pointsource_algs: comparison src/kernels/gaussian.rs

-:6105b5cd8d89
+:f0e8704d3f0e
 LocalAnalysis,
 GlobalAnalysis,
 Weighted,
 Bounded,
 };
-use alg_tools::mapping::Apply;
+use alg_tools::mapping::{
+Mapping,
+Instance,
+Differential,
+DifferentiableImpl,
+};
 use alg_tools::maputil::array_init;
+use crate::types::*;
 use crate::fourier::Fourier;
 use super::base::*;
 use super::ball_indicator::CubeIndicator;
 /// Storage presentation of the the anisotropic gaussian kernel of `variance` $σ^2$.
 }
 }
 #[replace_float_literals(S::Type::cast_from(literal))]
-impl<'a, S, const N : usize> Apply<&'a Loc<S::Type, N>> for Gaussian<S, N>
+impl<'a, S, const N : usize> Mapping<Loc<S::Type, N>> for Gaussian<S, N>
-where S : Constant {
+where
-type Output = S::Type;
+S : Constant
+{
+type Codomain = S::Type;
 // This is not normalised to neither to have value 1 at zero or integral 1
 // (unless the cut-off ε=0).
 #[inline]
-fn apply(&self, x : &'a Loc<S::Type, N>) -> Self::Output {
+fn apply<I : Instance<Loc<S::Type, N>>>(&self, x : I) -> Self::Codomain {
-let d_squared = x.norm2_squared();
+let d_squared = x.eval(|x| x.norm2_squared());
 let σ2 = self.variance.value();
 let scale = self.scale();
 (-d_squared / (2.0 * σ2)).exp() / scale
 }
 }
-impl<S, const N : usize> Apply<Loc<S::Type, N>> for Gaussian<S, N>
+#[replace_float_literals(S::Type::cast_from(literal))]
-where S : Constant {
+impl<'a, S, const N : usize> DifferentiableImpl<Loc<S::Type, N>> for Gaussian<S, N>
-type Output = S::Type;
+where S : Constant {
-// This is not normalised to neither to have value 1 at zero or integral 1
+type Derivative = Loc<S::Type, N>;
-// (unless the cut-off ε=0).
 #[inline]
-fn apply(&self, x : Loc<S::Type, N>) -> Self::Output {
+fn differential_impl<I : Instance<Loc<S::Type, N>>>(&self, x0 : I) -> Self::Derivative {
-self.apply(&x)
+let x = x0.cow();
+let f = -self.apply(&*x) / self.variance.value();
+*x * f
+}
+}
+// To calculate the the Lipschitz factors, we consider
+// f(t)    = e^{-t²/2}
+// f'(t)   = -t f(t)       which has max at t=1 by f''(t)=0
+// f''(t)  = (t²-1)f(t)    which has max at t=√3 by f'''(t)=0
+// f'''(t) = -(t³-3t)
+// So f has the Lipschitz factor L=f'(1), and f' has the Lipschitz factor L'=f''(√3).
+//
+// Now g(x) = Cf(‖x‖/σ) for a scaling factor C is the Gaussian.
+// Thus ‖g(x)-g(y)‖ = C‖f(‖x‖/σ)-f(‖y‖/σ)‖ ≤ (C/σ)L‖x-y‖,
+// so g has the Lipschitz factor (C/σ)f'(1) = (C/σ)exp(-0.5).
+//
+// Also ∇g(x)= Cx/(σ‖x‖)f'(‖x‖/σ)       (*)
+//            = -(C/σ²)xf(‖x‖/σ)
+//            = -C/σ (x/σ) f(‖x/σ‖)
+// ∇²g(x) = -(C/σ)[Id/σ f(‖x‖/σ) + x ⊗ x/(σ²‖x‖) f'(‖x‖/σ)]
+//        = (C/σ²)[-Id + x ⊗ x/σ²]f(‖x‖/σ).
+// Thus ‖∇²g(x)‖ = (C/σ²)‖-Id + x ⊗ x/σ²‖f(‖x‖/σ), where
+// ‖-Id + x ⊗ x/σ²‖ = ‖[-Id + x ⊗ x/σ²](x/‖x‖)‖ = |-1 + ‖x²/σ^2‖|.
+// This means that  ‖∇²g(x)‖ = (C/σ²)|f''(‖x‖/σ)|, which is maximised with ‖x‖/σ=√3.
+// Hence the Lipschitz factor of ∇g is (C/σ²)f''(√3) = (C/σ²)2e^{-3/2}.
+#[replace_float_literals(S::Type::cast_from(literal))]
+impl<S, const N : usize> Lipschitz<L2> for Gaussian<S, N>
+where S : Constant {
+type FloatType = S::Type;
+fn lipschitz_factor(&self, L2 : L2) -> Option<Self::FloatType> {
+Some((-0.5).exp() / (self.scale() * self.variance.value().sqrt()))
+}
+}
+#[replace_float_literals(S::Type::cast_from(literal))]
+impl<'a, S : Constant, const N : usize> Lipschitz<L2>
+for Differential<'a, Loc<S::Type, N>, Gaussian<S, N>> {
+type FloatType = S::Type;
+fn lipschitz_factor(&self, _l2 : L2) -> Option<S::Type> {
+let g = self.base_fn();
+let σ2 = g.variance.value();
+let scale = g.scale();
+Some(2.0*(-3.0/2.0).exp()/(σ2*scale))
+}
+}
+// From above, norm bounds on the differnential can be calculated as achieved
+// for f' at t=1, i.e., the bound is |f'(1)|.
+// For g then |C/σ f'(1)|.
+// It follows that the norm bounds on the differential are just the Lipschitz
+// factors of the undifferentiated function, given how the latter is calculed above.
+#[replace_float_literals(S::Type::cast_from(literal))]
+impl<'b, S : Constant, const N : usize> NormBounded<L2>
+for Differential<'b, Loc<S::Type, N>, Gaussian<S, N>> {
+type FloatType = S::Type;
+fn norm_bound(&self, _l2 : L2) -> S::Type {
+self.base_fn().lipschitz_factor(L2).unwrap()
+}
+}
+#[replace_float_literals(S::Type::cast_from(literal))]
+impl<'b, 'a, S : Constant, const N : usize> NormBounded<L2>
+for Differential<'b, Loc<S::Type, N>, &'a Gaussian<S, N>> {
+type FloatType = S::Type;
+fn norm_bound(&self, _l2 : L2) -> S::Type {
+self.base_fn().lipschitz_factor(L2).unwrap()
 }
 }
 #[replace_float_literals(S::Type::cast_from(literal))]
 /// where $a>0$ and $f$ is a gaussian kernel on $ℝ^n$.
 pub type BasicCutGaussian<C, S, const N : usize> = SupportProductFirst<CubeIndicator<C, N>,
 Gaussian<S, N>>;
-/// This implements $χ\_{[-b, b]^n} \* (f χ\_{[-a, a]^n})$
+/// This implements $g := χ\_{[-b, b]^n} \* (f χ\_{[-a, a]^n})$ where $a,b>0$ and $f$ is
-/// where $a,b>0$ and $f$ is a gaussian kernel on $ℝ^n$.
+/// a gaussian kernel on $ℝ^n$. For an expression for $g$, see Lemma 3.9 in the manuscript.
 #[replace_float_literals(F::cast_from(literal))]
-impl<'a, F : Float, R, C, S, const N : usize> Apply<&'a Loc<F, N>>
+impl<'a, F : Float, R, C, S, const N : usize> Mapping<Loc<F, N>>
 for Convolution<CubeIndicator<R, N>, BasicCutGaussian<C, S, N>>
 where R : Constant<Type=F>,
 C : Constant<Type=F>,
 S : Constant<Type=F> {
-type Output = F;
+type Codomain = F;
 #[inline]
-fn apply(&self, y : &'a Loc<F, N>) -> F {
+fn apply<I : Instance<Loc<F, N>>>(&self, y : I) -> F {
 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 t = F::SQRT_2 * σ;
 let c = 0.5; // 1/(σ√(2π) * σ√(π/2) = 1/2
 // This is just a product of one-dimensional versions
-y.product_map(|x| {
+y.cow().product_map(|x| {
 let c1 = -(a.min(b + x)); //(-a).max(-x-b);
 let c2 = a.min(b - x);
 if c1 >= c2 {
 0.0
 } else {
 }
 })
 }
 }
-impl<F : Float, R, C, S, const N : usize> Apply<Loc<F, N>>
+/// This implements the differential of $g := χ\_{[-b, b]^n} \* (f χ\_{[-a, a]^n})$ where $a,b>0$
-for Convolution<CubeIndicator<R, N>, BasicCutGaussian<C, S, N>>
+/// and $f$ is a gaussian kernel on $ℝ^n$. For an expression for the value of $g$, from which the
-where R : Constant<Type=F>,
+/// derivative readily arises (at points of differentiability), see Lemma 3.9 in the manuscript.
-C : Constant<Type=F>,
+#[replace_float_literals(F::cast_from(literal))]
-S : Constant<Type=F> {
+impl<'a, F : Float, R, C, S, const N : usize> DifferentiableImpl<Loc<F, N>>
+for Convolution<CubeIndicator<R, N>, BasicCutGaussian<C, S, N>>
-type Output = F;
+where R : Constant<Type=F>,
+C : Constant<Type=F>,
-#[inline]
+S : Constant<Type=F> {
-fn apply(&self, y : Loc<F, N>) -> F {
-self.apply(&y)
+type Derivative = Loc<F, N>;
-}
-}
+/// Although implemented, this function is not differentiable.
+#[inline]
+fn differential_impl<I : Instance<Loc<F, N>>>(&self, y0 : I) -> Loc<F, N> {
+let Convolution(ref ind,
+SupportProductFirst(ref cut,
+ref gaussian)) = self;
+let y = y0.cow();
+let a = cut.r.value();
+let b = ind.r.value();
+let σ = gaussian.variance.value().sqrt();
+let t = F::SQRT_2 * σ;
+let c = 0.5; // 1/(σ√(2π) * σ√(π/2) = 1/2
+let c_mul_erf_scale_div_t = c * F::FRAC_2_SQRT_PI / t;
+// Calculate the values for all component functions of the
+// product. This is just the loop from apply above.
+let unscaled_vs = y.map(|x| {
+let c1 = -(a.min(b + x)); //(-a).max(-x-b);
+let c2 = a.min(b - x);
+if c1 >= c2 {
+0.0
+} else {
+let e1 = F::cast_from(erf((c1 / t).as_()));
+let e2 = F::cast_from(erf((c2 / t).as_()));
+debug_assert!(e2 >= e1);
+c * (e2 - e1)
+}
+});
+// This computes the gradient for each coordinate
+product_differential(&*y, &unscaled_vs, |x| {
+let c1 = -(a.min(b + x)); //(-a).max(-x-b);
+let c2 = a.min(b - x);
+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 (the minus comes from inside c_1 or c_2, and changes the
+// order of de2 and de1 in the final calculation).
+let de1 = if b + x < a {
+(-((b+x)/t).powi(2)).exp()
+} else {
+0.0
+};
+let de2 = if b - x < a {
+(-((b-x)/t).powi(2)).exp()
+} else {
+0.0
+};
+c_mul_erf_scale_div_t * (de1 - de2)
+}
+})
+}
+}
+#[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;
+#[inline]
+fn lipschitz_factor(&self, L2 : L2) -> Option<Self::FloatType> {
+self.lipschitz_factor(L1).map(|l1| l1 * <S::Type>::cast_from(N).sqrt())
+}
+}
+*/
 impl<F : Float, R, C, S, const N : usize>
 Convolution<CubeIndicator<R, N>, BasicCutGaussian<C, S, N>>
 where R : Constant<Type=F>,
 C : Constant<Type=F>,

Mercurial > repos > pointsource_algs / file comparison

comparison: src/kernels/gaussian.rs

src/kernels/gaussian.rs