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Persistence Learning

This code implements the kernel(s) for persistence diagrams proposed in the following two publications. Please use the provided BibTeX entries when citing our work.

    author    = {R.~Reininghaus and U.~Bauer and S.~Huber and R.~Kwitt},
    title     = {A Stable Multi-scale Kernel for Topological Machine Learning},
    booktitle = {CVPR},
    year      = {2015}}
    author    = {R.~Kwitt and S.~Huber and M.~Niethammer and W.~Lin and U.~Bauer},
    title     = {Statistical Topological Data Analysis - A Kernel Perspective},
    booktitle = {NIPS},
    year      = {2015}}



The core of the code is in diagram_distance.cpp and depends on DIPHA which is included as a submodule. After you have checked out the repository via

git clone

you can checkout the submodule(s) via

git pull
git submodule update --init --recursive   

Once this has finished, change into the code/diagram_distance directory and create a build directory, then use cmake or ccmake to configure the build process, e.g.,

cd code/dipha-pss
mkdir build
cd build
cmake ...

If you want to build diagram_distance without support for the Fast-Gauss-Transform, simply take the standard settings as they are. In case you do not have MPI support on your machine, you can install, e.g., OpenMPI. On MacOS (using homebrew) this can simply be done via

brew install open-mpi

Compiling support for the Fast-Gauss-Transform

To enable support for kernel computation via the (improved) Fast-Gauss-Transform we need the figtree library. This can be obtained from the original author Vlad I. Morariu from here or you use our fork (which contains a few small fixes to eliminate compiler warnings) which was checked-out during the git submodule update into code/external/figtree.

Next, compile figtree to build a static library. This is done, since we will call (in our experiments) diagram_distance from MATLAB and we want to avoid having to set the library path. To compile figtree simply type

cd code/external/figtree
make FIGTREE_LIB_TYPE=static

Once this is done, we can configure dipha-pss to use the library. This is done by entering the cmake GUI, enabling the USE_FGT flag and then setting the correct paths for the FIGTREE_INCLUDE and FIGTREE_LIB variable. In our example, using the command line this would look like (from scratch)

cd dipha-pss
mkdir build
cd build
cmake .. -DUSE_FGT=ON \
  -DFIGTREE_LIB=../../external/figtree/lib \

This will compile diagram_distance and enable the --use_fgt option.

Compiling DIPHA

You will need to compile DIPHA (i.e., the submodule that we checked out earlier) in case you want to compute your own persistence diagrams. DIPHA also uses cmake, so the process should be fairly simple. The standard process would look like

cd code/external/dipha
mkdir build
cd build
cmake ..


In the following, we show some examples which reproduce some of the results in the CVPR and NIPS paper. Note that these examples use MATLAB code (also contained in the repository).


We start with a simple timing experiment, where we do not use persistence diagrams computed from data, but simple create random persistence diagrams for measuring performance of the kernel.

cd 'code/matlab';
stat = pl_test_timing('/tmp/test', 50:50:200, 20, 0, 1);

This will run diagram_distance (1) with and (2) without support for FGT and output timing results (in seconds) for both variants. In particular, we start with diagrams of 50 random points and increase this to 200 in steps of 50 points. In every run, 20 such diagrams are created, resulting in 20x20 Gram matrices. The output is written to /tmp/test which is created in case it does not exist.

Averaging PSS feature maps

In this example, we take a large (random) sample of points from a double annulus, then draw a couple of small random samples from that collection, compute persistence diagrams (from the small samples) and eventually average the corresponding PSS feature maps. The full collection of points and three exemplary random samples are shown in the figure below:


This is a good example to illustrate how to compute persistence diagrams from distance matrices using DIPHA. In particular, for each random sample, the distance matrix simply is the pairwise Euclidean distance between all the points in each random sample.

The full functionality is implemented in the MATLAB function pl_experiment_pss_average.m. To produce the results from the NIPS 2015 paper (see reference above), we additionally provide a .mat file pl_experiment_pss_average_NIPS15.mat which you can load and pass to the script. This sets the configuration (e.g., radii of annuli, seed, etc.) we used in the paper. We run the script as follows:

cd 'code/matlab';
out_dir = '/tmp/out';
load ../../data/pl_experiment_pss_average_NIPS15.mat
result = pl_experiment_pss_average(pl_experiment_pss_average_NIPS15, out_dir);

This writes all output files to /tmp/out including (1) the persistence diagrams, (2) the distance matrices, (3) plots of all the samples, (4) PSS feature maps and (5) the average PSS feature map. The computed persistence diagrams are also available as part of the result structure that is returned by the script.

Simple classification with SVMs

In this demonstration, we will create toy data from the annuli as before, however, this time the objective is to distinguish samples drawn from a single annulus and samples drawn from a double-annulus based on their persistence diagrams.

First, we create the sample data. We will use /tmp/ as our output directory.

out_dir = '/tmp';
cd code/matlab;

for i=1:10
    % create filename
    filename = fullfile(out_dir, sprintf('dmat_%.3d.dipha', cnt));
    % Draw 100 points, center [0, 0], inner radius = 1, outer radius = 2
    points = pl_sample_annulus([0,0], 1, 2, 100, -1)';
    D = squareform(pdist(points));
    save_distance_matrix(D, filename);
for i=1:10
    filename = fullfile(out_dir, sprintf('dmat_%.3d.dipha', cnt));
    % Draw 100 points, two centers [0, 1] and [0, -1]
    points = pl_sample_linked_annuli( ...
        100, [0 1], 1, 1.5, [0 -1], 0.5, 1, -1);
    D = squareform(pdist(points));
    save_distance_matrix(D, filename);

Next, we compute persistence diagrams (by hand), using DIPHA. The inputs are (as before) the distance matrices we just created from the point samples. We use a simple bash script (e.g., to do this job. Just set the variable DIPHA_BINARY to the correct path to the dipha binary.

for i in `seq 1 20`; do
    SRC_FILE=`printf "dmat_%.3d.dipha" ${i}`
    DST_FILE=`printf "dmat_%.3d.pd" ${i}`
    CMD="${DIPHA_BINARY} --upper_dim 2 ${SRC_FILE} ${DST_FILE}"

Move this script to /tmp/ (since this was the output directory in the MATLAB code) and execute it:

cd /tmp/
chmod +x

We can now use the PSS kernel to compute the Gram matrix, i.e., the matrix of pairwise kernel evaluations between all created persistence diagrams. This can be done very easily, since diagram_distance accepts a ASCII file as input, where all persistence diagrams are listed (one per line). In /tmp/ we create such a list via

cd /tmp/
find . -name 'dmat*.pd' > diagrams.list

Finally, we execute diagram_distance and compute features up to dimension two (set via ---dim). In our example, we compute the kernel for 1-dimensional features and set the time parameter of the kernel (set via --time) to 0.1.

cd code/dipha-pss/build/bin
./diagram_distance --inner_product --time 0.1 --dim 1 /tmp/diagrams.list > /tmp/kernel.txt

The kernel matrix, saved as /tmp/kernel.txt can now be used, e.g., to train a SVM classifier. We will use libsvm for that purpose, in particular, the MATLAB interface to libsvm (see the libsvm documentation on how to compile the MATLAB interface).

labels = [ones(10,1);ones(10,1)*2]; % Create labels for training
pos = randsample(1:20,15); % Indices of diagrams used for training
neg = setdiff(1:20,pos);   % Indices of diagrams used for testing
model = svmtrain(labels(pos),[(1:length(pos))' kernel(pos,pos)], '-t 4 -c 1');
[pred,acc,~] = svmpredict(labels(neg), [(1:5)' kernel(neg,pos)], model);

Ideally, we get an accuracy of 100%, simply because the problem is also very easy. In the demonstrations that follows, we will use more realistic data. This example just illustrates the basic pipeline when we want to use the kernel in a classification setup.

Using shapes as input data

In both the CVPR and the NIPS paper, we experiment with persistence diagrams obtained from surfaces of 3D shapes. In particular, filtrations are computed via sublevel sets of a function defined on a simplical complex (given by the triangulated surface mesh of the 3D shape in that case).


In the following steps, we demonstrate a full processing pipeline to reproduce the results for one dataset of the NIPS paper. In particular, we go from 3D shapes, represented as surface meshes, to persistence diagrams and then compute a two-sample hypothesis test.

We provide the full dataset of 3D corpus callosum shapes from the NIPS paper for research purposes. The datasets of the CVPR paper (i.e., SHREC 2014) can be found online - The processing pipeline is very similar, except that the hypothesis test is replaced by a support vector machine for classification (similar to our previous experiment).

Additional 3rd party code

For the full pipeline to work, we will need some additional MATLAB code which will make our life easier when dealing with meshes. In particular, we need:

  1. STLRead
  2. iso2mesh
  3. (Scale-Invariant) Heat-Kernel Signature

The script pl_setup.m expects these software packages to be available under code/external .


The data that we use are segmentations of the corpus callosum, i.e., a structure in our brain that connects the two hemispheres. These segmentations are binary masks (in 3D) for which we also have a surface mesh available (i.e., part of the output of the segmentation process). The data can be found at:

  • Download raw meshes (STL files)
  • Download preprocessed meshes for MATLAB
  • Download meta data (i.e., subject information)

If you are not specifically interested in processing the meshes from scratch, we recommend using the MATLAB data, since all meshes have been checked already.

Processing pipeline

We use the pre-processed meshes in this example. The functions that will be used within our scripts are:

  • utilities/pl_mmd.m
  • utilities/pl_normalize_kernel.m
  • utilities/pl_mesh2dipha.m
  • utilities/pl_mesh2hks.m
  • experiments/pl_experiment_OASIS_run_dipha.m
  • experiments/pl_experiment_OASIS_run_mmd.m

First, we download the MATLAB data from the provided link and save the .mat file OASIS_cc.mat, e.g., at /tmp/OASIS_cc.mat. To compute simplicial complexes, we then use the MATLAB function pl_experiment_OASIS_run_dipha.m in the following way:

pl_experiment_OASIS_run_dipha('/tmp/OASIS_cc.mat', 'OASIS_cc', 'cc', '/tmp/output')

This will process all meshes in the .mat file and write the simplicial complexes, as well as the persistence diagrams into /tmp/output with all files prefixed by cc_.

Second, we group all subjects according to the desired membership (here: demented vs. non-demented at the first visit), compute the kernel and finally run the kernel two-sample test. The grouping and visit information is available as meta-data and should be extracted into /tmp/output. You will also need to configure the two-sample test and provide the options (in the form of a MATLAB struct) to pl_experiment_OASIS_run_mmd.m. An exemplary option file (to reproduce the results of the NIPS paper is provided in the data directory).

load ../data/options_pl_experiment_OASIS_run_dipha.mat
[K,pval] = pl_experiment_OASIS_run_mmd('/tmp/OASIS_cc.mat','OASIS_cc', 'cc', ...
  '/tmp/dipha/Group.txt', ...
  '/tmp/dipha/Visit.txt', ...

Mesh preprocessing

In case you don't want to use the preprocessed meshes with already computed Heat-Kernel signatures (e.g., when you want to set the Heat-Kernel signature times yourself), unpack the raw data, e.g., to /tmp/output and also save the meta-data, e.g., at /tmp/output/.

subjects = pl_experiment_OASIS_subjects('/tmp/output/Subjects.txt');

Next, edit the pl_mesh2hks.m MATLAB script and set the desired parameters. Then, execute

OASIS_cc = pl_mesh2hks('/tmp/output/raw_OASIS_cc', subjects, 1000);

The final parameter 1000 is the scaling of the mesh (to avoid numerical instabilities). This produces the same result that is already available in the MATLAB file OASIS_cc.mat.


Persistence Learning






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