--- a/src/kernels/gaussian.rs Tue Aug 01 10:25:09 2023 +0300
+++ b/src/kernels/gaussian.rs Mon Feb 17 13:54:53 2025 -0500
@@ -17,9 +17,15 @@
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;
@@ -58,28 +64,104 @@
#[replace_float_literals(S::Type::cast_from(literal))]
-impl<'a, S, const N : usize> Apply<&'a Loc<S::Type, N>> for Gaussian<S, N>
-where S : Constant {
- type Output = S::Type;
+impl<'a, S, const N : usize> Mapping<Loc<S::Type, N>> for Gaussian<S, N>
+where
+ 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 {
- let d_squared = x.norm2_squared();
+ fn apply<I : Instance<Loc<S::Type, N>>>(&self, x : I) -> Self::Codomain {
+ 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))]
+impl<'a, S, const N : usize> DifferentiableImpl<Loc<S::Type, N>> for Gaussian<S, N>
+where S : Constant {
+ type Derivative = Loc<S::Type, N>;
+
+ #[inline]
+ fn differential_impl<I : Instance<Loc<S::Type, N>>>(&self, x0 : I) -> Self::Derivative {
+ 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 Output = 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 : Loc<S::Type, N>) -> Self::Output {
- self.apply(&x)
+ 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()
}
}
@@ -169,19 +251,19 @@
Gaussian<S, N>>;
-/// This implements $χ\_{[-b, b]^n} \* (f χ\_{[-a, a]^n})$
-/// where $a,b>0$ and $f$ is a gaussian kernel on $ℝ^n$.
+/// This implements $g := χ\_{[-b, b]^n} \* (f χ\_{[-a, a]^n})$ where $a,b>0$ and $f$ is
+/// 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;
@@ -192,7 +274,7 @@
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 {
@@ -207,20 +289,143 @@
}
}
-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 $a,b>0$
+/// and $f$ is a gaussian kernel on $ℝ^n$. For an expression for the value of $g$, from which the
+/// derivative readily arises (at points of differentiability), see Lemma 3.9 in the manuscript.
+#[replace_float_literals(F::cast_from(literal))]
+impl<'a, F : Float, R, C, S, const N : usize> DifferentiableImpl<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 Derivative = Loc<F, N>;
+ /// Although implemented, this function is not differentiable.
#[inline]
- fn apply(&self, y : Loc<F, N>) -> F {
- self.apply(&y)
+ 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>,