Which Way Does It Go?

2026-01-11 · Source: Agus’s Substack · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Data Science & Analytics, Mathematics & Computational Sciences · Depth: Advanced, long

Summary

Traditional geometric frameworks, including coordinate, similarity, partition, and manifold geometries, inherently assume symmetry, meaning relations like distance or kernel similarity are bidirectional. However, many real-world systems, such as causality, citation networks, supply chains, and attention mechanisms, exhibit asymmetric or directional structures where order and direction fundamentally alter meaning. This article introduces directional geometry to address these limitations, emphasizing that enforcing symmetry on inherently asymmetric data discards critical information. It highlights the decomposition of any square matrix R into a symmetric part S = ½(R + Rᵀ) and an antisymmetric part A = ½(R − Rᵀ), where A captures the directional component. The post explores directed graphs, asymmetric distances (quasimetrics), causality, and transformer attention as examples where directionality is paramount, demonstrating how symmetric tools fail to capture these essential properties. Practical code examples using the `geomlearn` library illustrate building and analyzing directed graphs, measuring asymmetry, and applying the S+A decomposition.

Key takeaway

For machine learning engineers building models for systems with inherent directionality, such as causal graphs or time-series data, you must explicitly account for asymmetry. Relying solely on symmetric geometric tools will discard crucial information about influence, temporal order, and information flow. Integrate directional concepts like directed graphs, quasimetrics, and the symmetric/antisymmetric matrix decomposition into your feature engineering and model architectures to capture the full complexity of your data.

Key insights

Symmetric geometric models are insufficient for systems where direction, order, or causality are fundamental.

Principles

• Symmetry is a modeling assumption, not a universal fact.
• Directional structure is encoded by the antisymmetric component of a relation.
• Correlation is symmetric; causation is not.

Method

Decompose any square matrix R into symmetric S = ½(R + Rᵀ) and antisymmetric A = ½(R − Rᵀ) parts to reveal directional structure.

In practice

• Use directed graphs for reachability and influence analysis.
• Employ quasimetrics for asymmetric distance calculations.
• Analyze attention matrices for query-key asymmetry.

Topics

• Directional Geometry
• Symmetric-Antisymmetric Decomposition
• Directed Graphs
• Causality
• Attention Mechanisms

Code references

• asudjianto-xml/geomlearn

Best for: AI Scientist, Machine Learning Engineer, Data Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by Agus’s Substack.