Introduction

This library aims to provide XLA/JAX based Python implementations for various algorithms related to:

  • Sparse approximation

  • Compressive sensing

  • Linear operators

Bulk of this library is built using functional programming techniques which is critical for the generation of efficient numerical codes for CPU and GPU architectures.

Functional Programming

Functional Programming is a programming paradigm where computer programs are constructed by applying and composing functions. Functions define a tree of expressions which map values to other values (akin to mathematical functions) rather than a sequence of iterative statements. Some famous languages based on functional programming are Haskell and Common Lisp. A key idea in functional programming is a pure function. A pure function has following properties:

  • The return values are identical for identical arguments.

  • The function has no side-effects (no mutation of local static variables, non-local variables, etc.).

XLA is a domain-specific compiler for linear algebra. XLA uses JIT (just-in-time) compilation techniques to analyze the structure of a numerical algorithm written using it. It then specializes the algorithm for actual runtime dimensions and types of parameters involved, fuses multiple operations together and emits efficient native machine code for devices like CPUs, GPUs and custom accelerators (like Google TPUs).

JAX is a front-end for XLA and Autograd with a NumPy inspired API. Unlike NumPy, JAX arrays are always immutable. While x[0] = 10 is perfectly fine in NumPy as arrays are mutable, the equivalent functional code in JAX is x = x.at[0].set(10).

Linear Operators

Efficient linear operator implementations provide much faster computations compared to direct matrix vector multiplication. A framework for building and composing linear operators has been provided in cr.sparse.lop. Functionality includes:

  • Basic operators: identity, matrix, diagonal, zero, flipud, sum, pad_zeros, symmetrize, restriction, etc.

  • Signal processing: fourier_basis_1d, dirac_fourier_basis_1d, etc.

  • Random dictionaries: gaussian_dict, rademacher_dict, random_onb_dict, random_orthonormal_rows_dict, etc.

  • Operator algebra: neg, scale, add, subtract, compose, transpose, hermitian, hcat, etc.

  • Additional utilites

Greedy Sparse Recovery/Approximation Algorithms

JAX based implementations for the following algorithms are included.

  • Orthogonal Matching Pursuit

  • Compressive Sampling Matching Pursuit

  • Subspace Pursuit

  • Iterative Hard Thresholding

  • Hard Thresholding Pursuit

Convex Optimization based Recovery Algorithms

A port of yall1 (Your algorithms for \(\ell_1\)) has been provided. It provides alternating directions method of multipliers based solutions for basis pursuit, basis pursuit denoising, basis pursuit with inequality constraints, their non-negative counterparts and other variants.

Evaluation Framework

The library also provides

  • Various simple dictionaries and sensing matrices

  • Sample data generation utilities

  • Framework for evaluation of sparse recovery algorithms

Installation

Basic installation from PYPI:

python -m pip install cr-sparse

Installation from GitHub:

python -m pip install git+https://github.com/carnotresearch/cr-sparse.git