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Example_5_beach_evolution
 
 
Example_8_1_small_multilayer
 
 
Example_8_2_Image_data
 
 
Example_8_3_Pavia_center
 
 
Example_SM3_SBM
 
 
Subroutines
 
 
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README.md
 
 

README.md

MultilayerSSL

This repository contains the codes to our publication:

K. Bergermann, M. Stoll and T. Volkmer
Semi-supervised learning for aggregated multilayer graphs using diffuse interface methods and fast matrix vector products
SIAM Journal on Mathematics of Data Science, 3(2):758-785, 2021.

Multilayer-SSL-NFFT-Examples Version 2.0, 2020-10-07

Multilayer-SSL-NFFT-Examples is a collection of MATLAB code that performs semi-supervised learning based on a diffuse interface approach on multilayer graphs with acceleration for large image datasets using the NFFT-based fast summation.

The collection consists of the following files:

README

  • This file.

COPYING

  • A copy of the GPL v2 license.

Example_5_beach_evolution/beach_eighth_evolution.m

  • Generates the images of Figure 2 visualizing the evolution of a graph based phase-field simulation for image segmentation of the example from Figure 3 based on a 4-class 2-layer Graph Allen-Cahn classification scheme [1, Algorithm 6.1] using the power mean Laplacian with p=1 and the NFFT-based fast summation for the eigeninformation computations.

Example_8_1_SBM/SBM_example1.m

  • Section 8.1 numerical experiment on 3-layer stochastic block model (SBM) graphs with each layer carrying the information about one class for the Allen-Cahn multiclass classification scheme [1, Algorithm 6.1] using the power mean Laplacian with two and three layers respectively as well as the three single layer graphs.

Example_8_1_SBM/SBM_example2.m

  • Section 8.1 numerical experiment on 2-layer stochastic block model (SBM) graphs with and without one noisy layer for the Allen-Cahn multiclass classification scheme [1, Algorithm 6.1] using the power mean Laplacian.

Example_8_2_small_multilayer/*_multiple_rng.m

  • Section 8.2 numerical experiment on various small but high-dimensional single layer and multilayer data sets for the Allen-Cahn multiclass classification scheme [1, Algorithm 6.1] using the power mean Laplacian.

Example_8_3_Image_data/Fig5_single_beach_image_full.m

  • Section 8.3 numerical experiment on image data for the Allen-Cahn multiclass classification scheme [1, Algorithm 6.1] using the power mean Laplacian with a splitting into 2 layers (RGB color + XY pixel coordinates) using the NFFT-based fast summation for the eigeninformation computations.

Example_8_3_Image_data/Fig6_Fig7_two_beach_images_transfer.m

  • Section 8.3 numerical experiment on transfer learning from one image to another for the Allen-Cahn multiclass classification scheme [1, Algorithm 6.1] using the power mean Laplacian with a splitting into 2 layers (RGB color + XY pixel coordinates) using the NFFT-based fast summation for the eigeninformation computations.

Example_8_4_Pavia_center/Pavia_center*.m

  • Section 8.4 numerical experiment on the Pavia center data set (hyperspectral image classification) for the Allen-Cahn multiclass classification scheme [1, Algorithm 6.1] using the power mean Laplacian using the NFFT-based fast summation for the eigeninformation computations.

Subroutines/convexity_splitting_vector_modified_fast.m

  • Semi-supervised classification scheme using the Allen-Cahn model with a smooth potential and convexity splitting, which realize steps 2-4 of [1, Algorithm 6.1].

Subroutines/dist2.m

  • Computes Euclidean distance matrix between two matrices.

Subroutines/fastsumAdjacencyEigs.m

  • MATLAB code to approximate the largest eigenvalues of a graph adjacency matrix based on fastsumAdjacencySetup.m and MATLAB's EIGS function.

Subroutines/fastsumAdjacencySetup.m

  • MATLAB code to setup fast evaluation of matrix-vector products with the graph adjacency matrix or its normalized version using the NFFT3 toolbox.

Subroutines/fh_pml.m

  • Calculates the arithmetic mean of a cell of size [1,T] applied to a vector w.

Subroutines/gen_arnoldi_for_power_of_a_matrix_times_a_vector_modified.m

  • Computes an M-orthogonal basis of the generalized Krylov subspace K_k(P, Q, v) = {v, P \ (Q v), ..., (P \ Q)^k v)} well as the generalized Arnoldi decomposition of P and Q.

Subroutines/generate_sbm_graph.m

  • Generates a random graph following the prescribed stochastic block model (SBM) distribution.

Subroutines/MV_fastsum_T_layers.m

  • Calculates the arithmetic mean of a cell of NFFT-fastsum objects applied to a vector x

Subroutines/power_of_a_matrix.m

  • Calculates the power of a matrix based on its spectral decomposition.

Subroutines/sample_idx_per_class.m

  • Randomly selects a given equal percentage/number of elements of label information for each class.

Subroutines/fastsum

For all MATLAB source files, please refer to the descriptions inside the files for further information.

REFERENCE [1] - Kai Bergermann, Martin Stoll, and Toni Volkmer. Semi-supervised Learning for Aggregated Multilayer Graphs Using Diffuse Interface Methods and Fast Matrix Vector Products. SIAM Journal on Mathematics of Data Science, 3(2):758-785, 2021. Available at https://doi.org/10.1137/20M1352028

[2] - Dominik Alfke, Daniel Potts, Martin Stoll, Toni Volkmer. NFFT meets Krylov methods: Fast matrix-vector products for the graph Laplacian of fully connected networks. Front. Appl. Math. Stat. 4:61, 2018. Available at http://dx.doi.org/10.1007/s00211-016-0861-7

PREREQUISITES

  • This software has been tested in MATLAB R2018a, but it should run in older versions as well.
  • The MATLAB interface of the NFFT-based fast summation has been compiled for x64 Windows/Linux/macosX using GCC with flag -march=haswell. Therefore, it may not work on older CPUs (below Intel i3/i5/i7-4xxx or AMD Excavator/4th gen Bulldozer) as well as on some Intel Atom/Pentium CPUs. If you want to compile it yourself, please follow these instructions:
    • Download and unpack the toolbox from https://www-user.tu-chemnitz.de/~potts/nfft/
    • Run the 'configure' script with option --enable-openmp --with-matlab=/PATH/TO/MATLAB/HOME/FOLDER.
    • Run 'make' and optionally 'make check'.
    • Copy the compiled MATLAB interface matlab/fastsum/fastsummex.mex* to the Multilayer-SSL-NFFT-Examples/Subroutines/fastsum/ folder.

LICENSE

  • Most of this software (cf. the header of each file) is distributed under the GNU General Public License v2. See COPYING for the full license text. If that file is not available, see http://www.gnu.org/licenses/.

AUTHORS

  • Dominik Alfke, TU Chemnitz, dominik.alfke(at)math.tu-chemnitz.de
  • Kai Bergermann, TU Chemnitz, kai.bergermann(at)math.tu-chemnitz.de
  • Toni Volkmer, TU Chemnitz, toni.volkmer(at)math.tu-chemnitz.de

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