Manifold Regularized Nuclear Norm Matrix Group Sparse Classifier for Robust Image Classification

Abstract

Robust representation‐based image classification must cope with complex mixed degradations, including global structural variations (e.g., severe non‐linear illumination) and localized contiguous corruptions (e.g., block occlusions or disguises). While existing matrix‐based low‐rank regression methods effectively model structural residuals without breaking the intrinsic two‐dimensional spatial correlations of pixels, they consistently neglect the underlying geometric relationships among training samples. This oversight leads to dense and weakly discriminative representation coefficients, particularly in under‐determined small sample size (SSS) scenarios. To address this limitation, we propose the Manifold Regularized Nuclear Norm Matrix Group Sparse Classifier (MNMGSC), a unified convex matrix optimization framework. MNMGSC jointly enforces a low‐rank residual via a nuclear‐norm penalty, explicitly models contiguous occlusion via a matrix ℓ2,1$$ {\mathrm{\ell}}_{2,1} $$ structural penalty, promotes class‐wise discriminative coefficients via a group‐sparse ℓ2,1$$ {\mathrm{\ell}}_{2,1} $$ ‐norm, and ensures geometric consistency via a Graph Laplacian regularizer. This mathematical design explicitly yields a formal ℓ2,1$$ {\mathrm{\ell}}_{2,1} $$ ‐spectrum that bridges sparse and collaborative representation paradigms. To efficiently solve the proposed non‐smooth model, we develop a Nesterov‐accelerated Alternating Direction Method of Multipliers (ADMM) and provide explicit theoretical guarantees, including a Group Sparse Low‐Rank Restricted Isometry Property (GS‐LRIP) recovery error bound and convergence proof. Evaluated extensively on the ORL, Extended Yale B, AR Face, and MNIST databases against both the unregularized baseline NMGSC and recent state‐of‐the‐art matrix/tensor representation methods, MNMGSC demonstrates statistically significant performance improvements ( p<0.05$$ p<0.05 $$ ), particularly under severe non‐linear degradations and highly limited training samples.