Learning in Infinitesimal Non-Compositional Sketches

2026-07-16 · Source: Machine Learning · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Expert, quick

Summary

Learning in Infinitesimal Non-Compositional Sketches (LINCS) is a categorical framework developed to address non-compositionality in machine learning problems. Non-compositionality is defined as the failure of diagrams to factor through quotient sketches, lifted to the tangent category setting, rather than an arithmetic error. Machine learning problems are specified as sketches, which are graphs with commutativity conditions, limit cones, and colimit cocones, generalizing scalarized loss functions. LINCS defines the base defect as an obstruction to factorization and then applies a tangent lift to obtain the LINCS obstruction, assessing if infinitesimal perturbations preserve compositionality. The framework introduces Tangent Learning Sketches and an INC endofunctor, creating a tower of factorization problems. Machine learning is formulated as searching for a coalgebraic fixed point where successive tangent unfoldings stabilize, with existence proven via the Aczel--Mendler theorem. Experimental evaluation is underway across deep learning, large language models, and reinforcement learning.

Key takeaway

For AI scientists exploring foundational ML theory, this categorical framework offers a novel perspective on non-compositionality, reframing learning as a search for a coalgebraic fixed point. You should consider how LINCS's tangent lift approach could inform the design of more robust models by addressing fundamental factorization failures. This theoretical shift provides a new lens for analyzing model behavior beyond traditional loss function scalarization.

Key insights

LINCS frames machine learning as repairing non-compositionality using categorical tangent lifts to find a coalgebraic fixed point.

Principles

• Non-compositionality is a universal factorization problem.
• Infinitesimal perturbations define compositionality constraints.
• ML can be a search for a coalgebraic fixed point.

Method

LINCS defines learning problems as sketches, identifies non-compositionality via factorization obstruction, then applies a tangent lift to iterate this process, seeking a stable coalgebraic fixed point.

In practice

• Apply LINCS to deep learning architectures.
• Evaluate LINCS in large language models.
• Test LINCS in reinforcement learning settings.

Topics

• Categorical Frameworks
• Non-Compositionality
• Machine Learning Theory
• Tangent Categories
• Coalgebraic Fixed Points
• Deep Learning Applications

Best for: Research Scientist, AI Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by Machine Learning.