Topological robustness in fuzzy support vector machines: a Kikkawa-Suzuki and Ulam-Hyers stability perspective
Journal of Decision Making and Healthcare, Volume 3, Issue 1, April 2026, Pages: 46–61
RAMAN CHAUDHARY
Department of Mathematical Sciences, Indian Institute of Technology (BHU) Varanasi, Varanasi 221005, Uttar Pradesh, India
Abstract
We analyse the robustness of Fuzzy Linear Support Vector Machines (FSVMs) through the lens of nonlinear functional analysis. The central idea is to recast support vector selection in the FSVM dual as a multi-valued fixed point problem, posed on the compact feasible region under the Euclidean metric. Provided the kernel matrix \(Q\) is strictly positive definite, the projected gradient selection mapping acts as a multi-valued contraction of Nadler type [1], and consequently satisfies the Kikkawa-Suzuki condition [2]. Fixed points therefore exist. A retraction-displacement argument produces generalized Ulam-Hyers stability: for \(\epsilon\)-approximate solver outputs of the same FSVM instance, the displacement to an exact fixed point is controlled by a continuous increasing function of \(\epsilon\). Separately, the Petrușel-Rus data dependence framework [3] bounds the Hausdorff distance between two fixed point sets, proportionally to the perturbation magnitude \(\epsilon\) measured in \(\ell^\infty\). Notably, this bound demonstrates that the fixed point sets of two distinct FSVM instances remain Hausdorff-close, and the estimate is entirely deterministic. All three results are topological certificates, fundamentally distinct from probabilistic generalisation bounds. Our analysis is confined to linear kernels throughout. We include small-scale numerical experiments measuring support vector set stability via Jaccard similarity, a discrete proxy for the continuous Hausdorff bound; results are broadly consistent with the theoretical predictions, though the metric gap means the comparison is heuristic rather than direct.
References
- S. B. Nadler, Jr. Multi-valued contraction mappings. Pacific Journal of Mathematics, 30(2):475-488, 1969.
- M. Kikkawa and T. Suzuki. Three fixed point theorems for generalized contractions with constants in complete metric spaces. Nonlinear Analysis: Theory, Methods & Applications, 69(9):2942-2949, 2008.
- I. A. Rus, A. Petrușel, and A. Sîntămărian. Data dependence of the fixed point set of some multivalued weakly Picard operators. Nonlinear Analysis, 52:1947–1959, 2003.
Cite this Article as
Raman Chaudhary, Topological robustness in fuzzy support vector machines: a Kikkawa-Suzuki and Ulam-Hyers stability perspective, Journal of Decision Making and Healthcare, 3(1), 46–61, 2026