Optimal Multiscale Learning of Linear Operators

Two recent studies on arXiv explore the limits of learning linear operators and physical learning methods in linear circuits, shedding light on the challenges and opportunities in these areas.

A study submitted on June 15, 2026, examines the statistical and computational limits of learning bounded linear operators between Sobolev spaces from noisy data^[1]. The researchers recast the problem as an infinite-dimensional matrix regression problem in wavelet coordinates and constructed a finite-resolution blockwise least-squares estimator to achieve minimax rates. The analysis reveals a nonuniform local estimation difficulty across scales. Another study on arXiv, submitted around the same time, investigates the coercivity and local convergence of physical learning methods in linear circuits^[2]. The researchers analyzed methods such as Equilibrium Propagation (EP), Adjoint Coupled Learning (AL), and Coupled Learning (CL), finding that EP and AL perform gradient descent on a natural loss function, while CL follows modified dynamics with an additional cubic correction. A coercivity condition was identified, expressed as a rank condition on a matrix built from the network's incidence structure.