Robust PCA Example

Robust PCA is an awesome relatively new method for factoring a matrix into a low rank component and a sparse component. This enables really neat applications for outlier detection, or models that are robust to outliers.



In [209]:

    
%matplotlib inline

Make Some Toy Data



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import matplotlib.pyplot as plt
import numpy as np



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def mk_rot_mat(rad=np.pi / 4):
    rot = np.array([[np.cos(rad),-np.sin(rad)], [np.sin(rad), np.cos(rad)]])
    return rot



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rot_mat = mk_rot_mat( np.pi / 4)
x = np.random.randn(100) * 5
y = np.random.randn(100)
points = np.vstack([y,x])



In [213]:

    
rotated = np.dot(points.T, rot_mat).T

Add Some Outliers to Make Life Difficult



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outliers = np.tile([15,-10], 10).reshape((-1,2))



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pts = np.vstack([rotated.T, outliers]).T

Compute SVD on both the clean data and the outliery data



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U,s,Vt = np.linalg.svd(rotated)
U_n,s_n,Vt_n = np.linalg.svd(pts)

Just 10 outliers can really screw up our line fit!



In [217]:

    
plt.ylim([-20,20])
plt.xlim([-20,20])
plt.scatter(*pts)
pca_line = np.dot(U[0].reshape((2,1)), np.array([-20,20]).reshape((1,2)))
plt.plot(*pca_line)
rpca_line = np.dot(U_n[0].reshape((2,1)), np.array([-20,20]).reshape((1,2)))
plt.plot(*rpca_line, c='r')









    Out[217]:





[<matplotlib.lines.Line2D at 0x10ba22d90>]

Now the robust pca version!



In [218]:

    
import tga



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reload(tga)









    Out[219]:





<module 'tga' from 'tga.pyc'>



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import logging
logger = logging.getLogger(tga.__name__)
logger.setLevel(logging.INFO)

Factor the matrix into L (low rank) and S (sparse) parts



In [227]:

    
X = pts.copy()
v = tga.tga(X.T, eps=1e-5, k=1, p=0.0)

And have a look at this!



In [228]:

    
plt.ylim([-20,20])
plt.xlim([-20,20])
plt.scatter(*pts)
tga_line = np.dot(v[0].reshape((2,1)), np.array([-20,20]).reshape((1,2)))
plt.plot(*tga_line)
#plt.scatter(*L, c='red')









    Out[228]:





[<matplotlib.lines.Line2D at 0x10c7dbf10>]



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