Pure Mathematics for Theoretical Computer Science

Polynomial graph filters Ha(S) = ΣKk =0 akSk are local, scalable, and central to spectral graph neural computation, but their output may vary sharply when the graph shift S is perturbed. This paper develops a finite-radius perturbation calculus that requires neither commutativity nor eigengaps. For ∥S∥2 ≤ ρ and ∥E∥2 ≤ ε, we prove the certificate ∥Ha(S+E)−Ha(S)∥2 ≤ ΣKk =1 |ak|(ρ +ε)k−ρk, show equality for aligned positive polynomials, derive an explicit second-order remainder, and propagate the certificate through multilayer graph neural networks. The bound induces a sensitivity-weighted quadratic design whose unique minimizer is available in closed form. Controlled analysis uses three spectral responses, degree K = 8, and 2,304 filter evaluations on Erd˝os–Rényi, stochastic-block, Barabási– Albert, and random-geometric graphs. Relative to unconstrained least squares, the proposed design reduces the mean finite-radius certificate by 74.9%, the first-order coefficient sensitivity by 72.9%, and observed operator sensitivity by 6.9%. It matches accuracy-controlled ridge filtering within 0.3% in observed sensitivity while yielding a 7.3% smaller certificate. The result is a compact, auditable robustness layer for graph filters and polynomial graph-convolution banks.

This study investigates the effectiveness of variable selection techniques in linear regression models under grouped structures and correlation among predictors. Specifically, it evaluates and compares the performance of three prominent methods: LASSO, Elastic Net, and OSCAR. The simulation study spans multiple scenarios, including varying correlation levels and sample sizes, and utilizes key metrics such as Mean Squared Error (MSE), True Positive Rate (TPR), False Positive Rate (FPR), and Grouping Accuracy. The results reveal the superior performance of OSCAR, particularly in grouped settings, where it consistently achieves lower error rates and better variable selection accuracy. A real data application using the prostate cancer dataset further supports the empirical advantages of OSCAR over its counterparts, especially in scenarios involving correlated and grouped predictors. The findings provide strong evidence in favor of OSCAR as a reliable tool for robust regression modeling.