Introduction#

Construct a data matrix $X$ with correlated features; compute condition number $\kappa(X)$ and $\kappa(X^\top X)$; solve least squares via normal equations vs. QR to demonstrate error amplification.

Purpose#

Illustrate why normal equations fail for ill-conditioned data; show that $\kappa(X^\top X) = \kappa(X)^2$ causes error to explode.

Importance#

Foundational motivation for using QR/SVD in regression; explains why scikit-learn warns on ill-conditioned matrices.

What this example demonstrates#

• Construct $X$ with exponentially decaying singular values (high correlation).

• Compute $\kappa(X)$ and $\kappa(X^\top X) = \kappa(X)^2$.

• Solve via normal equations: $G = X^\top X$, $w = G^{-1} X^\top y$.

• Solve via QR factorization: $w$ from $R w = Q^\top y$.

• Compare residuals and error growth on perturbed data.

Background#

Gram matrix $G = X^\top X$ is square and symmetric but inherits (squared) conditioning from $X$. Gaussian elimination on $G$ amplifies errors by $\kappa(G)^2 = \kappa(X)^4$ in worst case.

Historical context#

Golub & Kahan (1965) proved $\kappa(X^\top X) = \kappa(X)^2$; advocated QR as practical solution.

History#

Recognition of this issue led to LINPACK (1979) and LAPACK (1992) emphasis on QR/SVD.

Prevalence in ML#

Statisticians and practitioners routinely encounter this; feature correlation is common in real data.

Notes#

• Problem is fundamental to linear algebra, not implementation-specific.

• Data standardization (mean 0, std 1) helps but doesn’t eliminate correlation.

• Large-scale regression ($n, d > 10^6$) exacerbates conditioning issues.

Connection to ML#

Ridge regression adds $\lambda I$ to shift eigenvalues; reduces effective condition number. Feature engineering (orthogonalization, PCA preprocessing) also helps.

Connection to Linear Algebra Theory#

$\kappa(X^\top X) = \kappa(X)^2$ is exact; QR avoids Gram matrix entirely.

Pedagogical Significance#

Concrete demonstration of why theory (backward error analysis) matters in practice.

References#

1. Golub, G. H., & Kahan, W. (1965). Calculating the singular values and pseudo-inverse of a matrix.

2. Golub, G. H., & Van Loan, C. F. (2013). Matrix Computations (4th ed.).

3. Higham, N. J. (2002). Accuracy and Stability of Numerical Algorithms (2nd ed.).

Solution (Python)#

import numpy as np
np.random.seed(20)

# Create ill-conditioned data (correlated features)
n, d = 100, 10
# Generate features with exponentially decaying singular values
U, _ = np.linalg.qr(np.random.randn(n, n))
s = np.logspace(0, -2, d)
X = U[:n, :d] @ np.diag(s)

# True solution and target
w_true = np.random.randn(d)
y = X @ w_true + 0.001 * np.random.randn(n)

# Condition numbers
kappa_X = np.linalg.cond(X)
G = X.T @ X
kappa_G = np.linalg.cond(G)

# Solve via normal equations
w_ne = np.linalg.solve(G, X.T @ y)
residual_ne = np.linalg.norm(X @ w_ne - y)

# Solve via QR
Q, R = np.linalg.qr(X, mode='reduced')
w_qr = np.linalg.solve(R, Q.T @ y)
residual_qr = np.linalg.norm(X @ w_qr - y)

# Solve via SVD (most stable)
U_svd, s_svd, Vt = np.linalg.svd(X, full_matrices=False)
w_svd = Vt.T @ (np.linalg.solve(np.diag(s_svd), U_svd.T @ y))
residual_svd = np.linalg.norm(X @ w_svd - y)

print("Condition numbers:")
print("  kappa(X) =", round(kappa_X, 2))
print("  kappa(X^T X) =", round(kappa_G, 2))
print("  kappa(X^T X) / kappa(X)^2 =", round(kappa_G / kappa_X**2, 4), "(should be ~1)")
print("\nResiduals:")
print("  Normal equations:", round(residual_ne, 8))
print("  QR:", round(residual_qr, 8))
print("  SVD:", round(residual_svd, 8))
print("\nSolution errors:")
print("  NE error:", round(np.linalg.norm(w_ne - w_true), 4))
print("  QR error:", round(np.linalg.norm(w_qr - w_true), 4))
print("  SVD error:", round(np.linalg.norm(w_svd - w_true), 4))

Introduction#

Solve the same least squares problem via QR and normal equations; compute backward and forward errors; verify that QR is backward-stable while normal equations magnify error.

Purpose#

Demonstrate backward error analysis in practice; show that backward-stable algorithm (QR) guarantees small forward error even on ill-conditioned problems.

Importance#

Foundation for trusting numerical algorithms; explains why stable algorithms are worth the computational cost.

What this example demonstrates#

• Solve $Ax = b$ via QR factorization and normal equations.

• Compute backward error: $\frac{\| A \tilde{x} - b \|}{\| b \|}$.

• Compute forward error: $\frac{\| \tilde{x} - x^* \|}{\| x^* \|}$.

• Verify $\text{forward error} \approx \kappa(A) \times \text{backward error}$ for both methods.

Background#

Backward-stable algorithm: computed $\tilde{x}$ is exact solution to $(A + \Delta A) \tilde{x} = b$ with small $\Delta A$.

Historical context#

Wilkinson (1961) formalized backward error analysis; revolutionary impact on numerical linear algebra.

History#

Modern algorithms designed to be backward-stable (QR, SVD, Cholesky); guarantees via theorems in Higham (2002).

Prevalence in ML#

Used in numerical software certification; backward error commonly reported in solvers.

Notes#

• QR backward error: $O(\varepsilon_{\text{mach}} \| A \|)$ (small).

• Normal equations backward error: $O(\varepsilon_{\text{mach}} \| A \|^2 / \min(\sigma_i))$ (amplified).

Connection to ML#

Backward stability ensures that model parameters are not polluted by rounding error; prediction is numerically sound.

Connection to Linear Algebra Theory#

Backward error analysis decouples algorithm stability from problem conditioning.

Pedagogical Significance#

Shows that algorithm design (not just theory) determines practical accuracy.

References#

1. Wilkinson, J. H. (1961). Error analysis of direct methods of matrix inversion.

2. Higham, N. J. (2002). Accuracy and Stability of Numerical Algorithms (2nd ed.).

3. Golub & Van Loan (2013). Matrix Computations.

Solution (Python)#

import numpy as np
np.random.seed(21)

# Ill-conditioned system
n = 50
U, _ = np.linalg.qr(np.random.randn(n, n))
s = np.logspace(0, -3, n)  # kappa ~ 1000
A = U @ np.diag(s) @ U.T
A = (A + A.T) / 2  # Ensure symmetry
x_true = np.random.randn(n)
b = A @ x_true

# Solve via QR
Q, R = np.linalg.qr(A)
x_qr = np.linalg.solve(R, Q.T @ b)

# Solve via normal equations
G = A.T @ A
x_ne = np.linalg.solve(G, A.T @ b)

# Direct solve (reference)
x_ref = np.linalg.solve(A, b)

# Backward errors
residual_qr = np.linalg.norm(A @ x_qr - b)
residual_ne = np.linalg.norm(A @ x_ne - b)
backward_error_qr = residual_qr / np.linalg.norm(b)
backward_error_ne = residual_ne / np.linalg.norm(b)

# Forward errors
forward_error_qr = np.linalg.norm(x_qr - x_ref) / np.linalg.norm(x_ref)
forward_error_ne = np.linalg.norm(x_ne - x_ref) / np.linalg.norm(x_ref)

# Condition number
kappa = np.linalg.cond(A)

# Verify: forward_error ≈ kappa × backward_error
pred_forward_qr = kappa * backward_error_qr
pred_forward_ne = kappa * backward_error_ne

print("Condition number kappa(A):", round(kappa, 0))
print("\nQR Factorization:")
print("  Backward error:", format(backward_error_qr, '.2e'))
print("  Observed forward error:", format(forward_error_qr, '.2e'))
print("  Predicted forward error:", format(pred_forward_qr, '.2e'))
print("  Ratio (obs/pred):", round(forward_error_qr / pred_forward_qr, 2))
print("\nNormal Equations:")
print("  Backward error:", format(backward_error_ne, '.2e'))
print("  Observed forward error:", format(forward_error_ne, '.2e'))
print("  Predicted forward error:", format(pred_forward_ne, '.2e'))
print("  Ratio (obs/pred):", round(forward_error_ne / pred_forward_ne, 2))
print("\nConclusion: QR is backward-stable; NE backward error is larger.")

Introduction#

Demonstrate catastrophic cancellation when subtracting nearby numbers; show how orthogonal transformations avoid this.

Purpose#

Illustrate fundamental floating-point pitfall; motivate use of orthogonal algorithms (QR, SVD, Householder).

Importance#

Concrete example of why naive implementations fail; builds intuition for numerical stability.

What this example demonstrates#

• Compute $c = a - b$ where $a \approx b$ (many leading digits cancel).

• Compare relative error in $c$ vs. in $a, b$ individually.

• Show that orthogonal transformations (e.g., Householder reflection) preserve norms and avoid subtraction.

Background#

Subtraction of nearly equal numbers is inherently unstable; cannot be fixed by higher precision alone.

Historical context#

Recognized early in computer arithmetic; led to development of Householder reflections (1958) and other orthogonal methods.

History#

Modern best practice: avoid subtraction of near-equal numbers; use transformation-based algorithms (QR, SVD).

Prevalence in ML#

Common issue in numerical gradient computation, vector normalization, and Gram–Schmidt orthogonalization.

Notes#

• Relative error in $c$ can be $O(1)$ even if $a, b$ are accurate.

• Classical Gram–Schmidt suffers; modified Gram–Schmidt fixes by reorthogonalization.

Connection to ML#

Adversarial robustness and numerical gradient computation are vulnerable to cancellation; careful implementation required.

Connection to Linear Algebra Theory#

Orthogonal transformations (Householder, Givens) have condition number $\kappa = 1$; stable regardless of input.

Pedagogical Significance#

Demonstrates limit of forward-stable algorithms; backward stability (computing exact solution to nearby problem) is more achievable.

References#

1. Wilkinson, J. H. (1961). Error analysis of direct methods of matrix inversion.

2. Higham, N. J. (2002). Accuracy and Stability of Numerical Algorithms (2nd ed.).

3. Goldberg, D. (1991). What every computer scientist should know about floating-point arithmetic.

Solution (Python)#

import numpy as np

# Example 1: Catastrophic cancellation
a = 1.0 + 1e-16
b = 1.0
c = a - b
print("Example: Subtraction of nearly equal numbers")
print("  a = 1.0 + 1e-16 =", repr(a))
print("  b = 1.0 =", repr(b))
print("  c = a - b =", repr(c))
print("  Exact: 1e-16, Computed: ", c)
print("  Relative error:", abs(c - 1e-16) / 1e-16 if c != 0 else "inf")

# Example 2: Loss of significance in dot product
print("\nExample: Loss of significance in inner product")
np.random.seed(22)
x = np.array([1e8, 1.0, 1.0])
y = np.array([1e-8, 1.0, 1.0])

# Naive: compute separately then subtract
xy_naive = x[0] * y[0] + x[1] * y[1] + x[2] * y[2]  # 1 + 1 + 1 = 3

# More stable: accumulate with smaller terms first
xy_stable = x[2] * y[2] + x[1] * y[1] + x[0] * y[0]  # Reverse order

print("  Inner product (x · y):")
print("    Naive order: {:.15f}".format(xy_naive))
print("    Reverse order: {:.15f}".format(xy_stable))
print("    NumPy dot: {:.15f}".format(np.dot(x, y)))

# Example 3: Orthogonal transformation vs. subtraction
print("\nExample: Orthogonal transformation avoids cancellation")
# Create orthogonal matrix (preserve norms)
theta = np.pi / 6
Q = np.array([[np.cos(theta), -np.sin(theta)],
              [np.sin(theta), np.cos(theta)]])
v = np.array([1.0, 0.0])
v_rot = Q @ v
print("  ||v|| =", np.linalg.norm(v))
print("  ||Q v|| =", np.linalg.norm(v_rot), "(should equal ||v||)")
print("  Orthogonal transformation preserves norms: stable!")

Introduction#

Construct an ill-conditioned SPD system; solve via unpreconditioned CG and preconditioned CG with diagonal preconditioner; measure iteration count reduction.

Purpose#

Show quantitatively how preconditioning reduces condition number and accelerates convergence.

Importance#

Essential technique for large-scale optimization; directly translates to practical speedups.

What this example demonstrates#

• Construct ill-conditioned SPD matrix (exponentially decaying eigenvalues).

• Solve via CG (count iterations to convergence).

• Construct diagonal (Jacobi) preconditioner $M = \text{diag}(A)$.

• Solve via preconditioned CG (count iterations).

• Compute effective $\kappa(M^{-1} A)$ and verify iteration count reduction.

Background#

CG iteration count $\approx \sqrt{\kappa(A)}$; preconditioning reduces $\kappa \to \kappa(M^{-1} A)$.

Historical context#

Preconditioning recognized as essential by 1970s; became standard in iterative solvers.

History#

Standard in PETSc, Trilinos, scipy.sparse.linalg (all offer preconditioning options).

Prevalence in ML#

L-BFGS uses quasi-Newton preconditioning; preconditioned SGD in modern optimizers.

Notes#

• Iteration count $\approx \sqrt{\kappa}$; reducing $\kappa$ by 100 reduces iterations by $\sqrt{100} = 10$.

• Preconditioner setup cost must be amortized over multiple solves.

Connection to ML#

Optimization accelerators: Newton’s method, natural gradient, adaptive methods all precondition the gradient.

Connection to Linear Algebra Theory#

Preconditioning clusters eigenvalues; standard result in iterative methods.

Pedagogical Significance#

Demonstrates power of structure exploitation (knowing $M \approx A$) to improve algorithm complexity.

References#

1. Axelsson, O. (1994). Iterative Solution Methods.

2. Nocedal, J., & Wright, S. J. (2006). Numerical Optimization (2nd ed.).

3. Saad, Y. (2003). Iterative Methods for Sparse Linear Systems (2nd ed.).

Solution (Python)#

import numpy as np
import scipy.sparse as sp
import scipy.sparse.linalg as spla
np.random.seed(23)

# Create ill-conditioned SPD matrix
n = 200
U, _ = np.linalg.qr(np.random.randn(n, n))
s = np.logspace(0, -3, n)  # kappa ~ 1000
A = U @ np.diag(s) @ U.T
A = (A + A.T) / 2
b = np.random.randn(n)

# Unpreconditioned CG
residuals_unprecond = []
def callback_un(x):
    residuals_unprecond.append(np.linalg.norm(A @ x - b))
x_un, info_un = spla.cg(A, b, tol=1e-8, callback=callback_un, maxiter=n)

# Preconditioned CG with diagonal (Jacobi) preconditioner
M_diag = np.diag(1.0 / np.diag(A))
M_diag_sparse = sp.diags(np.diag(M_diag))
residuals_precond = []
def callback_pre(x):
    residuals_precond.append(np.linalg.norm(A @ x - b))
x_pre, info_pre = spla.cg(A, b, M=M_diag_sparse, tol=1e-8, 
                           callback=callback_pre, maxiter=n)

kappa_A = np.linalg.cond(A)
# Approximate kappa(M^{-1} A)
AM = M_diag @ A
kappa_AM = np.linalg.cond(AM)

print("Condition numbers:")
print("  kappa(A) =", round(kappa_A, 0))
print("  kappa(M^{-1} A) ≈", round(kappa_AM, 0), "(after diagonal preconditioning)")
print("\nCG Convergence:")
print("  Unpreconditioned iterations:", len(residuals_unprecond))
print("  Preconditioned iterations:", len(residuals_precond))
print("  Speedup factor:", round(len(residuals_unprecond) / len(residuals_precond), 1))
print("  Predicted speedup (~sqrt(kappa)):", round(np.sqrt(kappa_A / kappa_AM), 1))
print("\nBoth solutions satisfy A x = b:",
      np.allclose(A @ x_un, b) and np.allclose(A @ x_pre, b))

Introduction#

Construct an ill-posed rectangular system (underdetermined or rank-deficient); solve via Tikhonov regularization with varying $\lambda$; plot condition number reduction and solution norm vs. residual (L-curve).

Purpose#

Demonstrate how regularization shifts small singular values and reduces effective condition number.

Importance#

Core technique in inverse problems, ridge regression, and ill-posed ML problems.

What this example demonstrates#

• Construct rank-deficient or ill-posed system (exponentially decaying singular values).

• Solve regularized system $(A^\top A + \lambda I) w = A^\top b$ for range of $\lambda$.

• Compute effective condition number: $\kappa_\lambda = (\sigma_1^2 + \lambda) / (\sigma_n^2 + \lambda)$.

• Plot L-curve: solution norm vs. residual; optimal $\lambda$ at corner.

• Verify that small $\lambda$ gives large solution, large $\lambda$ gives small solution but large residual.

Background#

Tikhonov adds penalty; shifts eigenvalues $\sigma_i^2 \to \sigma_i^2 + \lambda$.

Historical context#

Tikhonov (1963) for ill-posed problems; Hansen (1998) developed regularization parameter selection tools.

History#

Standard in inverse problems (deblurring, tomography); ridge regression standard in ML.

Prevalence in ML#

Ridge regression, elastic net, dropout, early stopping all act as implicit regularization.

Notes#

• Optimal $\lambda$ balances fit (small residual) and solution norm (small solution).

• L-curve method (Golub et al. 1979): plot $\log(\| r \|)$ vs. $\log(\| w \|)$; corner indicates optimal $\lambda$.

Connection to ML#

Bias-variance trade-off: underfitting vs. overfitting; regularization parameter selection critical.

Connection to Linear Algebra Theory#

Spectral filtering: Tikhonov filters small singular values; LSQR via early stopping does implicit filtering.

Pedagogical Significance#

Shows how problem structure (ill-posedness) guides algorithm design (regularization).

References#

1. Tikhonov, A. N. (1963). On the solution of ill-posed problems and regularized methods.

2. Hansen, P. C. (1998). Rank-deficient and Discrete Ill-Posed Problems.

3. Golub, G. H., Van Loan, C. F., & Milanfar, P. (1999). The total least squares problem.

Solution (Python)#

import numpy as np
import matplotlib.pyplot as plt
np.random.seed(24)

# Construct ill-posed system
m, n = 100, 40
U, _ = np.linalg.qr(np.random.randn(m, m))
V, _ = np.linalg.qr(np.random.randn(n, n))
s = np.exp(-np.linspace(0, 4, n))  # Exponentially decaying singular values
A = U[:m, :n] @ np.diag(s) @ V.T

# True solution and noisy data
x_true = np.zeros(n)
x_true[:5] = [10, 8, 6, 4, 2]
b_clean = A @ x_true
noise_level = 0.01
b = b_clean + noise_level * np.random.randn(m)

# Solve Tikhonov for range of lambda
lams = np.logspace(-6, 2, 50)
sol_norms = []
residuals = []
cond_numbers = []

for lam in lams:
    G = A.T @ A + lam * np.eye(n)
    w = np.linalg.solve(G, A.T @ b)
    sol_norms.append(np.linalg.norm(w))
    residuals.append(np.linalg.norm(A @ w - b))
    cond_numbers.append(np.linalg.cond(G))

sol_norms = np.array(sol_norms)
residuals = np.array(residuals)
cond_numbers = np.array(cond_numbers)

# Find corner of L-curve (optimal lambda via curvature)
log_res = np.log(residuals)
log_norm = np.log(sol_norms)
curvature = np.gradient(np.gradient(log_res) / np.gradient(log_norm))
opt_idx = np.argmax(curvature)
lam_opt = lams[opt_idx]

# Solve with optimal lambda
G_opt = A.T @ A + lam_opt * np.eye(n)
w_opt = np.linalg.solve(G_opt, A.T @ b)

print("Regularization parameter selection (L-curve method):")
print("  Optimal lambda:", format(lam_opt, '.2e'))
print("  At optimal lambda:")
print("    Solution norm:", round(np.linalg.norm(w_opt), 4))
print("    Residual norm:", round(np.linalg.norm(A @ w_opt - b), 4))
print("    Condition number:", round(np.linalg.cond(G_opt), 2))
print("\nFor comparison:")
print("  Smallest lambda: cond =", round(cond_numbers[0], 0), ", residual =", 
      format(residuals[0], '.2e'))
print("  Largest lambda: cond =", round(cond_numbers[-1], 0), ", residual =", 
      format(residuals[-1], '.2e'))
print("\nTrue solution norm:", round(np.linalg.norm(x_true), 4))
print("Solution error at optimal lambda:", round(np.linalg.norm(w_opt - x_true), 4))