src/prox_penalty/radon_squared.rs

/*!
Solver for the point source localisation problem using a simplified forward-backward splitting method.

Instead of the $𝒟$-norm of `fb.rs`, this uses a standard Radon norm for the proximal map.
*/

use super::{InsertionConfig, ProxPenalty, ProxTerm, StepLengthBound, StepLengthBoundPD};
use crate::dataterm::QuadraticDataTerm;
use crate::forward_model::ForwardModel;
use crate::measures::merging::SpikeMerging;
use crate::measures::{DeltaMeasure, DiscreteMeasure, Radon};
use crate::prox_penalty::StepLengthBoundValue;
use crate::regularisation::RegTerm;
use crate::types::*;
use alg_tools::bounds::MinMaxMapping;
use alg_tools::error::DynResult;
use alg_tools::instance::{Instance, Space};
use alg_tools::iterate::{AlgIterator, AlgIteratorIteration};
use alg_tools::linops::BoundedLinear;
use alg_tools::nalgebra_support::ToNalgebraRealField;
use alg_tools::norms::{Norm, L2};
use anyhow::ensure;
use nalgebra::DVector;
use numeric_literals::replace_float_literals;
use serde::{Deserialize, Serialize};

/// Radon-norm squared proximal penalty

#[derive(Copy, Clone, Serialize, Deserialize)]
pub struct RadonSquared;

#[replace_float_literals(F::cast_from(literal))]
impl<Domain, F, M, Reg> ProxPenalty<Domain, M, Reg, F> for RadonSquared
where
    Domain: Space + Clone + PartialEq + 'static,
    for<'a> &'a Domain: Instance<Domain>,
    F: Float + ToNalgebraRealField,
    M: MinMaxMapping<Domain, F>,
    Reg: RegTerm<Domain, F>,
    DiscreteMeasure<Domain, F>: SpikeMerging<F>,
{
    type ReturnMapping = M;

    fn prox_type() -> ProxTerm {
        ProxTerm::RadonSquared
    }

    fn insert_and_reweigh<I>(
        &self,
        μ: &mut DiscreteMeasure<Domain, F>,
        τv: &mut M,
        μ_base: &DiscreteMeasure<Domain, F>,
        ν_delta: Option<&DiscreteMeasure<Domain, F>>,
        τ: F,
        ε: F,
        config: &InsertionConfig<F>,
        reg: &Reg,
        _state: &AlgIteratorIteration<I>,
        stats: &mut IterInfo<F>,
    ) -> DynResult<(Option<Self::ReturnMapping>, bool)>
    where
        I: AlgIterator,
    {
        let mut y = μ_base.masses_dvector();

        ensure!(μ_base.len() <= μ.len());

        'i_and_w: for i in 0..=1 {
            // Optimise weights
            if μ.len() > 0 {
                // Form finite-dimensional subproblem. The subproblem references to the original μ^k
                // from the beginning of the iteration are all contained in the immutable c and g.
                // TODO: observe negation of -τv after switch from minus_τv: finite-dimensional
                // problems have not yet been updated to sign change.
                let g̃ = DVector::from_iterator(
                    μ.len(),
                    μ.iter_locations()
                        .map(|ζ| -F::to_nalgebra_mixed(τv.apply(ζ))),
                );
                let mut x = μ.masses_dvector();
                y.extend(std::iter::repeat(0.0.to_nalgebra_mixed()).take(0.max(x.len() - y.len())));
                assert_eq!(y.len(), x.len());
                // Solve finite-dimensional subproblem.
                // TODO: This assumes that ν_delta has no common locations with μ-μ_base, to
                // ignore it.
                stats.inner_iters += reg.solve_findim_l1squared(&y, &g̃, τ, &mut x, ε, config);

                // Update masses of μ based on solution of finite-dimensional subproblem.
                μ.set_masses_dvector(&x);
            }

            if i > 0 {
                // Simple debugging test to see if more inserts would be needed. Doesn't seem so.
                //let n = μ.dist_matching(μ_base);
                //println!("{:?}", reg.find_tolerance_violation_slack(τv, τ, ε, false, config, n));
                break 'i_and_w;
            }

            // Calculate ‖μ - μ_base‖_ℳ
            // TODO: This assumes that ν_delta has no common locations with μ-μ_base.
            let n = μ.dist_matching(μ_base) + ν_delta.map_or(0.0, |ν| ν.norm(Radon));

            // Find a spike to insert, if needed.
            // This only check the overall tolerances, not tolerances on support of μ-μ_base or μ,
            // which are supposed to have been guaranteed by the finite-dimensional weight optimisation.
            match reg.find_tolerance_violation_slack(τv, τ, ε, false, config, n) {
                None => break 'i_and_w,
                Some((ξ, _v_ξ, _in_bounds)) => {
                    // Weight is found out by running the finite-dimensional optimisation algorithm
                    // above
                    *μ += DeltaMeasure { x: ξ, α: 0.0 };
                    stats.inserted += 1;
                }
            };
        }

        Ok((None, true))
    }

    fn merge_spikes(
        &self,
        μ: &mut DiscreteMeasure<Domain, F>,
        τv: &mut M,
        μ_base: &DiscreteMeasure<Domain, F>,
        ν_delta: Option<&DiscreteMeasure<Domain, F>>,
        τ: F,
        ε: F,
        config: &InsertionConfig<F>,
        reg: &Reg,
        fitness: Option<impl Fn(&DiscreteMeasure<Domain, F>) -> F>,
    ) -> usize {
        if config.fitness_merging {
            if let Some(f) = fitness {
                return μ.merge_spikes_fitness(config.merging, f, |&v| v).1;
            }
        }
        μ.merge_spikes(config.merging, |μ_candidate| {
            // Important: μ_candidate's new points are afterwards,
            // and do not conflict with μ_base.
            // TODO: could simplify to requiring μ_base instead of μ_radon.
            // but may complicate with sliding base's exgtra points that need to be
            // after μ_candidate's extra points.
            // TODO: doesn't seem to work, maybe need to merge μ_base as well?
            // Although that doesn't seem to make sense.
            let μ_radon = match ν_delta {
                None => μ_candidate.sub_matching(μ_base),
                Some(ν) => μ_candidate.sub_matching(μ_base) - ν,
            };
            reg.verify_merge_candidate_radonsq(τv, μ_candidate, τ, ε, &config, &μ_radon)
            //let n = μ_candidate.dist_matching(μ_base);
            //reg.find_tolerance_violation_slack(τv, τ, ε, false, config, n).is_none()
        })
    }
}

#[replace_float_literals(F::cast_from(literal))]
impl<'a, F, A, Domain> StepLengthBound<F, QuadraticDataTerm<F, Domain, A>> for RadonSquared
where
    F: Float + ToNalgebraRealField,
    Domain: Space + Norm<Radon, F>,
    A: ForwardModel<Domain, F> + BoundedLinear<Domain, Radon, L2, F>,
{
    fn step_length_bound(&self, f: &QuadraticDataTerm<F, Domain, A>) -> StepLengthBoundValue<F> {
        // TODO: direct squared calculation
        match f.operator().opnorm_bound(Radon, L2) {
            Err(_) => StepLengthBoundValue::Failure,
            Ok(l) => StepLengthBoundValue::LipschitzFactor(l.powi(2)),
        }
    }
}

#[replace_float_literals(F::cast_from(literal))]
impl<'a, F, A, Domain> StepLengthBoundPD<F, A, DiscreteMeasure<Domain, F>> for RadonSquared
where
    Domain: Space + Clone + PartialEq + 'static,
    F: Float + ToNalgebraRealField,
    A: BoundedLinear<DiscreteMeasure<Domain, F>, Radon, L2, F>,
{
    fn step_length_bound_pd(&self, opA: &A) -> DynResult<F> {
        opA.opnorm_bound(Radon, L2)
    }
}