表題番号：2025R-034
日付：2026/03/31
研究課題実数間類推関係：数学的定義と機械学習における実用的応用
| 研究者所属（当時） | 資格 | 氏名 | |
|---|---|---|---|
| （代表者） | 理工学術院 大学院情報生産システム研究科 | 教授 | ルパージュ イヴ |
| （連携研究者） | University of Lisbon, Portugal | 教授 | Miguel Couceiro |
- 研究成果概要
- Background:
Analogy, through analogy-making or the analogical inference principle, is a fundamental cognitive process. Nowadays AI uses vector representations which necessitate parallel processing or GPU architectures. Until now, analogy has been confined to symbolic AI with Boolean or arithmetic analogy. The PI recently proposed a formalisation of analogy between numbers that has the potential to be applied to vector representations used in AI.
Goals:
(a) Consolidate the theoretical foundations of numerical analogy through adequate definitions and necessary formalization for machine learning;
(b) Demonstrate the efficiency of numerical analogy in several AI tasks using vector representations.
Methods:
(a.1) Propose extensions of formalization, study relations with machine learning, determine bounds and approximations for analogical powers (main feature in numerical analogy);
(a.2) Design fast approximation algorithms and implement vectorization for GPU devices;
(b.1) Demonstrate usefulness of numerical analogy in static word embedding spaces;
(b.2) Demonstrate potential of numerical analogy in image processing.
Results:
The grant enabled significant progress in the formalization of numerical analogy: crucial results have been obtained on bounds, approximations, and learnable functions. Fast computation of analogical powers, supported by these theoretical results, allowed to get promising results in lexical representation and in image reconstruction and classification.
All these advances point to the possibility of fundamental breakthroughs in analogical inference and its application to core machine learning tasks such as classification. In particular, they open the door to applications in large language models for more effective language learning from less training data.
(a.1.i) Characterization of a class of functions compatible with the analogical inference principle, results presented in paper [1] (published);
(a.1.ii) Premiminary study of the relation between analogy and PAC-learnability, first results presented in paper [5] (submitted).
(a.2.i) Mathematical results in determination of bounds, results presented in paper [2] (submitted);
(a.2.ii) Fast approximate computation of analogical powers
- Determination of analytical formulae, inspection of quality of approximations, results presented in paper [3] (submitted);
- Tabulation techniques for the approximation of analogical powers, report under preparation.
(b.1) Successful demonstration of the efficiency of numerical analogy as a tool for the analysis of word analogies and for the enforcement of analogical structure in word representations through the design and implementation of analogy-based loss functions, results presented in paper [6] and journal paper in preparation.
(b.2) Successful demonstration of the potential of numerical analogy in image reconstruction and image classification through the definition of analogical pooling, results presented in paper [4] (submitted).