Unrolling a Graph-Laplacian Denoiser Realizes Only Compositions of Polynomial Graph Filters

This paper proves that an unrolled graph-Laplacian denoiser with a order-K Taylor system-matrix expansion followed by m fixed conjugate-gradient steps cannot learn beyond polynomial graph filters: for every coefficient setting, its output is P(Ψ)y, with degree at most K(m−1). In the practical configuration K=10 and m=3, the reachable family is only 17-dimensional inside the 21-dimensional polynomial class P20, making it a measure-zero subset rather than an expanded hypothesis space. At standard initialization, the spectral response is leaky, approaching the nonzero floor 1/(K+1); for measured graph conditioning κ(Ψ)=107, achieving 10−2 relative accuracy requires K≥488, far above the usual order. Experiments on TAMPERE17 grayscale image patches show that with 17 parameters, the unrolled TSE+CG model reaches 26.65±1.67 dB test PSNR, while a direct degree-16 Bernstein graph-filter parameterization reaches 27.52±0.60 dB and optimizes more reliably. The Bernstein form spans the full polynomial class, can represent the target filter exactly, and offers a convex training objective with optional nonexpansiveness guarantees. The central conclusion is that learning solver coefficients changes parameterization and optimization dynamics, not the underlying spectral capacity; changing the graph operator itself is required to escape this polynomial-filter class.