Spectral and Pseudospectral Approximation of Finite-Interaction-Range Operators in Doubling Metric Measure Spaces

What the math gives to ML

The paper gives a constructive localization principle: global spectral and pseudospectral behavior of a finite-range operator can be estimated from rectangular finite sections on metric balls, with an explicit error that decays as O(1/L). This is transferable to sparse graph layers, banded sequence operators, and state-space models whose linear propagation or Jacobian operators have bounded interaction range. The most useful adaptation is a cheap local pseudospectral monitor or regularizer: compute smallest singular values of local sections instead of forming the full operator, then use the O(1/L) margin to detect or suppress unstable modes. The method is especially attractive for irregular graphs and long sequences where global spectral computation is expensive but local windows remain small.