PURE HYPERSPHERE / JOINT CUTAWAYONE CENTER · ONE RADIUSSPARSE LINEAR INPUT MAP

01 / EQ. 02Joint input–label mapz(x,y) = x̄ ⊗ e_yx̄ = [xᵀ,b]ᵀ · sparse linear input

02 / EQ. 01Fixed regular-simplex labelsΣ_ye_y = 0 · ‖e_y‖₂ = 1e_yᵀe_y′ = −1/(K−1), y≠y′

03 / EQ. 03–04One decision hypersurface∂B(c,ρ) = {z : ‖z−c‖₂² = ρ}one center c · one scalar squared radius ρ

04 / EQ. 17–19Constraint-active supportdᵢ⁺−ρ ≥ −ε · ρ+m_out−dᵢ⁻ ≥ −εm_rank+dᵢ⁺−dᵢ⁻ ≥ −ε

05 / EQ. 05Factored joint distanced(x,y)=‖x̄‖₂²+‖A‖_F²−2x̄ᵀAe_yA = reshape(c) · full sparse evaluation

06 / EQ. 07–10Score · prediction · membershipŷ(x) = arg min_y d(x,y)s=ρ−d · inside=1{d(x,ŷ)≤ρ}

inside · correct pairboundary · constraint-active supportoutside · wrong pair

DRAG ORBIT · WHEEL SCALE · R RESET

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