gam 0.3.65

//! Penalized least-squares solver and Gaussian fast paths.
//!
//! Owns:
//! - `GaussianFixedCache` — `XᵀWX`/`XᵀW(y−offset)` cache for the
//!   Gaussian-Identity short-circuit that the REML outer loop reuses across
//!   smoothing-parameter candidates.
//! - `SparseXtwxPrecomputed` — the sparse-pattern-aligned twin of the above
//!   for designs that take the sparse-native PIRLS path.
//! - `solve_penalized_least_squares_implicit` — identity/Gaussian implicit
//!   PLS, dense and sparse-native paths.

use super::loop_driver::max_symmetric_asymmetry;
use super::{
    FIXED_STABILIZATION_RIDGE, PirlsPenalty, PirlsWorkspace, SparseXtWxCache, StablePLSResult,
    WorkingReparamTransform, calculate_edf_from_sparse_factor,
    calculate_edfwithworkspace_from_factor, ensure_sparse_positive_definitewithridge,
    solve_sparse_spd,
};
use crate::estimate::EstimationError;
use crate::faer_ndarray::{FaerLinalgError, array1_to_col_matmut};
use crate::linalg::utils::{StableSolver, array_is_finite};
use crate::matrix::{DesignMatrix, LinearOperator, SymmetricMatrix};
use crate::types::{Coefficients, LinkFunction};
use faer::sparse::SparseColMat;
use ndarray::{Array1, Array2, ArrayView1, ShapeBuilder};
use std::sync::Arc;

/// Reusable `XᵀWX` and `XᵀW(y − offset)` for Gaussian + Identity REML fits.
///
/// The Gaussian-identity P-IRLS short-circuit solves a single linear system
/// `(XᵀWX + Σ λ_k S_k + ρ·I) β = XᵀW(y − offset)`. The right-hand-side matrix
/// and vector are independent of the smoothing parameters `λ`, so when the
/// outer REML loop evaluates the same problem at many `(λ_1, …, λ_k)`
/// candidates we only need to assemble them **once** before the loop and
/// reuse them inside every inner PIRLS call.
///
/// Stored in *original* coordinates (no Qs rotation applied). When the
/// inner solver uses a `WorkingReparamTransform`, it conjugates / projects
/// these matrices on the fly — that step is O(p³) / O(p²), independent of N.
#[derive(Debug)]
pub struct GaussianFixedCache {
    /// `XᵀWX` in the original coefficient basis. Symmetric, p × p.
    pub xtwx_orig: Array2<f64>,
    /// `XᵀW(y − offset)` in the original basis. Length p.
    pub xtwy_orig: Array1<f64>,
    /// `(y − offset)ᵀW(y − offset)`.
    ///
    /// Together with `xtwx_orig` and `xtwy_orig`, this is the last scalar
    /// sufficient statistic needed to evaluate the Gaussian penalized RSS
    /// exactly at any λ without re-streaming the rows.
    pub centered_weighted_y_sq: f64,
    /// `XᵀWX` precomputed for the sparse path, aligned with the symbolic
    /// pattern of `SparseXtWxCache::new(x)` on the original sparse design.
    /// `None` when the design has no sparse form (e.g. dense-only fits).
    ///
    /// The sparse REML path rebuilds `H = XᵀWX + Sλ + δI` per outer
    /// evaluation. For Gaussian-Identity the weights never change, so the
    /// `XᵀWX` contribution is invariant across the outer loop and can be
    /// scattered from this cached values vector instead of re-doing the
    /// O(nnz²/n) SpGEMM each call.
    pub xtwx_sparse_orig: Option<Arc<SparseXtwxPrecomputed>>,
}

/// Precomputed numerical values of `XᵀWX` aligned with the symbolic pattern
/// that `SparseXtWxCache::new(x)` produces on its first call. Two such caches
/// built from the same sparse `x` produce byte-identical symbolic patterns
/// (faer's `sparse_sparse_matmul_symbolic` is deterministic), so the cached
/// values can be installed back into a fresh `SparseXtWxCache` for the same
/// `x` without rerunning the SpGEMM.
///
/// We snapshot the symbolic pattern (`col_ptr` / `row_idx`) alongside the
/// values so the consumer can verify pattern equivalence and fall through to
/// the per-call recomputation if anything diverges (e.g. an `x` with a
/// different symbolic shape sneaks in).
#[derive(Debug, Clone)]
pub struct SparseXtwxPrecomputed {
    pub xtwx_symbolic_col_ptr: Vec<usize>,
    pub xtwx_symbolic_row_idx: Vec<usize>,
    pub xtwxvalues: Vec<f64>,
}

impl SparseXtwxPrecomputed {
    /// Build the precomputed `XᵀWX` value layout for `x` at the given
    /// `weights`. The output reuses the same construction path the inner
    /// PIRLS workspace uses, so it lands in exactly the symbolic pattern
    /// the consumer expects.
    pub fn build(
        x: &SparseColMat<usize, f64>,
        weights: &Array1<f64>,
    ) -> Result<Self, EstimationError> {
        let mut cache = SparseXtWxCache::new(x)?;
        cache.compute_numeric(x, weights)?;
        Ok(Self {
            xtwx_symbolic_col_ptr: cache.xtwx_symbolic.col_ptr().to_vec(),
            xtwx_symbolic_row_idx: cache.xtwx_symbolic.row_idx().to_vec(),
            xtwxvalues: cache.xtwxvalues,
        })
    }
}

/// Identity-link solver that operates in original or QS-transformed coordinates
/// without materializing X·Qs.  When the design is sparse and `qs` is `None`
/// (sparse-native path), uses sparse Cholesky for O(nnz^{1.5}) cost instead
/// of the O(p³) dense Cholesky.
pub(super) fn solve_penalized_least_squares_implicit(
    x_original: &DesignMatrix,
    transform: Option<&WorkingReparamTransform>,
    z: ArrayView1<f64>,
    weights: ArrayView1<f64>,
    offset: ArrayView1<f64>,
    penalty: &PirlsPenalty,
    workspace: &mut PirlsWorkspace,
    y: ArrayView1<f64>,
    link_function: LinkFunction,
    gaussian_fixed_cache: Option<&GaussianFixedCache>,
) -> Result<(StablePLSResult, usize), EstimationError> {
    let p_dim = penalty.dim();

    // ── Sparse-native fast path ──────────────────────────────────────────
    // When design is sparse and we are in original coordinates (qs = None),
    // assemble the penalized Hessian in sparse format and solve with sparse
    // Cholesky.  This avoids O(p²) dense X'WX and O(p³) dense factorization.
    if transform.is_none()
        && let Some(x_sparse) = x_original.as_sparse()
    {
        let PirlsPenalty::Dense { s_transformed, .. } = penalty else {
            crate::bail_invalid_estim!(
                "sparse-native PIRLS requires a dense transformed penalty matrix"
            );
        };
        let weights_owned = weights.to_owned();

        // Gaussian-Identity fast path: the inner sparse `XᵀWX` is invariant
        // across the outer REML loop because the IRLS weights are constant
        // (W = priorweights). The cached values land in the inner workspace
        // and bypass the per-eval SpGEMM.
        let precomputed_xtwx =
            gaussian_fixed_cache.and_then(|c| c.xtwx_sparse_orig.as_ref().map(|arc| arc.as_ref()));

        // 1. Sparse penalized Hessian: H = X'diag(w)X + S_λ + ridge·I.
        //    The Cholesky factor is reused from the SPD check so we avoid
        //    factorizing the same matrix twice.
        let (h_sparse, factor, ridge_used) = ensure_sparse_positive_definitewithridge(|ridge| {
            let ridge = if ridge == 0.0 {
                FIXED_STABILIZATION_RIDGE
            } else {
                ridge
            };
            workspace.assemble_sparse_penalized_hessian(
                x_sparse,
                &weights_owned,
                s_transformed,
                ridge,
                precomputed_xtwx,
            )
        })?;

        // 2. RHS = X'W(z - offset) + S_λ μ + ridge_used · μ.
        // The `ridge_used · μ` term matches the diagonal ridge added to
        // the Hessian in step 1, keeping the augmented system a
        // Tikhonov regularization centered at the prior mean target
        // rather than at zero (see `prior_mean_target` field docs).
        let mut wz = z.to_owned();
        wz -= &offset;
        wz *= &weights_owned;
        let mut rhs = x_original.transpose_vector_multiply(&wz);
        rhs += penalty.linear_shift();
        if ridge_used > 0.0 {
            let prior_mean_target = penalty.prior_mean_target();
            if prior_mean_target.len() == rhs.len() {
                rhs.scaled_add(ridge_used, prior_mean_target);
            }
        }

        // 3. Sparse Cholesky solve (factor reused from step 1)
        let betavec = solve_sparse_spd(&factor, &rhs)?;

        // 4. EDF — reuse the sparse Cholesky factor from step 1 to avoid a
        // second O(nnz·…) factorization of the identical penalized Hessian.
        let h_sym = SymmetricMatrix::Sparse(h_sparse);
        let edf = calculate_edf_from_sparse_factor(&factor, penalty)?;

        // 5. Fitted values and scale
        let fitted_vals = {
            let xb = x_original.apply(&betavec);
            let mut f = xb;
            f += &offset;
            f
        };
        let standard_deviation = match link_function {
            LinkFunction::Identity => {
                let residuals = &y - &fitted_vals;
                let weighted_rss: f64 = weights
                    .iter()
                    .zip(residuals.iter())
                    .map(|(&w, &r)| w * r * r)
                    .sum();
                let effective_n = y.len() as f64;
                (weighted_rss / (effective_n - edf).max(1.0)).sqrt()
            }
            _ => 1.0,
        };

        return Ok((
            StablePLSResult {
                beta: Coefficients::new(betavec),
                penalized_hessian: h_sym,
                edf,
                standard_deviation,
                ridge_used,
            },
            p_dim,
        ));
    }

    // ── Dense / QS-rotated path ──────────────────────────────────────────

    // 1. Prepare weighted buffers
    if workspace.wz.len() != z.len() {
        workspace.wz = Array1::zeros(z.len());
    }
    workspace.wz.assign(&z);
    workspace.wz -= &offset;
    workspace.wz *= &weights;

    // 2. Form X'WX: compute in original coordinates, then rotate by Qs.
    //
    // Gaussian + Identity REML reuses a precomputed `XᵀWX` (the weights and
    // design never change across the outer loop in that family), so when the
    // caller supplied a `GaussianFixedCache` we skip the O(N·p²) dense
    // assembly here and adopt the cached matrix as-is.
    let weights_owned = weights.to_owned();
    let xtwx_orig = if let Some(cache) = gaussian_fixed_cache {
        // Cache hit: weights and design are invariant for Gaussian-Identity
        // across the outer REML loop, so adopt the precomputed XᵀWX directly
        // and avoid the O(N·p²) dense assembly entirely.
        let p = x_original.ncols();
        if cache.xtwx_orig.nrows() != p || cache.xtwx_orig.ncols() != p {
            return Err(EstimationError::InvalidInput(format!(
                "GaussianFixedCache XᵀWX shape {}×{} does not match design p={}",
                cache.xtwx_orig.nrows(),
                cache.xtwx_orig.ncols(),
                p,
            )));
        }
        cache.xtwx_orig.clone()
    } else {
        match x_original {
            // Only materialized dense designs can use the shared dense assembly path.
            // Lazy operator-backed dense designs route to diag_xtw_x like sparse.
            DesignMatrix::Dense(x_dense) if x_dense.is_materialized_dense() => {
                let p = x_dense.ncols();
                let x_dense = x_dense.to_dense_arc();
                if workspace.hessian_buf.nrows() != p || workspace.hessian_buf.ncols() != p {
                    workspace.hessian_buf = Array2::zeros((p, p).f());
                } else {
                    workspace.hessian_buf.fill(0.0);
                }
                PirlsWorkspace::add_dense_xtwx_signed(
                    &weights_owned,
                    &mut workspace.weighted_x_chunk,
                    x_dense.as_ref(),
                    &mut workspace.hessian_buf,
                );
                std::mem::take(&mut workspace.hessian_buf)
            }
            _ => {
                // Operator-form fallback: sparse designs and lazy operator-backed
                // dense designs cannot be densified, so route through the signed
                // XᵀWX operator.
                crate::matrix::xt_diag_x_signed(
                    x_original,
                    crate::matrix::SignedWeightsView::from_array(&weights_owned),
                )
                .map(|h| h.to_dense())
                .map_err(EstimationError::InvalidInput)?
            }
        }
    };
    let xtwx_orig_asym = max_symmetric_asymmetry(&xtwx_orig);
    let xtwx_transformed = if let Some(transform) = transform {
        transform.conjugate_matrix(&xtwx_orig)
    } else {
        xtwx_orig
    };
    let mut penalized_hessian = xtwx_transformed.clone();
    penalty.add_to_hessian(&mut penalized_hessian);

    // 3. Form X'Wz: compute in original coordinates, then rotate.
    //    With the Gaussian-Identity cache `z = y` and `wz = W·(y − offset)`
    //    is identical across outer iterations, so reuse the precomputed
    //    `XᵀW(y − offset)` directly.
    let xtwy_orig = if let Some(cache) = gaussian_fixed_cache {
        assert_eq!(
            cache.xtwy_orig.len(),
            x_original.ncols(),
            "GaussianFixedCache XᵀW(y−offset) length must match design p"
        );
        cache.xtwy_orig.clone()
    } else {
        x_original.transpose_vector_multiply(&workspace.wz)
    };
    if workspace.vec_buf_p.len() != p_dim {
        workspace.vec_buf_p = Array1::zeros(p_dim);
    }
    if let Some(transform) = transform {
        workspace
            .vec_buf_p
            .assign(&transform.apply_transpose(&xtwy_orig));
    } else {
        workspace.vec_buf_p.assign(&xtwy_orig);
    }
    workspace.vec_buf_p += penalty.linear_shift();

    {
        let xtwx_asym = max_symmetric_asymmetry(&xtwx_transformed);
        let penalty_asym = match penalty {
            PirlsPenalty::Dense { s_transformed, .. } => max_symmetric_asymmetry(s_transformed),
            PirlsPenalty::Diagonal { .. } => 0.0,
        };
        let total_asym = max_symmetric_asymmetry(&penalized_hessian);
        assert!(
            total_asym <= 1e-8,
            "implicit PLS penalized Hessian asymmetry too large: total={total_asym:.3e}, xtwx_orig={xtwx_orig_asym:.3e}, xtwx={xtwx_asym:.3e}, penalty={penalty_asym:.3e}, tol={:.3e}",
            1e-8
        );
    }

    // 4. Ridge stabilization. Augment both sides by the ridge so the
    // stabilization is a Tikhonov regularization centered at the prior
    // mean target: (H + δI) β = r + δ μ. The prior_mean_target is zero
    // when no penalty block carries a non-zero prior mean, so this is a
    // no-op in the common case but recovers `β = μ` exactly on
    // X'WX = 0 / X'Wz = 0 problems where the data carries no information.
    let nugget = FIXED_STABILIZATION_RIDGE;
    let mut regularizedhessian = penalized_hessian.clone();
    if nugget > 0.0 {
        for i in 0..p_dim {
            regularizedhessian[[i, i]] += nugget;
        }
    }
    let ridge_used = nugget;

    // 5. Solve
    if workspace.rhs_full.len() != p_dim {
        workspace.rhs_full = Array1::zeros(p_dim);
    }
    workspace.rhs_full.assign(&workspace.vec_buf_p);
    if nugget > 0.0 {
        let prior_mean_target = penalty.prior_mean_target();
        if prior_mean_target.len() == p_dim {
            workspace.rhs_full.scaled_add(nugget, prior_mean_target);
        }
    }
    let factor = StableSolver::new("pirls implicit pls")
        .factorize(&regularizedhessian)
        .map_err(EstimationError::LinearSystemSolveFailed)?;
    let mut rhsview = array1_to_col_matmut(&mut workspace.rhs_full);
    factor.solve_in_place(rhsview.as_mut());
    if !array_is_finite(&workspace.rhs_full) {
        return Err(EstimationError::LinearSystemSolveFailed(
            FaerLinalgError::FactorizationFailed {
                context: "PIRLS implicit PLS non-finite solve",
            },
        ));
    }
    let betavec = workspace.rhs_full.clone();

    // 6. EDF — reuse the factor already produced in step 5 to avoid a second
    // O(p³) factorization of the identical regularized Hessian.
    let edf = calculate_edfwithworkspace_from_factor(&factor, penalty, workspace)?;

    // 7. Scale (composed: eta = offset + X Qs beta)
    let qbeta = if let Some(transform) = transform {
        transform.apply(&betavec)
    } else {
        betavec.clone()
    };
    let xqbeta = x_original.apply(&qbeta);
    let mut fitted = xqbeta;
    fitted += &offset;
    let standard_deviation = match link_function {
        LinkFunction::Identity => {
            let residuals = &y - &fitted;
            let weighted_rss: f64 = weights
                .iter()
                .zip(residuals.iter())
                .map(|(&w, &r)| w * r * r)
                .sum();
            let effective_n = y.len() as f64;
            (weighted_rss / (effective_n - edf).max(1.0)).sqrt()
        }
        _ => 1.0,
    };

    Ok((
        StablePLSResult {
            beta: Coefficients::new(betavec),
            penalized_hessian: SymmetricMatrix::Dense(penalized_hessian),
            edf,
            standard_deviation,
            ridge_used,
        },
        p_dim,
    ))
}