src/prox_penalty/wave.rs

/*!
Basic proximal penalty based on convolution operators $𝒟$.
 */

use numeric_literals::replace_float_literals;
use nalgebra::DVector;
use colored::Colorize;

use alg_tools::types::*;
use alg_tools::loc::Loc;
use alg_tools::mapping::{Mapping, RealMapping};
use alg_tools::nalgebra_support::ToNalgebraRealField;
use alg_tools::norms::Linfinity;
use alg_tools::iterate::{
    AlgIteratorIteration,
    AlgIterator,
};
use alg_tools::bisection_tree::{
    BTFN,
    PreBTFN,
    Bounds,
    BTSearch,
    SupportGenerator,
    LocalAnalysis,
    BothGenerators,
};
use crate::measures::{
    RNDM,
    DeltaMeasure,
    Radon,
};
use crate::measures::merging::{
    SpikeMerging,
};
use crate::seminorms::DiscreteMeasureOp;
use crate::types::{
    IterInfo,
};
use crate::measures::merging::SpikeMergingMethod;
use crate::regularisation::RegTerm;
use super::{ProxPenalty, FBGenericConfig};

#[replace_float_literals(F::cast_from(literal))]
impl<F, GA, BTA, S, Reg, 𝒟, G𝒟, K, const N : usize>
ProxPenalty<F, BTFN<F, GA, BTA, N>, Reg, N> for 𝒟
where
    F : Float + ToNalgebraRealField,
    GA : SupportGenerator<F, N, SupportType = S, Id = usize> + Clone,
    BTA : BTSearch<F, N, Data=usize, Agg=Bounds<F>>,
    S: RealMapping<F, N> + LocalAnalysis<F, Bounds<F>, N>,
    G𝒟 : SupportGenerator<F, N, SupportType = K, Id = usize> + Clone,
    𝒟 : DiscreteMeasureOp<Loc<F, N>, F, PreCodomain = PreBTFN<F, G𝒟, N>>,
    𝒟::Codomain : RealMapping<F, N>,
    K : RealMapping<F, N> + LocalAnalysis<F, Bounds<F>, N>,
    Reg : RegTerm<F, N>,
    RNDM<F, N> : SpikeMerging<F>,
{
    type ReturnMapping = BTFN<F, BothGenerators<GA, G𝒟>, BTA, N>;

    fn insert_and_reweigh<I>(
        &self,
        μ : &mut RNDM<F, N>,
        τv : &mut BTFN<F, GA, BTA, N>,
        μ_base : &RNDM<F, N>,
        ν_delta: Option<&RNDM<F, N>>,
        τ : F,
        ε : F,
        config : &FBGenericConfig<F>,
        reg : &Reg,
        state : &AlgIteratorIteration<I>,
        stats : &mut IterInfo<F, N>,
    ) -> (Option<BTFN<F, BothGenerators<GA, G𝒟>, BTA, N>>, bool)
    where
        I : AlgIterator
    {

        // TODO: is this inefficient to do in every iteration?
        let op𝒟norm = self.opnorm_bound(Radon, Linfinity);

        // Maximum insertion count and measure difference calculation depend on insertion style.
        let (max_insertions, warn_insertions) = match (state.iteration(), config.bootstrap_insertions) {
            (i, Some((l, k))) if i <= l => (k, false),
            _ => (config.max_insertions, !state.is_quiet()),
        };

        let ω0 = match ν_delta {
            None => self.apply(μ_base),
            Some(ν) => self.apply(μ_base + ν),
        };

        // Add points to support until within error tolerance or maximum insertion count reached.
        let mut count = 0;
        let (within_tolerances, d) = 'insertion: loop {
            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 Ã = self.findim_matrix(μ.iter_locations());
                let g̃ = DVector::from_iterator(μ.len(),
                                            μ.iter_locations()
                                                .map(|ζ| ω0.apply(ζ) - τv.apply(ζ))
                                                .map(F::to_nalgebra_mixed));
                let mut x = μ.masses_dvector();

                // The gradient of the forward component of the inner objective is C^*𝒟Cx - g̃.
                // We have |C^*𝒟Cx|_2 = sup_{|z|_2 ≤ 1} ⟨z, C^*𝒟Cx⟩ = sup_{|z|_2 ≤ 1} ⟨Cz|𝒟Cx⟩
                // ≤ sup_{|z|_2 ≤ 1} |Cz|_ℳ |𝒟Cx|_∞ ≤  sup_{|z|_2 ≤ 1} |Cz|_ℳ |𝒟| |Cx|_ℳ
                // ≤ sup_{|z|_2 ≤ 1} |z|_1 |𝒟| |x|_1 ≤ sup_{|z|_2 ≤ 1} n |z|_2 |𝒟| |x|_2
                // = n |𝒟| |x|_2, where n is the number of points. Therefore
                let Ã_normest = op𝒟norm * F::cast_from(μ.len());

                // Solve finite-dimensional subproblem.
                stats.inner_iters += reg.solve_findim(&Ã, &g̃, τ, &mut x, Ã_normest, ε, config);

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

            // Form d = τv + 𝒟μ - ω0 = τv + 𝒟(μ - μ^k) for checking the proximate optimality
            // conditions in the predual space, and finding new points for insertion, if necessary.
            let mut d = &*τv + match ν_delta {
                None => self.preapply(μ.sub_matching(μ_base)),
                Some(ν) => self.preapply(μ.sub_matching(μ_base) - ν)
            };

            // If no merging heuristic is used, let's be more conservative about spike insertion,
            // and skip it after first round. If merging is done, being more greedy about spike
            // insertion also seems to improve performance.
            let skip_by_rough_check = if let SpikeMergingMethod::None = config.merging {
                false
            } else {
                count > 0
            };

            // Find a spike to insert, if needed
            let (ξ, _v_ξ, in_bounds) =  match reg.find_tolerance_violation(
                &mut d, τ, ε, skip_by_rough_check, config
            ) {
                None => break 'insertion (true, d),
                Some(res) => res,
            };

            // Break if maximum insertion count reached
            if count >= max_insertions {
                break 'insertion (in_bounds, d)
            }

            // No point in optimising the weight here; the finite-dimensional algorithm is fast.
            *μ += DeltaMeasure { x : ξ, α : 0.0 };
            count += 1;
            stats.inserted += 1;
        };

        if !within_tolerances && warn_insertions {
            // Complain (but continue) if we failed to get within tolerances
            // by inserting more points.
            let err = format!("Maximum insertions reached without achieving \
                                subproblem solution tolerance");
            println!("{}", err.red());
        }

        (Some(d), within_tolerances)
    }

    fn merge_spikes(
        &self,
        μ : &mut RNDM<F, N>,
        τv : &mut BTFN<F, GA, BTA, N>,
        μ_base : &RNDM<F, N>,
        τ : F,
        ε : F,
        config : &FBGenericConfig<F>,
        reg : &Reg,
    ) -> usize
    {
        μ.merge_spikes(config.merging, |μ_candidate| {
            let mut d = &*τv + self.preapply(μ_candidate.sub_matching(μ_base));
            reg.verify_merge_candidate(&mut d, μ_candidate, τ, ε, config)
        })
    }
}