src/sliding_pdps.rs

/*!
Solver for the point source localisation problem using a sliding
primal-dual proximal splitting method.
*/

use numeric_literals::replace_float_literals;
use serde::{Serialize, Deserialize};
//use colored::Colorize;
//use nalgebra::{DVector, DMatrix};
use std::iter::Iterator;

use alg_tools::iterate::AlgIteratorFactory;
use alg_tools::euclidean::Euclidean;
use alg_tools::sets::Cube;
use alg_tools::loc::Loc;
use alg_tools::mapping::{Mapping, DifferentiableRealMapping, Instance};
use alg_tools::norms::Norm;
use alg_tools::direct_product::Pair;
use alg_tools::bisection_tree::{
    BTFN,
    PreBTFN,
    Bounds,
    BTNodeLookup,
    BTNode,
    BTSearch,
    P2Minimise,
    SupportGenerator,
    LocalAnalysis,
    //Bounded,
};
use alg_tools::mapping::RealMapping;
use alg_tools::nalgebra_support::ToNalgebraRealField;
use alg_tools::linops::{
    BoundedLinear, AXPY, GEMV, Adjointable, IdOp,
};
use alg_tools::convex::{Conjugable, Prox};
use alg_tools::norms::{L2, Linfinity, PairNorm};

use crate::types::*;
use crate::measures::{DiscreteMeasure, Radon, RNDM};
use crate::measures::merging::SpikeMerging;
use crate::forward_model::{
    ForwardModel,
    AdjointProductPairBoundedBy,
    LipschitzValues,
};
// use crate::transport::TransportLipschitz;
use crate::seminorms::DiscreteMeasureOp;
//use crate::tolerance::Tolerance;
use crate::plot::{
    SeqPlotter,
    Plotting,
    PlotLookup
};
use crate::fb::*;
use crate::regularisation::SlidingRegTerm;
// use crate::dataterm::L2Squared;
use crate::sliding_fb::{
    TransportConfig,
    TransportStepLength,
    initial_transport,
    aposteriori_transport,
};
use crate::dataterm::{calculate_residual, calculate_residual2};

/// Settings for [`pointsource_sliding_pdps_pair`].
#[derive(Clone, Copy, Eq, PartialEq, Serialize, Deserialize, Debug)]
#[serde(default)]
pub struct SlidingPDPSConfig<F : Float> {
    /// Primal step length scaling.
    pub τ0 : F,
    /// Primal step length scaling.
    pub σp0 : F,
    /// Dual step length scaling.
    pub σd0 : F,
    /// Transport parameters
    pub transport : TransportConfig<F>,
    /// Generic parameters
    pub insertion : FBGenericConfig<F>,
}

#[replace_float_literals(F::cast_from(literal))]
impl<F : Float> Default for SlidingPDPSConfig<F> {
    fn default() -> Self {
        let τ0 = 0.99;
        SlidingPDPSConfig {
            τ0,
            σd0 : 0.1,
            σp0 : 0.99,
            transport : Default::default(),
            insertion : Default::default()
        }
    }
}

type MeasureZ<F, Z, const N : usize> = Pair<RNDM<F, N>, Z>;

/// Iteratively solve the pointsource localisation with an additional variable
/// using sliding primal-dual proximal splitting
///
/// The parametrisation is as for [`crate::forward_pdps::pointsource_forward_pdps_pair`].
#[replace_float_literals(F::cast_from(literal))]
pub fn pointsource_sliding_pdps_pair<
    'a, F, I, A, GA, 𝒟, BTA, BT𝒟, G𝒟, S, K, Reg, Z, R, Y, /*KOpM, */ KOpZ, H, const N : usize
>(
    opA : &'a A,
    b : &A::Observable,
    reg : Reg,
    op𝒟 : &'a 𝒟,
    config : &SlidingPDPSConfig<F>,
    iterator : I,
    mut plotter : SeqPlotter<F, N>,
    //opKμ : KOpM,
    opKz : &KOpZ,
    fnR : &R,
    fnH : &H,
    mut z : Z,
    mut y : Y,
) -> MeasureZ<F, Z, N>
where
    F : Float + ToNalgebraRealField,
    I : AlgIteratorFactory<IterInfo<F, N>>,
    for<'b> &'b A::Observable : std::ops::Neg<Output=A::Observable> + Instance<A::Observable>,
    for<'b> A::Preadjoint<'b> : LipschitzValues<FloatType=F>,
    BTFN<F, GA, BTA, N> : DifferentiableRealMapping<F, N>,
    GA : SupportGenerator<F, N, SupportType = S, Id = usize> + Clone,
    A : ForwardModel<
            MeasureZ<F, Z, N>,
            F,
            PairNorm<Radon, L2, L2>,
            PreadjointCodomain = Pair<BTFN<F, GA, BTA, N>, Z>,
        >
        + AdjointProductPairBoundedBy<MeasureZ<F, Z, N>, 𝒟, IdOp<Z>, FloatType=F>,
    BTA : BTSearch<F, N, Data=usize, Agg=Bounds<F>>,
    G𝒟 : SupportGenerator<F, N, SupportType = K, Id = usize> + Clone,
    𝒟 : DiscreteMeasureOp<Loc<F, N>, F, PreCodomain = PreBTFN<F, G𝒟, N>,
                                        Codomain = BTFN<F, G𝒟, BT𝒟, N>>,
    BT𝒟 : BTSearch<F, N, Data=usize, Agg=Bounds<F>>,
    S: RealMapping<F, N> + LocalAnalysis<F, Bounds<F>, N>
        + DifferentiableRealMapping<F, N>,
    K: RealMapping<F, N> + LocalAnalysis<F, Bounds<F>, N>,
        //+ Differentiable<Loc<F, N>, Derivative=Loc<F,N>>,
    BTNodeLookup: BTNode<F, usize, Bounds<F>, N>,
    Cube<F, N>: P2Minimise<Loc<F, N>, F>,
    PlotLookup : Plotting<N>,
    RNDM<F, N> : SpikeMerging<F>,
    Reg : SlidingRegTerm<F, N>,
    // KOpM : Linear<RNDM<F, N>, Codomain=Y>
    //     + GEMV<F, RNDM<F, N>>
    //     + Preadjointable<
    //         RNDM<F, N>, Y,
    //         PreadjointCodomain = BTFN<F, GA, BTA, N>,
    //     >
    //     + TransportLipschitz<L2Squared, FloatType=F>
    //     + AdjointProductBoundedBy<RNDM<F, N>, 𝒟, FloatType=F>,
    // for<'b> KOpM::Preadjoint<'b> : GEMV<F, Y>,
    // Since Z is Hilbert, we may just as well use adjoints for K_z.
    KOpZ : BoundedLinear<Z, L2, L2, F, Codomain=Y>
        + GEMV<F, Z>
        + Adjointable<Z, Y, AdjointCodomain = Z>,
    for<'b> KOpZ::Adjoint<'b> : GEMV<F, Y>,
    Y : AXPY<F> + Euclidean<F, Output=Y> + Clone + ClosedAdd,
    for<'b> &'b Y : Instance<Y>,
    Z : AXPY<F, Owned=Z> + Euclidean<F, Output=Z> + Clone + Norm<F, L2>,
    for<'b> &'b Z : Instance<Z>,
    R : Prox<Z, Codomain=F>,
    H : Conjugable<Y, F, Codomain=F>,
    for<'b> H::Conjugate<'b> : Prox<Y>,
{

    // Check parameters
    assert!(config.τ0 > 0.0 &&
            config.τ0 < 1.0 &&
            config.σp0 > 0.0 &&
            config.σp0 < 1.0 &&
            config.σd0 > 0.0 &&
            config.σp0 * config.σd0 <= 1.0,
            "Invalid step length parameters");
    config.transport.check();

    // Initialise iterates
    let mut μ = DiscreteMeasure::new();
    let mut γ1 = DiscreteMeasure::new();
    let mut residual = calculate_residual(Pair(&μ, &z), opA, b);
    let zero_z = z.similar_origin();

    // Set up parameters
    let op𝒟norm = op𝒟.opnorm_bound(Radon, Linfinity);
    // TODO: maybe this PairNorm doesn't make sense here?
    let opAnorm = opA.opnorm_bound(PairNorm(Radon, L2, L2), L2);
    let bigθ = 0.0; //opKμ.transport_lipschitz_factor(L2Squared);
    let bigM = 0.0; //opKμ.adjoint_product_bound(&op𝒟).unwrap().sqrt();
    let nKz = opKz.opnorm_bound(L2, L2);
    let ℓ = 0.0;
    let opIdZ = IdOp::new();
    let (l, l_z) = opA.adjoint_product_pair_bound(&op𝒟, &opIdZ).unwrap();
    // We need to satisfy
    //
    //     τσ_dM(1-σ_p L_z)/(1 - τ L) + [σ_p L_z + σ_pσ_d‖K_z‖^2] < 1
    //                                  ^^^^^^^^^^^^^^^^^^^^^^^^^
    // with 1 > σ_p L_z and 1 > τ L.
    //
    // To do so, we first solve σ_p and σ_d from standard PDPS step length condition
    // ^^^^^ < 1. then we solve τ from  the rest.
    let σ_d = config.σd0 / nKz;
    let σ_p = config.σp0 / (l_z + config.σd0 * nKz);
    // Observe that = 1 - ^^^^^^^^^^^^^^^^^^^^^ = 1 - σ_{p,0}
    // We get the condition τσ_d M (1-σ_p L_z) < (1-σ_{p,0})*(1-τ L)
    // ⟺ τ [ σ_d M (1-σ_p L_z) + (1-σ_{p,0}) L ] < (1-σ_{p,0})
    let φ = 1.0 - config.σp0;
    let a = 1.0 - σ_p * l_z;
    let τ = config.τ0 * φ / ( σ_d * bigM * a + φ * l );
    let ψ = 1.0 - τ * l;
    let β = σ_p * config.σd0 * nKz / a; // σ_p * σ_d * (nKz * nK_z) / a;
    assert!(β < 1.0);
    // Now we need κ‖K_μ(π_♯^1 - π_♯^0)γ‖^2 ≤ (1/θ - τ[ℓ_v + ℓ]) ∫ c_2 dγ for κ defined as:
    let κ = τ * σ_d * ψ / ((1.0 - β) * ψ - τ * σ_d * bigM);
    //  The factor two in the manuscript disappears due to the definition of 𝚹 being
    // for ‖x-y‖₂² instead of c_2(x, y)=‖x-y‖₂²/2.
    let calculate_θ = |ℓ_v, max_transport| {
        config.transport.θ0 / (τ*(ℓ + ℓ_v) + κ * bigθ * max_transport)
    };
    let mut θ_or_adaptive = match opA.preadjoint().value_diff_unit_lipschitz_factor() {
        // We only estimate w (the uniform Lipschitz for of v), if we also estimate ℓ_v
        // (the uniform Lipschitz factor of ∇v).
        // We assume that the residual is decreasing.
        Some(ℓ_v0) => TransportStepLength::AdaptiveMax{
            l: ℓ_v0 * b.norm2(),
            max_transport : 0.0,
            g : calculate_θ
        },
        None => TransportStepLength::FullyAdaptive{
            l : 0.0,
            max_transport : 0.0,
            g : calculate_θ
        },
    };
    // Acceleration is not currently supported
    // let γ = dataterm.factor_of_strong_convexity();
    let ω = 1.0;

    // We multiply tolerance by τ for FB since our subproblems depending on tolerances are scaled
    // by τ compared to the conditional gradient approach.
    let tolerance = config.insertion.tolerance * τ * reg.tolerance_scaling();
    let mut ε = tolerance.initial();

    let starH = fnH.conjugate();

    // Statistics
    let full_stats = |residual : &A::Observable, μ : &RNDM<F, N>, z : &Z, ε, stats| IterInfo {
        value : residual.norm2_squared_div2() + fnR.apply(z)
                + reg.apply(μ) + fnH.apply(/* opKμ.apply(μ) + */ opKz.apply(z)),
        n_spikes : μ.len(),
        ε,
        // postprocessing: config.insertion.postprocessing.then(|| μ.clone()),
        .. stats
    };
    let mut stats = IterInfo::new();

    // Run the algorithm
    for state in iterator.iter_init(|| full_stats(&residual, &μ, &z, ε, stats.clone())) {
        // Calculate initial transport
        let Pair(v, _) = opA.preadjoint().apply(&residual);
        //opKμ.preadjoint().apply_add(&mut v, y);
        let z_base = z.clone();
        // We want to proceed as in Example 4.12 but with v and v̆ as in §5.
        // With A(ν, z) = A_μ ν + A_z z, following Example 5.1, we have
        // P_ℳ[F'(ν, z) + Ξ(ν, z, y)]= A_ν^*[A_ν ν + A_z z] + K_μ ν = A_ν^*A(ν, z) + K_μ ν,
        // where A_ν^* becomes a multiplier.
        // This is much easier with K_μ = 0, which is the only reason why are enforcing it.
        // TODO: Write a version of initial_transport that can deal with K_μ ≠ 0.
 
        let (μ_base_masses, mut μ_base_minus_γ0) = initial_transport(
            &mut γ1, &mut μ, |ν| opA.apply(Pair(ν, &z)),
            ε, τ, &mut θ_or_adaptive, opAnorm,
            v, &config.transport,
        );

        // Solve finite-dimensional subproblem several times until the dual variable for the
        // regularisation term conforms to the assumptions made for the transport above.
        let (d, _within_tolerances, Pair(τv̆, τz̆)) = 'adapt_transport: loop {
            // Calculate τv̆ = τA_*(A[μ_transported + μ_transported_base]-b)
            let residual_μ̆ = calculate_residual2(Pair(&γ1, &z),
                                                 Pair(&μ_base_minus_γ0, &zero_z),
                                                 opA, b);
            let Pair(τv̆, τz) = opA.preadjoint().apply(residual_μ̆ * τ);
            // opKμ.preadjoint().gemv(&mut τv̆, τ, y, 1.0);

            // Construct μ^{k+1} by solving finite-dimensional subproblems and insert new spikes.
            let (d, within_tolerances) = insert_and_reweigh(
                &mut μ, &τv̆, &γ1, Some(&μ_base_minus_γ0),
                op𝒟, op𝒟norm,
                τ, ε, &config.insertion,
                &reg, &state, &mut stats,
            );

            // A posteriori transport adaptation.
            // TODO: this does not properly treat v^{k+1} - v̆^k that depends on z^{k+1}!
            if aposteriori_transport(
                &mut γ1, &mut μ, &mut μ_base_minus_γ0, &μ_base_masses,
                ε, &config.transport
            ) {
                break 'adapt_transport (d, within_tolerances, Pair(τv̆, τz))
            }
        };

        stats.untransported_fraction = Some({
            assert_eq!(μ_base_masses.len(), γ1.len());
            let (a, b) = stats.untransported_fraction.unwrap_or((0.0, 0.0));
            let source = μ_base_masses.iter().map(|v| v.abs()).sum();
            (a + μ_base_minus_γ0.norm(Radon), b + source)
        });
        stats.transport_error = Some({
            assert_eq!(μ_base_masses.len(), γ1.len());
            let (a, b) = stats.transport_error.unwrap_or((0.0, 0.0));
            (a + μ.dist_matching(&γ1), b + γ1.norm(Radon))
        });

        // // Merge spikes.
        // // This expects the prune below to prune γ.
        // // TODO: This may not work correctly in all cases.
        // let ins = &config.insertion;
        // if ins.merge_now(&state) {
        //     if let SpikeMergingMethod::None = ins.merging {
        //     } else {
        //         stats.merged += μ.merge_spikes(ins.merging, |μ_candidate| {
        //             let ν = μ_candidate.sub_matching(&γ1)-&μ_base_minus_γ0;
        //             let mut d = &τv̆ + op𝒟.preapply(ν);
        //             reg.verify_merge_candidate(&mut d, μ_candidate, τ, ε, ins)
        //         });
        //     }
        // }

        // Prune spikes with zero weight. To maintain correct ordering between μ and γ1, also the
        // latter needs to be pruned when μ is.
        // TODO: This could do with a two-vector Vec::retain to avoid copies.
        let μ_new = DiscreteMeasure::from_iter(μ.iter_spikes().filter(|δ| δ.α != F::ZERO).cloned());
        if μ_new.len() != μ.len() {
            let mut μ_iter = μ.iter_spikes();
            γ1.prune_by(|_| μ_iter.next().unwrap().α != F::ZERO);
            stats.pruned += μ.len() - μ_new.len();
            μ = μ_new;
        }

        // Do z variable primal update
        z.axpy(-σ_p/τ, τz̆, 1.0); // TODO: simplify nasty factors
        opKz.adjoint().gemv(&mut z, -σ_p, &y, 1.0);
        z = fnR.prox(σ_p, z);
        // Do dual update
        // opKμ.gemv(&mut y, σ_d*(1.0 + ω), &μ, 1.0);    // y = y + σ_d K[(1+ω)(μ,z)^{k+1}]
        opKz.gemv(&mut y, σ_d*(1.0 + ω), &z, 1.0);
        // opKμ.gemv(&mut y, -σ_d*ω, μ_base, 1.0);// y = y + σ_d K[(1+ω)(μ,z)^{k+1} - ω (μ,z)^k]-b
        opKz.gemv(&mut y, -σ_d*ω, z_base, 1.0);// y = y + σ_d K[(1+ω)(μ,z)^{k+1} - ω (μ,z)^k]-b
        y = starH.prox(σ_d, y);

        // Update residual
        residual = calculate_residual(Pair(&μ, &z), opA, b);

        // Update step length parameters
        // let ω = pdpsconfig.acceleration.accelerate(&mut τ, &mut σ, γ);

        // Give statistics if requested
        let iter = state.iteration();
        stats.this_iters += 1;

        state.if_verbose(|| {
            plotter.plot_spikes(iter, Some(&d), Some(&τv̆), &μ);
            full_stats(&residual, &μ, &z, ε, std::mem::replace(&mut stats, IterInfo::new()))
        });

        // Update main tolerance for next iteration
        ε = tolerance.update(ε, iter);
    }

    let fit = |μ̃ : &RNDM<F, N>| {
        (opA.apply(Pair(μ̃, &z))-b).norm2_squared_div2()
        //+ fnR.apply(z) + reg.apply(μ)
        + fnH.apply(/* opKμ.apply(&μ̃) + */ opKz.apply(&z))
    };

    μ.merge_spikes_fitness(config.insertion.merging, fit, |&v| v);
    μ.prune();
    Pair(μ, z)
}