| Aspect | Ordinary Least Squares (OLS) | Least‑Absolute‑Deviations (LAD) | |--------|------------------------------|---------------------------------| | | Minimize ∑ ( yᵢ − Xᵢβ )² | Minimize ∑ | yᵢ − Xᵢβ | | | Loss function | Quadratic (smooth, differentiable) | Linear (non‑smooth at 0) | | Sensitivity to outliers | High (outliers pull the fit) | Low – each outlier contributes linearly | | Statistical interpretation | MLE under Gaussian errors | MLE under Laplace (double‑exponential) errors | | Closed‑form solution | Yes (β = (XᵀX)⁻¹Xᵀy) | No – requires linear programming / iterative methods |
| Language | Package / Library | Main Algorithm(s) | Install | Minimal Example | |----------|-------------------|-------------------|---------|-----------------| | | fastLAD (CRAN) | Barrodale‑Roberts + CD fallback | install.packages("fastLAD") | r library(fastLAD) <br> fit <- fastLAD(y ~ X, data = mydata) <br> summary(fit) | | Python | statsmodels.robust (LAD) + pyLAD (GitHub) | IRLS (statsmodels) & ADMM (pyLAD) | pip install statsmodels pyLAD | python import statsmodels.api as sm <br> X = sm.add_constant(X) <br> model = sm.RLM(y, X, M=sm.robust.norms.L1()) <br> res = model.fit() | | Python (scikit‑learn compatible) | sklearn.linear_model.LADRegression (experimental, 2024) | Coordinate Descent + warm‑start | pip install scikit-learn==1.5 | python from sklearn.linear_model import LADRegression <br> lad = LADRegression() <br> lad.fit(X, y) | | MATLAB | ladfit (File Exchange) | ADMM | addpath('ladfit') | matlab [beta, stats] = ladfit(X, y); | | Julia | LAD.jl | IRLS + Proximal Gradient | using Pkg; Pkg.add("LAD") | julia using LAD; β = lad(X, y) | | C++ / CUDA | fastlad (open‑source, GitHub) | Parallel ADMM & GPU kernels | Clone & cmake . && make | See repo README for a 5‑line C++ call. | fastlad
print("Estimated coefficients (first 5):", model.coef_[:5]) print("RMSE vs true:", np.sqrt(np.mean((model.coef_ - beta_true)**2))) | Aspect | Ordinary Least Squares (OLS) |
The Rise of Fastlad: How This Emerging Trend is Revolutionizing the Way We Live and Work - fastLAD(y ~ X
# Simulated data with outliers np.random.seed(0) n, p = 200_000, 20 X = np.random.randn(n, p) beta_true = np.random.randn(p) y = X @ beta_true + np.random.laplace(scale=0.5, size=n)