src/linear_selftrain.py

"""
Core library for reproducing the linear-regression theory of
"Why Self-Training Helps and Hurts: Denoising vs. Signal Forgetting" (arXiv 2602.14029).

Implements:
  * Algorithm 1 (iterative self-training / self-distillation, ridgeless & ridge)
  * Theorem 3.2  single-spike deterministic risk recursion  (B*_t, V*_t)
  * Theorem 3.6  multi-spike deterministic risk recursion
  * iGCV estimator (eq. 11-12)  and the survival / suppression factors

Notation matches the paper:
  y = x'beta + eps,   x ~ N(0, Sigma),   eps ~ N(0, sigma^2)   [noise only at t=0]
  rho = p / n,   tau = rho - 1   (ridgeless effective regularization)
  Prediction risk  R(bhat) = (bhat - beta)' Sigma (bhat - beta)
"""
import numpy as np


# ---------------------------------------------------------------------------
# Estimators & Algorithm 1
# ---------------------------------------------------------------------------
def ridgeless_fit(X, Y):
    """Minimum-norm least squares  (X'X)^+ X'Y.

    In the overparameterized regime (n < p, which is our entire setting) the
    min-norm interpolator has the closed form  X'(XX')^-1 Y, requiring only an
    n x n solve -- far faster than SVD-based lstsq.  Falls back to lstsq if the
    Gram matrix is singular or n >= p.
    """
    n, p = X.shape
    if n < p:
        G = X @ X.T                      # n x n
        try:
            z = np.linalg.solve(G, Y)
            return X.T @ z
        except np.linalg.LinAlgError:
            pass
    beta, *_ = np.linalg.lstsq(X, Y, rcond=None)
    return beta


def project_rowspace(X, V):
    """Apply P = X^+ X (orthogonal projection onto row space of X) to columns of V.
    P V = X'(XX')^-1 X V  when n < p.  V may be 1-D or 2-D."""
    n, p = X.shape
    if n < p:
        G = X @ X.T
        return X.T @ np.linalg.solve(G, X @ V)
    # underparameterized: P = I
    return V


def ridge_fit(X, Y, lam):
    """Ridge:  (X'X + n*lam I)^-1 X'Y."""
    n, p = X.shape
    if lam == 0.0:
        return ridgeless_fit(X, Y)
    A = X.T @ X + n * lam * np.eye(p)
    return np.linalg.solve(A, X.T @ Y)


class SpikedCov:
    """Structured spiked covariance  Sigma = sum_j (s_j-1) u_j u_j' + I.
    Provides O(n p) sampling and O(p) quadratic forms (no dense p x p matmul).
    U columns are the orthonormal spike directions; `spikes` their strengths."""
    def __init__(self, p, spikes, U):
        self.p = p
        self.spikes = np.asarray(spikes, float)
        self.U = U                      # (p, k)
        self.a = np.sqrt(self.spikes) - 1.0   # sqrt-eigval offset

    def sample(self, m, rng):
        Z = rng.standard_normal((m, self.p))
        return Z + (Z @ self.U) * self.a @ self.U.T   # X ~ N(0, Sigma)

    def quad(self, d):                  # d' Sigma d
        Ud = self.U.T @ d
        return float(d @ d + ((self.spikes - 1.0) * Ud) @ Ud)


class DiagCov:
    """Diagonal covariance Sigma = diag(v). O(n p) sampling."""
    def __init__(self, v):
        self.v = np.asarray(v, float)
        self.sq = np.sqrt(self.v)
        self.p = len(self.v)

    def sample(self, m, rng):
        return rng.standard_normal((m, self.p)) * self.sq

    def quad(self, d):
        return float((d * self.v) @ d)


def iterative_self_train(Sigma_sqrt, beta, n, sigma, T, lam=0.0, rng=None,
                         return_betas=False, cov=None):
    """
    Run Algorithm 1 for T iterations (t = 0 .. T).

    t=0 : fit on noisy data (Y0 = X0 beta + eps).
    t>=1: fresh X_t, noiseless pseudo-labels Y_t = X_t bhat_{t-1}, refit.

    Sigma_sqrt : (p,p) symmetric square-root of the feature covariance Sigma.
    Returns array of prediction risks R_t, shape (T+1,).  Optionally the betas.
    """
    if rng is None:
        rng = np.random.default_rng()
    p = beta.shape[0]

    if cov is not None:
        gen_X = lambda m: cov.sample(m, rng)
        def risk(bhat):
            return cov.quad(bhat - beta)
    else:
        Sigma = Sigma_sqrt @ Sigma_sqrt  # Sigma_sqrt is symmetric
        gen_X = lambda m: rng.standard_normal((m, p)) @ Sigma_sqrt
        def risk(bhat):
            d = bhat - beta
            return float(d @ (Sigma @ d))

    # t = 0 : noisy fit
    X0 = gen_X(n)
    eps = sigma * rng.standard_normal(n)
    Y0 = X0 @ beta + eps
    bhat = ridge_fit(X0, Y0, lam) if lam > 0 else ridgeless_fit(X0, Y0)

    risks = [risk(bhat)]
    betas = [bhat.copy()]

    for t in range(1, T + 1):
        Xt = gen_X(n)
        Yt = Xt @ bhat                       # noiseless pseudo-labels
        bhat = ridge_fit(Xt, Yt, lam) if lam > 0 else ridgeless_fit(Xt, Yt)
        risks.append(risk(bhat))
        betas.append(bhat.copy())

    risks = np.array(risks)
    return (risks, betas) if return_betas else risks


def simulate_risk(Sigma_sqrt, beta, n, sigma, T, lam=0.0, trials=10, seed=0, cov=None):
    """Monte-Carlo prediction risk R_t averaged over `trials`."""
    rng = np.random.default_rng(seed)
    acc = np.zeros(T + 1)
    sq = np.zeros(T + 1)
    for _ in range(trials):
        r = iterative_self_train(Sigma_sqrt, beta, n, sigma, T, lam=lam, rng=rng, cov=cov)
        acc += r
        sq += r ** 2
    mean = acc / trials
    std = np.sqrt(np.maximum(sq / trials - mean ** 2, 0.0))
    return mean, std


# ---------------------------------------------------------------------------
# Theorem 3.2  --  single-spike deterministic recursion
# ---------------------------------------------------------------------------
def spiked_theory(s, rho, r2, sigma2, T):
    """
    Deterministic risk R*_t = B*_t + V*_t for the single-spike model (Thm 3.2).

    Sigma = (s-1) u1 u1' + I,  beta = r u1  with r^2 = r2.
    tau = rho - 1.
    Returns dict with arrays B, V, R (length T+1) and 'survival' factor.
    """
    tau = rho - 1.0
    kappa = s / (s + tau)                       # contraction / survival factor
    survival = kappa ** (np.arange(T + 1) + 1)  # (s/(s+tau))^{t+1}

    B = r2 * s * (1.0 - survival) ** 2          # eq (4)

    V = np.zeros(T + 1)
    V[0] = sigma2 / tau + (tau * s / (s + tau) ** 2) * r2
    for t in range(1, T + 1):
        # eq (5):  V_t = V_{t-1}/(1+tau) + tau r^2 s^{2t+1}/(s+tau)^{2(t+1)}
        V[t] = V[t - 1] / (1.0 + tau) + tau * r2 * s ** (2 * t + 1) / (s + tau) ** (2 * (t + 1))

    return {"B": B, "V": V, "R": B + V, "tau": tau, "kappa": kappa,
            "survival": survival}


def multi_spike_theory(spikes, r2s, rho, sigma2, T):
    """
    Multi-spike deterministic recursion (Thm 3.6).
    spikes : list of spike strengths s_1..s_k (each > 1)
    r2s    : list of signal powers r_j^2 along each spike direction.
    Returns dict with B, V, R arrays and per-direction survival factors.
    """
    spikes = np.asarray(spikes, float)
    r2s = np.asarray(r2s, float)
    tau = rho - 1.0
    tt = np.arange(T + 1)

    # eq (6): B_t = sum_j r_j^2 s_j (1 - (s_j/(s_j+tau))^{t+1})^2
    B = np.zeros(T + 1)
    survivals = {}
    for j, (s, rj2) in enumerate(zip(spikes, r2s)):
        surv = (s / (s + tau)) ** (tt + 1)
        survivals[j] = surv
        B += rj2 * s * (1.0 - surv) ** 2

    # eq (7): V recursion
    V = np.zeros(T + 1)
    V[0] = sigma2 / tau + np.sum(tau * r2s * spikes / (spikes + tau) ** 2)
    for t in range(1, T + 1):
        inject = np.sum(tau * r2s * spikes ** (2 * t + 1) / (spikes + tau) ** (2 * (t + 1)))
        V[t] = V[t - 1] / (1.0 + tau) + inject

    return {"B": B, "V": V, "R": B + V, "tau": tau, "survivals": survivals,
            "kappas": spikes / (spikes + tau)}


def general_diag_theory(eigs, beta, rho, sigma2, T, lam=0.0):
    """
    General deterministic-equivalent recursion (Section 4, Thm 4.2 / eq 10) for a
    diagonal feature covariance Sigma = diag(eigs), identical across iterations,
    with aspect ratio rho = p/n_t fixed (so tau_t = tau constant).  Returns the
    deterministic prediction risk R*_t decomposed into systematic + stochastic.

    Because Sigma is diagonal and constant, Q_t = Q = diag(q_i), q_i=lam_i/(lam_i+tau),
    and every trace reduces to a 1-D sum over eigenvalues.
    """
    eigs = np.asarray(eigs, float)
    beta = np.asarray(beta, float)
    p = len(eigs)

    # solve fixed point (8):  1/rho = (1/p) sum_i lam_i/(lam_i+tau) + lam/tau
    def fp(tau):
        return (np.mean(eigs / (eigs + tau)) + lam / tau) - 1.0 / rho
    lo, hi = 1e-8, 1e8
    for _ in range(200):
        mid = np.sqrt(lo * hi)
        if fp(mid) > 0:   # decreasing in tau
            lo = mid
        else:
            hi = mid
    tau = np.sqrt(lo * hi)

    q = eigs / (eigs + tau)                       # Q diagonal
    L = lam / tau + (tau / p) * np.sum(eigs / (eigs + tau) ** 2)

    # deterministic effective noise D^2_t
    D = np.zeros(T + 1)
    D[0] = (sigma2 + tau ** 2 * np.sum(beta ** 2 * eigs / (eigs + tau) ** 2)) / L
    for t in range(1, T + 1):
        term1 = tau ** 2 * np.sum(beta ** 2 * q ** (2 * t + 1) / (eigs + tau))
        term2 = 0.0
        for h in range(t):
            trace = np.sum(q ** (2 * (t - h)) / (eigs + tau) ** 2)
            term2 += (D[h] / p) * trace
        D[t] = (term1 + tau ** 2 * term2) / L

    # deterministic risk R*_t (eq 10)
    B = np.zeros(T + 1)   # systematic
    V = np.zeros(T + 1)   # stochastic
    for t in range(T + 1):
        B[t] = np.sum(eigs * (q ** (t + 1) - 1.0) ** 2 * beta ** 2)
        acc = 0.0
        for h in range(t + 1):
            trace = np.sum(q ** (2 * (t - h)) * eigs ** 2 / (eigs + tau) ** 2)
            acc += (D[h] / p) * trace
        V[t] = acc
    return {"B": B, "V": V, "R": B + V, "tau": tau, "D": D}


def build_spiked_covariance(p, spikes, dirs=None):
    """
    Sigma = sum_j (s_j - 1) u_j u_j' + I_p.  Returns (Sigma_sqrt, U) with U the
    spike eigenvectors (columns).  dirs: optional (p,k) orthonormal directions;
    default = first k canonical basis vectors.
    """
    spikes = np.asarray(spikes, float)
    k = len(spikes)
    if dirs is None:
        U = np.zeros((p, k))
        for j in range(k):
            U[j, j] = 1.0
    else:
        U = dirs
    # eigen-decomposition is trivial: Sigma_sqrt = I + sum_j (sqrt(s_j)-1) u_j u_j'
    Sigma_sqrt = np.eye(p)
    for j in range(k):
        uj = U[:, j]
        Sigma_sqrt += (np.sqrt(spikes[j]) - 1.0) * np.outer(uj, uj)
    return Sigma_sqrt, U


# ---------------------------------------------------------------------------
# iGCV estimator (Section 4.2, eq. 11-12)
# ---------------------------------------------------------------------------
def igcv_trajectory(Sigma_sqrt, beta, n, sigma, T, lam=0.0, rng=None, cov=None):
    """
    One trial: returns (true_risk[t], igcv_est[t]) for t=0..T.

    iGCV (eq 12) estimates R(bhat_t) + sigma^2 using ONLY the initial noisy
    dataset D0 and the cumulative projection A_t = P_t...P_1.
    We report igcv_est - sigma^2 as the estimate of R(bhat_t).
    """
    if rng is None:
        rng = np.random.default_rng()
    p = beta.shape[0]
    if cov is not None:
        gen_X = lambda m: cov.sample(m, rng)
        risk = lambda bhat: cov.quad(bhat - beta)
    else:
        Sigma = Sigma_sqrt @ Sigma_sqrt
        gen_X = lambda m: rng.standard_normal((m, p)) @ Sigma_sqrt
        def risk(bhat):
            d = bhat - beta
            return float(d @ (Sigma @ d))

    # Initial ridge fit on the noisy data D0. lam>0 keeps the GCV correction
    # well-conditioned (the interpolating ridgeless fit has zero residuals, so
    # the leave-one-out correction 1 - tr(H)/n0 degenerates; the paper uses the
    # ridge / pseudoinverse-continuity profile, Hastie et al. 2022; Patil 2021).
    # The self-training iterations t>=1 are always ridgeless row-space projections.
    lam0 = lam if lam > 0 else 1e-3
    X0 = gen_X(n)
    eps = sigma * rng.standard_normal(n)
    Y0 = X0 @ beta + eps
    bhat0 = ridge_fit(X0, Y0, lam0)

    # smoother  H = X0 (X0'X0/n + lam0 I)^-1 X0' / n0  (n0 x n0);  X0 @ C = H
    G = X0.T @ X0 / n
    Ginv = np.linalg.inv(G + lam0 * np.eye(p))
    C = Ginv @ X0.T / n                       # p x n0
    H = X0 @ C                                # n0 x n0
    denom = 1.0 - np.trace(H) / n
    resid0 = Y0 - X0 @ bhat0                  # y_i - x_i' bhat0  (nonzero for lam0>0)

    # Trajectory:  bhat_t = P_t ... P_1 bhat0 = A_t bhat0.
    A_bhat0 = bhat0.copy()
    A_C = C.copy()
    true_risk, igcv = [], []
    for t in range(0, T + 1):
        if t >= 1:
            Xt = gen_X(n)
            A_C = project_rowspace(Xt, A_C)          # P_t A_{t-1} C
            A_bhat0 = project_rowspace(Xt, A_bhat0)  # P_t A_{t-1} bhat0
        true_risk.append(risk(A_bhat0))
        # leverage multiplier M_t (eq 11) and corrected residual iGCV (eq 12)
        Mt = (np.trace(X0 @ A_C) / n) / denom
        corr = (Y0 - X0 @ A_bhat0) + resid0 * Mt
        igcv.append(float(np.mean(corr ** 2)) - sigma ** 2)

    return np.array(true_risk), np.array(igcv)