Dual-Scale Fuzzy Boundary Abstention for Selective Prototype

Classification

Ajoy Kanti1,* Das Suman Das2

1 Associate Professor, Department of Mathematics, Tripura University, Agartala-799022, Tripura, India

2 Assistant Professor Grade II (Mathematics), Department of Education, National Institute of Technology Calicut, Kozhikode-673601,

Kerala, India

Received: November 20, 2025 Revised: January 22, 2026 Accepted: March 01, 2026 ⋆ Corresponding author

ABSTRACT

A classifier can be accurate on average and still be unreliable near regions in which competing classes overlap.

We study this problem as selective fuzzy classification: for x ∈ Rd, the decision is either a label by(x) ∈ {1, . . . ,K}

or abstention ⊥. The proposed dual-scale fuzzy boundary index (DFBI) decomposes local ambiguity into a

distributed term Bmass(x) and an extremal term Bpress(x). The former quantifies similarity-weighted contradictory

neighborhood mass, whereas the latter compares the strongest opposing and supporting fuzzy relations. Their

geometric fusion qDFBI(x) = {Bmass(x)Bpress(x)}1/2 ∈ [0,1] induces the selective map gθc (x) = 1{qDFBI(x) ≤ θc},

where the empirical validation quantile θc targets coverage c. The analysis is entirely numerical rather than graphical

and uses repeated stratified splits, selective accuracy, macro-F1, error capture, relative risk reduction, AURC, AUGRC,

paired bootstrap intervals, ablation, neighborhood sensitivity, and a nonlinear stress test. At nominal c = 0.90, the

mean selective-accuracy vector is ¯a = (0.9354,0.9882,0.9847,0.9744) for Iris, Wine, Breast Cancer, and Digits,

while the corresponding error-capture vector is ¯e=(0.5726,0.7528,0.8093,0.7937). Relative to membership-margin

uncertainty, Δa = (0.0211,0.0022,0.0194,0.0446). Thus the gain is not obtained by changing the base classifier

f ; it follows from a local fuzzy acceptance policy ( f ,gθc ) whose uncertainty ordering is informed by boundary

geometry.

Keywords: Fuzzy rough sets Selective classification Reject option Uncertainty Fuzzy neighborhood Prototype

classification Risk–coverage analysis

1. INTRODUCTION

Fuzzy set theory replaces a binary membership statement

x ∈ A with a grade μA(x) ∈ [0,1] [1]. Rough sets express a

different form of uncertainty through lower and upper approximations,

customarily satisfying A ⊆ A ⊆ A under a crisp

indiscernibility relation [2]. Fuzzy–rough models combine

these ideas by allowing the relation and/or the approximation

grades to be valued in [0,1] [3, 4]. For classification, this

boundary perspective motivates a decision problem beyond

ordinary error Pr[ f (X) ̸= Y]: when the evidence around a

query x is locally contradictory, should the system expose

f (x) = by(x) at all?

The reject-option formulation enlarges the action space from

Y to Y ∪{⊥}, where ⊥ denotes abstention. Classical analysis

characterizes recognition error against rejection under

posterior and cost assumptions [5], whereas modern selective

classification couples a classifier f with a selector g : Rd →

{0,1} [6, 7]. Writing ℓ( f (X),Y) = 1{ f (X) ̸=Y}, coverage

and selective risk may be summarized as C (g) = E[g(X)]