Ondrian - 2 months ago 19

Python Question

I am using cython to compute a pairwise distance matrix using a custom metric as a faster alternative to scipy.spatial.distance.pdist.

My metric has the form

`def mymetric(u,v,w):`

np.sum(w * (1 - np.abs(np.abs(u - v) / np.pi - 1))**2)

and the pairwise distance using scipy can be computed as

`x = sp.spatial.distance.pdist(r, metric=lambda u, v: mymetric(u, v, w))`

Here,

`r`

`m`

`n`

`m`

`n`

`w`

`n`

Since in my problem

`m`

`m = 2000`

`n = 10`

I implemented a simple function in cython that computes the pairwise distance and immediately got very promising results -- speedup of over 500x.

`import numpy as np`

cimport numpy as np

import cython

from libc.math cimport fabs, M_PI

@cython.wraparound(False)

@cython.boundscheck(False)

def pairwise_distance(np.ndarray[np.double_t, ndim=2] r, np.ndarray[np.double_t, ndim=1] w):

cdef int i, j, k, c, size

cdef np.ndarray[np.double_t, ndim=1] ans

size = r.shape[0] * (r.shape[0] - 1) / 2

ans = np.zeros(size, dtype=r.dtype)

c = -1

for i in range(r.shape[0]):

for j in range(i + 1, r.shape[0]):

c += 1

for k in range(r.shape[1]):

ans[c] += w[k] * (1.0 - fabs(fabs(r[i, k] - r[j, k]) / M_PI - 1.0))**2.0

return ans

I wanted to speed up the computation some more using OpenMP, however, the following solution is roughly 3 times slower than the serial version.

`import numpy as np`

cimport numpy as np

import cython

from cython.parallel import prange, parallel

cimport openmp

from libc.math cimport fabs, M_PI

@cython.wraparound(False)

@cython.boundscheck(False)

def pairwise_distance_omp(np.ndarray[np.double_t, ndim=2] r, np.ndarray[np.double_t, ndim=1] w):

cdef int i, j, k, c, size, m, n

cdef np.double_t a

cdef np.ndarray[np.double_t, ndim=1] ans

m = r.shape[0]

n = r.shape[1]

size = m * (m - 1) / 2

ans = np.zeros(size, dtype=r.dtype)

with nogil, parallel(num_threads=8):

for i in prange(m, schedule='dynamic'):

for j in range(i + 1, m):

c = i * (m - 1) - i * (i + 1) / 2 + j - 1

for k in range(n):

ans[c] += w[k] * (1.0 - fabs(fabs(r[i, k] - r[j, k]) / M_PI - 1.0))**2.0

return ans

I don't know why is it actually slower, but I tried to introduce the following changes.

`ans`

`import numpy as np`

cimport numpy as np

import cython

from cython.parallel import prange, parallel

cimport openmp

from libc.math cimport fabs, M_PI

from libc.stdlib cimport malloc, free

@cython.wraparound(False)

@cython.boundscheck(False)

def pairwise_distance_omp_2(np.ndarray[np.double_t, ndim=2] r, np.ndarray[np.double_t, ndim=1] w):

cdef int k, l, c, m, n

cdef Py_ssize_t i, j, d

cdef size_t size

cdef int *ci, *cj

cdef np.ndarray[np.double_t, ndim=1, mode="c"] ans

cdef np.ndarray[np.double_t, ndim=2, mode="c"] data

cdef np.ndarray[np.double_t, ndim=1, mode="c"] weight

data = np.ascontiguousarray(r, dtype=np.float64)

weight = np.ascontiguousarray(w, dtype=np.float64)

m = r.shape[0]

n = r.shape[1]

size = m * (m - 1) / 2

ans = np.zeros(size, dtype=r.dtype)

cj = <int*> malloc(size * sizeof(int))

ci = <int*> malloc(size * sizeof(int))

c = -1

for i in range(m):

for j in range(i + 1, m):

c += 1

ci[c] = i

cj[c] = j

with nogil, parallel(num_threads=8):

for d in prange(size, schedule='guided'):

for k in range(n):

ans[d] += weight[k] * (1.0 - fabs(fabs(data[ci[d], k] - data[cj[d], k]) / M_PI - 1.0))**2.0

return ans

For all functions, I am using the following

`.pyxbld`

`def make_ext(modname, pyxfilename):`

from distutils.extension import Extension

return Extension(name=modname,

sources=[pyxfilename],

extra_compile_args=['-O3', '-march=native', '-ffast-math', '-fopenmp'],

extra_link_args=['-fopenmp'],

)

I have zero experience with cython and know only basics of C. I would appreciate any suggestion of what may be the cause of this unexpected behavior, or even, how to rephrase my question better.

`@cython.cdivision(True)`

@cython.wraparound(False)

@cython.boundscheck(False)

def pairwise_distance_2(np.ndarray[np.double_t, ndim=2] r, np.ndarray[np.double_t, ndim=1] w):

cdef int i, j, k, c, size

cdef np.ndarray[np.double_t, ndim=1] ans

cdef np.double_t accumulator, tmp

size = r.shape[0] * (r.shape[0] - 1) / 2

ans = np.zeros(size, dtype=r.dtype)

c = -1

for i in range(r.shape[0]):

for j in range(i + 1, r.shape[0]):

c += 1

accumulator = 0

for k in range(r.shape[1]):

tmp = (1.0 - fabs(fabs(r[i, k] - r[j, k]) / M_PI - 1.0))

accumulator += w[k] * (tmp*tmp)

ans[c] = accumulator

return ans

`@cython.cdivision(True)`

@cython.wraparound(False)

@cython.boundscheck(False)

def pairwise_distance_omp_2d(np.ndarray[np.double_t, ndim=2] r, np.ndarray[np.double_t, ndim=1] w):

cdef int i, j, k, c, size, m, n

cdef np.ndarray[np.double_t, ndim=1] ans

cdef np.double_t accumulator, tmp

m = r.shape[0]

n = r.shape[1]

size = m * (m - 1) / 2

ans = np.zeros(size, dtype=r.dtype)

with nogil, parallel(num_threads=8):

for i in prange(m, schedule='dynamic'):

for j in range(i + 1, m):

c = i * (m - 1) - i * (i + 1) / 2 + j - 1

accumulator = 0

for k in range(n):

tmp = (1.0 - fabs(fabs(r[i, k] - r[j, k]) / M_PI - 1.0))

ans[c] += w[k] * (tmp*tmp)

return ans

When I try to apply the

`accumulator`

`Error compiling Cython file:`

------------------------------------------------------------

...

c = i * (m - 1) - i * (i + 1) / 2 + j - 1

accumulator = 0

for k in range(n):

tmp = (1.0 - fabs(fabs(r[i, k] - r[j, k]) / M_PI - 1.0))

accumulator += w[k] * (tmp*tmp)

ans[c] = accumulator

^

------------------------------------------------------------

pdist.pyx:207:36: Cannot read reduction variable in loop body

Full code:

`@cython.cdivision(True)`

@cython.wraparound(False)

@cython.boundscheck(False)

def pairwise_distance_omp(np.ndarray[np.double_t, ndim=2] r, np.ndarray[np.double_t, ndim=1] w):

cdef int i, j, k, c, size, m, n

cdef np.ndarray[np.double_t, ndim=1] ans

cdef np.double_t accumulator, tmp

m = r.shape[0]

n = r.shape[1]

size = m * (m - 1) / 2

ans = np.zeros(size, dtype=r.dtype)

with nogil, parallel(num_threads=8):

for i in prange(m, schedule='dynamic'):

for j in range(i + 1, m):

c = i * (m - 1) - i * (i + 1) / 2 + j - 1

accumulator = 0

for k in range(n):

tmp = (1.0 - fabs(fabs(r[i, k] - r[j, k]) / M_PI - 1.0))

accumulator += w[k] * (tmp*tmp)

ans[c] = accumulator

return ans

Answer

I haven't timed this myself so it's possible this might not help too much, however:

If you run `cython -a`

to get an annotated version of your initial attempt (`pairwise_distance_omp`

) you'll find the `ans[c] += ...`

line is yellow, suggesting it's got Python overhead. A look at that the C corresponding to that line suggests that it's checking for divide by zero. One key part of it starts:

```
if (unlikely(M_PI == 0)) {
```

You know this will never be true (and in any case you'd probably live with NaN values rather than an exception if it was). You can avoid this check by adding the following extra decorator to the function:

```
@cython.cdivision(True)
# other decorators
def pairwise_distance_omp # etc...
```

This cuts out quite a bit of C code, including bits that have to be run in a single thread. The flip-side is that most of that code should never be run, and the compiler should probably be able to work that out, so it isn't clear how much difference that will make.

**Second suggestion:**

```
# at the top
cdef np.double_t accumulator, tmp
# further down later in the loop:
c = i * (m - 1) - i * (i + 1) / 2 + j - 1
accumulator = 0
for k in range(r.shape[1]):
tmp = (1.0 - fabs(fabs(r[i, k] - r[j, k]) / M_PI - 1.0))
accumulator += w[k] * (tmp*tmp)
ans[c] = accumulator
```

This has two advantages hopefully: 1) `tmp*tmp`

should probably be quicker than floating point exponent to the power of 2. 2) You avoid reading from the `ans`

array, which might be a bit slow because the compiler always has to be careful that some other thread hasn't changed it (even though you know it shouldn't have).