Inductive bias, not physics: what neural renormalization-group and machine-learned holographic maps can and cannot identify
Year: 2026
Type: preprint
Abstract: Neural renormalization-group (RG) methods and machine-learned holographic maps train a network on lattice samples or on a lattice action and read physics off the result: coarse couplings, critical exponents, or an emergent bulk geometry. We ask which of these outputs the training objective and the readout themselves determine. For a bijection trained by a divergence to a factorized target with a free coarse coupling, every admissible coupling attains zero loss through a smooth optimal-transport map that respects every symmetry acting identically on field and latents, so neither more training nor an equivariant architecture selects the coupling. Restricting the class can: in the Gaussian chain a zero-loss linear map that is local below the coarse correlation length has the decimation coupling, the threshold is sharp, and on the square lattice no finite-range linear map reaches zero loss. A Gaussian kernel on a network graph yields a distance bounded above and below by the graph distance whatever the data, and exact wavelet latents of a power-law Gaussian give an isotropic hyperbolic geometry except along chains through the wavelet's jumps, which a free-fermion computation ties to the anisotropy reported for the exact holographic mapping. We then re-examine three published pipelines. A hyperbolicity test of neural-RG holography is passed by kernels built without boundary data. In RGFlow at the reported sizes, general linear maps reach zero loss at a continuum of couplings, whereas in a small, unconverged φ⁴ computation RGFlow's own RealNVP class selects a coupling through its Jacobian structure, close to the decimation value in the Gaussian sector. In inverse RG of the three-dimensional spin glass, with a ten-parameter stand-in upscaler on small lattices, the one clean upscaler that closes under blocking reproduced the forward Monte Carlo RG fixed point of its training pair, non-closing ones drifted, and the lower training loss went to a non-closing upscaler in both training pairs. The sharp results are Gaussian. We close with diagnostics, each with a recipe.
Author: Pei, Yan Ru
Language: English
Subfield: Condensed Matter Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225457055