Grade Discovery as an Identifiability Diagnostic for Clifford Valued Features

TMLR Paper9234 Authors

Abstract: Clifford valued models can represent scalars, vectors, bivectors and pseudoscalars in one algebra, but current models usually assume the geometric type of each feature is known prior to training. This assumption is not always identifiable from the data. For example, rotation only evidence makes scalar and pseudoscalar transformation laws indistinguishable, whereas reflections reveal the difference. We study grade discovery as an identifiability diagnostic, where grade means the Clifford algebra type of a feature channel, such as scalar, vector, bivector or pseudoscalar. The method estimates this type from paired observations before and after known transformations, assigns a soft weight over candidate grades with the same coordinate dimension and fits this weight with a least-squares equivariant prediction loss. We prove that the loss recovers the correct type when the observed transformations separate the candidate transformation laws and that it remains uninformative when those laws agree on all observed transformations. In controlled three-dimensional experiments, rotation only evidence keeps the true type weight at one half, while rotations with reflections assign weight near one to the true type. Reflection frequency and loss landscape experiments show how often parity revealing transformations must appear and why they remove the flat ambiguity. Stress tests further show that inaccurate orthogonal transformation matrices and weak variance in separating directions reduce the diagnostic evidence. Finally, a differentiable soft gate and a minimal trainable prediction module show that the diagnostic can be optimized jointly with another parameter, although this experiment is not evidence of performance in full Clifford neural networks.

Submission Type: Regular submission (no more than 12 pages of main content)

Previous TMLR Submission Url: https://openreview.net/forum?id=hduY3XOoyF&nesting=2&sort=date-desc

Changes Since Last Submission: The previous version of this paper was desk rejected solely due to formatting issues (modified format). We have corrected our LaTeX document to strictly adhere to the official TMLR style template. The scientific content of the submission remains completely unchanged.

Assigned Action Editor: ~David_Rügamer1

Submission Number: 9234