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New parallelogram rule extends proportional analogies to Riemannian manifolds

By PulseAugur Editorial · [1 sources] ·

Researchers have introduced a new method for formalizing proportional analogies within Riemannian domains, extending the parallelogram rule traditionally used in Euclidean spaces. This novel approach is demonstrated on various non-Euclidean manifolds, including spheres, shape spaces, and manifolds of probability distributions. The work aims to broaden the application of proportional analogies beyond symbolic and vector-based domains. AI

IMPACT This research could enable more sophisticated analogical reasoning in AI systems operating in non-Euclidean data spaces.

RANK_REASON Academic paper published on arXiv. [lever_c_demoted from research: ic=1 ai=1.0]

Read on arXiv cs.AI →

AI-generated summary · Google Gemini · from 1 sources. How we write summaries →

New parallelogram rule extends proportional analogies to Riemannian manifolds

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Academic paper published on arXiv. [lever_c_demoted from research: ic=1 ai=1.0]
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COVERAGE [1]

  1. arXiv cs.AI TIER_1 English(EN) · Pierre-Alexandre Murena, Marcelo Hartmann ·

    A Generalized Parallelogram Rule for Proportional Analogies on Riemannian Manifolds

    arXiv:2608.14220v1 Announce Type: new Abstract: Analogies are quaternary relations of the form "a is to b as c is to d", usually denoted a : b :: c : d. This notion is formalized in particular with the notion of proportional analogy, which imposes some constraints on the valid an…

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