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A Unified Theoretic and Algorithmic Framework for Solving Multivariate Linear Model with $\ell^1$-norm Approximation

Zhi-Qiang Feng, Hong-Yan Zhanga, Ji Ma, Daniel Delahaye, Ruo-Shi Yang, Man Liang

TL;DR

This work addresses robust parameter estimation for the multivariate linear model by solving $\min_{{\mathbf{x}}} \|\mathbf{A}{\mathbf{x}}-\mathbf{b}\|_1$, introducing an equivalence theorem that expresses the optimal solution as $\mathbf{x}_{\mathrm{opt}}=\mathbf{A}^{\dagger}(\mathbf{b}+\mathbf{r}_{\mathrm{opt}})$ with a minimum $\ell^1$-norm residual $\mathbf{r}_{\mathrm{opt}}$. It then develops a unified REV-based framework and six practical algorithms (L1-GPRS, L1-TNIPM, L1-HP, L1-IST, L1-ADM, L1-POB) that rely on basic matrix operations, accompanied by pseudocode and open-source implementations. The paper demonstrates high-accuracy performance across noisy and noise-free regimes, and highlights how data redundancy and sparse noise influence accuracy and runtime. By releasing code on GitHub, it enables rapid adoption of $\ell^1$-norm optimization for MLM parameter estimation across sciences and engineering.

Abstract

It is a challenging problem that solving the \textit{multivariate linear model} (MLM) $\mathbf{A}\mathbf{x}=\mathbf{b}$ with the $\ell_1 $-norm approximation method such that $||\mathbf{A}\mathbf{x}-\mathbf{b}||_1$, the $\ell_1$-norm of the \textit{residual error vector} (REV), is minimized. In this work, our contributions lie in two aspects: firstly, the equivalence theorem for the structure of the $\ell_1$-norm optimal solution to the MLM is proposed and proved; secondly, a unified algorithmic framework for solving the MLM with $\ell_1$-norm optimization is proposed and six novel algorithms (L1-GPRS, L1-TNIPM, L1-HP, L1-IST, L1-ADM, L1-POB) are designed. There are three significant characteristics in the algorithms discussed: they are implemented with simple matrix operations which do not depend on specific optimization solvers; they are described with algorithmic pseudo-codes and implemented with Python and Octave/MATLAB which means easy usage; and the high accuracy and efficiency of our six new algorithms can be achieved successfully in the scenarios with different levels of data redundancy. We hope that the unified theoretic and algorithmic framework with source code released on GitHub could motivate the applications of the $\ell_1$-norm optimization for parameter estimation of MLM arising in science, technology, engineering, mathematics, economics, and so on.

A Unified Theoretic and Algorithmic Framework for Solving Multivariate Linear Model with $\ell^1$-norm Approximation

TL;DR

This work addresses robust parameter estimation for the multivariate linear model by solving , introducing an equivalence theorem that expresses the optimal solution as with a minimum -norm residual . It then develops a unified REV-based framework and six practical algorithms (L1-GPRS, L1-TNIPM, L1-HP, L1-IST, L1-ADM, L1-POB) that rely on basic matrix operations, accompanied by pseudocode and open-source implementations. The paper demonstrates high-accuracy performance across noisy and noise-free regimes, and highlights how data redundancy and sparse noise influence accuracy and runtime. By releasing code on GitHub, it enables rapid adoption of -norm optimization for MLM parameter estimation across sciences and engineering.

Abstract

It is a challenging problem that solving the \textit{multivariate linear model} (MLM) with the -norm approximation method such that , the -norm of the \textit{residual error vector} (REV), is minimized. In this work, our contributions lie in two aspects: firstly, the equivalence theorem for the structure of the -norm optimal solution to the MLM is proposed and proved; secondly, a unified algorithmic framework for solving the MLM with -norm optimization is proposed and six novel algorithms (L1-GPRS, L1-TNIPM, L1-HP, L1-IST, L1-ADM, L1-POB) are designed. There are three significant characteristics in the algorithms discussed: they are implemented with simple matrix operations which do not depend on specific optimization solvers; they are described with algorithmic pseudo-codes and implemented with Python and Octave/MATLAB which means easy usage; and the high accuracy and efficiency of our six new algorithms can be achieved successfully in the scenarios with different levels of data redundancy. We hope that the unified theoretic and algorithmic framework with source code released on GitHub could motivate the applications of the -norm optimization for parameter estimation of MLM arising in science, technology, engineering, mathematics, economics, and so on.

Paper Structure

This paper contains 34 sections, 4 theorems, 91 equations, 17 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Matrix Decomposition Induced by REV
  5. Minimum $\ell^1$-norm REV
  6. Concept
  7. Equivalence and Sparsity
  8. Perturbation Strategy
  9. Linear Programming and MLM
  10. Theoretic Framework of $\ell^1$-norm Approximation Solution to MLM
  11. Structure of $\ell^1$-norm Approximate Solution
  12. MLM and Penalized Least Squares Problem
  13. Algorithmic Framework of $\ell^1$-norm Approximation Solution to MLM
  14. Engineering of Available $\ell^1$-norm Optimization Methods for Solving the REV
  15. Linear Programming Method
  16. ...and 19 more sections

Key Result

Theorem 1

Let $\bm{Ax = b}$ and $\bm{Bx = b}$ be two different MLM. If $\exists \bm{P}\in \mathrm{GL}(n,\mathbb{R})=\left\{ \bm{Q}\in \mathbbm{R}^{{n}\times {n}}, \det(\bm{Q})\neq 0 \right\}$ such that $\bm{B = AP}$, then the ML1-REV of the two MLMs are the same.

Figures (17)

  • Figure 1: Unified theoretic and algorithmic framework for solving the MLM with $\ell_1$-norm optimization
  • Figure 2: Decompositon of the REV $\bm{r} = \bm{A}\bm{x} -\bm{b} = \left\langle {\bm{r}_*}, {\bm{r}_z} \right\rangle = \left\langle {\bm{r}[\mathscr{Z}]}, {\bm{r}[\mathscr{Z}^c]} \right\rangle$
  • Figure 3: Solving the $\ell^1$-approximation via perturbation.
  • Figure 4: Solving the $\bm{A}\bm{x}=\bm{b}$ with $\ell^1$-norm approximation via linear programming.
  • Figure 5: Solving the $\ell^1$-norm REV via linear programming.
  • ...and 12 more figures

Theorems & Definitions (4)

  • Theorem 1: Equivalence
  • Theorem 2: Sparsity
  • Theorem 3: Perturbation Strategy
  • Theorem 4