Relating eigenmodes when multiple soft physical modes are relevant

Establish a general framework to relate individual eigenmodes of the physical Hessian H to the learned constraints and to the stiff eigenmodes of the cost Hessian ℋ in adaptable linear resistor networks (and, more generally, steady-state physical networks that minimize dissipated power) when multiple soft eigenmodes of H are simultaneously relevant to the linear response. Specifically, determine how to map physical soft modes to task constraints and corresponding stiff cost modes in the multi-mode regime, extending the single-mode correspondence derived in the paper.

Background

The paper proves a direct relationship between the physical Hessian H (governing linear response of node voltages) and the cost Hessian ℋ (governing curvature of the cost landscape over adaptive conductances) for networks trained to satisfy linear constraints. In the single-constraint, single-soft-mode regime, the stiff cost mode aligns with squared edge voltage differences of the soft physical mode, enabling task inference from physical response alone.

However, when the trained network exhibits multiple low-lying soft modes of the physical Hessian that significantly contribute to the response, the simple one-to-one correspondence between a single soft physical mode and a single stiff cost mode no longer applies. The authors note this complication and defer a comprehensive treatment of the multi-mode case.