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src/kernels/hat_convolution.rs@7a953a87b6c1
src/kernels/hat_convolution.rs
Tue, 29 Nov 2022 15:36:12 +0200
- author
- Tuomo Valkonen <tuomov@iki.fi>
- date
- Tue, 29 Nov 2022 15:36:12 +0200
- changeset 2
- 7a953a87b6c1
- parent 0
- eb3c7813b67a
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fubar
//! Implementation of the convolution of two hat functions, //! and its convolution with a [`CubeIndicator`]. use numeric_literals::replace_float_literals; use serde::Serialize; use alg_tools::types::*; use alg_tools::norms::*; use alg_tools::loc::Loc; use alg_tools::sets::Cube; use alg_tools::bisection_tree::{ Support, Constant, Bounds, LocalAnalysis, GlobalAnalysis, Bounded, }; use alg_tools::mapping::Apply; use alg_tools::maputil::array_init; use super::base::*; use super::ball_indicator::CubeIndicator; /// Hat convolution kernel. /// /// This struct represents the function /// $$ /// f(x\_1, …, x\_n) = \prod\_{i=1}^n \frac{4}{σ} (h\*h)(x\_i/σ) /// $$ /// where the “hat function” $h(y)= \max(0, 1 - |2y|)$. /// The factor $4/σ$ normalises $∫ f d x = 1$. /// We have /// $$ /// (h*h)(y) = /// \begin{cases} /// \frac{2}{3} (y+1)^3 & -1<y\leq -\frac{1}{2}, \\\\ /// -2 y^3-2 y^2+\frac{1}{3} & -\frac{1}{2}<y\leq 0, \\\\ /// 2 y^3-2 y^2+\frac{1}{3} & 0<y<\frac{1}{2}, \\\\ /// -\frac{2}{3} (y-1)^3 & \frac{1}{2}\leq y<1. \\\\ /// \end{cases} /// $$ #[derive(Copy,Clone,Debug,Serialize,Eq)] pub struct HatConv<S : Constant, const N : usize> { /// The parameter $σ$ of the kernel. pub radius : S, } impl<S1, S2, const N : usize> PartialEq<HatConv<S2, N>> for HatConv<S1, N> where S1 : Constant, S2 : Constant<Type=S1::Type> { fn eq(&self, other : &HatConv<S2, N>) -> bool { self.radius.value() == other.radius.value() } } impl<'a, S, const N : usize> HatConv<S, N> where S : Constant { /// Returns the $σ$ parameter of the kernel. #[inline] pub fn radius(&self) -> S::Type { self.radius.value() } } impl<'a, S, const N : usize> Apply<&'a Loc<S::Type, N>> for HatConv<S, N> where S : Constant { type Output = S::Type; #[inline] fn apply(&self, y : &'a Loc<S::Type, N>) -> Self::Output { let σ = self.radius(); y.product_map(|x| { self.value_1d_σ1(x / σ) / σ }) } } impl<'a, S, const N : usize> Apply<Loc<S::Type, N>> for HatConv<S, N> where S : Constant { type Output = S::Type; #[inline] fn apply(&self, y : Loc<S::Type, N>) -> Self::Output { self.apply(&y) } } #[replace_float_literals(S::Type::cast_from(literal))] impl<'a, F : Float, S, const N : usize> HatConv<S, N> where S : Constant<Type=F> { /// Computes the value of the kernel for $n=1$ with $σ=1$. #[inline] fn value_1d_σ1(&self, x : F) -> F { let y = x.abs(); if y >= 1.0 { 0.0 } else if y > 0.5 { - (8.0/3.0) * (y - 1.0).powi(3) } else /* 0 ≤ y ≤ 0.5 */ { (4.0/3.0) + 8.0 * y * y * (y - 1.0) } } } impl<'a, S, const N : usize> Support<S::Type, N> for HatConv<S, N> where S : Constant { #[inline] fn support_hint(&self) -> Cube<S::Type,N> { let σ = self.radius(); array_init(|| [-σ, σ]).into() } #[inline] fn in_support(&self, y : &Loc<S::Type,N>) -> bool { let σ = self.radius(); y.iter().all(|x| x.abs() <= σ) } #[inline] fn bisection_hint(&self, cube : &Cube<S::Type, N>) -> [Option<S::Type>; N] { let σ = self.radius(); cube.map(|c, d| symmetric_peak_hint(σ, c, d)) } } #[replace_float_literals(S::Type::cast_from(literal))] impl<S, const N : usize> GlobalAnalysis<S::Type, Bounds<S::Type>> for HatConv<S, N> where S : Constant { #[inline] fn global_analysis(&self) -> Bounds<S::Type> { Bounds(0.0, self.apply(Loc::ORIGIN)) } } impl<S, const N : usize> LocalAnalysis<S::Type, Bounds<S::Type>, N> for HatConv<S, N> where S : Constant { #[inline] fn local_analysis(&self, cube : &Cube<S::Type, N>) -> Bounds<S::Type> { // The function is maximised/minimised where the 2-norm is minimised/maximised. let lower = self.apply(cube.maxnorm_point()); let upper = self.apply(cube.minnorm_point()); Bounds(lower, upper) } } #[replace_float_literals(C::Type::cast_from(literal))] impl<'a, C : Constant, const N : usize> Norm<C::Type, L1> for HatConv<C, N> { #[inline] fn norm(&self, _ : L1) -> C::Type { 1.0 } } #[replace_float_literals(C::Type::cast_from(literal))] impl<'a, C : Constant, const N : usize> Norm<C::Type, Linfinity> for HatConv<C, N> { #[inline] fn norm(&self, _ : Linfinity) -> C::Type { self.bounds().upper() } } #[replace_float_literals(F::cast_from(literal))] impl<'a, F : Float, R, C, const N : usize> Apply<&'a Loc<F, N>> for Convolution<CubeIndicator<R, N>, HatConv<C, N>> where R : Constant<Type=F>, C : Constant<Type=F> { type Output = F; #[inline] fn apply(&self, y : &'a Loc<F, N>) -> F { let Convolution(ref ind, ref hatconv) = self; let β = ind.r.value(); let σ = hatconv.radius(); // This is just a product of one-dimensional versions y.product_map(|x| { // With $u_σ(x) = u_1(x/σ)/σ$ the normalised hat convolution // we have // $$ // [χ_{-β,β} * u_σ](x) // = ∫_{x-β}^{x+β} u_σ(z) d z // = (1/σ)∫_{x-β}^{x+β} u_1(z/σ) d z // = ∫_{(x-β)/σ}^{(x+β)/σ} u_1(z) d z // = [χ_{-β/σ, β/σ} * u_1](x/σ) // $$ self.value_1d_σ1(x / σ, β / σ) }) } } impl<'a, F : Float, R, C, const N : usize> Apply<Loc<F, N>> for Convolution<CubeIndicator<R, N>, HatConv<C, N>> where R : Constant<Type=F>, C : Constant<Type=F> { type Output = F; #[inline] fn apply(&self, y : Loc<F, N>) -> F { self.apply(&y) } } #[replace_float_literals(F::cast_from(literal))] impl<F : Float, C, R, const N : usize> Convolution<CubeIndicator<R, N>, HatConv<C, N>> where R : Constant<Type=F>, C : Constant<Type=F> { #[inline] pub fn value_1d_σ1(&self, x : F, β : F) -> F { // The integration interval let a = x - β; let b = x + β; #[inline] fn pow4<F : Float>(x : F) -> F { let y = x * x; y * y } /// Integrate $f$, whose support is $[c, d]$, on $[a, b]$. /// If $b > d$, add $g()$ to the result. #[inline] fn i<F: Float>(a : F, b : F, c : F, d : F, f : impl Fn(F) -> F, g : impl Fn() -> F) -> F { if b < c { 0.0 } else if b <= d { if a <= c { f(b) - f(c) } else { f(b) - f(a) } } else /* b > d */ { g() + if a <= c { f(d) - f(c) } else if a < d { f(d) - f(a) } else { 0.0 } } } // Observe the factor 1/6 at the front from the antiderivatives below. // The factor 4 is from normalisation of the original function. (4.0/6.0) * i(a, b, -1.0, -0.5, // (2/3) (y+1)^3 on -1 < y ≤ - 1/2 // The antiderivative is (2/12)(y+1)^4 = (1/6)(y+1)^4 |y| pow4(y+1.0), || i(a, b, -0.5, 0.0, // -2 y^3 - 2 y^2 + 1/3 on -1/2 < y ≤ 0 // The antiderivative is -1/2 y^4 - 2/3 y^3 + 1/3 y |y| y*(-y*y*(y*3.0 + 4.0) + 2.0), || i(a, b, 0.0, 0.5, // 2 y^3 - 2 y^2 + 1/3 on 0 < y < 1/2 // The antiderivative is 1/2 y^4 - 2/3 y^3 + 1/3 y |y| y*(y*y*(y*3.0 - 4.0) + 2.0), || i(a, b, 0.5, 1.0, // -(2/3) (y-1)^3 on 1/2 < y ≤ 1 // The antiderivative is -(2/12)(y-1)^4 = -(1/6)(y-1)^4 |y| -pow4(y-1.0), || 0.0 ) ) ) ) } } impl<F : Float, R, C, const N : usize> Convolution<CubeIndicator<R, N>, HatConv<C, N>> where R : Constant<Type=F>, C : Constant<Type=F> { #[inline] fn get_r(&self) -> F { let Convolution(ref ind, ref hatconv) = self; ind.r.value() + hatconv.radius() } } impl<F : Float, R, C, const N : usize> Support<F, N> for Convolution<CubeIndicator<R, N>, HatConv<C, N>> where R : Constant<Type=F>, C : Constant<Type=F> { #[inline] fn support_hint(&self) -> Cube<F, N> { let r = self.get_r(); array_init(|| [-r, r]).into() } #[inline] fn in_support(&self, y : &Loc<F, N>) -> bool { let r = self.get_r(); y.iter().all(|x| x.abs() <= r) } #[inline] fn bisection_hint(&self, cube : &Cube<F, N>) -> [Option<F>; N] { // It is not difficult to verify that [`HatConv`] is C^2. // Therefore, so is [`Convolution<CubeIndicator<R, N>, HatConv<C, N>>`] so that a finer // subdivision for the hint than this is not particularly useful. let r = self.get_r(); cube.map(|c, d| symmetric_peak_hint(r, c, d)) } } impl<F : Float, R, C, const N : usize> GlobalAnalysis<F, Bounds<F>> for Convolution<CubeIndicator<R, N>, HatConv<C, N>> where R : Constant<Type=F>, C : Constant<Type=F> { #[inline] fn global_analysis(&self) -> Bounds<F> { Bounds(F::ZERO, self.apply(Loc::ORIGIN)) } } impl<F : Float, R, C, const N : usize> LocalAnalysis<F, Bounds<F>, N> for Convolution<CubeIndicator<R, N>, HatConv<C, N>> where R : Constant<Type=F>, C : Constant<Type=F> { #[inline] fn local_analysis(&self, cube : &Cube<F, N>) -> Bounds<F> { // The function is maximised/minimised where the absolute value is minimised/maximised. let lower = self.apply(cube.maxnorm_point()); let upper = self.apply(cube.minnorm_point()); //assert!(upper >= lower); if upper < lower { let Convolution(ref ind, ref hatconv) = self; let β = ind.r.value(); let σ = hatconv.radius(); eprintln!("WARNING: Hat convolution {β} {σ} upper bound {upper} < lower bound {lower} on {cube:?} with min-norm point {:?} and max-norm point {:?}", cube.minnorm_point(), cube.maxnorm_point()); Bounds(upper, lower) } else { Bounds(lower, upper) } } } /// This [`BoundedBy`] implementation bounds $u * u$ by $(ψ * ψ) u$ for $u$ a hat convolution and /// $ψ = χ_{[-a,a]^N}$ for some $a>0$. /// /// This is based on the general formula for bounding $(uχ) * (uχ)$ by $(ψ * ψ) u$, /// where we take $ψ = χ_{[-a,a]^N}$ and $χ = χ_{[-σ,σ]^N}$ for $σ$ the width of the hat /// convolution. #[replace_float_literals(F::cast_from(literal))] impl<F, C, S, const N : usize> BoundedBy<F, SupportProductFirst<AutoConvolution<CubeIndicator<S, N>>, HatConv<C, N>>> for AutoConvolution<HatConv<C, N>> where F : Float, C : Constant<Type=F>, S : Constant<Type=F> { fn bounding_factor( &self, kernel : &SupportProductFirst<AutoConvolution<CubeIndicator<S, N>>, HatConv<C, N>> ) -> Option<F> { // We use the comparison $ℱ[𝒜(ψ v)] ≤ L_1 ℱ[𝒜(ψ)u] ⟺ I_{v̂} v̂ ≤ L_1 û$ with // $ψ = χ_{[-w, w]}$ satisfying $supp v ⊂ [-w, w]$, i.e. $w ≥ σ$. Here $v̂ = ℱ[v]$ and // $I_{v̂} = ∫ v̂ d ξ. For this relationship to be valid, we need $v̂ ≥ 0$, which is guaranteed // by $v̂ = u_σ$ being an autoconvolution. With $u = v$, therefore $L_1 = I_v̂ = ∫ u_σ(ξ) d ξ$. let SupportProductFirst(AutoConvolution(ref ind), hatconv2) = kernel; let σ = self.0.radius(); let a = ind.r.value(); let bounding_1d = 4.0 / (3.0 * σ); // Check that the cutting indicator of the comparison // `SupportProductFirst<AutoConvolution<CubeIndicator<S, N>>, HatConv<C, N>>` // is wide enough, and that the hat convolution has the same radius as ours. if σ <= a && hatconv2 == &self.0 { Some(bounding_1d.powi(N as i32)) } else { // We cannot compare None } } } /// This [`BoundedBy`] implementation bounds $u * u$ by $u$ for $u$ a hat convolution. /// /// This is based on Example 3.3 in the manuscript. #[replace_float_literals(F::cast_from(literal))] impl<F, C, const N : usize> BoundedBy<F, HatConv<C, N>> for AutoConvolution<HatConv<C, N>> where F : Float, C : Constant<Type=F> { /// Returns an estimate of the factor $L_1$. /// /// Returns `None` if `kernel` does not have the same width as hat convolution that `self` /// is based on. fn bounding_factor( &self, kernel : &HatConv<C, N> ) -> Option<F> { if kernel == &self.0 { Some(1.0) } else { // We cannot compare None } } } #[cfg(test)] mod tests { use alg_tools::lingrid::linspace; use alg_tools::mapping::Apply; use alg_tools::norms::Linfinity; use alg_tools::loc::Loc; use crate::kernels::{BallIndicator, CubeIndicator, Convolution}; use super::HatConv; /// Tests numerically that [`HatConv<f64, 1>`] is monotone. #[test] fn hatconv_monotonicity() { let grid = linspace(0.0, 1.0, 100000); let hatconv : HatConv<f64, 1> = HatConv{ radius : 1.0 }; let mut vals = grid.into_iter().map(|t| hatconv.apply(Loc::from(t))); let first = vals.next().unwrap(); let monotone = vals.fold((first, true), |(prev, ok), t| (prev, ok && prev >= t)).1; assert!(monotone); } /// Tests numerically that [`Convolution<CubeIndicator<f64, 1>, HatConv<f64, 1>>`] is monotone. #[test] fn convolution_cubeind_hatconv_monotonicity() { let grid = linspace(-2.0, 0.0, 100000); let hatconv : Convolution<CubeIndicator<f64, 1>, HatConv<f64, 1>> = Convolution(BallIndicator { r : 0.5, exponent : Linfinity }, HatConv{ radius : 1.0 } ); let mut vals = grid.into_iter().map(|t| hatconv.apply(Loc::from(t))); let first = vals.next().unwrap(); let monotone = vals.fold((first, true), |(prev, ok), t| (prev, ok && prev <= t)).1; assert!(monotone); let grid = linspace(0.0, 2.0, 100000); let hatconv : Convolution<CubeIndicator<f64, 1>, HatConv<f64, 1>> = Convolution(BallIndicator { r : 0.5, exponent : Linfinity }, HatConv{ radius : 1.0 } ); let mut vals = grid.into_iter().map(|t| hatconv.apply(Loc::from(t))); let first = vals.next().unwrap(); let monotone = vals.fold((first, true), |(prev, ok), t| (prev, ok && prev >= t)).1; assert!(monotone); } }
