COSMOS Learns Neural Representations as Gauge Dynamics on the Grassmannian

A framework called COSMOS proposes a different way to think about what a neural network learns. Rather than adjusting Euclidean weight parameters, it treats training as the evolution of representation subspaces. In an operational study published in Neural Computing and Applications, Lucas Battú describes a system whose learnable content is a single unified global state built from orthonormal bases and Grassmann projectors, mathematical objects that describe oriented subspaces inside a higher-dimensional space.

Learning in COSMOS departs sharply from standard practice. Instead of gradient descent on weights, the framework advances through a single global conjugation-based update that evolves the Grassmann frames. Forward computation requires no gradients at all: inputs are simply projected onto the learned subspaces. Classification is read out through a geodesic-margin Class Manifold Region criterion, which measures principal angles between representations and class anchor subspaces, a geometric decision rule rather than a purely statistical one.

One of the most striking results concerns geometric integrity. Across controlled experiments on Fashion-MNIST and CIFAR-10, the orthonormality and projector structure of the learned state held at machine precision, independently of the retraction schedule, the scheme governing how updates are mapped back onto the manifold of subspaces. For a design whose guarantees depend on exact geometry, maintaining that structure throughout training is a notable measured property.

The empirical picture is nuanced. When COSMOS was parameter-matched against conventional baselines, its flat input interface fell below a two-layer multilayer perceptron, scoring 76.30 percent test accuracy on Fashion-MNIST against the MLP’s 88.28. A patchwise input interface, which processes images in local patches rather than as one flat vector, reached statistical parity with the MLP on both benchmarks: 88.21 versus 88.28 on Fashion-MNIST, and 51.62 versus 52.09 on CIFAR-10. The patchwise variant also outperformed a per-frame Riemannian SGD baseline, which managed 66.25 on Fashion-MNIST, indicating that how the geometric interface is wired to the input matters as much as the geometry itself.

Scalability was the central engineering challenge, and the blockwise conjugation update is the proposed answer. A dense update would require a matrix exponential costing roughly 4,000 seconds at a dimension of n = 4096. The blockwise scheme costs 3.8 milliseconds per block at the same size, a gap of about six orders of magnitude that makes the geometric core computationally workable.

The framework also turns information dilution in deep sequential arrangements into a direct measurement. The author measured the flow ratio and the class-manifold margin directly and observed that both vanish in collapsed deep chains, providing an explicit, quantitative picture of how deep geometric pipelines can fail to carry class-relevant information forward.

The author is explicit about scope. These results span two small-scale, vector-input image-classification benchmarks evaluated with CPU-scale models. COSMOS is positioned as a controlled benchmark for geometric consistency in subspace-based learning, not as a general framework for geometric neural architectures. The contribution is a testbed in which geometric claims can be checked exactly, not a demonstration that the approach can replace gradient-based deep learning at scale.

Even within those limits, the study combines a learning system whose geometric invariants held at machine precision, an update rule whose per-block cost was reduced from seconds scaled to milliseconds, and parameter-matched comparisons showing where the approach matches conventional networks and where it still trails. For researchers asking whether the geometry of representation spaces can be an operational, measurable part of machine learning, COSMOS offers a concrete data point.

Subject of Research: Geometric learning framework representing neural network training as subspace evolution on the Grassmannian

Article Title: COSMOS: continuous subspace manifold operating system learning as gauge dynamics on the Grassmannian

Article References: Battú, L. (2026). COSMOS: continuous subspace manifold operating system learning as gauge dynamics on the Grassmannian. Neural Computing and Applications, 38(20), Article 842. https://doi.org/10.1007/s00521-026-12581-3

Image Credits: AI Generated

DOI: 10.1007/s00521-026-12581-3

Keywords: COSMOS, Grassmannian, subspace learning, gauge dynamics, orthonormal bases, geometric deep learning, Fashion-MNIST, CIFAR-10, Riemannian optimization, machine learning, neural networks, information dilution