src/seminorms.rs

/*! This module implements the convolution operators $𝒟$.

The principal data type of the module is [`ConvolutionOp`] and the main abstraction
the trait [`DiscreteMeasureOp`].
*/

use crate::measures::{DeltaMeasure, DiscreteMeasure, Radon, SpikeIter, RNDM};
use alg_tools::bisection_tree::*;
use alg_tools::bounds::Bounded;
use alg_tools::error::DynResult;
use alg_tools::instance::Instance;
use alg_tools::iter::{FilterMapX, Mappable};
use alg_tools::linops::{BoundedLinear, Linear, Mapping};
use alg_tools::loc::Loc;
use alg_tools::mapping::RealMapping;
use alg_tools::nalgebra_support::ToNalgebraRealField;
use alg_tools::norms::{Linfinity, Norm, NormExponent};
use alg_tools::sets::Cube;
use alg_tools::types::*;
use itertools::Itertools;
use nalgebra::DMatrix;
use std::iter::Zip;
use std::marker::PhantomData;
use std::ops::RangeFrom;

/// Abstraction for operators $𝒟 ∈ 𝕃(𝒵(Ω); C_c(Ω))$.
///
/// Here $𝒵(Ω) ⊂ ℳ(Ω)$ is the space of sums of delta measures, presented by [`DiscreteMeasure`].
pub trait DiscreteMeasureOp<Domain, F>:
    BoundedLinear<DiscreteMeasure<Domain, F>, Radon, Linfinity, F>
where
    F: Float + ToNalgebraRealField,
    Domain: 'static + Clone + PartialEq,
{
    /// The output type of [`Self::preapply`].
    type PreCodomain;

    /// Creates a finite-dimensional presentatin of the operator restricted to a fixed support.
    ///
    /// <p>
    /// This returns the matrix $C_*𝒟C$, where $C ∈ 𝕃(ℝ^n; 𝒵(Ω))$, $Ca = ∑_{i=1}^n α_i δ_{x_i}$
    /// for a $x_1, …, x_n$ the coordinates given by the iterator `I`, and $a=(α_1,…,α_n)$.
    /// Here $C_* ∈ 𝕃(C_c(Ω); ℝ^n) $ stands for the preadjoint.
    /// </p>
    fn findim_matrix<'a, I>(&self, points: I) -> DMatrix<F::MixedType>
    where
        I: ExactSizeIterator<Item = &'a Domain> + Clone;

    /// [`Mapping`] that typically returns an uninitialised [`PreBTFN`]
    /// instead of a full [`BTFN`].
    fn preapply(&self, μ: DiscreteMeasure<Domain, F>) -> Self::PreCodomain;
}

// Blanket implementation of a measure as a linear functional over a predual
// (that by assumption is a linear functional over a measure).
/*impl<F, Domain, Predual> Linear<Predual>
for DiscreteMeasure<Domain, F>
where F : Float + ToNalgebraRealField,
      Predual : Linear<DiscreteMeasure<Domain, F>, Codomain=F> {
    type Codomain = F;

    #[inline]
    fn apply(&self, ω : &Predual) -> F {
        ω.apply(self)
    }
}*/

//
// Convolutions for discrete measures
//

/// A trait alias for simple convolution kernels.
pub trait SimpleConvolutionKernel<const N: usize, F: Float = f64>:
    RealMapping<N, F> + Support<N, F> + Bounded<F> + Clone + 'static
{
}

impl<T, F: Float, const N: usize> SimpleConvolutionKernel<N, F> for T where
    T: RealMapping<N, F> + Support<N, F> + Bounded<F> + Clone + 'static
{
}

/// [`SupportGenerator`] for [`ConvolutionOp`].
#[derive(Clone, Debug)]
pub struct ConvolutionSupportGenerator<K, const N: usize, F: Float = f64>
where
    K: SimpleConvolutionKernel<N, F>,
{
    kernel: K,
    centres: RNDM<N, F>,
}

impl<F: Float, K, const N: usize> ConvolutionSupportGenerator<K, N, F>
where
    K: SimpleConvolutionKernel<N, F>,
{
    /// Construct the convolution kernel corresponding to `δ`, i.e., one centered at `δ.x` and
    /// weighted by `δ.α`.
    #[inline]
    fn construct_kernel<'a>(
        &'a self,
        δ: &'a DeltaMeasure<Loc<N, F>, F>,
    ) -> Weighted<Shift<K, N, F>, F> {
        self.kernel.clone().shift(δ.x).weigh(δ.α)
    }

    /// This is a helper method for the implementation of [`ConvolutionSupportGenerator::all_data`].
    /// It filters out `δ` with zero weight, and otherwise returns the corresponding convolution
    /// kernel. The `id` is passed through as-is.
    #[inline]
    fn construct_kernel_and_id_filtered<'a>(
        &'a self,
        (id, δ): (usize, &'a DeltaMeasure<Loc<N, F>, F>),
    ) -> Option<(usize, Weighted<Shift<K, N, F>, F>)> {
        (δ.α != F::ZERO).then(|| (id.into(), self.construct_kernel(δ)))
    }
}

impl<F: Float, K, const N: usize> SupportGenerator<N, F> for ConvolutionSupportGenerator<K, N, F>
where
    K: SimpleConvolutionKernel<N, F>,
{
    type Id = usize;
    type SupportType = Weighted<Shift<K, N, F>, F>;
    type AllDataIter<'a> = FilterMapX<
        'a,
        Zip<RangeFrom<usize>, SpikeIter<'a, Loc<N, F>, F>>,
        Self,
        (Self::Id, Self::SupportType),
    >;

    #[inline]
    fn support_for(&self, d: Self::Id) -> Self::SupportType {
        self.construct_kernel(&self.centres[d])
    }

    #[inline]
    fn support_count(&self) -> usize {
        self.centres.len()
    }

    #[inline]
    fn all_data(&self) -> Self::AllDataIter<'_> {
        (0..)
            .zip(self.centres.iter_spikes())
            .filter_mapX(self, Self::construct_kernel_and_id_filtered)
    }
}

/// Representation of a convolution operator $𝒟$.
#[derive(Clone, Debug)]
pub struct ConvolutionOp<F, K, BT, const N: usize>
where
    F: Float + ToNalgebraRealField,
    BT: BTImpl<N, F, Data = usize>,
    K: SimpleConvolutionKernel<N, F>,
{
    /// Depth of the [`BT`] bisection tree for the outputs [`Mapping::apply`].
    depth: BT::Depth,
    /// Domain of the [`BT`] bisection tree for the outputs [`Mapping::apply`].
    domain: Cube<N, F>,
    /// The convolution kernel
    kernel: K,
    _phantoms: PhantomData<(F, BT)>,
}

impl<F, K, BT, const N: usize> ConvolutionOp<F, K, BT, N>
where
    F: Float + ToNalgebraRealField,
    BT: BTImpl<N, F, Data = usize>,
    K: SimpleConvolutionKernel<N, F>,
{
    /// Creates a new convolution operator $𝒟$ with `kernel` on `domain`.
    ///
    /// The output of [`Mapping::apply`] is a [`BT`] of given `depth`.
    pub fn new(depth: BT::Depth, domain: Cube<N, F>, kernel: K) -> Self {
        ConvolutionOp {
            depth: depth,
            domain: domain,
            kernel: kernel,
            _phantoms: PhantomData,
        }
    }

    /// Returns the support generator for this convolution operator.
    fn support_generator(&self, μ: RNDM<N, F>) -> ConvolutionSupportGenerator<K, N, F> {
        // TODO: can we avoid cloning μ?
        ConvolutionSupportGenerator {
            kernel: self.kernel.clone(),
            centres: μ,
        }
    }

    /// Returns a reference to the kernel of this convolution operator.
    pub fn kernel(&self) -> &K {
        &self.kernel
    }
}

impl<F, K, BT, const N: usize> Mapping<RNDM<N, F>> for ConvolutionOp<F, K, BT, N>
where
    F: Float + ToNalgebraRealField,
    BT: BTImpl<N, F, Data = usize>,
    K: SimpleConvolutionKernel<N, F>,
    Weighted<Shift<K, N, F>, F>: LocalAnalysis<F, BT::Agg, N>,
{
    type Codomain = BTFN<F, ConvolutionSupportGenerator<K, N, F>, BT, N>;

    fn apply<I>(&self, μ: I) -> Self::Codomain
    where
        I: Instance<RNDM<N, F>>,
    {
        let g = self.support_generator(μ.own());
        BTFN::construct(self.domain.clone(), self.depth, g)
    }
}

/// [`ConvolutionOp`]s as linear operators over [`DiscreteMeasure`]s.
impl<F, K, BT, const N: usize> Linear<RNDM<N, F>> for ConvolutionOp<F, K, BT, N>
where
    F: Float + ToNalgebraRealField,
    BT: BTImpl<N, F, Data = usize>,
    K: SimpleConvolutionKernel<N, F>,
    Weighted<Shift<K, N, F>, F>: LocalAnalysis<F, BT::Agg, N>,
{
}

impl<F, K, BT, const N: usize> BoundedLinear<RNDM<N, F>, Radon, Linfinity, F>
    for ConvolutionOp<F, K, BT, N>
where
    F: Float + ToNalgebraRealField,
    BT: BTImpl<N, F, Data = usize>,
    K: SimpleConvolutionKernel<N, F>,
    Weighted<Shift<K, N, F>, F>: LocalAnalysis<F, BT::Agg, N>,
{
    fn opnorm_bound(&self, _: Radon, _: Linfinity) -> DynResult<F> {
        // With μ = ∑_i α_i δ_{x_i}, we have
        // |𝒟μ|_∞
        // = sup_z |∑_i α_i φ(z - x_i)|
        // ≤ sup_z ∑_i |α_i| |φ(z - x_i)|
        // ≤ ∑_i |α_i| |φ|_∞
        // = |μ|_ℳ |φ|_∞
        Ok(self.kernel.bounds().uniform())
    }
}

impl<'a, F, K, BT, const N: usize> NormExponent for &'a ConvolutionOp<F, K, BT, N>
where
    F: Float + ToNalgebraRealField,
    BT: BTImpl<N, F, Data = usize>,
    K: SimpleConvolutionKernel<N, F>,
    Weighted<Shift<K, N, F>, F>: LocalAnalysis<F, BT::Agg, N>,
{
}

impl<'a, F, K, BT, const N: usize> Norm<&'a ConvolutionOp<F, K, BT, N>, F> for RNDM<N, F>
where
    F: Float + ToNalgebraRealField,
    BT: BTImpl<N, F, Data = usize>,
    K: SimpleConvolutionKernel<N, F>,
    Weighted<Shift<K, N, F>, F>: LocalAnalysis<F, BT::Agg, N>,
{
    fn norm(&self, op𝒟: &'a ConvolutionOp<F, K, BT, N>) -> F {
        self.apply(op𝒟.apply(self)).sqrt()
    }
}

impl<F, K, BT, const N: usize> DiscreteMeasureOp<Loc<N, F>, F> for ConvolutionOp<F, K, BT, N>
where
    F: Float + ToNalgebraRealField,
    BT: BTImpl<N, F, Data = usize>,
    K: SimpleConvolutionKernel<N, F>,
    Weighted<Shift<K, N, F>, F>: LocalAnalysis<F, BT::Agg, N>,
{
    type PreCodomain = PreBTFN<F, ConvolutionSupportGenerator<K, N, F>, N>;

    fn findim_matrix<'a, I>(&self, points: I) -> DMatrix<F::MixedType>
    where
        I: ExactSizeIterator<Item = &'a Loc<N, F>> + Clone,
    {
        // TODO: Preliminary implementation. It be best to use sparse matrices or
        // possibly explicit operators without matrices
        let n = points.len();
        let points_clone = points.clone();
        let pairs = points.cartesian_product(points_clone);
        let kernel = &self.kernel;
        let values = pairs.map(|(x, y)| kernel.apply(y - x).to_nalgebra_mixed());
        DMatrix::from_iterator(n, n, values)
    }

    /// A version of [`Mapping::apply`] that does not instantiate the [`BTFN`] codomain with
    /// a bisection tree, instead returning a [`PreBTFN`]. This can improve performance when
    /// the output is to be added as the right-hand-side operand to a proper BTFN.
    fn preapply(&self, μ: RNDM<N, F>) -> Self::PreCodomain {
        BTFN::new_pre(self.support_generator(μ))
    }
}

/// Generates an scalar operation (e.g. [`std::ops::Mul`], [`std::ops::Div`])
/// for [`ConvolutionSupportGenerator`].
macro_rules! make_convolutionsupportgenerator_scalarop_rhs {
    ($trait:ident, $fn:ident, $trait_assign:ident, $fn_assign:ident) => {
        impl<F: Float, K: SimpleConvolutionKernel<N, F>, const N: usize> std::ops::$trait_assign<F>
            for ConvolutionSupportGenerator<K, N, F>
        {
            fn $fn_assign(&mut self, t: F) {
                self.centres.$fn_assign(t);
            }
        }

        impl<F: Float, K: SimpleConvolutionKernel<N, F>, const N: usize> std::ops::$trait<F>
            for ConvolutionSupportGenerator<K, N, F>
        {
            type Output = ConvolutionSupportGenerator<K, N, F>;
            fn $fn(mut self, t: F) -> Self::Output {
                std::ops::$trait_assign::$fn_assign(&mut self.centres, t);
                self
            }
        }
        impl<'a, F: Float, K: SimpleConvolutionKernel<N, F>, const N: usize> std::ops::$trait<F>
            for &'a ConvolutionSupportGenerator<K, N, F>
        {
            type Output = ConvolutionSupportGenerator<K, N, F>;
            fn $fn(self, t: F) -> Self::Output {
                ConvolutionSupportGenerator {
                    kernel: self.kernel.clone(),
                    centres: (&self.centres).$fn(t),
                }
            }
        }
    };
}

make_convolutionsupportgenerator_scalarop_rhs!(Mul, mul, MulAssign, mul_assign);
make_convolutionsupportgenerator_scalarop_rhs!(Div, div, DivAssign, div_assign);

/// Generates an unary operation (e.g. [`std::ops::Neg`]) for [`ConvolutionSupportGenerator`].
macro_rules! make_convolutionsupportgenerator_unaryop {
    ($trait:ident, $fn:ident) => {
        impl<F: Float, K: SimpleConvolutionKernel<N, F>, const N: usize> std::ops::$trait
            for ConvolutionSupportGenerator<K, N, F>
        {
            type Output = ConvolutionSupportGenerator<K, N, F>;
            fn $fn(mut self) -> Self::Output {
                self.centres = self.centres.$fn();
                self
            }
        }

        impl<'a, F: Float, K: SimpleConvolutionKernel<N, F>, const N: usize> std::ops::$trait
            for &'a ConvolutionSupportGenerator<K, N, F>
        {
            type Output = ConvolutionSupportGenerator<K, N, F>;
            fn $fn(self) -> Self::Output {
                ConvolutionSupportGenerator {
                    kernel: self.kernel.clone(),
                    centres: (&self.centres).$fn(),
                }
            }
        }
    };
}

make_convolutionsupportgenerator_unaryop!(Neg, neg);