--- a/src/kernels/base.rs Tue Aug 01 10:25:09 2023 +0300
+++ b/src/kernels/base.rs Mon Feb 17 13:54:53 2025 -0500
@@ -14,15 +14,22 @@
GlobalAnalysis,
Bounded,
};
-use alg_tools::mapping::Apply;
-use alg_tools::maputil::{array_init, map2};
+use alg_tools::mapping::{
+ Mapping,
+ DifferentiableImpl,
+ DifferentiableMapping,
+ Differential,
+};
+use alg_tools::instance::{Instance, Space};
+use alg_tools::maputil::{array_init, map2, map1_indexed};
use alg_tools::sets::SetOrd;
use crate::fourier::Fourier;
+use crate::types::*;
/// Representation of the product of two kernels.
///
-/// The kernels typically implement [`Support`] and [`Mapping`][alg_tools::mapping::Mapping].
+/// The kernels typically implement [`Support`] and [`Mapping`].
///
/// The implementation [`Support`] only uses the [`Support::support_hint`] of the first parameter!
#[derive(Copy,Clone,Serialize,Debug)]
@@ -33,32 +40,94 @@
pub B
);
-impl<A, B, F : Float, const N : usize> Apply<Loc<F, N>>
+impl<A, B, F : Float, const N : usize> Mapping<Loc<F, N>>
+for SupportProductFirst<A, B>
+where
+ A : Mapping<Loc<F, N>, Codomain = F>,
+ B : Mapping<Loc<F, N>, Codomain = F>,
+{
+ type Codomain = F;
+
+ #[inline]
+ fn apply<I : Instance<Loc<F, N>>>(&self, x : I) -> Self::Codomain {
+ self.0.apply(x.ref_instance()) * self.1.apply(x)
+ }
+}
+
+impl<A, B, F : Float, const N : usize> DifferentiableImpl<Loc<F, N>>
for SupportProductFirst<A, B>
-where A : for<'a> Apply<&'a Loc<F, N>, Output=F>,
- B : for<'a> Apply<&'a Loc<F, N>, Output=F> {
- type Output = F;
+where
+ A : DifferentiableMapping<
+ Loc<F, N>,
+ DerivativeDomain=Loc<F, N>,
+ Codomain = F
+ >,
+ B : DifferentiableMapping<
+ Loc<F, N>,
+ DerivativeDomain=Loc<F, N>,
+ Codomain = F,
+ >
+{
+ type Derivative = Loc<F, N>;
+
#[inline]
- fn apply(&self, x : Loc<F, N>) -> Self::Output {
- self.0.apply(&x) * self.1.apply(&x)
+ fn differential_impl<I : Instance<Loc<F, N>>>(&self, x : I) -> Self::Derivative {
+ let xr = x.ref_instance();
+ self.0.differential(xr) * self.1.apply(xr) + self.1.differential(xr) * self.0.apply(x)
}
}
-impl<'a, A, B, F : Float, const N : usize> Apply<&'a Loc<F, N>>
+impl<A, B, M : Copy, F : Float> Lipschitz<M>
for SupportProductFirst<A, B>
-where A : Apply<&'a Loc<F, N>, Output=F>,
- B : Apply<&'a Loc<F, N>, Output=F> {
- type Output = F;
+where A : Lipschitz<M, FloatType = F> + Bounded<F>,
+ B : Lipschitz<M, FloatType = F> + Bounded<F> {
+ type FloatType = F;
#[inline]
- fn apply(&self, x : &'a Loc<F, N>) -> Self::Output {
- self.0.apply(x) * self.1.apply(x)
+ fn lipschitz_factor(&self, m : M) -> Option<F> {
+ // f(x)g(x) - f(y)g(y) = f(x)[g(x)-g(y)] - [f(y)-f(x)]g(y)
+ let &SupportProductFirst(ref f, ref g) = self;
+ f.lipschitz_factor(m).map(|l| l * g.bounds().uniform())
+ .zip(g.lipschitz_factor(m).map(|l| l * f.bounds().uniform()))
+ .map(|(a, b)| a + b)
}
}
+impl<'a, A, B, M : Copy, Domain, F : Float> Lipschitz<M>
+for Differential<'a, Domain, SupportProductFirst<A, B>>
+where
+ Domain : Space,
+ A : Clone + DifferentiableMapping<Domain> + Lipschitz<M, FloatType = F> + Bounded<F>,
+ B : Clone + DifferentiableMapping<Domain> + Lipschitz<M, FloatType = F> + Bounded<F>,
+ SupportProductFirst<A, B> : DifferentiableMapping<Domain>,
+ for<'b> A::Differential<'b> : Lipschitz<M, FloatType = F> + NormBounded<L2, FloatType=F>,
+ for<'b> B::Differential<'b> : Lipschitz<M, FloatType = F> + NormBounded<L2, FloatType=F>
+{
+ type FloatType = F;
+ #[inline]
+ fn lipschitz_factor(&self, m : M) -> Option<F> {
+ // ∇[gf] = f∇g + g∇f
+ // ⟹ ∇[gf](x) - ∇[gf](y) = f(x)∇g(x) + g(x)∇f(x) - f(y)∇g(y) + g(y)∇f(y)
+ // = f(x)[∇g(x)-∇g(y)] + g(x)∇f(x) - [f(y)-f(x)]∇g(y) + g(y)∇f(y)
+ // = f(x)[∇g(x)-∇g(y)] + g(x)[∇f(x)-∇f(y)]
+ // - [f(y)-f(x)]∇g(y) + [g(y)-g(x)]∇f(y)
+ let &SupportProductFirst(ref f, ref g) = self.base_fn();
+ let (df, dg) = (f.diff_ref(), g.diff_ref());
+ [
+ df.lipschitz_factor(m).map(|l| l * g.bounds().uniform()),
+ dg.lipschitz_factor(m).map(|l| l * f.bounds().uniform()),
+ f.lipschitz_factor(m).map(|l| l * dg.norm_bound(L2)),
+ g.lipschitz_factor(m).map(|l| l * df.norm_bound(L2))
+ ].into_iter().sum()
+ }
+}
+
+
impl<'a, A, B, F : Float, const N : usize> Support<F, N>
for SupportProductFirst<A, B>
-where A : Support<F, N>,
- B : Support<F, N> {
+where
+ A : Support<F, N>,
+ B : Support<F, N>
+{
#[inline]
fn support_hint(&self) -> Cube<F, N> {
self.0.support_hint()
@@ -97,7 +166,7 @@
/// Representation of the sum of two kernels
///
-/// The kernels typically implement [`Support`] and [`Mapping`][alg_tools::mapping::Mapping].
+/// The kernels typically implement [`Support`] and [`Mapping`].
///
/// The implementation [`Support`] only uses the [`Support::support_hint`] of the first parameter!
#[derive(Copy,Clone,Serialize,Debug)]
@@ -108,33 +177,48 @@
pub B
);
-impl<'a, A, B, F : Float, const N : usize> Apply<&'a Loc<F, N>>
+impl<'a, A, B, F : Float, const N : usize> Mapping<Loc<F, N>>
for SupportSum<A, B>
-where A : Apply<&'a Loc<F, N>, Output=F>,
- B : Apply<&'a Loc<F, N>, Output=F> {
- type Output = F;
+where
+ A : Mapping<Loc<F, N>, Codomain = F>,
+ B : Mapping<Loc<F, N>, Codomain = F>,
+{
+ type Codomain = F;
+
#[inline]
- fn apply(&self, x : &'a Loc<F, N>) -> Self::Output {
- self.0.apply(x) + self.1.apply(x)
+ fn apply<I : Instance<Loc<F, N>>>(&self, x : I) -> Self::Codomain {
+ self.0.apply(x.ref_instance()) + self.1.apply(x)
}
}
-impl<A, B, F : Float, const N : usize> Apply<Loc<F, N>>
+impl<'a, A, B, F : Float, const N : usize> DifferentiableImpl<Loc<F, N>>
for SupportSum<A, B>
-where A : for<'a> Apply<&'a Loc<F, N>, Output=F>,
- B : for<'a> Apply<&'a Loc<F, N>, Output=F> {
- type Output = F;
+where
+ A : DifferentiableMapping<
+ Loc<F, N>,
+ DerivativeDomain = Loc<F, N>
+ >,
+ B : DifferentiableMapping<
+ Loc<F, N>,
+ DerivativeDomain = Loc<F, N>,
+ >
+{
+
+ type Derivative = Loc<F, N>;
+
#[inline]
- fn apply(&self, x : Loc<F, N>) -> Self::Output {
- self.0.apply(&x) + self.1.apply(&x)
+ fn differential_impl<I : Instance<Loc<F, N>>>(&self, x : I) -> Self::Derivative {
+ self.0.differential(x.ref_instance()) + self.1.differential(x)
}
}
+
impl<'a, A, B, F : Float, const N : usize> Support<F, N>
for SupportSum<A, B>
where A : Support<F, N>,
B : Support<F, N>,
Cube<F, N> : SetOrd {
+
#[inline]
fn support_hint(&self) -> Cube<F, N> {
self.0.support_hint().common(&self.1.support_hint())
@@ -174,9 +258,42 @@
}
}
+impl<F : Float, M : Copy, A, B> Lipschitz<M> for SupportSum<A, B>
+where A : Lipschitz<M, FloatType = F>,
+ B : Lipschitz<M, FloatType = F> {
+ type FloatType = F;
+
+ fn lipschitz_factor(&self, m : M) -> Option<F> {
+ match (self.0.lipschitz_factor(m), self.1.lipschitz_factor(m)) {
+ (Some(l0), Some(l1)) => Some(l0 + l1),
+ _ => None
+ }
+ }
+}
+
+impl<'b, F : Float, M : Copy, A, B, Domain> Lipschitz<M>
+for Differential<'b, Domain, SupportSum<A, B>>
+where
+ Domain : Space,
+ A : Clone + DifferentiableMapping<Domain, Codomain=F>,
+ B : Clone + DifferentiableMapping<Domain, Codomain=F>,
+ SupportSum<A, B> : DifferentiableMapping<Domain, Codomain=F>,
+ for<'a> A :: Differential<'a> : Lipschitz<M, FloatType = F>,
+ for<'a> B :: Differential<'a> : Lipschitz<M, FloatType = F>
+{
+ type FloatType = F;
+
+ fn lipschitz_factor(&self, m : M) -> Option<F> {
+ let base = self.base_fn();
+ base.0.diff_ref().lipschitz_factor(m)
+ .zip(base.1.diff_ref().lipschitz_factor(m))
+ .map(|(a, b)| a + b)
+ }
+}
+
/// Representation of the convolution of two kernels.
///
-/// The kernels typically implement [`Support`]s and [`Mapping`][alg_tools::mapping::Mapping].
+/// The kernels typically implement [`Support`]s and [`Mapping`].
//
/// Trait implementations have to be on a case-by-case basis.
#[derive(Copy,Clone,Serialize,Debug,Eq,PartialEq)]
@@ -187,9 +304,46 @@
pub B
);
+impl<F : Float, M, A, B> Lipschitz<M> for Convolution<A, B>
+where A : Norm<F, L1> ,
+ B : Lipschitz<M, FloatType = F> {
+ type FloatType = F;
+
+ fn lipschitz_factor(&self, m : M) -> Option<F> {
+ // For [f * g](x) = ∫ f(x-y)g(y) dy we have
+ // [f * g](x) - [f * g](z) = ∫ [f(x-y)-f(z-y)]g(y) dy.
+ // Hence |[f * g](x) - [f * g](z)| ≤ ∫ |f(x-y)-f(z-y)|g(y)| dy.
+ // ≤ L|x-z| ∫ |g(y)| dy,
+ // where L is the Lipschitz factor of f.
+ self.1.lipschitz_factor(m).map(|l| l * self.0.norm(L1))
+ }
+}
+
+impl<'b, F : Float, M, A, B, Domain> Lipschitz<M>
+for Differential<'b, Domain, Convolution<A, B>>
+where
+ Domain : Space,
+ A : Clone + Norm<F, L1> ,
+ Convolution<A, B> : DifferentiableMapping<Domain, Codomain=F>,
+ B : Clone + DifferentiableMapping<Domain, Codomain=F>,
+ for<'a> B :: Differential<'a> : Lipschitz<M, FloatType = F>
+{
+ type FloatType = F;
+
+ fn lipschitz_factor(&self, m : M) -> Option<F> {
+ // For [f * g](x) = ∫ f(x-y)g(y) dy we have
+ // ∇[f * g](x) - ∇[f * g](z) = ∫ [∇f(x-y)-∇f(z-y)]g(y) dy.
+ // Hence |∇[f * g](x) - ∇[f * g](z)| ≤ ∫ |∇f(x-y)-∇f(z-y)|g(y)| dy.
+ // ≤ L|x-z| ∫ |g(y)| dy,
+ // where L is the Lipschitz factor of ∇f.
+ let base = self.base_fn();
+ base.1.diff_ref().lipschitz_factor(m).map(|l| l * base.0.norm(L1))
+ }
+}
+
/// Representation of the autoconvolution of a kernel.
///
-/// The kernel typically implements [`Support`] and [`Mapping`][alg_tools::mapping::Mapping].
+/// The kernel typically implements [`Support`] and [`Mapping`].
///
/// Trait implementations have to be on a case-by-case basis.
#[derive(Copy,Clone,Serialize,Debug,Eq,PartialEq)]
@@ -198,34 +352,130 @@
pub A
);
+impl<F : Float, M, C> Lipschitz<M> for AutoConvolution<C>
+where C : Lipschitz<M, FloatType = F> + Norm<F, L1> {
+ type FloatType = F;
+
+ fn lipschitz_factor(&self, m : M) -> Option<F> {
+ self.0.lipschitz_factor(m).map(|l| l * self.0.norm(L1))
+ }
+}
+
+impl<'b, F : Float, M, C, Domain> Lipschitz<M>
+for Differential<'b, Domain, AutoConvolution<C>>
+where
+ Domain : Space,
+ C : Clone + Norm<F, L1> + DifferentiableMapping<Domain, Codomain=F>,
+ AutoConvolution<C> : DifferentiableMapping<Domain, Codomain=F>,
+ for<'a> C :: Differential<'a> : Lipschitz<M, FloatType = F>
+{
+ type FloatType = F;
+
+ fn lipschitz_factor(&self, m : M) -> Option<F> {
+ let base = self.base_fn();
+ base.0.diff_ref().lipschitz_factor(m).map(|l| l * base.0.norm(L1))
+ }
+}
+
+
/// Representation a multi-dimensional product of a one-dimensional kernel.
///
/// For $G: ℝ → ℝ$, this is the function $F(x\_1, …, x\_n) := \prod_{i=1}^n G(x\_i)$.
-/// The kernel $G$ typically implements [`Support`] and [`Mapping`][alg_tools::mapping::Mapping]
+/// The kernel $G$ typically implements [`Support`] and [`Mapping`]
/// on [`Loc<F, 1>`]. Then the product implements them on [`Loc<F, N>`].
#[derive(Copy,Clone,Serialize,Debug,Eq,PartialEq)]
+#[allow(dead_code)]
struct UniformProduct<G, const N : usize>(
/// The one-dimensional kernel
G
);
-impl<'a, G, F : Float, const N : usize> Apply<&'a Loc<F, N>>
+impl<'a, G, F : Float, const N : usize> Mapping<Loc<F, N>>
for UniformProduct<G, N>
-where G : Apply<Loc<F, 1>, Output=F> {
- type Output = F;
+where
+ G : Mapping<Loc<F, 1>, Codomain = F>
+{
+ type Codomain = F;
+
#[inline]
- fn apply(&self, x : &'a Loc<F, N>) -> F {
- x.iter().map(|&y| self.0.apply(Loc([y]))).product()
+ fn apply<I : Instance<Loc<F, N>>>(&self, x : I) -> F {
+ x.cow().iter().map(|&y| self.0.apply(Loc([y]))).product()
+ }
+}
+
+
+
+impl<'a, G, F : Float, const N : usize> DifferentiableImpl<Loc<F, N>>
+for UniformProduct<G, N>
+where
+ G : DifferentiableMapping<
+ Loc<F, 1>,
+ DerivativeDomain = F,
+ Codomain = F,
+ >
+{
+ type Derivative = Loc<F, N>;
+
+ #[inline]
+ fn differential_impl<I : Instance<Loc<F, N>>>(&self, x0 : I) -> Loc<F, N> {
+ x0.eval(|x| {
+ let vs = x.map(|y| self.0.apply(Loc([y])));
+ product_differential(x, &vs, |y| self.0.differential(Loc([y])))
+ })
}
}
-impl<G, F : Float, const N : usize> Apply<Loc<F, N>>
-for UniformProduct<G, N>
-where G : Apply<Loc<F, 1>, Output=F> {
- type Output = F;
- #[inline]
- fn apply(&self, x : Loc<F, N>) -> F {
- x.into_iter().map(|y| self.0.apply(Loc([y]))).product()
+/// Helper function to calulate the differential of $f(x)=∏_{i=1}^N g(x_i)$.
+///
+/// The vector `x` is the location, `vs` consists of the values `g(x_i)`, and
+/// `gd` calculates the derivative `g'`.
+#[inline]
+pub(crate) fn product_differential<F : Float, G : Fn(F) -> F, const N : usize>(
+ x : &Loc<F, N>,
+ vs : &Loc<F, N>,
+ gd : G
+) -> Loc<F, N> {
+ map1_indexed(x, |i, &y| {
+ gd(y) * vs.iter()
+ .zip(0..)
+ .filter_map(|(v, j)| (j != i).then_some(*v))
+ .product()
+ }).into()
+}
+
+/// Helper function to calulate the Lipschitz factor of $∇f$ for $f(x)=∏_{i=1}^N g(x_i)$.
+///
+/// The parameter `bound` is a bound on $|g|_∞$, `lip` is a Lipschitz factor for $g$,
+/// `dbound` is a bound on $|∇g|_∞$, and `dlip` a Lipschitz factor for $∇g$.
+#[inline]
+pub(crate) fn product_differential_lipschitz_factor<F : Float, const N : usize>(
+ bound : F,
+ lip : F,
+ dbound : F,
+ dlip : F
+) -> F {
+ // For arbitrary ψ(x) = ∏_{i=1}^n ψ_i(x_i), we have
+ // ψ(x) - ψ(y) = ∑_i [ψ_i(x_i)-ψ_i(y_i)] ∏_{j ≠ i} ψ_j(x_j)
+ // by a simple recursive argument. In particular, if ψ_i=g for all i, j, we have
+ // |ψ(x) - ψ(y)| ≤ ∑_i L_g M_g^{n-1}|x-y|, where L_g is the Lipschitz factor of g, and
+ // M_g a bound on it.
+ //
+ // We also have in the general case ∇ψ(x) = ∑_i ∇ψ_i(x_i) ∏_{j ≠ i} ψ_j(x_j), whence
+ // using the previous formula for each i with f_i=∇ψ_i and f_j=ψ_j for j ≠ i, we get
+ // ∇ψ(x) - ∇ψ(y) = ∑_i[ ∇ψ_i(x_i)∏_{j ≠ i} ψ_j(x_j) - ∇ψ_i(y_i)∏_{j ≠ i} ψ_j(y_j)]
+ // = ∑_i[ [∇ψ_i(x_i) - ∇ψ_j(x_j)] ∏_{j ≠ i}ψ_j(x_j)
+ // + [∑_{k ≠ i} [ψ_k(x_k) - ∇ψ_k(x_k)] ∏_{j ≠ i, k}ψ_j(x_j)]∇ψ_i(x_i)].
+ // With $ψ_i=g for all i, j, it follows that
+ // |∇ψ(x) - ∇ψ(y)| ≤ ∑_i L_{∇g} M_g^{n-1} + ∑_{k ≠ i} L_g M_g^{n-2} M_{∇g}
+ // = n [L_{∇g} M_g^{n-1} + (n-1) L_g M_g^{n-2} M_{∇g}].
+ // = n M_g^{n-2}[L_{∇g} M_g + (n-1) L_g M_{∇g}].
+ if N >= 2 {
+ F::cast_from(N) * bound.powi((N-2) as i32)
+ * (dlip * bound + F::cast_from(N-1) * lip * dbound)
+ } else if N==1 {
+ dlip
+ } else {
+ panic!("Invalid dimension")
}
}
