How to speed up functions on numpy arrays

Question

I'm writing a code to fit a 2D function in (5, 5)-shaped numpy arrays. The fitting is done by maximizing with scipy.optimize.minimize, so a lot of functions get called a tremendous amount of times during the iterative process.

These are the first cProfile results:

Ordered by: internal time

ncalls  tottime  percall  cumtime  percall filename:lineno(function)
1392579   60.849    0.000   60.849    0.000 maxima.py:288(derfs)
1392578   34.243    0.000   34.243    0.000 maxima.py:279(dexp)
696289   31.186    0.000  152.278    0.000 maxima.py:322(ll_jac)
1392578   26.670    0.000   26.670    0.000 maxima.py:283(derf)
3639673   23.772    0.000   23.772    0.000 {method 'reduce' of 'numpy.ufunc' objects}
696290   10.440    0.000   53.965    0.000 maxima.py:308(logll)
1392579    8.882    0.000   69.731    0.000 maxima.py:295(lambda_g)
13202    5.216    0.000  215.145    0.016 lbfgsb.py:198(_minimize_lbfgsb)
3494647    4.468    0.000   32.576    0.000 fromnumeric.py:1621(sum)

And these are the functions

import numpy as np
from scipy.special import erf

def dexp(x0, sigma, x=np.arange(5)):
    xx1 = (x + 1 - x0)/sigma
    xx = (x - x0)/sigma
    return np.exp(-xx1*xx1) - np.exp(-xx*xx)

def derf(x0, sigma, x=np.arange(5)):
    return erf((x + 1 - x0) / sigma) - erf((x - x0) / sigma)

def derfs(x0, y0, sigma, xy=np.arange(5)):
    i = erf((xy + 1 - x0) / sigma) - erf((xy - x0) / sigma)
    j = erf((xy + 1 - y0) / sigma) - erf((xy - y0) / sigma)
    return i[:, np.newaxis] * j

def lambda_g(x0, y0, fwhm, factor=0.09*np.pi, f2=0.6):
    return factor * fwhm**2 * derfs(x0, y0, fwhm * f2)

Objective function and its gradient, that needs some optimizing as well:

def logll(parameters, *args):
    """ Log-likelihood function
    """
    A, x0, y0, bkg = parameters
    fwhm, area = args

    lambda_p = A * lambda_g(x0, y0, fwhm) + bkg
    return -np.sum(area * np.log(lambda_p) - lambda_p)

def ll_jac(parameters, *args):
    """ This is the Jacobian of the log-likelihood function
    """
    A, x0, y0, bkg = parameters
    fwhm, area = args

    derfx = derf(x0, fwhm*0.6)
    derfy = derf(y0, fwhm*0.6)
    lambda1 = lambda_g(x0, y0, fwhm)
    factor = 1 - area/(A * lambda1 + bkg)

    jac = np.zeros(4)
    # dL/d(A)
    # The derivative of lambda_g is lambda_g(A=1)
    jac[0] = np.sum(factor*lambda1)
    # dL/d(x0) and dL/d(y0)
    jac12 = -0.3*A*fwhm*np.sqrt(np.pi)
    jac[1] = jac12*np.sum(dexp(x0, fwhm * 0.6)[:, np.newaxis] * derfy*factor)
    jac[2] = jac12*np.sum(dexp(y0, fwhm * 0.6)[:, np.newaxis] * derfx*factor)
    # dL/d(bkg)
    jac[3] = np.sum(factor)

    return jac

On my PC,

%timeit dexp(2.3, 2.)
10000 loops, best of 3: 16 µs per loop

%timeit derf(2.3, 2.)
100000 loops, best of 3: 14 µs per loop

%timeit derfs(2.3, 2.2, 2.)
10000 loops, best of 3: 32.2 µs per loop

Is it possible to speed these up? What do you recommend me to do? cython?numba? Is there anything else I can try before diving into those modules?

For starters i would change your ** 2 to x*x (see stackoverflow.com/questions/1019740/…) — Stanton
– Stanton, Commented Aug 24, 2015 at 22:17
Can you post your objective function as well? Also, I wouldn't worry about replacing the squares with x * x, as it is handled by numpy and I'd guess it's pretty optimized. — benavente
– benavente, Commented Aug 24, 2015 at 22:25
Thank you both, I'm updating the question taking your suggestions into account — Federico Barabas
– Federico Barabas, Commented Aug 24, 2015 at 22:29
I don't think cython or numba will help because you aren't doing iterations in these functions. They are calling np.exp and erf which are already compiled code. — hpaulj
– hpaulj, Commented Aug 24, 2015 at 22:52
I agree, but the sum of np.exps and erfs are not compiled... right? Would it make any difference? — Federico Barabas
– Federico Barabas, Commented Aug 24, 2015 at 22:54

kikocorreoso · Accepted Answer · 2015-08-24 23:13:30Z

3

You can try to perform less operations. For instance you could do the following:

def dexp2(x0, sigma, x=np.arange(5)):
    a = (x - x0) / sigma
    return np.exp(-(a + 1/sigma)**2) - np.exp(-(a*a))

~35%-40% better.

def derf2(x0, sigma, x=np.arange(5)):
    a = (x - x0) / sigma
    return erf(a + 1 / sigma) - erf(a)

~50% better.

def derfs2(x0, y0, sigma, xy=np.arange(5)):
    a = (xy - x0) / sigma
    b = (xy - y0) / sigma
    i = erf(a + 1 / sigma) - erf(a)
    j = erf(b + 1 / sigma) - erf(b)
    return i[:, np.newaxis] * j

~50% better.

These are just micro optimizations. In general, if your inputs have this size I think numba and/or cython can't help too much. Maybe it is better to check which optimization method of the available ones suits better for your special case and try to initialise it in a clever way.

answered Aug 24, 2015 at 23:13

kikocorreoso

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2 Comments

Federico Barabas Over a year ago

With your suggestions and changing the optimization method to Newton-CG I reduced computation time 10-fold!

kikocorreoso Over a year ago

Great. Something similar happened to me some time ago with an optimization.

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