SPARCL: Spectral Partitioned Analytic Continual Learning

SPARCL, or Spectral Partitioned Analytic Continual Learning, targets forgetting in exemplar-free class-incremental learning without gradient updates. It argues that analytic methods such as ACIL still forget because new-task features alter the shared ridge operator, (R plus λI)⁻¹, diluting dominant eigenvalues and shifting old-class logits. SPARCL eigendecomposes the accumulated autocorrelation matrix, freezes old-class classifier coefficients in a high-energy core subspace, and applies recursive least-squares updates only in the residual subspace; optional orthogonal random projections restore residual capacity without disturbing the core. This yields an exact invariance guarantee for the core contribution to old logits, while bounding remaining drift by residual feature energy, and reduces computation through a residual-scale Woodbury solve. With a frozen ViT-B/16 encoder, 10-task streams, and no stored exemplars, SPARCL reaches 94.52 percent on CIFAR-100, 94.28 percent on CUB-200-2011, 83.85 percent on ImageNet-R, and 68.70 percent on ImageNet-A, outperforming the listed RanPAC and Fly-CL results under the same protocol. At a core-energy threshold of τ equals 0.95, it takes 11.8 seconds per task on CIFAR-100, positioning it between dense random-projection methods and Fly-CL’s faster sparse path. The central contribution is separating stable analytic memory from plastic adaptation, rather than treating the entire classifier as a single jointly updated ridge solution.