src/sliding_fb.rs

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

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

use crate::fb::*;
use crate::forward_model::{BoundedCurvature, BoundedCurvatureGuess};
use crate::measures::merging::SpikeMerging;
use crate::measures::{DeltaMeasure, DiscreteMeasure, Radon, RNDM};
use crate::plot::Plotter;
use crate::prox_penalty::{ProxPenalty, RadonSquared, StepLengthBound};
use crate::regularisation::SlidingRegTerm;
use crate::seminorms::DiscreteMeasureOp;
use crate::types::*;
use alg_tools::bounds::{Bounds, MinMaxMapping};
use alg_tools::error::DynResult;
use alg_tools::euclidean::Euclidean;
use alg_tools::instance::Space;
use alg_tools::iterate::AlgIteratorFactory;
use alg_tools::mapping::{DifferentiableMapping, DifferentiableRealMapping, Mapping, RealMapping};
use alg_tools::nalgebra_support::ToNalgebraRealField;
use alg_tools::norms::Norm;
use anyhow::ensure;
use std::ops::ControlFlow;

/// Transport settings for [`pointsource_sliding_fb_reg`].
#[derive(Clone, Copy, Eq, PartialEq, Serialize, Deserialize, Debug)]
#[serde(default)]
pub struct TransportConfig<F: Float> {
    /// Transport step length $θ$ normalised to $(0, 1)$.
    pub θ0: F,
    /// Unnormalised transport step length $θ$. Overrides θ0.
    pub τθ: Option<F>,
    /// Factor in $(0, 1)$ for decreasing transport to adapt to tolerance.
    pub adaptation: F,
    /// A posteriori transport tolerance multiplier
    pub tolerance_mult: F,
    /// Multiplier for rough estimate of ℓ_{∇v}. Should be ≥ 1.
    /// If explicit τθ is given, ℓ_{∇v} is one divided by this number times τθ.
    /// Otherwise, if  τθ relative to an estimate of ℓ_{∇v}, this number multiplies that estimate.
    pub ℓ_gradv_mult: F,
    /// maximum number of adaptation iterations, until cancelling transport.
    pub max_attempts: usize,
    /// Maximum number of failed transportations for a single source point
    pub max_fail: usize,
    /// Allow points to be transported partially.
    pub allow_partial_transport: bool,
    /// Use an alternative remainder control rule.
    pub alt_remainder_control: bool,
}

#[replace_float_literals(F::cast_from(literal))]
impl<F: Float> TransportConfig<F> {
    /// Check that the parameters are ok. Panics if not.
    pub fn check(&self) -> DynResult<()> {
        ensure!(self.θ0 > 0.0);
        ensure!(0.0 < self.adaptation && self.adaptation < 1.0);
        ensure!(self.tolerance_mult > 0.0);
        Ok(())
    }
}

#[replace_float_literals(F::cast_from(literal))]
impl<F: Float> Default for TransportConfig<F> {
    fn default() -> Self {
        TransportConfig {
            θ0: 0.99,
            τθ: None,
            adaptation: 0.9,
            allow_partial_transport: true,
            alt_remainder_control: false,
            tolerance_mult: 1e1,
            ℓ_gradv_mult: 3.0,
            max_attempts: 2,
            max_fail: usize::MAX,
        }
    }
}

/// Settings for [`pointsource_sliding_fb_reg`].
#[derive(Clone, Copy, Eq, PartialEq, Serialize, Deserialize, Debug)]
#[serde(default)]
pub struct SlidingFBConfig<F: Float> {
    /// Step length scaling
    pub τ0: F,
    // Auxiliary variable step length scaling for [`crate::sliding_pdps::pointsource_sliding_fb_pair`]
    pub σp0: F,
    /// Transport parameters
    pub transport: TransportConfig<F>,
    /// Generic parameters
    pub insertion: InsertionConfig<F>,
    /// Guess for curvature bound calculations.
    pub guess: BoundedCurvatureGuess,
}

#[replace_float_literals(F::cast_from(literal))]
impl<F: Float> Default for SlidingFBConfig<F> {
    fn default() -> Self {
        SlidingFBConfig {
            τ0: 0.99,
            σp0: 0.99,
            transport: Default::default(),
            insertion: Default::default(),
            guess: BoundedCurvatureGuess::BetterThanZero,
        }
    }
}

/// Internal type of adaptive transport step length calculation
#[derive(Clone, Debug, Serialize, Deserialize)]
pub enum TransportStepLength<F: Float> {
    /// Fixed, known step length
    #[allow(dead_code)]
    Fixed { τθ: F, ℓ_gradv: F },
    /// Simple step lengths that do not depend on maximum transport
    Simple { ℓ_gradv: F, τθ0: F, ℓ_base: F },
    /// Adaptive step length, only wrt. maximum transport.
    AdaptiveMax {
        ℓ_gradv: F,
        adaptive_max_transport: F,
        τθ0: F,
        ℓ_base: F,
        ℓ_base_max_transport: F,
    },
    /// Adaptive step length.
    FullyAdaptive {
        adaptive_ℓ_gradv: F,
        adaptive_max_transport: F,
        τθ0: F,
        ℓ_base: F,
        ℓ_base_max_transport: F,
    },
}

#[replace_float_literals(F::cast_from(literal))]
impl<F: Float> TransportStepLength<F> {
    fn get_ℓ_gradv(&self) -> F {
        use TransportStepLength::*;
        match *self {
            Fixed { ℓ_gradv, .. } => ℓ_gradv,
            Simple { ℓ_gradv, .. } => ℓ_gradv,
            AdaptiveMax { ℓ_gradv, .. } => ℓ_gradv,
            FullyAdaptive { adaptive_ℓ_gradv, .. } => adaptive_ℓ_gradv,
        }
    }

    fn new(
        maybe_ℓ_gradv_est: DynResult<F>,
        tconfig: &TransportConfig<F>,
        ℓ_base: F,
        ℓ_base_max_transport: F,
    ) -> Self {
        if let Some(τθ) = tconfig.τθ {
            TransportStepLength::Fixed { τθ, ℓ_gradv: 1.0 / (τθ * tconfig.ℓ_gradv_mult) }
        } else {
            match maybe_ℓ_gradv_est {
                Ok(ℓ_gradv_est) => {
                    if ℓ_base_max_transport == 0.0 {
                        TransportStepLength::Simple {
                            ℓ_gradv: tconfig.ℓ_gradv_mult * ℓ_gradv_est,
                            τθ0: tconfig.θ0,
                            ℓ_base,
                        }
                    } else {
                        TransportStepLength::AdaptiveMax {
                            ℓ_gradv: tconfig.ℓ_gradv_mult * ℓ_gradv_est,
                            adaptive_max_transport: 0.0,
                            τθ0: tconfig.θ0,
                            ℓ_base,
                            ℓ_base_max_transport,
                        }
                    }
                }
                Err(_) => TransportStepLength::FullyAdaptive {
                    adaptive_ℓ_gradv: 10.0 * F::EPSILON, // Start with something very small to estimate differentials
                    adaptive_max_transport: 0.0,
                    τθ0: tconfig.θ0,
                    ℓ_base,
                    ℓ_base_max_transport,
                },
            }
        }
    }
}

#[derive(Clone, Debug, Serialize)]
pub struct SingleTransport<Domain, F: Float> {
    /// Source point
    x: Domain,
    /// Target point
    y: Domain,
    /// Original mass
    α_μ_orig: F,
    /// Transported mass
    α_γ: F,
    /// Helper for pruning
    retain: bool,
    /// Fail count
    fail_count: usize,
    /// Contribution to remainder (temporary variable)
    excess: F,
}

#[derive(Clone, Debug, Serialize)]
pub struct Transport<Domain, F: Float> {
    vec: Vec<SingleTransport<Domain, F>>,
}

/// Whether partially transported points are allowed.
///
/// Partial transport can cause spike count explosion, so full or zero
/// transport is generally preferred. If this is set to `true`, different
/// transport adaptation heuristics will be used.
const MINIMAL_PARTIAL_TRANSPORT: bool = true;
const NEW_APPROACH: bool = true;

pub trait TransportProxPenalty<Domain, PreadjointCodomain, Reg, F = f64>:
    ProxPenalty<Domain, PreadjointCodomain, Reg, F>
where
    F: Float + ToNalgebraRealField,
    Reg: SlidingRegTerm<Domain, F>,
    Domain: Space + Clone,
{
    type TransportStepLength;

    /// Constrution of initial transport `γ1` from initial measure `μ` and `v=F'(μ)`
    /// with step lengh τ and transport step length `θ_or_adaptive`.
    fn initial_transport(
        &self,
        γ: &mut Transport<Domain, F>,
        μ: &DiscreteMeasure<Domain, F>,
        ε: F,
        τ: F,
        τθ_or_adaptive: &mut Self::TransportStepLength,
        v: &PreadjointCodomain,
        tconfig: &TransportConfig<F>,
    );

    /// A posteriori transport adaptation.
    fn aposteriori_transport(
        &self,
        γ: &mut Transport<Domain, F>,
        μ: &DiscreteMeasure<Domain, F>,
        μ̆: &DiscreteMeasure<Domain, F>,
        τv̆: &mut PreadjointCodomain,
        v: &mut PreadjointCodomain,
        extra: Option<F>,
        ε: F,
        τ: F,
        τθ_or_adaptive: &Self::TransportStepLength,
        reg: &Reg,
        tconfig: &TransportConfig<F>,
        rconfig: &RefinementSettings<F>,
        attempts: &mut usize,
    ) -> bool;

    fn get_transport_steplength(
        &self,
        lips: (DynResult<F>, DynResult<F>, DynResult<F>),
        tconfig: &TransportConfig<F>,
        ℓ_base: F,
        ℓ_base_max_transport: F,
    ) -> Self::TransportStepLength;
}

#[replace_float_literals(F::cast_from(literal))]
impl<F, M, Reg, const N: usize> TransportProxPenalty<Loc<N, F>, M, Reg, F> for RadonSquared
where
    RadonSquared: ProxPenalty<Loc<N, F>, M, Reg, F>,
    F: Float + ToNalgebraRealField,
    M: MinMaxMapping<Loc<N, F>, F> + DifferentiableRealMapping<N, F>,
    Reg: SlidingRegTerm<Loc<N, F>, F>,
    RNDM<N, F>: SpikeMerging<F>,
{
    type TransportStepLength = TransportStepLength<F>;

    fn get_transport_steplength(
        &self,
        (ℓ_F, maybe_ℓ_gradv, _maybe_transport_lip): (DynResult<F>, DynResult<F>, DynResult<F>),
        tconfig: &TransportConfig<F>,
        ℓ: F,
        ℓ_base_max_transport: F,
    ) -> Self::TransportStepLength {
        TransportStepLength::new(
            maybe_ℓ_gradv,
            tconfig,
            ℓ + ℓ_F.unwrap_or(0.0),
            ℓ_base_max_transport,
        )
    }

    fn initial_transport(
        &self,
        γ: &mut Transport<Loc<N, F>, F>,
        μ: &RNDM<N, F>,
        _ε: F,
        _τ: F,
        τθ_or_adaptive: &mut TransportStepLength<F>,
        v: &M,
        tconfig: &TransportConfig<F>,
    ) {
        γ.do_init_transport(v, μ, τθ_or_adaptive, tconfig);
    }

    fn aposteriori_transport(
        &self,
        γ: &mut Transport<Loc<N, F>, F>,
        μ: &RNDM<N, F>,
        μ̆: &RNDM<N, F>,
        τv̆: &mut M,
        v: &mut M,
        _extra: Option<F>,
        ε: F,
        τ: F,
        τθ_or_adaptive: &TransportStepLength<F>,
        reg: &Reg,
        tconfig: &TransportConfig<F>,
        _rconfig: &RefinementSettings<F>,
        attempts: &mut usize,
    ) -> bool {
        *attempts += 1;

        if *attempts > tconfig.max_attempts {
            // Previous round has set transport to zero
            return true;
        }

        let nΔ = μ.dist_matching(&μ̆);
        let all_ok = γ.do_new_aposteriori_transport(
            μ,
            τv̆,
            v,
            ε,
            τ,
            τθ_or_adaptive,
            reg,
            tconfig,
            |_, mass_zero, negated| {
                use std::cmp::Ordering::*;
                match (mass_zero, negated) {
                    (Less, _) => -nΔ,
                    (Greater, _) => nΔ,
                    (Equal, false) => nΔ, // pessimistic estimated for ω
                    (Equal, true) => -nΔ, // pessimistic estimated for -ω
                }
            },
        );

        if !all_ok {
            if *attempts >= tconfig.max_attempts {
                for ρ in γ.iter_mut() {
                    ρ.α_γ = 0.0;
                }
            }
        } else {
            for ρ in γ.iter_mut() {
                if ρ.α_γ == 0.0 {
                    ρ.fail_count += 1;
                } else {
                    ρ.fail_count = 0;
                }
            }
        }
        all_ok
    }
}

#[replace_float_literals(F::cast_from(literal))]
impl<F, M, Reg, 𝒟, O, const N: usize> TransportProxPenalty<Loc<N, F>, M, Reg, F> for 𝒟
where
    F: Float + ToNalgebraRealField,
    𝒟: DiscreteMeasureOp<Loc<N, F>, F>,
    𝒟::Codomain: RealMapping<N, F>,
    M: MinMaxMapping<Loc<N, F>, F> + DifferentiableRealMapping<N, F>,
    for<'a> &'a M: std::ops::Add<𝒟::PreCodomain, Output = O>,
    O: MinMaxMapping<Loc<N, F>, F>,
    Reg: SlidingRegTerm<Loc<N, F>, F>,
    Self: ProxPenalty<Loc<N, F>, M, Reg, F>,
    RNDM<N, F>: SpikeMerging<F>,
{
    type TransportStepLength = TransportStepLength<F>;

    fn get_transport_steplength(
        &self,
        (ℓ_F, maybe_ℓ_gradv, _maybe_transport_lip): (DynResult<F>, DynResult<F>, DynResult<F>),
        tconfig: &TransportConfig<F>,
        ℓ_base: F,
        ℓ_base_max_transport: F,
    ) -> Self::TransportStepLength {
        TransportStepLength::new(
            maybe_ℓ_gradv,
            tconfig,
            ℓ_base + ℓ_F.unwrap_or(0.0),
            ℓ_base_max_transport,
        )
    }

    fn initial_transport(
        &self,
        γ: &mut Transport<Loc<N, F>, F>,
        μ: &RNDM<N, F>,
        _ε: F,
        _τ: F,
        τθ_or_adaptive: &mut TransportStepLength<F>,
        v: &M,
        tconfig: &TransportConfig<F>,
    ) {
        γ.do_init_transport(v, μ, τθ_or_adaptive, tconfig);
    }

    fn aposteriori_transport(
        &self,
        γ: &mut Transport<Loc<N, F>, F>,
        μ: &RNDM<N, F>,
        μ̆: &RNDM<N, F>,
        τv̆: &mut M,
        v: &mut M,
        extra: Option<F>,
        ε: F,
        τ: F,
        τθ_or_adaptive: &TransportStepLength<F>,
        reg: &Reg,
        tconfig: &TransportConfig<F>,
        _rconfig: &RefinementSettings<F>,
        attempts: &mut usize,
    ) -> bool {
        *attempts += 1;

        if *attempts > tconfig.max_attempts {
            // Previous round has set transport to zero
            return true;
        }

        let all_ok = if NEW_APPROACH {
            let ω = self.apply(μ.sub_matching(&μ̆));
            γ.do_new_aposteriori_transport(
                μ,
                τv̆,
                v,
                ε,
                τ,
                τθ_or_adaptive,
                reg,
                tconfig,
                |x, _, _| ω.apply(x),
            )
        } else {
            let mut all_ok0 = true;

            // 1. If π_♯^1γ^{k+1} = γ1 has non-zero mass at some point y, but μ = μ^{k+1} does not,
            // then the ansatz ∇w̃_x(y) = w^{k+1}(y) may not be satisfied. So set the mass of γ1
            // at that point to zero, and retry.
            for (δ, ρ) in izip!(μ.iter_spikes(), γ.iter_mut()) {
                if δ.α == 0.0 && ρ.α_γ != 0.0 {
                    all_ok0 = false;
                    ρ.α_γ = 0.0;
                }
                // TODO: sign
            }

            // 2. Through bounding ∫ B_ω(y, z) dλ(x, y, z).
            //    through the estimate ≤ C ‖Δ‖‖γ^{k+1}‖ for Δ := μ^{k+1}-μ̆^k
            //    which holds for some some C if the convolution kernel in 𝒟 has Lipschitz gradient.

            let nγ = γ.norm(Radon);
            let nΔ = μ.dist_matching(&μ̆) + extra.unwrap_or(0.0);
            let t = ε * tconfig.tolerance_mult;
            if nγ * nΔ > t && *attempts >= tconfig.max_attempts {
                all_ok0 = false;
            } else if nγ * nΔ > t {
                // Since t/(nγ*nΔ)<1, and the constant tconfig.adaptation < 1,
                // this will guarantee that eventually ‖γ‖ decreases sufficiently that we
                // will not enter here.
                //*γ *= tconfig.adaptation * t / (nγ * nΔ);

                // We want a consistent behaviour that has the potential to set many weights to zero.
                // Therefore, we find the smallest uniform reduction `chg_one`, subtracted
                // from all weights, that achieves total `adapt` adaptation.
                let adapt_to = tconfig.adaptation * t / nΔ;
                let reduction_target = nγ - adapt_to;
                assert!(reduction_target > 0.0);
                if tconfig.allow_partial_transport {
                    if MINIMAL_PARTIAL_TRANSPORT {
                        // This reduces weights of transport, starting from … until `adapt` is
                        // exhausted. It will, therefore, only ever cause one extrap point insertion
                        // at the sources, unlike “full” partial transport.
                        //let refs = γ.vec.iter_mut().collect::<Vec<_>>();
                        //refs.sort_by(|ρ1, ρ2| ρ1.α_γ.abs().partial_cmp(&ρ2.α_γ.abs()).unwrap());
                        // let mut it = refs.into_iter();
                        //
                        // Maybe sort by differential norm
                        // let mut refs = γ
                        //     .vec
                        //     .iter_mut()
                        //     .map(|ρ| {
                        //         let val = v.differential(&ρ.x).norm2_squared();
                        //         (ρ, val)
                        //     })
                        //     .collect::<Vec<_>>();
                        // refs.sort_by(|(_, v1), (_, v2)| v2.partial_cmp(&v1).unwrap());
                        // let mut it = refs.into_iter().map(|(ρ, _)| ρ);
                        let mut it = γ.vec.iter_mut().rev();
                        let _unused = it.try_fold(reduction_target, |left, ρ| {
                            let w = ρ.α_γ.abs();
                            if left <= w {
                                ρ.α_γ = ρ.α_γ.signum() * (w - left);
                                ControlFlow::Break(())
                            } else {
                                ρ.α_γ = 0.0;
                                ControlFlow::Continue(left - w)
                            }
                        });
                    } else {
                        // This version equally reduces all weights. It causes partial transport, which
                        // has the problem that that we need to then adapt weights in both start and
                        // end points, in insert_and_reweigh, somtimes causing the number of spikes μ
                        // to explode.
                        let mut abs_weights = γ
                            .vec
                            .iter()
                            .map(|ρ| ρ.α_γ.abs())
                            .filter(|t| *t > F::EPSILON)
                            .collect::<Vec<F>>();
                        abs_weights.sort_by(|a, b| a.total_cmp(b));
                        let n = abs_weights.len();
                        // Cannot have partial transport; can cause spike count explosion
                        let chg = abs_weights.into_iter().zip((1..=n).rev()).try_fold(
                            0.0,
                            |smaller_total, (w, m)| {
                                let mf = F::cast_from(m);
                                let reduction = w * mf + smaller_total;
                                if reduction >= reduction_target {
                                    ControlFlow::Break((reduction_target - smaller_total) / mf)
                                } else {
                                    ControlFlow::Continue(smaller_total + w)
                                }
                            },
                        );
                        match chg {
                            ControlFlow::Continue(_) => γ.vec.iter_mut().for_each(|δ| δ.α_γ = 0.0),
                            ControlFlow::Break(chg_one) => γ.vec.iter_mut().for_each(|ρ| {
                                let t = ρ.α_γ.abs();
                                if t > 0.0 {
                                    if tconfig.allow_partial_transport {
                                        let new = (t - chg_one).max(0.0);
                                        ρ.α_γ = ρ.α_γ.signum() * new;
                                    }
                                }
                            }),
                        }
                    }
                } else {
                    // This version zeroes smallest weights, avoiding partial transport.
                    let mut abs_weights_idx = γ
                        .vec
                        .iter()
                        .map(|ρ| ρ.α_γ.abs())
                        .zip(0..)
                        .filter(|(w, _)| *w >= 0.0)
                        .collect::<Vec<(F, usize)>>();
                    abs_weights_idx.sort_by(|(a, _), (b, _)| a.total_cmp(b));

                    let mut left = reduction_target;

                    for (w, i) in abs_weights_idx {
                        left -= w;
                        let ρ = &mut γ.vec[i];
                        ρ.α_γ = 0.0;
                        if left < 0.0 {
                            break;
                        }
                    }
                }

                all_ok0 = false
            }
            all_ok0
        };

        if !all_ok {
            if *attempts >= tconfig.max_attempts {
                for ρ in γ.iter_mut() {
                    ρ.α_γ = 0.0;
                }
            }
        } else {
            for ρ in γ.iter_mut() {
                if ρ.α_γ == 0.0 {
                    ρ.fail_count += 1;
                } else {
                    ρ.fail_count = 0;
                }
            }
        }
        all_ok
    }
}

#[replace_float_literals(F::cast_from(literal))]
impl<const N: usize, F: Float> Transport<Loc<N, F>, F> {
    pub(crate) fn new() -> Self {
        Transport { vec: Vec::new() }
    }

    pub(crate) fn iter(&self) -> impl Iterator<Item = &'_ SingleTransport<Loc<N, F>, F>> {
        self.vec.iter()
    }

    pub(crate) fn iter_mut(
        &mut self,
    ) -> impl Iterator<Item = &'_ mut SingleTransport<Loc<N, F>, F>> {
        self.vec.iter_mut()
    }

    pub(crate) fn extend<I>(&mut self, it: I)
    where
        I: IntoIterator<Item = SingleTransport<Loc<N, F>, F>>,
    {
        self.vec.extend(it)
    }

    pub(crate) fn len(&self) -> usize {
        self.vec.len()
    }

    // pub(crate) fn dist_matching(&self, μ: &RNDM<N, F>) -> F {
    //     self.iter()
    //         .zip(μ.iter_spikes())
    //         .map(|(ρ, δ)| (ρ.α_γ - δ.α).abs())
    //         .sum()
    // }

    /// Construct `μ̆`, replacing the contents of `μ`.
    pub(crate) fn μ̆_into(&self, μ: &mut RNDM<N, F>) {
        assert!(self.len() <= μ.len());

        // First transported points
        for (δ, ρ) in izip!(μ.iter_spikes_mut(), self.iter()) {
            if ρ.α_γ.abs() > 0.0 {
                // Transport – transported point
                δ.α = ρ.α_γ;
                δ.x = ρ.y;
            } else {
                // No transport – original point
                δ.α = ρ.α_μ_orig;
                δ.x = ρ.x;
            }
        }

        // Then source points with partial transport
        let mut i = self.len();
        // This can cause the number of points to explode, so cannot have partial transport.
        for ρ in self.iter() {
            let α = ρ.α_μ_orig - ρ.α_γ;
            if ρ.α_γ.abs() > F::EPSILON && α.abs() > F::EPSILON {
                let δ = DeltaMeasure { α, x: ρ.x };
                if i < μ.len() {
                    μ[i] = δ;
                } else {
                    μ.push(δ)
                }
                i += 1;
            }
        }
        μ.truncate(i);
    }

    /// Returns $‖μ\^k - π\_♯\^0γ\^{k+1}‖$
    pub(crate) fn μ0_minus_γ0_radon(&self) -> F {
        self.vec.iter().map(|ρ| (ρ.α_μ_orig - ρ.α_γ).abs()).sum()
    }

    /// Returns $∫ c_2 d|γ|$
    pub(crate) fn c2integral(&self) -> F {
        self.vec
            .iter()
            .map(|ρ| ρ.y.dist2_squared(&ρ.x) / 2.0 * ρ.α_γ.abs())
            .sum()
    }

    pub(crate) fn get_transport_stats(&self, stats: &mut IterInfo<F>, μ: &RNDM<N, F>) {
        // TODO: This doesn't take into account μ[i].α becoming zero in the latest tranport
        // attempt, for i < self.len(), when a corresponding source term also exists with index
        // j ≥ self.len(). For now, we let that be reflected in the prune count.
        stats.inserted += μ.len() - self.len();

        let transp = stats.get_transport_mut();

        transp.dist = {
            let (a, b) = transp.dist;
            (a + self.c2integral(), b + self.norm(Radon))
        };
        transp.untransported_fraction = {
            let (a, b) = transp.untransported_fraction;
            let source = self.iter().map(|ρ| ρ.α_μ_orig.abs()).sum();
            (a + self.μ0_minus_γ0_radon(), b + source)
        };
        transp.transport_error = {
            let (a, b) = transp.transport_error;
            //(a + self.dist_matching(&μ), b + self.norm(Radon))

            // This ignores points that have been not transported at all, to only calculate
            // destnation error; untransported_fraction accounts for not being able to transport
            // at all.
            self.iter()
                .zip(μ.iter_spikes())
                .fold((a, b), |(a, b), (ρ, δ)| {
                    let transported = ρ.α_γ.abs();
                    if transported > F::EPSILON {
                        (a + (ρ.α_γ - δ.α).abs(), b + transported)
                    } else {
                        (a, b)
                    }
                })
        };
    }

    /// Prune spikes with zero weight. To maintain correct ordering between μ and γ, also the
    /// latter needs to be pruned when μ is.
    pub(crate) fn prune_compat(&mut self, μ: &mut RNDM<N, F>, stats: &mut IterInfo<F>) {
        assert!(self.vec.len() <= μ.len());
        let old_len = μ.len();
        for (ρ, δ) in self.vec.iter_mut().zip(μ.iter_spikes()) {
            ρ.retain = δ.α.abs() > F::EPSILON;
        }
        μ.prune_by(|δ| δ.α.abs() > F::EPSILON);
        stats.pruned += old_len - μ.len();
        self.vec.retain(|ρ| ρ.retain);
        assert!(self.vec.len() <= μ.len());
    }

    /// Helper for initial transport. Called from [`TransportProxPenalty::initial_transport`].
    pub(super) fn do_init_transport<D>(
        &mut self,
        v: &D,
        μ: &RNDM<N, F>,
        τθ_or_adaptive: &mut TransportStepLength<F>,
        tconfig: &TransportConfig<F>,
    ) where
        D: DifferentiableRealMapping<N, F>,
    {
        use TransportStepLength::*;

        // Initialise transport structure weights
        for (δ, ρ) in izip!(μ.iter_spikes(), self.iter_mut()) {
            ρ.α_μ_orig = δ.α;
            ρ.x = δ.x;
            ρ.y = δ.x; // Later updated if no fails.
            ρ.α_γ = if ρ.fail_count > tconfig.max_fail {
                0.0
            } else {
                // If old transport has opposing sign, the new transport will be none.
                if (ρ.α_γ > 0.0 && δ.α < 0.0) || (ρ.α_γ < 0.0 && δ.α > 0.0) {
                    0.0
                } else {
                    δ.α
                }
            };
        }

        let γ_prev_len = self.len();
        assert!(μ.len() >= γ_prev_len);
        self.extend(μ[γ_prev_len..].iter().map(|δ| SingleTransport {
            x: δ.x,
            y: δ.x, // Just something, will be filled properly in the next phase
            α_μ_orig: δ.α,
            α_γ: δ.α,
            retain: true,
            fail_count: 0,
            excess: 0.0,
        }));

        // Calculate transport rays.
        let simple_τθ = match *τθ_or_adaptive {
            Fixed { τθ, .. } => Some(τθ),
            Simple { ℓ_gradv, τθ0, ℓ_base } => Some(τθ0 / (ℓ_gradv + ℓ_base)),
            AdaptiveMax {
                ℓ_gradv,
                ref mut adaptive_max_transport,
                τθ0,
                ℓ_base,
                ℓ_base_max_transport,
            } => {
                *adaptive_max_transport = adaptive_max_transport.max(self.norm(Radon));
                Some(τθ0 / (ℓ_gradv + ℓ_base + ℓ_base_max_transport * *adaptive_max_transport))
            }
            FullyAdaptive {
                ref mut adaptive_ℓ_gradv,
                ref mut adaptive_max_transport,
                τθ0,
                ℓ_base,
                ℓ_base_max_transport,
            } => {
                *adaptive_max_transport = adaptive_max_transport.max(self.norm(Radon));
                let mut τθ = τθ0
                    / (*adaptive_ℓ_gradv + ℓ_base + ℓ_base_max_transport * *adaptive_max_transport);
                // Do two runs through the spikes to update θ, breaking if first run did not cause
                // a change.
                for _i in 0..=1 {
                    let mut changes = false;
                    for ρ in self.iter_mut() {
                        if ρ.fail_count < tconfig.max_fail {
                            let dv_x = v.differential(&ρ.x);
                            let g = &dv_x * (ρ.α_γ.signum() * τθ);
                            ρ.y = ρ.x - g;
                            let n = g.norm2();
                            if n >= F::EPSILON {
                                // Estimate Lipschitz factor of ∇v
                                let this_ℓ_gradv = (dv_x - v.differential(&ρ.y)).norm2() / n;
                                *adaptive_ℓ_gradv = adaptive_ℓ_gradv.max(this_ℓ_gradv);
                                τθ = τθ0
                                    / (*adaptive_ℓ_gradv
                                        + ℓ_base
                                        + ℓ_base_max_transport * *adaptive_max_transport);
                                changes = true
                            }
                        }
                    }
                    if !changes {
                        break;
                    }
                }
                None
            }
        };

        if let Some(τθ) = simple_τθ {
            for ρ in self.iter_mut() {
                if ρ.fail_count <= tconfig.max_fail {
                    ρ.y = ρ.x - v.differential(&ρ.x) * (ρ.α_γ.signum() * τθ);
                }
            }
        }
    }

    /// Helper for a posteriori transport error control.
    /// Called from [`TransportProxPenalty::aposteriori_transport`].
    fn do_new_aposteriori_transport<Reg, M>(
        &mut self,
        μ: &RNDM<N, F>,
        τv̆: &mut M,
        v: &mut M,
        ε: F,
        τ: F,
        τθ_or_adaptive: &TransportStepLength<F>,
        reg: &Reg,
        tconfig: &TransportConfig<F>,
        ω: impl Fn(&Loc<N, F>, std::cmp::Ordering, bool) -> F,
    ) -> bool
    where
        Reg: SlidingRegTerm<Loc<N, F>, F>,
        F: ToNalgebraRealField,
        M: MinMaxMapping<Loc<N, F>, F> + DifferentiableRealMapping<N, F>,
    {
        let τℓ_gradv2 = τ * τθ_or_adaptive.get_ℓ_gradv() / 2.0;
        let Bounds(α_lower, α_upper) = reg.subdiff_range();

        let (all_ok, m) = izip!(self.vec.iter_mut(), μ.iter_spikes()).fold(
            (true, 0.0),
            |(all_ok_so_far, total_excess), (ρ, δ)| {
                use std::cmp::Ordering::*;
                // NOTE: The tolerances ε are commented out, because they will in any case
                // be consumed by `t` below by suitably large choice of `tolerance_mult`.
                // Hence, we simply implicitly adapt the `tolerance_mult` be commenting out
                // the `ε` here. That way, `d` is in its entirely multiplied by τ, as is `t`,
                // making all factors independent of τ.
                let maybe_excess = match ρ.α_γ.total_cmp(&0.0) {
                    Greater => (δ.α >= 0.0).then(|| {
                        let gvx = v.differential(&ρ.x);
                        let d = if tconfig.alt_remainder_control {
                            let τv̆y = τv̆.apply(&ρ.y);
                            (/*ε +*/τ * α_upper + ω(&ρ.x, ρ.α_γ.total_cmp(&ρ.α_μ_orig), false))
                                + (τv̆y + τ * gvx.dot(&ρ.x - &ρ.y))
                                - τℓ_gradv2 * ρ.y.dist2_squared(&ρ.x)
                        } else {
                            (/*2.0 * ε*/-ω(&ρ.y, δ.α.total_cmp(&ρ.α_γ), true)
                                + ω(&ρ.x, ρ.α_γ.total_cmp(&ρ.α_μ_orig), false))
                                + τ * gvx.dot(&ρ.x - &ρ.y)
                                - τℓ_gradv2 * ρ.y.dist2_squared(&ρ.x)
                        };
                        d * ρ.α_γ
                    }),
                    Less => (δ.α <= 0.0).then(|| {
                        let gvx = v.differential(&ρ.x);
                        let d = if tconfig.alt_remainder_control {
                            let τv̆y = τv̆.apply(&ρ.y);
                            (/*ε*/-τ * α_lower - ω(&ρ.x, ρ.α_γ.total_cmp(&ρ.α_μ_orig), true))
                                - (τv̆y + τ * gvx.dot(&ρ.x - &ρ.y))
                                - τℓ_gradv2 * ρ.y.dist2_squared(&ρ.x)
                        } else {
                            (/*2.0 * ε +*/ω(&ρ.y, δ.α.total_cmp(&ρ.α_γ), false)
                                - ω(&ρ.x, ρ.α_γ.total_cmp(&ρ.α_μ_orig), true))
                                - τ * gvx.dot(&ρ.x - &ρ.y)
                                - τℓ_gradv2 * ρ.y.dist2_squared(&ρ.x)
                        };
                        d * (-ρ.α_γ)
                    }),
                    Equal => Some(0.0),
                };
                match maybe_excess {
                    None => {
                        ρ.α_γ = 0.0;
                        ρ.excess = 0.0;
                        (false, total_excess)
                    }
                    Some(e) => {
                        ρ.excess = e;
                        (all_ok_so_far, total_excess + e)
                    }
                }
            },
        );

        let t = τ * ε * tconfig.tolerance_mult;

        if m > t {
            let mut it = self.vec.iter_mut().rev().filter(|ρ| ρ.excess > 0.0);
            let reduction_target = m - tconfig.adaptation * t;
            let _unused = it.try_fold(reduction_target, |left, ρ| {
                let d = ρ.excess;
                if d >= left {
                    if tconfig.allow_partial_transport {
                        ρ.α_γ *= (d - left) / d;
                    } else {
                        ρ.α_γ = 0.0;
                    }
                    ControlFlow::Break(())
                } else {
                    ρ.α_γ = 0.0;
                    ControlFlow::Continue(left - d)
                }
            });
            false
        } else {
            all_ok
        }
    }
}

impl<const N: usize, F: Float> Norm<Radon, F> for Transport<Loc<N, F>, F> {
    fn norm(&self, _: Radon) -> F {
        self.iter().map(|ρ| ρ.α_γ.abs()).sum()
    }
}

impl<const N: usize, F: Float> MulAssign<F> for Transport<Loc<N, F>, F> {
    fn mul_assign(&mut self, factor: F) {
        for ρ in self.iter_mut() {
            ρ.α_γ *= factor;
        }
    }
}

/// Iteratively solve the pointsource localisation problem using sliding forward-backward
/// splitting
///
/// The parametrisation is as for [`pointsource_fb_reg`].
/// Inertia is currently not supported.
#[replace_float_literals(F::cast_from(literal))]
pub fn pointsource_sliding_fb_reg<F, I, Dat, Reg, Plot, P, const N: usize>(
    f: &Dat,
    reg: &Reg,
    prox_penalty: &P,
    config: &SlidingFBConfig<F>,
    iterator: I,
    mut plotter: Plot,
    μ0: Option<RNDM<N, F>>,
) -> DynResult<RNDM<N, F>>
where
    F: Float + ToNalgebraRealField,
    I: AlgIteratorFactory<IterInfo<F>>,
    Dat: DifferentiableMapping<RNDM<N, F>, Codomain = F> + BoundedCurvature<F>,
    Dat::DerivativeDomain:
        DifferentiableRealMapping<N, F> + ClosedMul<F> + MinMaxMapping<Loc<N, F>, F>,
    //for<'a> Dat::Differential<'a>: Lipschitz<&'a P, FloatType = F>,
    RNDM<N, F>: SpikeMerging<F>,
    Reg: SlidingRegTerm<Loc<N, F>, F>,
    P: TransportProxPenalty<Loc<N, F>, Dat::DerivativeDomain, Reg, F> + StepLengthBound<F, Dat>,
    Plot: Plotter<P::ReturnMapping, Dat::DerivativeDomain, RNDM<N, F>>,
{
    // Check parameters
    ensure!(config.τ0 > 0.0, "Invalid step length parameter");
    config.transport.check()?;

    // Initialise iterates
    let mut μ = μ0.unwrap_or_else(|| DiscreteMeasure::new());
    let mut γ = Transport::new();

    // Set up parameters
    // let opAnorm = opA.opnorm_bound(Radon, L2);
    //let max_transport = config.max_transport.scale
    //                    * reg.radon_norm_bound(b.norm2_squared() / 2.0);
    //let ℓ = opA.transport.lipschitz_factor(L2Squared) * max_transport;
    let τ = config.τ0 / prox_penalty.step_length_bound(&f)?;

    let mut τθ_or_adaptive = prox_penalty.get_transport_steplength(
        f.curvature_bound_components(config.guess),
        &config.transport,
        0.0,
        0.0,
    );
    // 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();

    // Statistics
    let full_stats = |μ: &RNDM<N, F>, ε, stats| IterInfo {
        value: f.apply(μ) + reg.apply(μ),
        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(&μ, ε, stats.clone())) {
        let mut v = f.differential(&μ);

        // Calculate initial transport
        prox_penalty.initial_transport(
            &mut γ,
            &μ,
            ε,
            τ,
            &mut τθ_or_adaptive,
            &v,
            &config.transport,
        );

        let mut attempts = 0;

        // Solve finite-dimensional subproblem several times until the dual variable for the
        // regularisation term conforms to the assumptions made for the transport above.
        let (mut τv̆, μ̆) = 'adapt_transport: loop {
            // Set initial guess for μ=μ^{k+1}.
            γ.μ̆_into(&mut μ);
            let μ̆ = μ.clone();

            // Calculate τv̆ = τA_*(A[μ_transported + μ_transported_base]-b)
            //let residual_μ̆ = calculate_residual2(&γ1, &μ0_minus_γ0, opA, b);
            // TODO: this could be optimised by doing the differential like the
            // old residual2.
            // NOTE: This assumes that μ = γ1
            let mut τv̆ = f.differential(&μ̆) * τ;

            // Construct μ^{k+1} by solving finite-dimensional subproblems and insert new spikes.
            prox_penalty.insert_and_reweigh(
                &mut μ,
                &mut τv̆,
                τ,
                ε,
                &config.insertion,
                &reg,
                &state,
                &mut stats,
            )?;

            // A posteriori transport adaptation.
            if prox_penalty.aposteriori_transport(
                &mut γ,
                &μ,
                &μ̆,
                &mut τv̆,
                &mut v,
                None,
                ε,
                τ,
                &τθ_or_adaptive,
                reg,
                &config.transport,
                &config.insertion.refinement,
                &mut attempts,
            ) {
                break 'adapt_transport (τv̆, μ̆);
            }

            stats.get_transport_mut().readjustment_iters += 1;
        };

        γ.get_transport_stats(&mut stats, &μ);

        // Merge spikes.
        // This crucially expects the merge routine to be stable with respect to spike locations,
        // and not to performing any pruning. That is be to done below simultaneously for γ.
        config.insertion.if_merge_now(&state, |ins| {
            let m = prox_penalty.merge_spikes(
                &mut μ,
                &mut τv̆,
                &μ̆,
                τ,
                ε,
                ins,
                &reg,
                Some(|μ̃: &RNDM<N, F>| f.apply(μ̃)),
            );
            //if m > 0 {
            stats.merged += m;
            //v = f.differential(&μ);
            //}
        });

        γ.prune_compat(&mut μ, &mut stats);

        // Do extra weight optimisation step heuristic
        for _ in 1..config.insertion.extra_weight_optimisation_steps {
            τv̆ = f.differential(&μ) * τ;
            prox_penalty.reweigh(
                &mut μ,
                &mut τv̆,
                τ,
                ε,
                &config.insertion,
                &reg,
                &state,
                &mut stats,
            )?;
        }

        if config.insertion.extra_weight_optimisation_steps > 0 {
            γ.prune_compat(&mut μ, &mut stats);
        }

        let iter = state.iteration();
        stats.this_iters += 1;

        // Give statistics if requested
        state.if_verbose(|| {
            plotter.plot_spikes(iter, None, Some(&τv̆), &μ);
            full_stats(&μ, ε, std::mem::replace(&mut stats, IterInfo::new()))
        });

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

    //postprocess(μ, &config.insertion, f)
    postprocess(μ, &config.insertion, |μ̃| f.apply(μ̃))
}