Fast Fast Hadamard Transform
FFHT (Fast Fast Hadamard Transform) is a library that provides a heavily optimized C99 implementation of the Fast Hadamard Transform. FFHT also provides a thin Python wrapper that allows to perform the Fast Hadamard Transform on one-dimensional NumPy arrays.
The Hadamard Transform is a linear orthogonal map defined on real vectors whose length is a power of two. For the precise definition, see the Wikipedia entry. The Hadamard Transform has been recently used a lot in various machine learning and numerical algorithms.
FFHT uses AVX to speed up the computation.
The header file fht.h exports two functions: int fht_float(float *buf, int log_n) and int fht_double(double *buf, int log_n). The
only difference between them is the type of vector entries. So, in what follows,
we describe how the version for floats fht_float works.
The function fht_float takes two parameters:
bufis a pointer to the data on which one needs to perform the Fast Hadamard Transform.log_nis the binary logarithm of the length ofbuffer. That is, the length is equal to2^log_n.
The return value is -1 if the input is invalid and is zero otherwise.
A header-only version of the library is provided in fht_header_only.h.
In addition to the Fast Hadamard Transform, we provide two auxiliary programs:
test_float and test_double, which are implemented in C99. The exhaustively
test and benchmark the library.
FFHT has been tested on 64-bit versions of Linux, OS X and Windows (the latter is via Cygwin).
To install the Python package, run python setup.py install. The script
example.py shows how to use FFHT from Python.
Benchmarks
Below are the times for the Fast Hadamard Transform for vectors of various lengths. The benchmarks were run on a machine with Intel Core i7-6700K and 2133 MHz DDR4 RAM. We compare FFHT, FFTW 3.3.6, and fht by Nicolas Barbey.
Let us stress that FFTW is a great versatile tool, and the authors of FFTW did not try to optimize the performace of the Fast Hadamard Transform. On the other hand, FFHT does one thing (the Fast Hadamard Transform), but does it extremely well.
| Vector size | FFHT (float) | FFHT (double) | FFTW 3.3.6 (float) | FFTW 3.3.6 (double) | fht (float) | fht (double) |
|---|---|---|---|---|---|---|
| 210 | 0.31 us | 0.49 us | 4.48 us | 7.72 us | 17.4 us | 19.3 us |
| 220 | 0.68 ms | 1.39 ms | 8.81 ms | 17.07 ms | 29.8 ms | 35.0 ms |
| 227 | 0.22 s | 0.50 s | 2.08 s | 3.57 s | 6.89 s | 7.49 s |
Troubleshooting
For some versions of OS X the native clang compiler (that mimicks gcc) may
not recognize the availability of AVX. A solution for this problem is to use a
genuine gcc (say from Homebrew) or to use -march=corei7-avx
instead of -march=native for compiler flags.
A symptom of the above happening is the undefined macros __AVX__.
Related Work
FFHT has been created as a part of FALCONN: a library for similarity search over high-dimensional data. FALCONN's underlying algorithms are described and analyzed in the following research paper:
Alexandr Andoni, Piotr Indyk, Thijs Laarhoven, Ilya Razenshteyn and Ludwig Schmidt, "Practical and Optimal LSH for Angular Distance", NIPS 2015, full version available at arXiv:1509.02897
This is the right paper to cite, if you use FFHT for your research projects.
Acknowledgments
We thank Ruslan Savchenko for useful discussions.
Thanks to:
- Clement Canonne
- Michal Forisek
- Rati Gelashvili
- Daniel Grier
- Dhiraj Holden
- Justin Holmgren
- Aleksandar Ivanovic
- Vladislav Isenbaev
- Jacob Kogler
- Ilya Kornakov
- Anton Lapshin
- Rio LaVigne
- Oleg Martynov
- Linar Mikeev
- Cameron Musco
- Sam Park
- Sunoo Park
- Amelia Perry
- Andrew Sabisch
- Abhishek Sarkar
- Ruslan Savchenko
- Vadim Semenov
- Arman Yessenamanov
for helping us with testing FFHT.