如何在R语言中使用非欧几里得（拟）范数实现线性回归并计算系数？

Hey there! Great questions—let's break down how to work with non-Euclidean (quasi-)norms and distance-based linear regression in R.

Absolutely! Traditional ordinary least squares (OLS) regression relies on the L₂ (Euclidean) norm, minimizing the sum of squared residuals. But R has robust tools to adapt this to other norms or quasi-norms—this just involves reframing the regression as an optimization problem with a custom loss function instead of the standard squared loss. Whether you're working with L₁, L₀, Minkowski, or even more specialized distances, you can either use pre-built packages or roll your own solution with optimization functions.

Below are practical approaches for common non-Euclidean distance/norm scenarios:

• L₁ Norm (Manhattan Distance) Regression (Least Absolute Deviation, LAD)
  This minimizes the sum of absolute residuals, making it far more robust to outliers than OLS. Two straightforward ways to implement this:

  # Option 1: Use quantreg (LAD is equivalent to 0.5 quantile regression)
  library(quantreg)
  data(mtcars)
  lad_model <- rq(mpg ~ wt + hp, data = mtcars, tau = 0.5)
  summary(lad_model)
  
  # Option 2: Robust regression with MASS package
  library(MASS)
  # Adjust psi function to prioritize L₁ behavior; psi.hampel leans into absolute residuals
  rlm_lad <- rlm(mpg ~ wt + hp, data = mtcars, psi = psi.hampel, maxit = 100)
  summary(rlm_lad)
  

• L₀ Norm Regression (Sparse Regression)
  The L₀ norm counts non-zero residuals, so this focuses on finding the sparest set of coefficients that fit the data. Since exact L₀ minimization is computationally hard, we use approximations like Lasso (L₁ regularization pushed to its limit):

  library(glmnet)
  # Prepare matrix input (glmnet expects this structured format)
  x <- model.matrix(mpg ~ wt + hp - 1, data = mtcars) # -1 removes intercept if unnecessary
  y <- mtcars$mpg
  
  # Fit Lasso model (alpha=1 specifies Lasso regularization)
  lasso_fit <- glmnet(x, y, alpha = 1)
  # Use cross-validation to find the optimal penalty strength (lambda)
  cv_lasso <- cv.glmnet(x, y, alpha = 1)
  # Extract coefficients at the lambda that minimizes cross-validation error
  coef(cv_lasso, s = "lambda.min")
  

• Custom Non-Euclidean Distance/Quasi-Norm Regression
  For specialized norms (e.g., Minkowski distance with p=1.5, or custom distance metrics), define your loss function and use R's optimization tools like optim():

  # Example: Minkowski distance with p=1.5 (minimize sum of |residuals|^1.5)
  minkowski_loss <- function(beta, x, y, p) {
    y_pred <- x %*% beta
    sum(abs(y - y_pred)^p)
  }
  
  # Prepare data (add intercept term manually)
  x <- cbind(1, mtcars$wt, mtcars$hp)
  y <- mtcars$mpg
  # Start with OLS coefficients as a reasonable initial guess
  init_beta <- lm(y ~ x[,2] + x[,3])$coefficients
  
  # Optimize using BFGS method for smooth loss functions
  opt_result <- optim(par = init_beta, 
                      fn = minkowski_loss,
                      x = x, y = y, p = 1.5,
                      method = "BFGS")
  
  # Print the resulting regression coefficients
  cat("Minkowski (p=1.5) Regression Coefficients:\n")
  print(opt_result$par)
  

• Mahalanobis Distance Regression
  For regression that accounts for covariance between variables (Mahalanobis distance), you can whiten the data first then run OLS, or embed the covariance inverse into your loss function. A quick implementation:

  # Whiten data using inverse square root of the covariance matrix
  cov_mat <- cov(mtcars[,c("wt", "hp")])
  inv_sqrt_cov <- solve(chol(cov_mat))
  whitened_x <- as.matrix(mtcars[,c("wt", "hp")]) %*% inv_sqrt_cov
  
  # Run OLS on whitened data (equivalent to Mahalanobis distance regression)
  mahal_model <- lm(mpg ~ whitened_x, data = mtcars)
  summary(mahal_model)
  

内容的提问来源于stack exchange，提问作者Marko Karbevski