src/norms.rs

/*!
Norms, projections, etc.
*/

use serde::Serialize;
use std::marker::PhantomData;
use crate::types::*;
use crate::euclidean::*;
use crate::mapping::{Mapping, Space, Instance, ArithmeticTrue};

//
// Abstract norms
//

#[derive(Copy,Clone,Debug)]
/// Helper structure to convert a [`NormExponent`] into a [`Mapping`]
pub struct NormMapping<F : Float, E : NormExponent>{
    pub(crate) exponent : E,
    _phantoms : PhantomData<F>
}

/// An exponent for norms.
///
// Just a collection of desirable attributes for a marker type
pub trait NormExponent : Copy + Send + Sync + 'static {
    /// Return the norm as a mappin
    fn as_mapping<F : Float>(self) -> NormMapping<F, Self> {
        NormMapping{ exponent : self, _phantoms : PhantomData }
    }
}

/// Exponent type for the 1-[`Norm`].
#[derive(Copy,Debug,Clone,Serialize,Eq,PartialEq)]
pub struct L1;
impl NormExponent for L1 {}

/// Exponent type for the 2-[`Norm`].
#[derive(Copy,Debug,Clone,Serialize,Eq,PartialEq)]
pub struct L2;
impl NormExponent for L2 {}

/// Exponent type for the ∞-[`Norm`].
#[derive(Copy,Debug,Clone,Serialize,Eq,PartialEq)]
pub struct Linfinity;
impl NormExponent for Linfinity {}

/// Exponent type for 2,1-[`Norm`].
/// (1-norm over a domain Ω, 2-norm of a vector at each point of the domain.)
#[derive(Copy,Debug,Clone,Serialize,Eq,PartialEq)]
pub struct L21;
impl NormExponent for L21 {}

/// Norms for pairs (a, b). ‖(a,b)‖ = ‖(‖a‖_A, ‖b‖_B)‖_J
/// For use with [`crate::direct_product::Pair`]
#[derive(Copy,Debug,Clone,Serialize,Eq,PartialEq)]
pub struct PairNorm<A, B, J>(pub A, pub B, pub J);

impl<A, B, J> NormExponent for PairNorm<A, B, J>
where A : NormExponent, B : NormExponent, J : NormExponent {}


/// A Huber/Moreau–Yosida smoothed [`L1`] norm. (Not a norm itself.)
///
/// The parameter γ of this type is the smoothing factor. Zero means no smoothing, and higher
/// values more smoothing. Behaviour with γ < 0 is undefined.
#[derive(Copy,Debug,Clone,Serialize,Eq,PartialEq)]
pub struct HuberL1<F : Float>(pub F);
impl<F : Float> NormExponent for HuberL1<F> {}

/// A Huber/Moreau–Yosida smoothed [`L21`] norm. (Not a norm itself.)
///
/// The parameter γ of this type is the smoothing factor. Zero means no smoothing, and higher
/// values more smoothing. Behaviour with γ < 0 is undefined.
#[derive(Copy,Debug,Clone,Serialize,Eq,PartialEq)]
pub struct HuberL21<F : Float>(pub F);
impl<F : Float> NormExponent for HuberL21<F> {}


/// A normed space (type) with exponent or other type `Exponent` for the norm.
///
/// Use as
/// ```
/// # use alg_tools::norms::{Norm, L1, L2, Linfinity};
/// # use alg_tools::loc::Loc;
/// let x = Loc([1.0, 2.0, 3.0]);
///
/// println!("{}, {} {}", x.norm(L1), x.norm(L2), x.norm(Linfinity))
/// ```
pub trait Norm<F : Num, Exponent : NormExponent> {
    /// Calculate the norm.
    fn norm(&self, _p : Exponent) -> F;
}

/// Indicates that the `Self`-[`Norm`] is dominated by the `Exponent`-`Norm` on the space
/// `Elem` with the corresponding field `F`.
pub trait Dominated<F : Num, Exponent : NormExponent, Elem> {
    /// Indicates the factor $c$ for the inequality $‖x‖ ≤ C ‖x‖_p$.
    fn norm_factor(&self, p : Exponent) -> F;
    /// Given a norm-value $‖x‖_p$, calculates $C‖x‖_p$ such that $‖x‖ ≤ C‖x‖_p$
    #[inline]
    fn from_norm(&self, p_norm : F, p : Exponent) -> F {
        p_norm * self.norm_factor(p)
    }
}

/// Trait for distances with respect to a norm.
pub trait Dist<F : Num, Exponent : NormExponent> : Norm<F, Exponent> + Space {
    /// Calculate the distance
    fn dist<I : Instance<Self>>(&self, other : I, _p : Exponent) -> F;
}

/// Trait for Euclidean projections to the `Exponent`-[`Norm`]-ball.
///
/// Use as
/// ```
/// # use alg_tools::norms::{Projection, L2, Linfinity};
/// # use alg_tools::loc::Loc;
/// let x = Loc([1.0, 2.0, 3.0]);
///
/// println!("{:?}, {:?}", x.proj_ball(1.0, L2), x.proj_ball(0.5, Linfinity));
/// ```
pub trait Projection<F : Num, Exponent : NormExponent> : Norm<F, Exponent> + Sized
where F : Float {
    /// Projection of `self` to the `q`-norm-ball of radius ρ.
    fn proj_ball(mut self, ρ : F, q : Exponent) -> Self {
        self.proj_ball_mut(ρ, q);
        self
    }

    /// In-place projection of `self` to the `q`-norm-ball of radius ρ.
    fn proj_ball_mut(&mut self, ρ : F, q : Exponent);
}

/*impl<F : Float, E : Euclidean<F>> Norm<F, L2> for E {
    #[inline]
    fn norm(&self, _p : L2) -> F { self.norm2() }

    fn dist(&self, other : &Self, _p : L2) -> F { self.dist2(other) }
}*/

impl<F : Float, E : Euclidean<F> + Norm<F, L2>> Projection<F, L2> for E {
    #[inline]
    fn proj_ball(self, ρ : F, _p : L2) -> Self { self.proj_ball2(ρ) }

    #[inline]
    fn proj_ball_mut(&mut self, ρ : F, _p : L2) { self.proj_ball2_mut(ρ) }
}

impl<F : Float> HuberL1<F> {
    fn apply(self, xnsq : F) -> F {
        let HuberL1(γ) = self;
        let xn = xnsq.sqrt();
        if γ == F::ZERO {
            xn
        } else {
            if xn > γ {
                xn-γ / F::TWO
            } else if xn<(-γ) {
                -xn-γ / F::TWO
            } else {
                xnsq / (F::TWO * γ)
            }
        }
    }
}

impl<F : Float, E : Euclidean<F>> Norm<F, HuberL1<F>> for E {
    fn norm(&self, huber : HuberL1<F>) -> F {
        huber.apply(self.norm2_squared())
    }
}

impl<F : Float, E : Euclidean<F>> Dist<F, HuberL1<F>> for E {
    fn dist<I : Instance<Self>>(&self, other : I, huber : HuberL1<F>) -> F {
        huber.apply(self.dist2_squared(other))
    }
}

// impl<F : Float, E : Norm<F, L2>> Norm<F, L21> for Vec<E> {
//     fn norm(&self, _l21 : L21) -> F {
//         self.iter().map(|e| e.norm(L2)).sum()
//     }
// }

// impl<F : Float, E : Dist<F, L2>> Dist<F, L21> for Vec<E> {
//     fn dist<I : Instance<Self>>(&self, other : I, _l21 : L21) -> F {
//         self.iter().zip(other.iter()).map(|(e, g)| e.dist(g, L2)).sum()
//     }
// }

impl<E, F, Domain> Mapping<Domain> for NormMapping<F, E>
where
    F : Float,
    E : NormExponent,
    Domain : Space + Norm<F, E>,
{
    type Codomain = F;
    type ArithmeticOptIn = ArithmeticTrue;

    #[inline]
    fn apply<I : Instance<Domain>>(&self, x : I) -> F {
        x.eval(|r| r.norm(self.exponent))
    }
}

pub trait Normed<F : Num = f64> : Space + Norm<F, Self::NormExp> {
    type NormExp : NormExponent;

    fn norm_exponent(&self) -> Self::NormExp;

    #[inline]
    fn norm_(&self) -> F {
        self.norm(self.norm_exponent())
    }

    // fn similar_origin(&self) -> Self;

    fn is_zero(&self) -> bool;
}

pub trait HasDual<F : Num = f64> : Normed<F> {
    type DualSpace : Normed<F>;
}

/// Automatically implemented trait for reflexive spaces
pub trait Reflexive<F : Num = f64> : HasDual<F>
where
    Self::DualSpace : HasDual<F, DualSpace = Self>
{ }

impl<F : Num, X : HasDual<F>> Reflexive<F> for X
where
    X::DualSpace : HasDual<F, DualSpace = X>
{ }

pub trait HasDualExponent : NormExponent {
    type DualExp : NormExponent;

    fn dual_exponent(&self) -> Self::DualExp;
}

impl HasDualExponent for L2 {
    type DualExp = L2;
    
    #[inline]
    fn dual_exponent(&self) -> Self::DualExp {
        L2
    }
}

impl HasDualExponent for L1 {
    type DualExp = Linfinity;
    
    #[inline]
    fn dual_exponent(&self) -> Self::DualExp {
        Linfinity
    }
}


impl HasDualExponent for Linfinity {
    type DualExp = L1;
    
    #[inline]
    fn dual_exponent(&self) -> Self::DualExp {
        L1
    }
}