Really Fast Persistence Kernels On GPUs

There is an increasing appetite for using the persistent homology of data as features for machine learning. For support vector machines in particular, there have been numerous definitions of persistence kernels that allow SVMs to be trained on, and then to classify, persistence diagrams. One notable example is the so-called heat kernel of Reininghaus et al.

A persistence kernel K evaluated on persistence diagrams X and Y tends to have a simple definition, but one that may involve computing some numerical value for each pair of points in the cartesian product X×Y. If one needs to compute a lot of kernel values K(Xᵢ, Xⱼ), for example if one has a lot of data objects to classify, and each of the Xⱼ's are relatively large, one is in a bind.

One approach is to exploit that the problem is data-parallel across the Xᵢ's and for example distribute the computation of the kernel values across tens or hundreds of CPUs (do ask me if you want me to make public an OpenMPI implementation that has been tested succesfully on hundreds of CPUs).

Computing K(X, Y) is also a task that lends itself nicely to GPU computations. And I wanted to finally get my hands dirty with GPU programming. Hence RFPKOG.

See https://nonempty.org/software/rfpkog/ for more information.

About

The heat kernel (AKA PSSK) is currently the only well-supported one. There is experimental support for the arctan version of the PWGK.

The code is licensed under the GNU Lesser General Public License, version 3.0.

Normalization and GUDHI compatibility

Important: the normalization prefactor and the bandwidth parameter in GUDHI's CPU implementation uses a different convention than the sigma parameter in RFPKOG. RFPKOG follows the paper's convention. Specifically, if K'_σ denotes GUDHI's kernel with bandwidth parameter σ, then RFPKOG with parameter σ is

K_σ(X, Y) = K'_{2*sqrt(σ)}/(2*sqrt(2πσ))

Installation

RFPKOG needs only:

• A C++ compiler that supports C++17. C++11 will suffice with very small modifications to the code. Debian/Ubuntu package name g++.

• OpenCL 1.2 or above. Typically dependent on your GPU hardware.

• Meson for building. Debian/Ubuntu package name meson.

To build and install:

meson setup -Doptimization=2 -Dprefix=/where/you/want/rfpkog/to/go build
cd build
meson compile
meson install 

You can also build and run RFPKOG without installing it. The location of the OpenCL code (confusingly also called a "kernel"… can we stop naming more things kernels already?) is hardcoded based on the installation directory, but you can override it by setting the environment variable RFPKOG_KERNEL_FILE_NAME.

Running

Run with --help to see a help text. An example invocation is

rfpkog --kernel heat -d 1 --devices 0 --platform 0 -f 1.0 -s 1.0 list_1.txt list_2.txt

See rfpkog --help for more information.

Multiple GPUs are supported. The set of pairs to compute is parallelized over the GPUs.

Benchmarks

RFPKOG was informally benchmarked on randomly generated persistence diagrams of various sizes. Comparisons with a straight-forward C++ implementation for CPUs, and with GUDHI's CPU implementation were made.

The following table refers to crude benchmarks for a single pair of persistence diagrams, each one of the given size.

Backend \ PD sizes	10k	100k	500k
CPU reference	1.0 s	104 s	—
CPU reference (double precision)	1.4 s	144 s	—
GUDHI CPU reference (double precision)	8.5 s	—	—
Intel UHD Graphics 630	0.1 s	11.4 s	—
Intel UHD Graphics 630 (double precision)	0.4 s	37.6	—
Nvidia GeForce GTX 1650	0.02 s	1.2 s	30 s
Nvidia GeForce GTX 1650 (double precision)	0.2 s	13.8 s	—
Nvidia Tesla T4	—	0.9 s	19 s
Nvidia Tesla T4 (double precision)	—	7.0 s	—

The CPU benchmarks, in italics above, were run on an Intel i7-9750H CPU. As a rough estimate, the current version provides a speedup factor of 100 from that CPU to a decent GPU.

To do

• Allow splitting really big persistence diagrams across GPUs and into parts.

• More careful selection of local work sizes.

• Turn the code into a library.

• ASCII input, and binary output.

• Tests.

• Better error handling.

• Support for more operating systems.

• Support for more kernels, like persistence weighted Gaussian kernels.

• More robust and standard options parsing.

Maybe to do

• Combine with the aforementioned MPI support to use a lot of GPUs.