--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/convex.rs Mon Feb 03 19:22:16 2025 -0500
@@ -0,0 +1,290 @@
+/*!
+Some convex analysis basics
+*/
+
+use std::marker::PhantomData;
+use crate::types::*;
+use crate::mapping::{Mapping, Space};
+use crate::linops::IdOp;
+use crate::instance::{Instance, InstanceMut, DecompositionMut};
+use crate::operator_arithmetic::{Constant, Weighted};
+use crate::norms::*;
+
+/// Trait for convex mappings. Has no features, just serves as a constraint
+///
+/// TODO: should constrain `Mapping::Codomain` to implement a partial order,
+/// but this makes everything complicated with little benefit.
+pub trait ConvexMapping<Domain : Space, F : Num = f64> : Mapping<Domain, Codomain = F>
+{}
+
+/// Trait for mappings with a Fenchel conjugate
+///
+/// The conjugate type has to implement [`ConvexMapping`], but a `Conjugable` mapping need
+/// not be convex.
+pub trait Conjugable<Domain : HasDual<F>, F : Num = f64> : Mapping<Domain> {
+ type Conjugate<'a> : ConvexMapping<Domain::DualSpace, F> where Self : 'a;
+
+ fn conjugate(&self) -> Self::Conjugate<'_>;
+}
+
+/// Trait for mappings with a Fenchel preconjugate
+///
+/// In contrast to [`Conjugable`], the preconjugate need not implement [`ConvexMapping`],
+/// but a `Preconjugable` mapping has to be convex.
+pub trait Preconjugable<Domain, Predual, F : Num = f64> : ConvexMapping<Domain, F>
+where
+ Domain : Space,
+ Predual : HasDual<F>
+{
+ type Preconjugate<'a> : Mapping<Predual> where Self : 'a;
+
+ fn preconjugate(&self) -> Self::Preconjugate<'_>;
+}
+
+/// Trait for mappings with a proximap map
+///
+/// The conjugate type has to implement [`ConvexMapping`], but a `Conjugable` mapping need
+/// not be convex.
+pub trait Prox<Domain : Space> : Mapping<Domain> {
+ type Prox<'a> : Mapping<Domain, Codomain=Domain> where Self : 'a;
+
+ /// Returns a proximal mapping with weight τ
+ fn prox_mapping(&self, τ : Self::Codomain) -> Self::Prox<'_>;
+
+ /// Calculate the proximal mapping with weight τ
+ fn prox<I : Instance<Domain>>(&self, τ : Self::Codomain, z : I) -> Domain {
+ self.prox_mapping(τ).apply(z)
+ }
+
+ /// Calculate the proximal mapping with weight τ in-place
+ fn prox_mut<'b>(&self, τ : Self::Codomain, y : &'b mut Domain)
+ where
+ &'b mut Domain : InstanceMut<Domain>,
+ Domain:: Decomp : DecompositionMut<Domain>,
+ for<'a> &'a Domain : Instance<Domain>,
+ {
+ *y = self.prox(τ, &*y);
+ }
+}
+
+
+/// Constraint to the unit ball of the norm described by `E`.
+pub struct NormConstraint<F : Float, E : NormExponent> {
+ radius : F,
+ norm : NormMapping<F, E>,
+}
+
+impl<Domain, E, F> ConvexMapping<Domain, F> for NormMapping<F, E>
+where
+ Domain : Space,
+ E : NormExponent,
+ F : Float,
+ Self : Mapping<Domain, Codomain=F>
+{}
+
+
+impl<F, E, Domain> Mapping<Domain> for NormConstraint<F, E>
+where
+ Domain : Space + Norm<F, E>,
+ F : Float,
+ E : NormExponent,
+{
+ type Codomain = F;
+
+ fn apply<I : Instance<Domain>>(&self, d : I) -> F {
+ if d.eval(|x| x.norm(self.norm.exponent)) <= self.radius {
+ F::ZERO
+ } else {
+ F::INFINITY
+ }
+ }
+}
+
+impl<Domain, E, F> ConvexMapping<Domain, F> for NormConstraint<F, E>
+where
+ Domain : Space,
+ E : NormExponent,
+ F : Float,
+ Self : Mapping<Domain, Codomain=F>
+{}
+
+
+impl<E, F, Domain> Conjugable<Domain, F> for NormMapping<F, E>
+where
+ E : HasDualExponent,
+ F : Float,
+ Domain : HasDual<F> + Norm<F, E> + Space,
+ <Domain as HasDual<F>>::DualSpace : Norm<F, E::DualExp>
+{
+ type Conjugate<'a> = NormConstraint<F, E::DualExp> where Self : 'a;
+
+ fn conjugate(&self) -> Self::Conjugate<'_> {
+ NormConstraint {
+ radius : F::ONE,
+ norm : self.exponent.dual_exponent().as_mapping()
+ }
+ }
+}
+
+impl<C, E, F, Domain> Conjugable<Domain, F> for Weighted<NormMapping<F, E>, C>
+where
+ C : Constant<Type = F>,
+ E : HasDualExponent,
+ F : Float,
+ Domain : HasDual<F> + Norm<F, E> + Space,
+ <Domain as HasDual<F>>::DualSpace : Norm<F, E::DualExp>
+{
+ type Conjugate<'a> = NormConstraint<F, E::DualExp> where Self : 'a;
+
+ fn conjugate(&self) -> Self::Conjugate<'_> {
+ NormConstraint {
+ radius : self.weight.value(),
+ norm : self.base_fn.exponent.dual_exponent().as_mapping()
+ }
+ }
+}
+
+impl<Domain, E, F> Prox<Domain> for NormConstraint<F, E>
+where
+ Domain : Space + Norm<F, E>,
+ E : NormExponent,
+ F : Float,
+ NormProjection<F, E> : Mapping<Domain, Codomain=Domain>,
+{
+ type Prox<'a> = NormProjection<F, E> where Self : 'a;
+
+ #[inline]
+ fn prox_mapping(&self, _τ : Self::Codomain) -> Self::Prox<'_> {
+ assert!(self.radius >= F::ZERO);
+ NormProjection{ radius : self.radius, exponent : self.norm.exponent }
+ }
+}
+
+/// Projection to the unit ball of the norm described by `E`.
+pub struct NormProjection<F : Float, E : NormExponent> {
+ radius : F,
+ exponent : E,
+}
+
+/*
+impl<F, Domain> Mapping<Domain> for NormProjection<F, L2>
+where
+ Domain : Space + Euclidean<F> + std::ops::MulAssign<F>,
+ F : Float,
+{
+ type Codomain = Domain;
+
+ fn apply<I : Instance<Domain>>(&self, d : I) -> Domain {
+ d.own().proj_ball2(self.radius)
+ }
+}
+*/
+
+impl<F, E, Domain> Mapping<Domain> for NormProjection<F, E>
+where
+ Domain : Space + Projection<F, E>,
+ F : Float,
+ E : NormExponent,
+{
+ type Codomain = Domain;
+
+ fn apply<I : Instance<Domain>>(&self, d : I) -> Domain {
+ d.own().proj_ball(self.radius, self.exponent)
+ }
+}
+
+
+/// The zero mapping
+pub struct Zero<Domain : Space, F : Num>(PhantomData<(Domain, F)>);
+
+impl<Domain : Space, F : Num> Zero<Domain, F> {
+ pub fn new() -> Self {
+ Zero(PhantomData)
+ }
+}
+
+impl<Domain : Space, F : Num> Mapping<Domain> for Zero<Domain, F> {
+ type Codomain = F;
+
+ /// Compute the value of `self` at `x`.
+ fn apply<I : Instance<Domain>>(&self, _x : I) -> Self::Codomain {
+ F::ZERO
+ }
+}
+
+impl<Domain : Space, F : Num> ConvexMapping<Domain, F> for Zero<Domain, F> { }
+
+
+impl<Domain : HasDual<F>, F : Float> Conjugable<Domain, F> for Zero<Domain, F> {
+ type Conjugate<'a> = ZeroIndicator<Domain::DualSpace, F> where Self : 'a;
+
+ #[inline]
+ fn conjugate(&self) -> Self::Conjugate<'_> {
+ ZeroIndicator::new()
+ }
+}
+
+impl<Domain, Predual, F : Float> Preconjugable<Domain, Predual, F> for Zero<Domain, F>
+where
+ Domain : Space,
+ Predual : HasDual<F>
+{
+ type Preconjugate<'a> = ZeroIndicator<Predual, F> where Self : 'a;
+
+ #[inline]
+ fn preconjugate(&self) -> Self::Preconjugate<'_> {
+ ZeroIndicator::new()
+ }
+}
+
+impl<Domain : Space + Clone, F : Num> Prox<Domain> for Zero<Domain, F> {
+ type Prox<'a> = IdOp<Domain> where Self : 'a;
+
+ #[inline]
+ fn prox_mapping(&self, _τ : Self::Codomain) -> Self::Prox<'_> {
+ IdOp::new()
+ }
+}
+
+
+/// The zero indicator
+pub struct ZeroIndicator<Domain : Space, F : Num>(PhantomData<(Domain, F)>);
+
+impl<Domain : Space, F : Num> ZeroIndicator<Domain, F> {
+ pub fn new() -> Self {
+ ZeroIndicator(PhantomData)
+ }
+}
+
+impl<Domain : Normed<F>, F : Float> Mapping<Domain> for ZeroIndicator<Domain, F> {
+ type Codomain = F;
+
+ /// Compute the value of `self` at `x`.
+ fn apply<I : Instance<Domain>>(&self, x : I) -> Self::Codomain {
+ x.eval(|x̃| if x̃.is_zero() { F::ZERO } else { F::INFINITY })
+ }
+}
+
+impl<Domain : Normed<F>, F : Float> ConvexMapping<Domain, F> for ZeroIndicator<Domain, F> { }
+
+impl<Domain : HasDual<F>, F : Float> Conjugable<Domain, F> for ZeroIndicator<Domain, F> {
+ type Conjugate<'a> = Zero<Domain::DualSpace, F> where Self : 'a;
+
+ #[inline]
+ fn conjugate(&self) -> Self::Conjugate<'_> {
+ Zero::new()
+ }
+}
+
+impl<Domain, Predual, F : Float> Preconjugable<Domain, Predual, F> for ZeroIndicator<Domain, F>
+where
+ Domain : Normed<F>,
+ Predual : HasDual<F>
+{
+ type Preconjugate<'a> = Zero<Predual, F> where Self : 'a;
+
+ #[inline]
+ fn preconjugate(&self) -> Self::Preconjugate<'_> {
+ Zero::new()
+ }
+}