Geometric Frameworks in Machine Learning

The field of machine learning is moving towards a greater emphasis on geometric frameworks, which provide a more interpretable and efficient way of understanding complex data. This shift is driven by the need to better understand how neural networks learn to perform discrete computations on continuous manifolds. Recent work has focused on developing geometric frameworks that can decompose network computation into discretising continuous input features and performing logical operations on these discretised variables. Notably, the integration of differential geometry with machine learning has led to the development of novel frameworks such as Fiber Bundle Networks, which provide clear geometric interpretability. Another significant area of research is the development of self-supervised learning methods that can efficiently capture class identity in physiological time-series data. These methods have been shown to be effective in learning representations that can distinguish semantic classes and improve transfer learning. Some noteworthy papers in this area include: Emergent Riemannian geometry over learning discrete computations on continuous manifolds, which provides a geometric framework for understanding how neural networks learn to perform discrete computations on continuous manifolds. Fiber Bundle Networks: A Geometric Machine Learning Paradigm, which introduces a novel machine learning framework integrating differential geometry with machine learning. The Geometry of Intelligence: Deterministic Functional Topology as a Foundation for Real-World Perception, which develops a deterministic functional-topological framework for real-world perception.

Sources

Emergent Riemannian geometry over learning discrete computations on continuous manifolds

Self-Supervised Dynamical System Representations for Physiological Time-Series

Fiber Bundle Networks: A Geometric Machine Learning Paradigm

Know Thyself by Knowing Others: Learning Neuron Identity from Population Context

A variational method for curve extraction with curvature-dependent energies

RNNs perform task computations by dynamically warping neural representations

The Geometry of Intelligence: Deterministic Functional Topology as a Foundation for Real-World Perception

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