Is there a way to make numpy.argmin() as fast as min()?

Question

I\'m trying to find the minimum array indices along one dimension of a very large 2D numpy array. I\'m finding that this is very slow (already tried speeding it up with bott

Answer 1

In [1]: import numpy as np

In [2]: a = np.random.rand(3000, 16000)

In [3]: %timeit a.min(axis=0)
1 loops, best of 3: 421 ms per loop

In [4]: %timeit a.argmin(axis=0)
1 loops, best of 3: 1.95 s per loop

In [5]: %timeit a.min(axis=1)
1 loops, best of 3: 302 ms per loop

In [6]: %timeit a.argmin(axis=1)
1 loops, best of 3: 303 ms per loop

In [7]: %timeit a.T.argmin(axis=1)
1 loops, best of 3: 1.78 s per loop

In [8]: %timeit np.asfortranarray(a).argmin(axis=0)
1 loops, best of 3: 1.97 s per loop

In [9]: b = np.asfortranarray(a)

In [10]: %timeit b.argmin(axis=0)
1 loops, best of 3: 329 ms per loop

Maybe min is smart enough to do its job sequentially over the array (hence with cache locality), and argmin is jumping around the array (causing a lot of cache misses)?

Anyway, if you're willing to keep randvals as a Fortran-ordered array from the start, it'll be faster, though copying into Fortran-ordered doesn't help.

Answer 2

I was just hitting the same problem, and found the large difference in performance when axis 0 is selected for finding the minimum. As suggested, the problem seems to be related to internally copying the array.

I devised a rather simple-minded workaround using Cython that simultaneously establishes the minimum values and their position in a 2D numpy array along a given axis. Note that for axis = 0, the algorithm works on an array of columns (maximum number specified by the constant blocksize - here set equivalent to 8 kByte of data) simultaneously to make good use of the cache:

%%cython -c=-O2 -c=-march=native

import numpy as np
cimport numpy as np 
cimport cython
from libc.stdint cimport uint32_t

@cython.boundscheck(False)
@cython.wraparound(False)
cdef void _minargmin_2d_64_axis_0(uint32_t[:] minloc, double[:] minimum, double[:, :] arr) nogil:
    """
    Find the minimum and argmin for a 2D array of 64-bit floats along axis 0
    Parameters:
    -----------
    arr: numpy_array, dtype=np.float64, shape=(m, n)
       The array for which the minima should be computed.
    minloc: numpy_array, dtype=np.uint32, shape=(n)
       Stores the rows where the minima occur for each column.
    minimum: numpy_array, dtype=np.float64, shape=(n)
       The minima along each column.
    """

    # Columns of the matrix are accessed in blocks to increase cache performance.
    # Specify the number of columns here:

    DEF blocksize = 1024

    cdef int i, j, k
    cdef double minim[blocksize]
    cdef uint32_t minimloc[blocksize]

    # Read blocks of data to make good use of the cache

    cdef int blocks
    blocks  = arr.shape[1] / blocksize
    remains = arr.shape[1] % blocksize

    for i in xrange(0, blocksize * blocks, blocksize):

        for k in xrange(blocksize):
            minim[k]    = arr[0, i + k]
            minimloc[k] = 0

        for j in xrange(1, arr.shape[0]):

            for k in xrange(blocksize):

                if arr[j, i + k] < minim[k]:
                    minim[k] = arr[j, i + k]
                    minimloc[k] = j

        for k in xrange(blocksize):
            minimum[i + k] = minim[k]
            minloc[i + k]  = minimloc[k]

    # Work on the final 'incomplete' block

    i = blocksize * blocks

    for k in xrange(remains):
        minim[k]    = arr[0, i + k]
        minimloc[k] = 0

    for j in xrange(1, arr.shape[0]):

        for k in xrange(remains):

            if arr[j, i + k] < minim[k]:
                minim[k] = arr[j, i + k]
                minimloc[k] = j

    for k in xrange(remains):
        minimum[i + k] = minim[k]
        minloc[i + k]  = minimloc[k]

    # Done!


@cython.boundscheck(False)
@cython.wraparound(False)
cdef void _minargmin_2d_64_axis_1(uint32_t[:] minloc, double[:] minimum, double[:, :] arr) nogil:
    """
    Find the minimum and argmin for a 2D array of 64-bit floats along axis 1
    Parameters:
    -----------
    arr: numpy_array, dtype=np.float64, shape=(m, n)
       The array for which the minima should be computed.
    minloc: numpy_array, dtype=np.uint32, shape=(m)
       Stores the rows where the minima occur for each row.
    minimum: numpy_array, dtype=np.float64, shape=(m)
       The minima along each row.
    """
    cdef int i
    cdef int j
    cdef double minim
    cdef uint32_t minimloc

    for i in xrange(arr.shape[0]):
        minim = arr[i, 0]
        minimloc = 0

        for j in xrange(1, arr.shape[1]):
            if arr[i, j] < minim:
                minim = arr[i, j]
                minimloc = j

        minimum[i] = minim
        minloc[i]  = minimloc


@cython.boundscheck(False)
@cython.wraparound(False)
cdef void _minargmin_2d_32_axis_0(uint32_t[:] minloc, float[:] minimum, float[:, :] arr) nogil:
    """
    Find the minimum and argmin for a 2D array of 32-bit floats along axis 0
    Parameters:
    -----------
    arr: numpy_array, dtype=np.float32, shape=(m, n)
       The array for which the minima should be computed.
    minloc: numpy_array, dtype=np.uint32, shape=(n)
       Stores the rows where the minima occur for each column.
    minimum: numpy_array, dtype=np.float32, shape=(n)
       The minima along each column.
    """

    # Columns of the matrix are accessed in blocks to increase cache performance.
    # Specify the number of columns here:  

    DEF blocksize = 2048

    cdef int i, j, k
    cdef float minim[blocksize]
    cdef uint32_t minimloc[blocksize]

    # Read blocks of data to make good use of the cache

    cdef int blocks
    blocks  = arr.shape[1] / blocksize
    remains = arr.shape[1] % blocksize

    for i in xrange(0, blocksize * blocks, blocksize):

        for k in xrange(blocksize):
            minim[k]    = arr[0, i + k]
            minimloc[k] = 0

        for j in xrange(1, arr.shape[0]):

            for k in xrange(blocksize):

                if arr[j, i + k] < minim[k]:
                    minim[k] = arr[j, i + k]
                    minimloc[k] = j

        for k in xrange(blocksize):
            minimum[i + k] = minim[k]
            minloc[i + k]  = minimloc[k]

    # Work on the final 'incomplete' block

    i = blocksize * blocks

    for k in xrange(remains):
        minim[k]    = arr[0, i + k]
        minimloc[k] = 0

    for j in xrange(1, arr.shape[0]):

        for k in xrange(remains):

            if arr[j, i + k] < minim[k]:
                minim[k] = arr[j, i + k]
                minimloc[k] = j

    for k in xrange(remains):
        minimum[i + k] = minim[k]
        minloc[i + k]  = minimloc[k]

    # Done!

@cython.boundscheck(False)
@cython.wraparound(False)
cdef void _minargmin_2d_32_axis_1(uint32_t[:] minloc, float[:] minimum, float[:, :] arr) nogil:
    """
    Find the minimum and argmin for a 2D array of 32-bit floats along axis 1
    Parameters:
    -----------
    arr: numpy_array, dtype=np.float32, shape=(m, n)
       The array for which the minima should be computed.
    minloc: numpy_array, dtype=np.uint32, shape=(m)
       Stores the rows where the minima occur for each row.
    minimum: numpy_array, dtype=np.float32, shape=(m)
       The minima along each row.
    """
    cdef int i
    cdef int j
    cdef float minim
    cdef uint32_t minimloc

    for i in xrange(arr.shape[0]):
        minim = arr[i, 0]
        minimloc = 0

        for j in xrange(1, arr.shape[1]):
            if arr[i, j] < minim:
                minim = arr[i, j]
                minimloc = j

        minimum[i] = minim
        minloc[i]  = minimloc

def Min_Argmin(array_2d, axis = 1):
    """
    Find the minima and corresponding locations (argmin) of a two-dimensional
    numpy array of floats along a given axis simultaneously, and returns them
    as a tuple of arrays (min_2d, argmin_2d).

    (Note: float16 arrays will be internally transformed to float32 arrays.)
    ----------
    array_2d: numpy_array, dtype=np.float32 or np.float64, shape=(m, n)
       The array for which the minima should be computed.
    axis : int, 0 or 1 (default) : 
        The axis along which minima are computed.
    min_2d: numpy_array, dtype=np.uint8, shape=(m) or shape=(n):
       The array where the minima along the given axis are stored.
    argmin_2d:
       The array storing the row/column where the minimum occurs.
    """

    # Sanity checks:
    if len(array_2d.shape) != 2:
        raise IndexError('Min_Argmin: Number of dimensions of array must be 2')

    if not (axis == 0 or axis == 1):
        raise ValueError('Min_Argmin: Invalid axis specified')

    arr_type = array_2d.dtype

    if not arr_type in ('float16', 'float32', 'float64'):
        raise ValueError('Min_Argmin: Not a float array')

    # Transform float16 arrays
    if arr_type == 'float16':
        array_2d = array_2d.astype(np.float32)

    # Run analysis

    if arr_type == 'float64':

        # Double accuracy

        min_array    = np.zeros(array_2d.shape[1 - axis], dtype = np.float64)
        argmin_array = np.zeros(array_2d.shape[1 - axis], dtype = np.uint32)

        if (axis == 0):
            _minargmin_2d_64_axis_0(argmin_array, min_array, array_2d)

        else:
            _minargmin_2d_64_axis_1(argmin_array, min_array, array_2d)

    else:

        # Single accuracy

        min_array    = np.zeros(array_2d.shape[1 - axis], dtype = np.float32)
        argmin_array = np.zeros(array_2d.shape[1 - axis], dtype = np.uint32)

        if (axis == 0):
            _minargmin_2d_32_axis_0(argmin_array, min_array, array_2d)

        else:
            _minargmin_2d_32_axis_1(argmin_array, min_array, array_2d)

        # Transform back to float16 type as necessary

        if arr_type == 'float16':
            min_array = min_array.astype(np.float16)


    # Return results
    return min_array, argmin_array

The code can be placed and compiled in an IPython notebook cell after loading Cython support:

%load_ext Cython

and then called in the form:

min_array, argmin_array = Min_Argmin(two_dim_array, axis = 0 or 1)

Timing example:

random_array = np.random.rand(20000, 20000).astype(np.float32)

%timeit min_array, argmin_array = Min_Argmin(random_array, axis = 0)
%timeit min_array, argmin_array = Min_Argmin(random_array, axis = 1)

1 loops, best of 3: 405 ms per loop
1 loops, best of 3: 307 ms per loop

For comparison:

%%timeit 
min_array    = random_array.min(axis = 0)
argmin_array = random_array.argmin(axis = 0)

1 loops, best of 3: 10.3 s per loop

%%timeit 
min_array    = random_array.min(axis = 1)
argmin_array = random_array.argmin(axis = 1)

1 loops, best of 3: 423 ms per loop

So, there is a significant speedup for axis = 0 (and still a small advantage for axis = 1, if one is interested in both minimum and location).

Answer 3

I just took a look at the source code, and while I don't fully understand why things are being done the way they are, this is what happens:

1. np.min is basically a call to np.minimum.reduce.

2. np.argmin first moves the axis you want to operate on to the end of the shape tuple, then makes it a contiguous array, which of course triggers a copy of the full array unless the axis was the last one to begin with.

Since a copy is being made, you can get creative and try to instantiate cheaper arrays:

a = np.random.rand(1000, 2000)

def fast_argmin_axis_0(a):
    matches = np.nonzero((a == np.min(a, axis=0)).ravel())[0]
    rows, cols = np.unravel_index(matches, a.shape)
    argmin_array = np.empty(a.shape[1], dtype=np.intp)
    argmin_array[cols] = rows
    return argmin_array

In [8]: np.argmin(a, axis=0)
Out[8]: array([230, 532, 815, ..., 670, 702, 989], dtype=int64)

In [9]: fast_argmin_axis_0(a)
Out[9]: array([230, 532, 815, ..., 670, 702, 989], dtype=int64)

In [10]: %timeit np.argmin(a, axis=0)
10 loops, best of 3: 27.3 ms per loop

In [11]: %timeit fast_argmin_axis_0(a)
100 loops, best of 3: 15 ms per loop

I wouldn't go as far as calling the current implementation a bug, since there may be good reasons for numpy doing what it does the way it does it, but that this kind of trickery can speed up what should be a highly optimized function, strongly suggests that things could be done better.