Multiscale Approximation

Introduction

This is an infrastructure for scattered data multiscale approximation. This readme describes the code and how to run it. Numerical examples from our paper and how to run them can be found in NumericalExamples.

The code is modular, and designed for customization. It is possible to choose and add new methods:

• Manifold - The range of the approximated function.
• Approximation method - Currently implemented quasi-interpolation, multiscale approach, naive kernel interpolation,...
• Data structure - The algorithm of storing and querying the data sites.
• Data sites generation - The method of choosing the sampling sites.

How to run?

Install Requirements

pip install -r requirements.txt

For flexible options, run runner.py:

# Approximate 3 scales both multiscale and single scale
python runner.py -m rotations -f ExampleFunctions.euler -s -n 3

# For help:
python runner.py -h

To run experiments from the paper, run:

python -m NumericalExamples.{{EXPERIMENT_NAME}}

Design

Main experiment

The main multiscale logic is in Experiment, in the multiscale_approximation(). One can run an experiment from the runner, which is flexible, or run the examples from the paper in NumericalExamples.

Config

The module Config contains the config object that holds the configurations for the current experiment. config is an instance of the Config class. It allows to set_base_config, and renew to the base config. It loads its default values from defaults. In order to update the config by a differences dict, use the method update_config_with_diff.

Approximation methods

• Quasi performs averaging using RBF coefficients.
  • $Q^Mf(x):=av_M(\Phi(x),f(\Xi))$
• Moving is a moving least squares that promises polynomial reproduction. It is based on the PolynomialReproduction module.

Manifolds

• AbstractManifold.py is the base class for the manifolds. It implements naively some required APIs for a manifold.
• Circle.py is the $S^1$ single dimensional sphere manifold.
  • The suggested average is geodesic.
  • Exp-Log pair is
    • $exp(x,y)={{x+y}\over{|x+y|}}$
    • $log(x,y)={y\over{|<x,y>|}}-x$
• SymmetricPositiveDefinite.py is the $SPD$ manifold.
  • Averages using KarcherMean.py.
  • Visualizes using Visualization.py
  • Exp-Log pair is
    • $exp(x,y)=x^{1/2}EXP(x^{-1/2}yx^{-1/2})x^{1/2}$
    • $log(x,y)=x^{1/2}LOG(x^{-1/2}yx^{-1/2})x^{1/2}$

Tools

• Utils.py has some util functions:
  • caching
  • operations on functions
  • grid operations
  • plotting
  • RBF functions
  • output directory management
• KarcherMean.py - calculates weighted Karcher mean.
• Visualization.py - Ellipsoid visualization of SPD matrices.