nucleation

LIT-1254: Information loss under coarse graining: A geometric approach

1 min read

Properties5
StatusActive — read 2026-10-20 (NOTE-972) in full, from arXiv v2; worth reading for the identity that coarse-graining an exponential family loses exactly the expected conditional covariance of its energy operators, for its consequence under an exact RG step (the loss is the Hessian of an analytic constant, so the singular part of the Fisher metric survives; THEORY-742), and for the exact diffusion case, where the metric scales as n^(1−μ) by Hermite order. Read it knowing that "relevant directions are exactly maintained" is true only of the singular part, that on its own scalings the preserved directions are those with RG eigenvalue above d/2 rather than above 0, that the contraction of irrelevant directions (Eq. 21) is a scaling argument backed by the numerics of Machta et al. (2013) and not derived, that the drift is exactly preserved only at the Gaussian fixed point, and that no learning dynamics appear.
Version2
Tagsinformation-geometry, natural-sciences, information-theory
Cited as “paper” byNOTE-972 — Raju, Machta and Sethna — Information loss under coarse graining
Cited as “source” byTHEORY-742 — For a Boltzmann distribution linear in its couplings, observed through any coarse-graining map, the Fisher information the coarse data lose about the bare couplings is the expected conditional covariance of the energy operators given the coarse state; when the renormalized Hamiltonian stays in the family, the parameter map is linear and the additive constant of the step is analytic, that loss is the constant's Hessian, so the singular part of the metric passes unchanged to the coarse data. In the hopping model of diffusion at its Gaussian fixed point, keeping every n-th position of a trajectory of fixed duration multiplies the metric by n^(1−μ) in the μ-th Hermite coordinate: the drift's information is kept exactly, the variance's falls as 1/n, and higher shape parameters' faster

Graph View

Raju, A., Machta, B. B. & Sethna, J. P. (2018). Information loss under coarse graining: A geometric approach. Physical Review E 98(5): 052112. doi:10.1103/PhysRevE.98.052112. Preprint: arXiv:1710.05787 (16 October 2017).

Read in full in NOTE-972. THEORY-742 states what it proves.

Why this is here

A2 recommended it at high priority as a bridge from information-preserving abstraction to the geometry of distinguishability, asking whether learning accessibility can be related to the contraction of distinguishable directions under coarse-graining. Bears on QUESTION-217, and on THEORY-194 and THEORY-202.

Backlinks