Figure 2: A complex-sampled hypervector of dimension $d=1000$ samples initialized to a Gaussian phase distribution around the unit circle: $\phi \sim \mathcal{N}(\mu,\,\sigma^{2})$, where here $\mu = 0$ and $\sigma = \frac{\pi}{7}$. We experimented with various $\sigma$ values.
  • Figure 3: Result of one sample-wise binding operation between two complex hypervectors.
  • Figure 4: Result of fractional power encoding for one sample within a complex hypervector.

Theorems & Definitions (1)

  • Definition 1