Corrected Block Sparse Spectral Optimization: Exact Models, Finite Identification, and Local Dynamics

Key points

• For A-SQ we separate the historical full-polar iteration from a safeguarded Riemannian ascent and prove, for the latter, whole-sequence convergence via a sufficient-ascent lemma, a relative-error lemma, and the Kurdyka-Lojasiewicz property, which applies because the soft-threshold objective and the Stiefel constraint are semi-algebraic.1
• The post-identification map is ordinary power iteration on the active covariance for a rank-one hard threshold, and the Frechet derivative of a coupled polar map for component-specific supports; a radius-based safe-screening rule follows from the same local analysis.1
• On modern-AI panels (GPT-2 residual stream, ViT CLS stream, Pitprops) the thresholded block-sparse components match dense PCA on reconstruction and improve on it in monosemanticity and class specialisation.1
• On three native-label scRNA datasets (Zheng17 FACS-sorted PBMCs, 68k donor-A with FACS-correlated labels, Tabula Muris) A-SQ beats dense PCA on cluster-recovery AMI by +0.023 to +0.044 at matched sparsity.1

Sources (1)

• [1]Corrected Block Sparse Spectral Optimization: Exact Models, Finite Identification, and Local Dynamics
  arXiv (AI, ML, NLP, CV, robotics, multi-agent) · Oct 19, 03:20 AM

  The block generalized-power method pairs threshold-based support selection with inexpensive Stiefel updates, but the standard block formulation conflates two nonequivalent objectives and leaves inactive components, threshold ties, and post-selection refitting ill-specified.

  For A-SQ we separate the historical full-polar iteration from a safeguarded Riemannian ascent and prove, for the latter, whole-sequence convergence via a sufficient-ascent lemma, a relative-error lemma, and the Kurdyka-Lojasiewicz property, which applies because the soft-threshold objective and the Stiefel constraint are semi-algebraic.

Extractive summary: sentences quoted from the sources.