--- a/src/kernels/gaussian.rs Tue Aug 01 10:32:12 2023 +0300
+++ b/src/kernels/gaussian.rs Thu Aug 29 00:00:00 2024 -0500
@@ -298,8 +298,8 @@
if c1 >= c2 {
0.0
} else {
- // erf'(z) = (2/√π)*exp(-z^2), and we get extra factor -1/√(2*σ) = -1/t
- // from the chain rule
+ // erf'(z) = (2/√π)*exp(-z^2), and we get extra factor -1/(√2*σ) = -1/t
+ // from the chain rule (the minus comes from inside c_1 or c_2).
let de1 = (-(c1/t).powi(2)).exp();
let de2 = (-(c2/t).powi(2)).exp();
c_div_t * (de1 - de2)
@@ -323,15 +323,70 @@
}
#[replace_float_literals(F::cast_from(literal))]
+impl<'a, F : Float, R, C, S, const N : usize> Lipschitz<L1>
+for Convolution<CubeIndicator<R, N>, BasicCutGaussian<C, S, N>>
+where R : Constant<Type=F>,
+ C : Constant<Type=F>,
+ S : Constant<Type=F> {
+ type FloatType = F;
+
+ fn lipschitz_factor(&self, L1 : L1) -> Option<F> {
+ // To get the product Lipschitz factor, we note that for any ψ_i, we have
+ // ∏_{i=1}^N φ_i(x_i) - ∏_{i=1}^N φ_i(y_i)
+ // = [φ_1(x_1)-φ_1(y_1)] ∏_{i=2}^N φ_i(x_i)
+ // + φ_1(y_1)[ ∏_{i=2}^N φ_i(x_i) - ∏_{i=2}^N φ_i(y_i)]
+ // = ∑_{j=1}^N [φ_j(x_j)-φ_j(y_j)]∏_{i > j} φ_i(x_i) ∏_{i < j} φ_i(y_i)
+ // Thus
+ // |∏_{i=1}^N φ_i(x_i) - ∏_{i=1}^N φ_i(y_i)|
+ // ≤ ∑_{j=1}^N |φ_j(x_j)-φ_j(y_j)| ∏_{j ≠ i} \max_i |φ_i|
+ //
+ // Thus we need 1D Lipschitz factors, and the maximum for φ = θ * ψ.
+ //
+ // We have
+ // θ * ψ(x) = 0 if c_1(x) ≥ c_2(x)
+ // = (1/2)[erf(c_2(x)/(√2σ)) - erf(c_1(x)/(√2σ))] if c_1(x) < c_2(x),
+ // where c_1(x) = max{-x-b,-a} = -min{b+x,a} and c_2(x)=min{b-x,a}, C is the Gaussian
+ // normalisation factor, and erf(s) = (2/√π) ∫_0^s e^{-t^2} dt.
+ // Thus, if c_1(x) < c_2(x) and c_1(y) < c_2(y), we have
+ // θ * ψ(x) - θ * ψ(y) = (1/√π)[∫_{c_1(x)/(√2σ)}^{c_1(y)/(√2σ) e^{-t^2} dt
+ // - ∫_{c_2(x)/(√2σ)}^{c_2(y)/(√2σ)] e^{-t^2} dt]
+ // Thus
+ // |θ * ψ(x) - θ * ψ(y)| ≤ (1/√π)/(√2σ)(|c_1(x)-c_1(y)|+|c_2(x)-c_2(y)|)
+ // ≤ 2(1/√π)/(√2σ)|x-y|
+ // ≤ √2/(√πσ)|x-y|.
+ //
+ // For the product we also need the value θ * ψ(0), which is
+ // (1/2)[erf(min{a,b}/(√2σ))-erf(max{-b,-a}/(√2σ)]
+ // = (1/2)[erf(min{a,b}/(√2σ))-erf(-min{a,b}/(√2σ))]
+ // = erf(min{a,b}/(√2σ))
+ //
+ // If c_1(x) ≥ c_2(x), then x ∉ [-(a+b), a+b]. If also y is outside that range,
+ // θ * ψ(x) = θ * ψ(y). If only y is in the range [-(a+b), a+b], we can replace
+ // x by -(a+b) or (a+b), either of which is closer to y and still θ * ψ(x)=0.
+ // Thus same calculations as above work for the Lipschitz factor.
+ let Convolution(ref ind,
+ SupportProductFirst(ref cut,
+ ref gaussian)) = self;
+ let a = cut.r.value();
+ let b = ind.r.value();
+ let σ = gaussian.variance.value().sqrt();
+ let π = F::PI;
+ let t = F::SQRT_2 * σ;
+ let l1d = F::SQRT_2 / (π.sqrt() * σ);
+ let e0 = F::cast_from(erf((a.min(b) / t).as_()));
+ Some(l1d * e0.powi(N as i32-1))
+ }
+}
+
impl<'a, F : Float, R, C, S, const N : usize> Lipschitz<L2>
for Convolution<CubeIndicator<R, N>, BasicCutGaussian<C, S, N>>
where R : Constant<Type=F>,
C : Constant<Type=F>,
S : Constant<Type=F> {
type FloatType = F;
-
- fn lipschitz_factor(&self, L2 : L2) -> Option<F> {
- todo!("This requirement some error function work.")
+ #[inline]
+ fn lipschitz_factor(&self, L2 : L2) -> Option<Self::FloatType> {
+ self.lipschitz_factor(L1).map(|l1| l1 * <S::Type>::cast_from(N).sqrt())
}
}
