LASSO Regression

LASSO adds the sum of absolute coefficient values (L1-norm) to the least squares cost function. This diamond-shaped constraint region forces coefficient estimates to hit the axes, producing exact zeros. Unlike ridge regression's L2 penalty, which only shrinks coefficients toward zero, LASSO's sharp corners enable true variable selection. The optimization problem minimizes: ||y - Xβ||²₂ + λ||β||₁, where λ controls the sparsity level.

In Volterra series modeling, the candidate regressor matrix contains hundreds of kernel terms across multiple nonlinear orders and memory depths. LASSO automatically identifies the dominant diagonal and cross-term kernels while zeroing out negligible contributions. This eliminates manual trial-and-error pruning and reduces a full 5th-order, 3-memory-depth Volterra model from 243 terms to typically 15-30 active coefficients.

The regularization parameter λ governs the tradeoff between sparsity and model fit:

• λ = 0: Reduces to ordinary least squares (all terms active)
• λ → ∞: All coefficients forced to zero
• Optimal λ: Selected via cross-validation or information criteria like BIC

The full regularization path shows coefficients entering and leaving the model as λ varies, providing insight into term importance ranking.

By pruning 80-90% of Volterra kernel terms, LASSO dramatically reduces the real-time computational burden in digital predistortion implementations. A sparse model with 20 coefficients requires far fewer multiply-accumulate operations per sample than a full 200-term model. This directly translates to lower FPGA resource utilization, reduced power consumption, and the ability to meet tight latency constraints in wideband 5G systems.

Volterra regressor matrices often exhibit high multicollinearity due to correlated polynomial and cross-product terms, leading to ill-conditioned normal equations. LASSO's L1 regularization implicitly improves the effective condition number by zeroing out redundant terms, producing more stable coefficient estimates that generalize better to unseen signals and resist noise amplification during model extraction.

LASSO is typically solved using coordinate descent, which iteratively optimizes one coefficient at a time while holding others fixed. The soft-thresholding update rule for each coefficient βⱼ is: βⱼ ← S(zⱼ, λ) / ||xⱼ||², where S() is the soft-thresholding operator and zⱼ is the partial residual. This algorithm scales efficiently to high-dimensional Volterra candidate sets and converges rapidly due to the sparse solution structure.