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FMCA

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Fast multiresolution covariance analysis

FMCA is a header only library for the multiresolution analysis of scattered data and kernel matrices. It is developed at the Università della Svizzera italiana in the research group of Michael Multerer.

Currently, the library features the construction of samplet bases and different versions of the pivoted Cholesky decomposition, as well as the fast samplet covariance compression introduced in Samplets: Construction and scattered data compression.

Different scaling distributions and samplets on a Sigma shaped point cloud may look for example like depicted below. What is this

Representing an exponential covariance kernel with respect to this basis and truncating small entries leads to a sparse matrix which can be factorized using nested dissection What is this The left panel shows the kernel matrix, the middle panel the reordered matrix and the right panel the Cholesky factor.

Installation

FMCA is header only and depends on Eigen. If Eigen is not installed, the build downloads it automatically. OpenMP is used whenever the compiler supports it.

Python module via pip

Thanks to pybind11, FMCA may be compiled into a Python module. Prebuilt wheels for Linux, macOS and Windows are on PyPI, so the easiest way is

python3 -m pip install fmca

To get the latest development version instead, install directly from GitHub (requires CMake ≥ 3.21, a C++17 compiler and Python ≥ 3.9):

python3 -m pip install git+https://github.com/muchip/fmca@master

To select a compiler, e.g. an OpenMP-capable GCC on macOS where Apple clang lacks OpenMP, set CXX before installing:

CXX=g++-15 python3 -m pip install git+https://github.com/muchip/fmca@master

Afterwards import FMCA works in that Python environment. Update to the latest commit with

python3 -m pip install --upgrade --no-cache-dir git+https://github.com/muchip/fmca@master

Building with CMake

For development, or to build the C++ tests, pybind11 needs to be installed (python3 -m pip install pybind11). Then

mkdir build
cd build
cmake -DCMAKE_BUILD_TYPE=Release ../
make

The compiled module and the example notebooks are located in build/py, the tests in build/tests. Add -DCMAKE_CXX_COMPILER=... to the cmake call to select a compiler; -DFMCA_BUILD_TESTS=OFF or -DFMCA_BUILD_PYTHON=OFF skip the respective parts.

Python interface

Points are passed as a dim x N array, one point per column, and data as an N x k array, one row per point. The samplet transform works on data in the order of the cluster tree, which toClusterOrder and toNaturalOrder take care of:

import numpy as np
import FMCA

pts = np.random.rand(2, 10000)          # 10000 points in 2D, one per column
ST = FMCA.SampletTree(pts, 3)           # samplet tree with 3 vanishing moments
f = np.sin(4 * pts[0]).reshape(-1, 1)   # data, one row per point

c = ST.sampletTransform(ST.toClusterOrder(f))           # samplet coefficients
g = ST.toNaturalOrder(ST.inverseSampletTransform(c))    # back to the data

Samplets

FMCA features a samplet basis, which can be used to localize a given signal in the frequency domain. Given for example a signal sampled at 100000 random locations, e.g., What is this

the first 500 coefficients of the transformed signal looks like this What is this

The example above can be found and modified in the jupyter notebook FMCA_Samplets

Denoising

Since samplets have vanishing moments, a smooth signal is represented by very few large coefficients, whereas white noise is spread evenly over all of them. Discarding the small coefficients thus removes most of the noise and almost none of the signal.

For an image, the samplet transform is applied to its columns and then to its rows, and every band of coefficients is thresholded on its own (BayesShrink), with the noise level estimated from the data. For the 600x512 image below, perturbed by Gaussian noise of standard deviation 0.1, the result is on par with standard wavelet denoising.

What is this

The denoising can be found and modified in the jupyter notebooks FMCA_SampletDenoising1D and FMCA_SampletImageDenoising.

The samplet coefficients moreover indicate where a function fails to be smooth, which can be fed back into the cluster tree. Refining the tree uniformly splits every cluster, no matter whether the data require it or not.

What is this

The Binev-DeVore algorithm instead activates the clusters carrying the largest amount of energy, until the energy left outside the tree drops below a prescribed tolerance. The leaves of the resulting adaptive tree form a partition of the point cloud which is fine only where the function varies.

In the example below, 100000 scattered points are partitioned into 128 clusters, while resolving the whole domain at the same finest scale would require 2048 uniform ones.

What is this

The tolerance is measured relative to the energy of the data and is therefore scale free. The example above can be found and modified in the jupyter notebook FMCA_SampletAdaptiveClustering.

Gaussian process learning

FMCA provides different variants of the pivoted (truncated) Cholesky decomposition, cp. On the low-rank approximation by the pivoted Cholesky decomposition and the references therein, that can be used for Gaussian process learning.

posterior mean (red) and posterior standard deviation (green) conditioned on the blue dots What is this

The example above can be found and modified in the jupyter notebook FMCA_GP.

Samplet Gaussian process filtering

A samplet matrix compression based approach is also available. It particular allows for filtering of the (compressed) kernel matrix, thus mitigating the very ill-conditioning of the kernel matrix.

What is thisWhat is this

For the Matern-3/2 kernel shown on the left, just considering the diagonal block associated to the 40 largest entries, shown on the right, leads to a relative approximation error of about 3e-5 of the kernel matrix in the Frobenius norm. Solving the associated system for the noisy data set shown below, leads to an effective denoising. The corresponding expectation is shown in orange.

What is this

This example can be found FMCA_Samplet_GP_Filtering.

Python notebooks

The notebooks below live in py/ and are copied next to the compiled module in build/py, so they can be run directly from there. They cover the samplet basis, the compression of signals, images and kernel matrices, and the data driven refinement of the cluster tree.

notebook what it shows
FMCA_Samplets samplet trees, what a samplet looks like in 1D and 2D, vanishing moments, orthogonality and sparsity of the transform, a signal in the samplet basis, and the compression of a kernel matrix
FMCA_SampletCompression1D a signal in the natural basis versus the samplet basis, coefficient decay and best N-term approximation
FMCA_SampletDenoising1D hard and soft thresholding of samplet coefficients, and the universal threshold
FMCA_SampletImageDenoising compression and denoising of an image with the separable samplet transform, compared with standard wavelets
FMCA_SampletKernelCompression the dense kernel matrix versus its samplet compression, accuracy against cost, and the scaling in N
FMCA_SampletAdaptiveClustering adaptive partitions of a point cloud obtained from the samplet coefficients
FMCA_Samplet_GP_Filtering filtering of the compressed kernel matrix for Gaussian process regression
FMCA_Samplet_KRR kernel ridge regression in the samplet basis
FMCA_GP Gaussian process learning based on the pivoted Cholesky decomposition
FMCA_Cholesky the different variants of the pivoted Cholesky decomposition
FMCA_H2Matrix H2-matrix construction and fast matrix-vector products
FMCA_LowRankBenchmarks low-rank benchmarks for adaptive joint distribution learning

Release files for fmca 0.1.2

For a detailed explanation of source distributions (sdists) and built distributions (wheels), please see the package formats documentation.

Source distribution (sdist)

Source distribution for fmca 0.1.2
File Size Uploaded
fmca-0.1.2.tar.gz 48.5 MB Details

Built distributions (wheels)

Table of built distributions (wheels) for fmca 0.1.2
File
fmca-0.1.2-cp315-cp315-win_amd64.whl CPython 3.15 CPython 3.15 Windows x86-64 Details
fmca-0.1.2-cp315-cp315-manylinux_2_27_x86_64.manylinux_2_28_x86_64.whl CPython 3.15 CPython 3.15 Linux glibc 2.27+ x86-64, Linux glibc 2.28+ x86-64 Details
fmca-0.1.2-cp315-cp315-manylinux_2_26_aarch64.manylinux_2_28_aarch64.whl CPython 3.15 CPython 3.15 Linux glibc 2.28+ ARM64, Linux glibc 2.26+ ARM64 Details
fmca-0.1.2-cp315-cp315-macosx_15_0_x86_64.whl CPython 3.15 CPython 3.15 macOS 15.0+ x86-64 Details
fmca-0.1.2-cp315-cp315-macosx_14_0_arm64.whl CPython 3.15 CPython 3.15 macOS 14.0+ ARM64 Details
fmca-0.1.2-cp314-cp314-win_amd64.whl CPython 3.14 CPython 3.14 Windows x86-64 Details
fmca-0.1.2-cp314-cp314-manylinux_2_27_x86_64.manylinux_2_28_x86_64.whl CPython 3.14 CPython 3.14 Linux glibc 2.28+ x86-64, Linux glibc 2.27+ x86-64 Details
fmca-0.1.2-cp314-cp314-manylinux_2_26_aarch64.manylinux_2_28_aarch64.whl CPython 3.14 CPython 3.14 Linux glibc 2.28+ ARM64, Linux glibc 2.26+ ARM64 Details
fmca-0.1.2-cp314-cp314-macosx_15_0_x86_64.whl CPython 3.14 CPython 3.14 macOS 15.0+ x86-64 Details
fmca-0.1.2-cp314-cp314-macosx_14_0_arm64.whl CPython 3.14 CPython 3.14 macOS 14.0+ ARM64 Details
fmca-0.1.2-cp313-cp313-win_amd64.whl CPython 3.13 CPython 3.13 Windows x86-64 Details
fmca-0.1.2-cp313-cp313-manylinux_2_27_x86_64.manylinux_2_28_x86_64.whl CPython 3.13 CPython 3.13 Linux glibc 2.27+ x86-64, Linux glibc 2.28+ x86-64 Details
fmca-0.1.2-cp313-cp313-manylinux_2_26_aarch64.manylinux_2_28_aarch64.whl CPython 3.13 CPython 3.13 Linux glibc 2.26+ ARM64, Linux glibc 2.28+ ARM64 Details
fmca-0.1.2-cp313-cp313-macosx_15_0_x86_64.whl CPython 3.13 CPython 3.13 macOS 15.0+ x86-64 Details
fmca-0.1.2-cp313-cp313-macosx_14_0_arm64.whl CPython 3.13 CPython 3.13 macOS 14.0+ ARM64 Details
fmca-0.1.2-cp312-cp312-win_amd64.whl CPython 3.12 CPython 3.12 Windows x86-64 Details
fmca-0.1.2-cp312-cp312-manylinux_2_27_x86_64.manylinux_2_28_x86_64.whl CPython 3.12 CPython 3.12 Linux glibc 2.27+ x86-64, Linux glibc 2.28+ x86-64 Details
fmca-0.1.2-cp312-cp312-manylinux_2_26_aarch64.manylinux_2_28_aarch64.whl CPython 3.12 CPython 3.12 Linux glibc 2.28+ ARM64, Linux glibc 2.26+ ARM64 Details
fmca-0.1.2-cp312-cp312-macosx_15_0_x86_64.whl CPython 3.12 CPython 3.12 macOS 15.0+ x86-64 Details
fmca-0.1.2-cp312-cp312-macosx_14_0_arm64.whl CPython 3.12 CPython 3.12 macOS 14.0+ ARM64 Details
fmca-0.1.2-cp311-cp311-win_amd64.whl CPython 3.11 CPython 3.11 Windows x86-64 Details
fmca-0.1.2-cp311-cp311-manylinux_2_27_x86_64.manylinux_2_28_x86_64.whl CPython 3.11 CPython 3.11 Linux glibc 2.27+ x86-64, Linux glibc 2.28+ x86-64 Details
fmca-0.1.2-cp311-cp311-manylinux_2_26_aarch64.manylinux_2_28_aarch64.whl CPython 3.11 CPython 3.11 Linux glibc 2.26+ ARM64, Linux glibc 2.28+ ARM64 Details
fmca-0.1.2-cp311-cp311-macosx_15_0_x86_64.whl CPython 3.11 CPython 3.11 macOS 15.0+ x86-64 Details
fmca-0.1.2-cp311-cp311-macosx_14_0_arm64.whl CPython 3.11 CPython 3.11 macOS 14.0+ ARM64 Details
fmca-0.1.2-cp310-cp310-win_amd64.whl CPython 3.10 CPython 3.10 Windows x86-64 Details
fmca-0.1.2-cp310-cp310-manylinux_2_27_x86_64.manylinux_2_28_x86_64.whl CPython 3.10 CPython 3.10 Linux glibc 2.27+ x86-64, Linux glibc 2.28+ x86-64 Details
fmca-0.1.2-cp310-cp310-manylinux_2_26_aarch64.manylinux_2_28_aarch64.whl CPython 3.10 CPython 3.10 Linux glibc 2.28+ ARM64, Linux glibc 2.26+ ARM64 Details
fmca-0.1.2-cp310-cp310-macosx_15_0_x86_64.whl CPython 3.10 CPython 3.10 macOS 15.0+ x86-64 Details
fmca-0.1.2-cp310-cp310-macosx_14_0_arm64.whl CPython 3.10 CPython 3.10 macOS 14.0+ ARM64 Details
fmca-0.1.2-cp39-cp39-win_amd64.whl CPython 3.9 CPython 3.9 Windows x86-64 Details
fmca-0.1.2-cp39-cp39-manylinux_2_27_x86_64.manylinux_2_28_x86_64.whl CPython 3.9 CPython 3.9 Linux glibc 2.27+ x86-64, Linux glibc 2.28+ x86-64 Details
fmca-0.1.2-cp39-cp39-manylinux_2_26_aarch64.manylinux_2_28_aarch64.whl CPython 3.9 CPython 3.9 Linux glibc 2.26+ ARM64, Linux glibc 2.28+ ARM64 Details
fmca-0.1.2-cp39-cp39-macosx_15_0_x86_64.whl CPython 3.9 CPython 3.9 macOS 15.0+ x86-64 Details
fmca-0.1.2-cp39-cp39-macosx_14_0_arm64.whl CPython 3.9 CPython 3.9 macOS 14.0+ ARM64 Details

Total release size: 70.4 MB

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Release history Release notifications | RSS feed

0.1.5

36 release files

0.1.4

36 release files

0.1.3

36 release files

This release

0.1.2 This release

36 release files

0.1.1

36 release files

0.1.0

36 release files

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