src/fb.rs

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

This corresponds to the manuscript

 * Valkonen T. - _Proximal methods for point source localisation_,
   [arXiv:2212.02991](https://arxiv.org/abs/2212.02991).

The main routine is [`pointsource_fb_reg`].

## Problem

<p>
Our objective is to solve
$$
    \min_{μ ∈ ℳ(Ω)}~ F_0(Aμ-b) + α \|μ\|_{ℳ(Ω)} + δ_{≥ 0}(μ),
$$
where $F_0(y)=\frac{1}{2}\|y\|_2^2$ and the forward operator $A \in 𝕃(ℳ(Ω); ℝ^n)$.
</p>

## Approach

<p>
As documented in more detail in the paper, on each step we approximately solve
$$
    \min_{μ ∈ ℳ(Ω)}~ F(x) + α \|μ\|_{ℳ(Ω)} + δ_{≥ 0}(x) + \frac{1}{2}\|μ-μ^k|_𝒟^2,
$$
where $𝒟: 𝕃(ℳ(Ω); C_c(Ω))$ is typically a convolution operator.
</p>

## Finite-dimensional subproblems.

With $C$ a projection from [`DiscreteMeasure`] to the weights, and $x^k$ such that $x^k=Cμ^k$, we
form the discretised linearised inner problem
<p>
$$
    \min_{x ∈ ℝ^n}~ τ\bigl(F(Cx^k) + [C^*∇F(Cx^k)]^⊤(x-x^k) + α {\vec 1}^⊤ x\bigr)
                    + δ_{≥ 0}(x) + \frac{1}{2}\|x-x^k\|_{C^*𝒟C}^2,
$$
equivalently
$$
    \begin{aligned}
    \min_x~ & τF(Cx^k) - τ[C^*∇F(Cx^k)]^⊤x^k + \frac{1}{2} (x^k)^⊤ C^*𝒟C x^k
            \\
            &
            - [C^*𝒟C x^k - τC^*∇F(Cx^k)]^⊤ x
            \\
            &
            + \frac{1}{2} x^⊤ C^*𝒟C x
            + τα {\vec 1}^⊤ x + δ_{≥ 0}(x),
    \end{aligned}
$$
In other words, we obtain the quadratic non-negativity constrained problem
$$
    \min_{x ∈ ℝ^n}~ \frac{1}{2} x^⊤ Ã x - b̃^⊤ x + c + τα {\vec 1}^⊤ x + δ_{≥ 0}(x).
$$
where
$$
   \begin{aligned}
    Ã & = C^*𝒟C,
    \\
    g̃ & = C^*𝒟C x^k - τ C^*∇F(Cx^k)
        = C^* 𝒟 μ^k - τ C^*A^*(Aμ^k - b)
    \\
    c & = τ F(Cx^k) - τ[C^*∇F(Cx^k)]^⊤x^k + \frac{1}{2} (x^k)^⊤ C^*𝒟C x^k
        \\
        &
        = \frac{τ}{2} \|Aμ^k-b\|^2 - τ[Aμ^k-b]^⊤Aμ^k + \frac{1}{2} \|μ_k\|_{𝒟}^2
        \\
        &
        = -\frac{τ}{2} \|Aμ^k-b\|^2 + τ[Aμ^k-b]^⊤ b + \frac{1}{2} \|μ_k\|_{𝒟}^2.
   \end{aligned}
$$
</p>

We solve this with either SSN or FB as determined by
[`InnerSettings`] in [`FBGenericConfig::inner`].
*/

use numeric_literals::replace_float_literals;
use serde::{Serialize, Deserialize};
use colored::Colorize;

use alg_tools::iterate::AlgIteratorFactory;
use alg_tools::euclidean::Euclidean;
use alg_tools::linops::{Mapping, GEMV};
use alg_tools::mapping::RealMapping;
use alg_tools::nalgebra_support::ToNalgebraRealField;
use alg_tools::instance::Instance;

use crate::types::*;
use crate::measures::{
    DiscreteMeasure,
    RNDM,
};
use crate::measures::merging::{
    SpikeMergingMethod,
    SpikeMerging,
};
use crate::forward_model::{
    ForwardModel,
    AdjointProductBoundedBy,
};
use crate::plot::{
    SeqPlotter,
    Plotting,
    PlotLookup
};
use crate::regularisation::RegTerm;
use crate::dataterm::{
    calculate_residual,
    L2Squared,
    DataTerm,
};
pub use crate::prox_penalty::{
    FBGenericConfig,
    ProxPenalty
};

/// Settings for [`pointsource_fb_reg`].
#[derive(Clone, Copy, Eq, PartialEq, Serialize, Deserialize, Debug)]
#[serde(default)]
pub struct FBConfig<F : Float> {
    /// Step length scaling
    pub τ0 : F,
    /// Generic parameters
    pub generic : FBGenericConfig<F>,
}

#[replace_float_literals(F::cast_from(literal))]
impl<F : Float> Default for FBConfig<F> {
    fn default() -> Self {
        FBConfig {
            τ0 : 0.99,
            generic : Default::default(),
        }
    }
}

pub(crate) fn prune_with_stats<F : Float, const N : usize>(
    μ : &mut RNDM<F, N>,
) -> usize {
    let n_before_prune = μ.len();
    μ.prune();
    debug_assert!(μ.len() <= n_before_prune);
    n_before_prune - μ.len()
}

#[replace_float_literals(F::cast_from(literal))]
pub(crate) fn postprocess<
    F : Float,
    V : Euclidean<F> + Clone,
    A : GEMV<F, RNDM<F, N>, Codomain = V>,
    D : DataTerm<F, V, N>,
    const N : usize
> (
    mut μ : RNDM<F, N>,
    config : &FBGenericConfig<F>,
    dataterm : D,
    opA : &A,
    b : &V,
) -> RNDM<F, N>
where
    RNDM<F, N> : SpikeMerging<F>,
    for<'a> &'a RNDM<F, N> : Instance<RNDM<F, N>>,
{
    μ.merge_spikes_fitness(config.merging,
                           |μ̃| dataterm.calculate_fit_op(μ̃, opA, b),
                           |&v| v);
    μ.prune();
    μ
}

/// Iteratively solve the pointsource localisation problem using forward-backward splitting.
///
/// The settings in `config` have their [respective documentation](FBConfig). `opA` is the
/// forward operator $A$, $b$ the observable, and $\lambda$ the regularisation weight.
/// The operator `op𝒟` is used for forming the proximal term. Typically it is a convolution
/// operator. Finally, the `iterator` is an outer loop verbosity and iteration count control
/// as documented in [`alg_tools::iterate`].
///
/// For details on the mathematical formulation, see the [module level](self) documentation.
///
/// The implementation relies on [`alg_tools::bisection_tree::BTFN`] presentations of
/// sums of simple functions usign bisection trees, and the related
/// [`alg_tools::bisection_tree::Aggregator`]s, to efficiently search for component functions
/// active at a specific points, and to maximise their sums. Through the implementation of the
/// [`alg_tools::bisection_tree::BT`] bisection trees, it also relies on the copy-on-write features
/// of [`std::sync::Arc`] to only update relevant parts of the bisection tree when adding functions.
///
/// Returns the final iterate.
#[replace_float_literals(F::cast_from(literal))]
pub fn pointsource_fb_reg<
    F, I, A, Reg, P, const N : usize
>(
    opA : &A,
    b : &A::Observable,
    reg : Reg,
    prox_penalty : &P,
    fbconfig : &FBConfig<F>,
    iterator : I,
    mut plotter : SeqPlotter<F, N>,
) -> RNDM<F, N>
where
    F : Float + ToNalgebraRealField,
    I : AlgIteratorFactory<IterInfo<F, N>>,
    for<'b> &'b A::Observable : std::ops::Neg<Output=A::Observable>,
    A : ForwardModel<RNDM<F, N>, F>
        + AdjointProductBoundedBy<RNDM<F, N>, P, FloatType=F>,
    A::PreadjointCodomain : RealMapping<F, N>,
    PlotLookup : Plotting<N>,
    RNDM<F, N> : SpikeMerging<F>,
    Reg : RegTerm<F, N>,
    P : ProxPenalty<F, A::PreadjointCodomain, Reg, N>,
{

    // Set up parameters
    let config = &fbconfig.generic;
    let τ = fbconfig.τ0/opA.adjoint_product_bound(prox_penalty).unwrap();
    // 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.tolerance * τ * reg.tolerance_scaling();
    let mut ε = tolerance.initial();

    // Initialise iterates
    let mut μ = DiscreteMeasure::new();
    let mut residual = -b;

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

    // Run the algorithm
    for state in iterator.iter_init(|| full_stats(&residual, &μ, ε, stats.clone())) {
        // Calculate smooth part of surrogate model.
        let mut τv = opA.preadjoint().apply(residual * τ);

        // Save current base point
        let μ_base = μ.clone();
            
        // Insert and reweigh
        let (maybe_d, _within_tolerances) = prox_penalty.insert_and_reweigh(
            &mut μ, &mut τv, &μ_base, None,
            τ, ε,
            config, &reg, &state, &mut stats
        );

        // Prune and possibly merge spikes
        if config.merge_now(&state) {
            stats.merged += prox_penalty.merge_spikes(&mut μ, &mut τv, &μ_base, τ, ε, config, &reg);
        }

        stats.pruned += prune_with_stats(&mut μ);

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

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

        // Give statistics if needed
        state.if_verbose(|| {
            plotter.plot_spikes(iter, maybe_d.as_ref(), Some(&τv), &μ);
            full_stats(&residual, &μ, ε, std::mem::replace(&mut stats, IterInfo::new()))
        });
        
        // Update main tolerance for next iteration
        ε = tolerance.update(ε, iter);
    }

    postprocess(μ, config, L2Squared, opA, b)
}

/// Iteratively solve the pointsource localisation problem using inertial forward-backward splitting.
///
/// The settings in `config` have their [respective documentation](FBConfig). `opA` is the
/// forward operator $A$, $b$ the observable, and $\lambda$ the regularisation weight.
/// The operator `op𝒟` is used for forming the proximal term. Typically it is a convolution
/// operator. Finally, the `iterator` is an outer loop verbosity and iteration count control
/// as documented in [`alg_tools::iterate`].
///
/// For details on the mathematical formulation, see the [module level](self) documentation.
///
/// The implementation relies on [`alg_tools::bisection_tree::BTFN`] presentations of
/// sums of simple functions usign bisection trees, and the related
/// [`alg_tools::bisection_tree::Aggregator`]s, to efficiently search for component functions
/// active at a specific points, and to maximise their sums. Through the implementation of the
/// [`alg_tools::bisection_tree::BT`] bisection trees, it also relies on the copy-on-write features
/// of [`std::sync::Arc`] to only update relevant parts of the bisection tree when adding functions.
///
/// Returns the final iterate.
#[replace_float_literals(F::cast_from(literal))]
pub fn pointsource_fista_reg<
    F, I, A, Reg, P, const N : usize
>(
    opA : &A,
    b : &A::Observable,
    reg : Reg,
    prox_penalty : &P,
    fbconfig : &FBConfig<F>,
    iterator : I,
    mut plotter : SeqPlotter<F, N>,
) -> RNDM<F, N>
where
    F : Float + ToNalgebraRealField,
    I : AlgIteratorFactory<IterInfo<F, N>>,
    for<'b> &'b A::Observable : std::ops::Neg<Output=A::Observable>,
    A : ForwardModel<RNDM<F, N>, F>
        + AdjointProductBoundedBy<RNDM<F, N>, P, FloatType=F>,
    A::PreadjointCodomain : RealMapping<F, N>,
    PlotLookup : Plotting<N>,
    RNDM<F, N> : SpikeMerging<F>,
    Reg : RegTerm<F, N>,
    P : ProxPenalty<F, A::PreadjointCodomain, Reg, N>,
{

    // Set up parameters
    let config = &fbconfig.generic;
    let τ = fbconfig.τ0/opA.adjoint_product_bound(prox_penalty).unwrap();
    let mut λ = 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.tolerance * τ * reg.tolerance_scaling();
    let mut ε = tolerance.initial();

    // Initialise iterates
    let mut μ = DiscreteMeasure::new();
    let mut μ_prev = DiscreteMeasure::new();
    let mut residual = -b;
    let mut warned_merging = false;

    // Statistics
    let full_stats = |ν : &RNDM<F, N>, ε, stats| IterInfo {
        value : L2Squared.calculate_fit_op(ν, opA, b) + reg.apply(ν),
        n_spikes : ν.len(),
        ε,
        // postprocessing: config.postprocessing.then(|| ν.clone()),
        .. stats
    };
    let mut stats = IterInfo::new();

    // Run the algorithm
    for state in iterator.iter_init(|| full_stats(&μ, ε, stats.clone())) {
        // Calculate smooth part of surrogate model.
        let mut τv = opA.preadjoint().apply(residual * τ);

        // Save current base point
        let μ_base = μ.clone();
            
        // Insert new spikes and reweigh
        let (maybe_d, _within_tolerances) = prox_penalty.insert_and_reweigh(
            &mut μ, &mut τv, &μ_base, None,
            τ, ε,
            config, &reg, &state, &mut stats
        );

        // (Do not) merge spikes.
        if config.merge_now(&state) {
            match config.merging {
                SpikeMergingMethod::None => { },
                _ => if !warned_merging {
                    let err = format!("Merging not supported for μFISTA");
                    println!("{}", err.red());
                    warned_merging = true;
                }
            }
        }

        // Update inertial prameters
        let λ_prev = λ;
        λ = 2.0 * λ_prev / ( λ_prev + (4.0 + λ_prev * λ_prev).sqrt() );
        let θ = λ / λ_prev - λ;

        // Perform inertial update on μ.
        // This computes μ ← (1 + θ) * μ - θ * μ_prev, pruning spikes where both μ
        // and μ_prev have zero weight. Since both have weights from the finite-dimensional
        // subproblem with a proximal projection step, this is likely to happen when the
        // spike is not needed. A copy of the pruned μ without artithmetic performed is
        // stored in μ_prev.
        let n_before_prune = μ.len();
        μ.pruning_sub(1.0 + θ, θ, &mut μ_prev);
        debug_assert!(μ.len() <= n_before_prune);
        stats.pruned += n_before_prune - μ.len();

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

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

        // Give statistics if needed
        state.if_verbose(|| {
            plotter.plot_spikes(iter, maybe_d.as_ref(), Some(&τv), &μ_prev);
            full_stats(&μ_prev, ε, std::mem::replace(&mut stats, IterInfo::new()))
        });

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

    postprocess(μ_prev, config, L2Squared, opA, b)
}