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We prove the Courtade-Kumar conjecture: among all Boolean functions $f\colon \{-1,1\}^n\to\{-1,1\}$, a dictator retains the most information about a uniformly random input observed through independent binary noise.
Fisher width measures the Gaussian width of a probe set after deformation by the local Fisher geometry. We study its evolution along learning trajectories and ask when training loss can serve as an effective coordinate for this quantity. We first derive an exact trace--shape factorization and a deterministic stability...
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