Information-Induced Training Geometry: Exact Reduction, Canonical Completion, and Structured Expressivity
Read the original on arXiv Machine Learning →The paper investigates how training data limits the geometry of an optimizer through the covectors that a specified information channel can observe. It establishes that, under affine‑invariant Riemannian geometry, a full‑column‑rank positive‑definite compression can be uniquely completed via a split‑Hadamard metric submetry, yielding exact variational reduction from the full geometry to the visible target. The resulting framework provides explicit formulas for pullback metrics, Gram matrices, and prior‑data shrinkage, and characterizes the gauge‑invariant rank stratification of the positive‑definite cone as the channel varies.
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