pointsource_algs: comparison src/kernels/base.rs

-:6105b5cd8d89
+:f0e8704d3f0e
 Bounds,
 LocalAnalysis,
 GlobalAnalysis,
 Bounded,
 };
-use alg_tools::mapping::Apply;
+use alg_tools::mapping::{
-use alg_tools::maputil::{array_init, map2};
+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)]
 pub struct SupportProductFirst<A, B>(
 /// First kernel
 pub A,
 /// Second kernel
 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 : for<'a> Apply<&'a Loc<F, N>, Output=F>,
+where
-B : for<'a> Apply<&'a Loc<F, N>, Output=F> {
+A : Mapping<Loc<F, N>, Codomain = F>,
-type Output = F;
+B : Mapping<Loc<F, N>, Codomain = F>,
-#[inline]
+{
-fn apply(&self, x : Loc<F, N>) -> Self::Output {
+type Codomain = F;
-self.0.apply(&x) * self.1.apply(&x)
-}
+#[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, A, B, F : Float, const N : usize> Apply<&'a Loc<F, N>>
+}
+}
+impl<A, B, F : Float, const N : usize> DifferentiableImpl<Loc<F, N>>
 for SupportProductFirst<A, B>
-where A : Apply<&'a Loc<F, N>, Output=F>,
+where
-B : Apply<&'a Loc<F, N>, Output=F> {
+A : DifferentiableMapping<
-type Output = F;
+Loc<F, N>,
-#[inline]
+DerivativeDomain=Loc<F, N>,
-fn apply(&self, x : &'a Loc<F, N>) -> Self::Output {
+Codomain = F
-self.0.apply(x) * self.1.apply(x)
+>,
-}
+B : DifferentiableMapping<
-}
+Loc<F, N>,
+DerivativeDomain=Loc<F, N>,
+Codomain = F,
+>
+{
+type Derivative = Loc<F, N>;
+#[inline]
+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, B, M : Copy, F : Float> Lipschitz<M>
+for SupportProductFirst<A, B>
+where A : Lipschitz<M, FloatType = F> + Bounded<F>,
+B : Lipschitz<M, FloatType = F> + Bounded<F> {
+type FloatType = F;
+#[inline]
+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>,
+where
-B : Support<F, N> {
+A : Support<F, N>,
+B : Support<F, N>
+{
 #[inline]
 fn support_hint(&self) -> Cube<F, N> {
 self.0.support_hint()
 }
 }
 }
 /// 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)]
 pub struct SupportSum<A, B>(
 /// First kernel
 pub A,
 /// Second kernel
 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>,
+where
-B : Apply<&'a Loc<F, N>, Output=F> {
+A : Mapping<Loc<F, N>, Codomain = F>,
-type Output = F;
+B : Mapping<Loc<F, N>, Codomain = F>,
-#[inline]
+{
-fn apply(&self, x : &'a Loc<F, N>) -> Self::Output {
+type Codomain = F;
-self.0.apply(x) + self.1.apply(x)
-}
+#[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> 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>,
+where
-B : for<'a> Apply<&'a Loc<F, N>, Output=F> {
+A : DifferentiableMapping<
-type Output = F;
+Loc<F, N>,
-#[inline]
+DerivativeDomain = Loc<F, N>
-fn apply(&self, x : Loc<F, N>) -> Self::Output {
+>,
-self.0.apply(&x) + self.1.apply(&x)
+B : DifferentiableMapping<
-}
+Loc<F, N>,
-}
+DerivativeDomain = Loc<F, N>,
+>
+{
+type Derivative = Loc<F, N>;
+#[inline]
+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())
 }
 fn local_analysis(&self, cube : &Cube<F, N>) -> Bounds<F> {
 self.0.local_analysis(cube) + self.1.local_analysis(cube)
 }
 }
+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)]
 pub struct Convolution<A, B>(
 /// First kernel
 pub A,
 /// Second kernel
 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)]
 pub struct AutoConvolution<A>(
 /// The kernel to be autoconvolved
 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> {
+where
-type Output = F;
+G : Mapping<Loc<F, 1>, Codomain = F>
-#[inline]
+{
-fn apply(&self, x : &'a Loc<F, N>) -> F {
+type Codomain = F;
-x.iter().map(|&y| self.0.apply(Loc([y]))).product()
-}
+#[inline]
-}
+fn apply<I : Instance<Loc<F, N>>>(&self, x : I) -> F {
+x.cow().iter().map(|&y| self.0.apply(Loc([y]))).product()
-impl<G, F : Float, const N : usize> Apply<Loc<F, N>>
+}
+}
+impl<'a, G, F : Float, const N : usize> DifferentiableImpl<Loc<F, N>>
 for UniformProduct<G, N>
-where G : Apply<Loc<F, 1>, Output=F> {
+where
-type Output = F;
+G : DifferentiableMapping<
-#[inline]
+Loc<F, 1>,
-fn apply(&self, x : Loc<F, N>) -> F {
+DerivativeDomain = F,
-x.into_iter().map(|y| self.0.apply(Loc([y]))).product()
+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])))
+})
+}
+}
+/// 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")
 }
 }
 impl<G, F : Float, const N : usize> Support<F, N>
 for UniformProduct<G, N>

Mercurial > repos > pointsource_algs / file comparison

comparison: src/kernels/base.rs

src/kernels/base.rs