SparseRationalLearning

A Julia package for sparse rational function regression learning.

Introduction

This package implements algorithms to learn rational function expressions P(x)/Q(x) from data, where P and Q are polynomials. Through sparsity constraints, the learned expressions are kept as simple and interpretable as possible.

Installation

# Navigate to the project directory and start Julia REPL
julia

# Activate the project environment
using Pkg
Pkg.activate(".")

# Install dependencies
Pkg.instantiate()

Quick Start

using SparseRationalLearning

# Prepare data
X = rand(100, 2) .* 5.0# 100 samples, 2 features
y = (1.5 .* X[:,1] .* X[:,2] .+ 0.8 .* X[:,1].^2) ./ (1.0 .+ 0.2 .* X[:,2].^2)

# Train using L1 method (fast)
a, b, labels = train_l1(X, y, 2, 1.0)

# View the resulting expression
println(sparse_rational_expression(a, b, labels, threshold=0.01))

# Calculate R² score
r2, _ = calculate_r2(X, y, a, b, 2)
println("R² = $r2")

API Reference

Training Functions

train_l0(X, y, degree, lambda; M=50.0, time_limit=60.0)

Sparse rational regression based on L0 norm (Mixed Integer Programming).

Parameters:

• X: Feature matrix (n_samples × n_features)
• y: Target vector
• degree: Maximum polynomial degree
• lambda: Sparsity regularization coefficient
• M: Big-M parameter
• time_limit: Solver time limit (seconds)

Returns: (a, b, labels) - numerator coefficients, denominator coefficients, basis function labels

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train_l1(X, y, degree, lambda)

Sparse rational regression based on L1 norm (Lasso).

Parameters:

• X: Feature matrix
• y: Target vector
• degree: Maximum polynomial degree
• lambda: L1 regularization coefficient

Returns: (a, b, labels)

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Utility Functions

Function	Description
sparse_rational_expression(a, b, labels; threshold)	Generate rational function expression string
calculate_r2(X, y_true, a, b, degree)	Calculate R² coefficient of determination
coefficients_threshold(a, b; threshold)	Truncate small coefficients to 0

Algorithm Comparison

Feature	L0 Method	L1 Method
Solver	HiGHS	SCS
Sparsity	More precise	Approximate
Speed	Slower	Fast
Use Case	Small-scale problems	Large-scale problems

Running Tests

using Pkg
Pkg.test()

Dependencies

• Julia >= 1.11
• Convex.jl - Convex optimization modeling
• HiGHS.jl - MIP solver
• SCS.jl - Conic solver
• Combinatorics.jl - Combinatorics utilities

License

MIT License

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