Model quality

Single-number layer quality signals derived from the spectrum. See the catalog for the exact fields.

The metrics

  • Weighted alpha — \(\alpha_{\mathrm{PL}}\,\log_{10}\lambda_{\max}\) (pl_alpha_weighted). The fitted power-law exponent \(\alpha_{\mathrm{PL}}\) scaled by the log of the largest eigenvalue, the “alpha-hat” combination of scale and shape [Martin et al., 2021]; it couples the tail exponent (smaller is heavier-tailed, typically better trained) with the spectral scale.

  • Randomized Wasserstein distance — \(\mathcal{W}_1(\lambda, \lambda^{\mathrm{rand}}) / \langle\lambda\rangle\) (w1_rand_distance). The Wasserstein-1 (earth-mover) distance \(\mathcal{W}_1\) between the ESD and its permutation null, divided by the shared mean eigenvalue \(\langle\lambda\rangle\). It measures how far the spectrum departs from a structureless baseline of the same weights, and the normalisation makes the index dimensionless and comparable across layers of different scale.

Conventions and pitfalls

  • Log base. pl_alpha_weighted uses \(\log_{10}\), matching the log-domain norms.

  • Scale invariance. Dividing by \(\bar{\lambda}\) makes w1_rand_distance invariant to rescaling the weights, so it reflects spectral shape rather than magnitude; it is \(0\) for a degenerate (zero-mean) spectrum and NaN when the spectrum is not measurable.

  • The exponent is fitted upstream. pl_alpha_weighted reuses pl_alpha from the power-law fit; its reliability inherits that fit’s.