Test_Optimization


Simulated Test Data


In [1]:
%%writefile Test/test.py

import numpy as np
import matplotlib.pyplot as plt

# Source matrix
a = ((np.arange(200)+1)-100)/100
a = np.concatenate((a,a,a,a,a), axis=0) 
b = np.sin((np.arange(1000)+1)/20)
S_test= np.vstack((b,a)).T
# Mixing matrix
A = np.array([0.291, 0.6557, -0.5439, 0.5572]).reshape((2, 2))
# test data
X_test = S_test @ A

def test(ic):
    np.random.seed(1)
    plt.subplot(121)
    plt.plot(np.arange(1000)+1, ic[:,0])
    plt.title("IC 1")
    plt.subplot(122)
    plt.plot(np.arange(1000)+1, ic[:,1])
    plt.title("IC 2")
pass


Overwriting Test/test.py

In [2]:
%run Test/test.py

True Source Components


In [3]:
test(S_test)


sklearn FastICA ICs


In [4]:
from sklearn.decomposition import FastICA
ica = FastICA(n_components=2)
S_sklearn= ica.fit_transform(X_test)

In [5]:
test(S_sklearn)


Real World Data (fMRI)

This is a subsample of real fMRI data from the 1000 Connectome Project.


In [6]:
%%writefile Test/fMRI.py

import pandas as pd
df = pd.read_csv('PC.csv')
df.drop('Unnamed: 0',1,inplace=True)
fMRI = df.as_matrix()


Overwriting Test/fMRI.py

In [7]:
%run Test/fMRI.py

In [8]:
fMRI.shape


Out[8]:
(30000, 20)

Test Algorithm Accuracy


In [9]:
%run Source/fastICA_0.py
%run Source/fastICA_1.py
%run Source/fastICA_3.py
%run Source/fastICA_scipy.py
%run Source/fastICA_jit.py
%run Source/fastICA_ne.py

In [10]:
test(fastICA_0(X_test,f = "logcosh",n_comp =2, alpha = 1,maxit = 200, tol = 0.0001)['S'])



In [11]:
test(fastICA_1(X_test,f = "logcosh",n_comp =2, alpha = 1,maxit = 200, tol = 0.0001)['S'])



In [12]:
test(fastICA_3(X_test,f = "logcosh",n_comp =2, alpha = 1,maxit = 200, tol = 0.0001)['S'])



In [13]:
test(fastICA_scipy(X_test,f = "logcosh",n_comp =2, alpha = 1,maxit = 200, tol = 0.0001)['S'])



In [14]:
test(fastICA_jit(X_test,f = "logcosh",n_comp =2, alpha = 1,maxit = 200, tol = 0.0001)['S'])



In [15]:
test(fastICA_ne(X_test,f = "logcosh",n_comp =2, alpha = 1,maxit = 200, tol = 0.0001)['S'])



In [16]:
%load_ext cython

In [17]:
%%cython -a

import numpy as np
import scipy.linalg 
from numpy import dot
import cython

@cython.cdivision(True)
@cython.wraparound(False)  
@cython.boundscheck(False)
cdef sym_decorrelation_cython(double[:,:] W):
    cdef double[:,:] K = dot(W,W.T)
        
    eigen = np.linalg.eigh(K)
    cdef double[:,:] dias = np.diag(1.0/np.sqrt(eigen[0])) 
    cdef double[:,:] u = eigen[1]
    
    return dot(dot(dot(u,dias),u.T),W)

@cython.wraparound(False)  
@cython.boundscheck(False)
cdef g_logcosh_cython(double[:,:] wx, double alpha):
    """derivatives of logcosh"""
    return np.tanh(np.multiply(alpha,wx))

@cython.wraparound(False)  
@cython.boundscheck(False)
cdef gprime_logcosh_cython(double[:,:] wx, double alpha):
    """second derivatives of logcosh"""
    return np.multiply(alpha,(1-np.square(np.tanh(np.multiply(alpha,wx)))))

@cython.cdivision(True)
@cython.wraparound(False)  
@cython.boundscheck(False)
cdef g_exp_cython(double[:,:] wx, double alpha):
    """derivatives of exp"""
    return wx * np.exp(-np.square(wx)/2)

@cython.cdivision(True)
@cython.wraparound(False)  
@cython.boundscheck(False)
cdef gprime_exp_cython(double[:,:] wx, double alpha):
    """second derivatives of exp"""
    return (1-np.square(wx)) * np.exp(-np.square(wx)/2)

@cython.cdivision(True)
@cython.wraparound(False)  
@cython.boundscheck(False)
def fastICA_cython(double[:,:] X, str f,double alpha, int n_comp,int maxit, double tol):
    """FastICA algorithm for several units"""
    cdef int n = X.shape[0]
    cdef int p = X.shape[1]
    #check if n_comp is valid
    if n_comp is None:
        n_comp = min(n,p)
    elif n_comp > min(n,p):
        print("n_comp is too large")
        n_comp = min(n,p)
        
    #centering
    #by subtracting the mean of each column of X (array).
    X = X - np.mean(X,axis=0)[None,:]
    X = X.T

    #whitening
    svd = np.linalg.svd(dot(X,X.T) / n)
    k = dot(np.diag(1/np.sqrt(svd[1])),svd[0].T)
    k = k[:n_comp,:] 
    X1 = dot(k,X)

    # initial random weght vector
    w_init = np.random.normal(size=(n_comp, n_comp))
    W = sym_decorrelation_cython(w_init)
    lim = 1
    it = 0
      
    # The FastICA algorithm
    while lim > tol and it < maxit :
        wx = dot(W,X1)
        if f == "logcosh":
            gwx = g_logcosh_cython(wx,alpha)
            g_wx = gprime_logcosh_cython(wx,alpha)
        elif f == "exp":
            gwx = g_exp_cython(wx,alpha)
            g_wx = gprime_exp_cython(wx,alpha)
        else:
            print("doesn't support this approximation negentropy function")
            
        W1 = np.dot(gwx,X1.T)/X1.shape[1] - np.dot(np.diag(g_wx.mean(axis=1)),W)
        W1 = sym_decorrelation_cython(W1)
        it = it +1
        lim = np.max(np.abs(np.abs(np.diag(dot(W1,W.T))) - 1.0))
        W = W1

        S = dot(W,X1)
        A = scipy.linalg.pinv2(dot(W,k))
        X_re = dot(A,S)
    return{'X':X1.T,'X_re':X_re.T,'A':A.T,'S':S.T}


Out[17]:
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In [18]:
test(fastICA_cython(X_test,f = "logcosh",n_comp =2, alpha = 1,maxit = 200, tol = 0.0001)['S'])


line_profiler


In [19]:
%%writefile Test/fastICA_00.py

import pandas as pd
import numpy as np
from sklearn import preprocessing

df = pd.read_csv('PC.csv')
df.drop('Unnamed: 0',1,inplace=True)
X = df.as_matrix()

def sym_decorrelation(W):
    """ Symmetric decorrelation """
    K = np.dot(W, W.T)
    s, u = np.linalg.eigh(K) 
    W = (u @ np.diag(1.0/np.sqrt(s)) @ u.T) @ W
    return W

def g_logcosh(wx,alpha):
    """derivatives of logcosh"""
    return np.tanh(alpha * wx)
def gprime_logcosh(wx,alpha):
    """second derivatives of logcosh"""
    return alpha * (1-np.square(np.tanh(alpha*wx)))
# exp
def g_exp(wx,alpha):
    """derivatives of exp"""
    return wx * np.exp(-np.square(wx)/2)
def gprime_exp(wx,alpha):
    """second derivatives of exp"""
    return (1-np.square(wx)) * np.exp(-np.square(wx)/2)

@profile
def fastICA_00(X, f,alpha=None, n_comp=None,maxit=200, tol=1e-04):
    """FastICA algorithm for several units"""
    n,p = X.shape
    #check if n_comp is valid
    if n_comp is None:
        n_comp = min(n,p)
    elif n_comp > min(n,p):
        print("n_comp is too large")
        n_comp = min(n,p)
        
    #centering
    #by subtracting the mean of each column of X (array).
    X = preprocessing.scale(X,axis = 0,with_std=False)
    X = X.T

    #whitening
    svd = np.linalg.svd(X @ (X.T) / n)
    k = np.diag(1/np.sqrt(svd[1])) @ (svd[0].T)
    k = k[:n_comp,:] 
    X1 = k @ X

    # initial random weght vector
    w_init = np.random.normal(size=(n_comp, n_comp))
    W = sym_decorrelation(w_init)
    lim = 1
    it = 0
    
    
    # The FastICA algorithm
    if f == "logcosh":
        while lim > tol and it < maxit :
            wx = W @ X1
            gwx = g_logcosh(wx,alpha)
            g_wx = gprime_logcosh(wx,alpha)
            W1 = np.dot(gwx,X1.T)/X1.shape[1] - np.dot(np.diag(g_wx.mean(axis=1)),W)
            W1 = sym_decorrelation(W1)
            it = it +1
            lim = np.max(np.abs(np.abs(np.diag(W1 @ W.T)) - 1.0))
            W = W1

        S = W @ X1
        A = np.linalg.inv(W @ k)
        X_re = A @ S
        return{'X':X1.T,'X_re':X_re.T,'A':A.T,'S':S.T}

    elif f == "exp":
        while lim > tol and it < maxit :
            wx = W @ X1
            gwx = g_exp(wx,alpha)
            g_wx = gprime_exp(wx,alpha)
            W1 = np.dot(gwx,X1.T)/X1.shape[1] - np.dot(np.diag(g_wx.mean(axis=1)),W)
            W1 = sym_decorrelation(W1)
            it = it +1
            lim = np.max(np.abs(np.abs(np.diag(W1 @ W.T)) - 1.0))
            W = W1

        S = W @ X1
        A = np.linalg.inv(W @ k)
        X_re = A @ S
        return{'X':X1.T,'X_re':X_re.T,'A':A.T,'S':S.T}

    else:
        print("doesn't support this approximation negentropy function")
        
fastICA_00(X,f = "logcosh",n_comp =20, alpha = 1,maxit = 200, tol = 0.0001)


Overwriting Test/fastICA_00.py

In [20]:
! pip install line_profiler


Requirement already satisfied: line_profiler in /opt/conda/lib/python3.5/site-packages
Requirement already satisfied: IPython>=0.13 in /opt/conda/lib/python3.5/site-packages (from line_profiler)
Requirement already satisfied: simplegeneric>0.8 in /opt/conda/lib/python3.5/site-packages (from IPython>=0.13->line_profiler)
Requirement already satisfied: pexpect; sys_platform != "win32" in /opt/conda/lib/python3.5/site-packages (from IPython>=0.13->line_profiler)
Requirement already satisfied: decorator in /opt/conda/lib/python3.5/site-packages (from IPython>=0.13->line_profiler)
Requirement already satisfied: traitlets in /opt/conda/lib/python3.5/site-packages (from IPython>=0.13->line_profiler)
Requirement already satisfied: setuptools>=18.5 in /opt/conda/lib/python3.5/site-packages/setuptools-23.0.0-py3.5.egg (from IPython>=0.13->line_profiler)
Requirement already satisfied: pickleshare in /opt/conda/lib/python3.5/site-packages (from IPython>=0.13->line_profiler)
Requirement already satisfied: six in /opt/conda/lib/python3.5/site-packages (from traitlets->IPython>=0.13->line_profiler)
Requirement already satisfied: ipython-genutils in /opt/conda/lib/python3.5/site-packages (from traitlets->IPython>=0.13->line_profiler)

In [21]:
!kernprof -l Test/fastICA_00.py


Wrote profile results to fastICA_00.py.lprof

In [22]:
!python -m line_profiler fastICA_00.py.lprof


Timer unit: 1e-06 s

Total time: 1.80312 s
File: Test/fastICA_00.py
Function: fastICA_00 at line 31

Line #      Hits         Time  Per Hit   % Time  Line Contents
==============================================================
    31                                           @profile
    32                                           def fastICA_00(X, f,alpha=None, n_comp=None,maxit=200, tol=1e-04):
    33                                               """FastICA algorithm for several units"""
    34         1            3      3.0      0.0      n,p = X.shape
    35                                               #check if n_comp is valid
    36         1            1      1.0      0.0      if n_comp is None:
    37                                                   n_comp = min(n,p)
    38         1            2      2.0      0.0      elif n_comp > min(n,p):
    39                                                   print("n_comp is too large")
    40                                                   n_comp = min(n,p)
    41                                                   
    42                                               #centering
    43                                               #by subtracting the mean of each column of X (array).
    44         1         2495   2495.0      0.1      X = preprocessing.scale(X,axis = 0,with_std=False)
    45         1            2      2.0      0.0      X = X.T
    46                                           
    47                                               #whitening
    48         1        21497  21497.0      1.2      svd = np.linalg.svd(X @ (X.T) / n)
    49         1           67     67.0      0.0      k = np.diag(1/np.sqrt(svd[1])) @ (svd[0].T)
    50         1            3      3.0      0.0      k = k[:n_comp,:] 
    51         1         1811   1811.0      0.1      X1 = k @ X
    52                                           
    53                                               # initial random weght vector
    54         1           60     60.0      0.0      w_init = np.random.normal(size=(n_comp, n_comp))
    55         1        30385  30385.0      1.7      W = sym_decorrelation(w_init)
    56         1            2      2.0      0.0      lim = 1
    57         1            1      1.0      0.0      it = 0
    58                                               
    59                                               
    60                                               # The FastICA algorithm
    61         1            1      1.0      0.0      if f == "logcosh":
    62        39          107      2.7      0.0          while lim > tol and it < maxit :
    63        38        41781   1099.5      2.3              wx = W @ X1
    64        38       781346  20561.7     43.3              gwx = g_logcosh(wx,alpha)
    65        38       841917  22155.7     46.7              g_wx = gprime_logcosh(wx,alpha)
    66        38        62263   1638.5      3.5              W1 = np.dot(gwx,X1.T)/X1.shape[1] - np.dot(np.diag(g_wx.mean(axis=1)),W)
    67        38        14799    389.4      0.8              W1 = sym_decorrelation(W1)
    68        38           85      2.2      0.0              it = it +1
    69        38         1829     48.1      0.1              lim = np.max(np.abs(np.abs(np.diag(W1 @ W.T)) - 1.0))
    70        38           70      1.8      0.0              W = W1
    71                                           
    72         1          726    726.0      0.0          S = W @ X1
    73         1          254    254.0      0.0          A = np.linalg.inv(W @ k)
    74         1         1612   1612.0      0.1          X_re = A @ S
    75         1            5      5.0      0.0          return{'X':X1.T,'X_re':X_re.T,'A':A.T,'S':S.T}
    76                                           
    77                                               elif f == "exp":
    78                                                   while lim > tol and it < maxit :
    79                                                       wx = W @ X1
    80                                                       gwx = g_exp(wx,alpha)
    81                                                       g_wx = gprime_exp(wx,alpha)
    82                                                       W1 = np.dot(gwx,X1.T)/X1.shape[1] - np.dot(np.diag(g_wx.mean(axis=1)),W)
    83                                                       W1 = sym_decorrelation(W1)
    84                                                       it = it +1
    85                                                       lim = np.max(np.abs(np.abs(np.diag(W1 @ W.T)) - 1.0))
    86                                                       W = W1
    87                                           
    88                                                   S = W @ X1
    89                                                   A = np.linalg.inv(W @ k)
    90                                                   X_re = A @ S
    91                                                   return{'X':X1.T,'X_re':X_re.T,'A':A.T,'S':S.T}
    92                                           
    93                                               else:
    94                                                   print("doesn't support this approximation negentropy function")

Bottlenecks

We used line profiler to identify the bottlenecks. From the result, it can be observed that the biggest bottleneck in the fastica function are "np.tanh" (in "g_logcosh" and "gprime_logcosh") and 'np.linalg.svd' function, which take about 90% percent of the time when running fastica function. The next bottleneck are 'preprocessing.scale'and matrix dot product.

Parallelism


In [23]:
import numpy as np
import numexpr as ne

In [24]:
# "big" array
TT = np.random.normal(size=(10**5,10**3))

In [25]:
%%time
res = np.tanh(TT)


CPU times: user 3.14 s, sys: 192 ms, total: 3.33 s
Wall time: 3.35 s

In [26]:
from concurrent.futures import ThreadPoolExecutor
import multiprocessing as mp

In [27]:
%%time
with ThreadPoolExecutor(max_workers=4) as pool:
    res = pool.map(np.tanh, [i for i in TT])


CPU times: user 9.64 s, sys: 3.19 s, total: 12.8 s
Wall time: 9.3 s

In [28]:
%%time
with mp.Pool(processes=4) as pool:
    res = pool.map(np.tanh, [i for i in TT])


CPU times: user 1.75 s, sys: 2.47 s, total: 4.22 s
Wall time: 4.71 s
  • Numexpr

In [29]:
ne.use_vml=False

In [30]:
%%time
res = ne.evaluate('tanh(TT)')


CPU times: user 560 ms, sys: 272 ms, total: 832 ms
Wall time: 238 ms

In [31]:
ne.use_vml=True

In [32]:
%%time
res = ne.evaluate('tanh(TT)')


CPU times: user 492 ms, sys: 188 ms, total: 680 ms
Wall time: 119 ms

VML can accelerate computations by a 50% using a single CPU.


In [33]:
ne.set_num_threads(1)


Out[33]:
8

In [34]:
%%time
res = ne.evaluate('tanh(TT)')


CPU times: user 488 ms, sys: 148 ms, total: 636 ms
Wall time: 635 ms

In [35]:
ne.detect_number_of_cores()


Out[35]:
8

In [36]:
ne.set_num_threads(ne.detect_number_of_cores())


Out[36]:
1

In [37]:
%%time
res = ne.evaluate('tanh(TT)')


CPU times: user 488 ms, sys: 224 ms, total: 712 ms
Wall time: 113 ms

More threads make computations faster

Time Comparison


In [38]:
%timeit -r2 -n4 fastICA_0(fMRI,f = "logcosh",n_comp =20, alpha = 1,maxit = 200, tol = 0.0001)


4 loops, best of 2: 2.34 s per loop

In [39]:
%timeit -r2 -n4 fastICA_1(fMRI,f = "logcosh",n_comp =20, alpha = 1,maxit = 200, tol = 0.0001)


4 loops, best of 2: 2.35 s per loop

In [40]:
%timeit -r2 -n4 fastICA_3(fMRI,f = "logcosh",n_comp =20, alpha = 1,maxit = 200, tol = 0.0001)


4 loops, best of 2: 45.5 ms per loop

In [41]:
%timeit -r2 -n4 fastICA_scipy(fMRI,f = "logcosh",n_comp =20, alpha = 1,maxit = 200, tol = 0.0001)


4 loops, best of 2: 2.69 s per loop

In [42]:
%timeit -r2 -n4 fastICA_jit(fMRI,f = "logcosh",n_comp =20, alpha = 1,maxit = 200, tol = 0.0001)


4 loops, best of 2: 47.3 ms per loop

In [43]:
%timeit -r2 -n4 fastICA_ne(fMRI,f = "logcosh",n_comp =20, alpha = 1,maxit = 200, tol = 0.0001)


4 loops, best of 2: 12.5 ms per loop

In [44]:
%timeit -r2 -n4 fastICA_cython(fMRI,f = "logcosh",n_comp =20, alpha = 1,maxit = 200, tol = 0.0001)


4 loops, best of 2: 3.22 s per loop