In [68]:

    

import sys
sys.path.append('../scripts/')


    

In [285]:

    

import matplotlib.pyplot as plt
import numpy as np
import numpy.random as npr
%matplotlib inline
plt.rcParams['font.family']='Serif'


    

In [2]:

    

npr.seed(0)


    

In [3]:

    

X = np.vstack((npr.randn(1000,3),
               npr.randn(1000,3)+5))


    

In [ ]:

    

X = np.vstack(([]))


    

In [423]:

    

import itertools
dim=3
radius=1
corners = [(np.array(i)*radius+radius)/2 for i in itertools.product([-1,1],repeat=dim)]


    

In [424]:

    

def coord_str(array):
    str_a = [str(int(elem)) for elem in array]
    return "(" + ", ".join(str_a) + ")"


    

In [425]:

    

from scipy.spatial.distance import pdist,squareform
plt.imshow(squareform(pdist(corners)),interpolation='none',cmap='Blues')
plt.colorbar()
#plt.axis('off')
plt.yticks(np.arange(len(corners)),[coord_str(c) for c in corners])
plt.xticks(np.arange(len(corners)),[coord_str(c) for c in corners],rotation=30)


    

    
Out[425]:





([<matplotlib.axis.XTick at 0x139dd72b0>,
  <matplotlib.axis.XTick at 0x139dcf588>,
  <matplotlib.axis.XTick at 0x139dd5f60>,
  <matplotlib.axis.XTick at 0x139e0a208>,
  <matplotlib.axis.XTick at 0x139f047b8>,
  <matplotlib.axis.XTick at 0x139f08208>,
  <matplotlib.axis.XTick at 0x139f08c18>,
  <matplotlib.axis.XTick at 0x139f0b668>],
 <a list of 8 Text xticklabel objects>)






    

In [419]:

    

for d in range(1,12):
    corners = [np.array(i) for i in itertools.product([-1,1],repeat=d)]
    plt.imshow(squareform(pdist(corners)),interpolation='none')
    plt.colorbar()
    #plt.yticks(np.arange(len(corners)),[str(c) for c in corners])
    plt.title('{0}D hypercube'.format(d))
    plt.savefig('../figures/hypercube-d={0}.jpg'.format(d),dpi=300)
    plt.close()


    

In [400]:

    

for d in range(1,13):
    corners = [np.array(i) for i in itertools.product([-1,1],repeat=d)]
    plt.imshow(squareform(pdist(corners)),interpolation='none',cmap='Blues')
    plt.axis('off')
    #plt.colorbar()
    #plt.yticks(np.arange(len(corners)),[str(c) for c in corners])
    #plt.title('{0}D hypercube'.format(d))
    plt.savefig('../figures/blue-hypercube-nolabel-d={0}.jpg'.format(d),dpi=300)
    plt.close()


    

In [ ]:

    




    

In [182]:

    

corners = [np.array(i) for i in itertools.product([-1,1],repeat=13)]
plt.hist(pdist(corners),bins=10);


    

In [183]:

    

len(set(pdist(corners)))


    

    
Out[183]:





13

In [ ]:

    

print(corners[0])


    

In [279]:

    

n=100
X = np.vstack([npr.randn(n,dim) + c for c in corners])
Y = np.hstack([np.ones(n)*r for r in range(len(corners))])
X.shape,Y.shape


    

    
Out[279]:





((800, 3), (800,))

In [240]:

    

def generate_hypercube(dim=3,radius=4,n_tot=100):
    corners = [np.array(i)*radius for i in itertools.product([-1,1],repeat=dim)]
    n = int(n_tot / len(corners))
    X = np.vstack([npr.randn(n,dim) + c for c in corners])
    Y = np.hstack([np.ones(n)*r for r in range(len(corners))])
    return X,Y


    

In [318]:

    

X,Y = generate_hypercube(4,n_tot=1400)


    

In [319]:

    

from mpl_toolkits.mplot3d import Axes3D
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.scatter(X[:,0],X[:,1],X[:,2],c=Y,linewidths=0,alpha=0.5)


    

    
Out[319]:





<mpl_toolkits.mplot3d.art3d.Path3DCollection at 0x11a87d198>






    

In [335]:

    

plt.scatter(X[:,0],X[:,1],c=Y,linewidths=0,alpha=0.5)


    

    
Out[335]:





<matplotlib.collections.PathCollection at 0x11cb9d7f0>






    

In [321]:

    

from sklearn.decomposition import PCA
pca = PCA()
pca.fit(X)


    

    
Out[321]:





PCA(copy=True, n_components=None, whiten=False)

In [322]:

    

pca.explained_variance_ratio_


    

    
Out[322]:





array([ 0.25703139,  0.24887468,  0.24822486,  0.24586907])

In [323]:

    

X_ = pca.transform(X)[:,:2]


    

In [359]:

    

plt.scatter(X_[:,0],X_[:,1],c=Y,linewidths=0,alpha=0.5)
plt.tick_params(labelleft='off',
    labelbottom='off') # labels along the bottom edge are off


    

In [445]:

    

ds = list(range(3,7))
#algs = []
results = dict()
num_algs = 3
for i,d in enumerate(ds):
    X,Y = generate_hypercube(d,n_tot=1400)
    pca = PCA()
    pca.fit(X)
    X_ = pca.transform(X)[:,:2]
    plt.subplot(len(ds),num_algs,(num_algs*i)+1)
    #plt.title('PCA: {0}D hypercube'.format(d))
    if i == 0:
        plt.title('PCA')
    plt.scatter(X_[:,0],X_[:,1],c=Y,linewidths=0,alpha=0.5,s=3)
    plt.tick_params(labelleft='off',labelbottom='off')
    #plt.xlabel('PC1')
    #plt.ylabel('PC2')
    plt.ylabel('{0}D'.format(d))
    
    #X_tpe = X_tsne = X_
    
    tpe = TPE('complete')
    X_tpe = tpe.fit_transform(X)
    plt.subplot(len(ds),num_algs,(num_algs*i)+2)
    if i == 0:
        plt.title('TPE')
    #plt.title('TPE: {0}D hypercube'.format(d))
    plt.scatter(X_tpe[:,0],X_tpe[:,1],c=Y,linewidths=0,alpha=0.5,s=3)
    plt.tick_params(labelleft='off',labelbottom='off')
    
    tsne = TSNE()
    X_tsne = tsne.fit_transform(X)
    plt.subplot(len(ds),num_algs,(num_algs*i)+3)
    if i == 0:
        plt.title('t-SNE')
    plt.scatter(X_tsne[:,0],X_tsne[:,1],c=Y,linewidths=0,alpha=0.5,s=3)
    plt.tick_params(labelleft='off',labelbottom='off')
    
    results[d] = (X_,X_tpe,X_tsne)
    
plt.tight_layout()
plt.savefig('blah-complete.pdf')


    

In [446]:

    

for d in results:
    for a in results[d]:
        _,Y = generate_hypercube(d,n_tot=1400)
        print(one_nn_class_baseline(a,Y))
    print()


    

    




0.954285714286
0.999285714286
1.0

0.665948275862
0.999281609195
1.0

0.524709302326
0.99636627907
0.998546511628

0.33556547619
0.997023809524
1.0

In [482]:

    

d = 6
X_pca,X_tpe,X_tsne = results[d]
X,Y = generate_hypercube(d,n_tot=1400)
cluster_centers = np.array([np.mean(X[Y==y],0) for y in set(Y)])
cluster_centers_pca = np.array([np.mean(X_pca[Y==y],0) for y in set(Y)])
cluster_centers_tpe = np.array([np.mean(X_tpe[Y==y],0) for y in set(Y)])
cluster_centers_tsne = np.array([np.mean(X_tsne[Y==y],0) for y in set(Y)])


    

In [499]:

    

plt.scatter(pdist(cluster_centers_pca),pdist(cluster_centers),alpha=0.5,linewidths=0)
plt.xlabel("Inter-cluster distances in PCA embedding")
plt.ylabel("Ground-truth inter-cluster distances")
plt.title('PCA')
plt.figure()

plt.scatter(pdist(cluster_centers_tpe),pdist(cluster_centers),alpha=0.5,linewidths=0)
plt.xlabel("Inter-cluster distances in Approx. TPE embedding")
plt.ylabel("Ground-truth inter-cluster distances")
plt.title('Approx. TPE')

plt.figure()

plt.scatter(pdist(cluster_centers_tsne),pdist(cluster_centers),alpha=0.5,linewidths=0)
plt.xlabel("Inter-cluster distances in t-SNE embedding")
plt.ylabel("Ground-truth inter-cluster distances")
plt.title('t-SNE')
plt.figure()


    

    
Out[499]:





<matplotlib.figure.Figure at 0x13a3c3be0>






    













    













    













    






<matplotlib.figure.Figure at 0x13a3c3be0>

In [501]:

    

from sklearn.metrics import r2_score
r2_score(pdist(cluster_centers),pdist(cluster_centers_pca))


    

    
Out[501]:





-5.4105023128029384

In [502]:

    

r2_score(pdist(cluster_centers),pdist(cluster_centers_tpe))


    

    
Out[502]:





-14.074424698056925

In [503]:

    

r2_score(pdist(cluster_centers),pdist(cluster_centers_tsne))


    

    
Out[503]:





-44.05004443285263

In [487]:

    

plt.imshow(squareform(pdist(cluster_centers)),interpolation='none',cmap='Blues')
plt.title('Ground-truth inter-cluster distances')
plt.figure()
plt.imshow(squareform(pdist(cluster_centers_pca)),interpolation='none',cmap='Blues')
plt.title('PCA inter-cluster distances')
plt.figure()
plt.imshow(squareform(pdist(cluster_centers_tpe)),interpolation='none',cmap='Blues')
plt.title('TPE inter-cluster distances')
plt.figure()
plt.imshow(squareform(pdist(cluster_centers_tsne)),interpolation='none',cmap='Blues')
plt.title('t-SNE inter-cluster distances')


    

    
Out[487]:





<matplotlib.text.Text at 0x136be02b0>






    













    













    













    

In [508]:

    

from sklearn.manifold import MDS
mds = MDS(dissimilarity='precomputed')
mds_centers = mds.fit_transform(squareform(pdist(cluster_centers)))


    

In [509]:

    

r2_score(pdist(cluster_centers),pdist(mds_centers))


    

    
Out[509]:





-1.9765479419190863

In [510]:

    

plt.imshow(squareform(pdist(mds_centers)),interpolation='none',cmap='Blues')
plt.title('Directly optimized inter-cluster distances')


    

    
Out[510]:





<matplotlib.text.Text at 0x13a0bb240>






    

In [513]:

    

plt.scatter(pdist(mds_centers),pdist(cluster_centers),alpha=0.5,linewidths=0)
plt.xlabel("Directly optimized inter-cluster distances (2D)")
plt.ylabel("Ground-truth inter-cluster distances (6D)")
plt.figure()


    

    
Out[513]:





<matplotlib.figure.Figure at 0x13a38f0f0>






    













    






<matplotlib.figure.Figure at 0x13a38f0f0>

In [340]:

    

for i,d in enumerate(ds):
    print(i,d)


    

    




0 3
1 4
2 5
3 6

In [331]:

    

len(set(Y))


    

    
Out[331]:





16

In [332]:

    

2**6


    

    
Out[332]:





64

In [329]:

    

2**7


    

    
Out[329]:





128

In [297]:

    

X_.shape,Y.shape


    

    
Out[297]:





((800, 2), (800,))

In [298]:

    

from sklearn.decomposition import FastICA
ica = FastICA(n_components=2)
X_ica = ica.fit_transform(X)
plt.scatter(X_ica[:,0],X_ica[:,1],c=Y,linewidths=0,alpha=0.5)


    

    
Out[298]:





<matplotlib.collections.PathCollection at 0x112060b70>






    

In [299]:

    

from sklearn import neighbors
def one_nn_baseline(X,Y):
    one_nn_X = neighbors.kneighbors_graph(X,2)
    one_nn_Y = neighbors.kneighbors_graph(Y,2)
    sames = 0
    for i in range(len(X)):
        neighbor_X = one_nn_X[i].indices[one_nn_X[i].indices!=i][0]
        neighbor_Y = one_nn_Y[i].indices[one_nn_Y[i].indices!=i][0]
        if neighbor_X == neighbor_Y:
            sames+=1
    return 1.0*sames / len(X)

def one_nn_class_baseline(X,Y):
    one_nn = neighbors.kneighbors_graph(X,2)
    inds = np.zeros(len(X),dtype=int)
    for i in range(len(X)):
        inds[i] = [ind for ind in one_nn[i].indices if ind != i][0]
    preds = Y[inds]
    return 1.0*sum(preds==Y) / len(Y)


    

In [300]:

    

one_nn_baseline(X_,X),one_nn_class_baseline(X_,Y)


    

    
Out[300]:





(0.2125, 0.85875000000000001)

In [301]:

    

one_nn_class_baseline(X_ica,Y)


    

    
Out[301]:





0.85875000000000001

In [302]:

    

# example, if we have clusters centered on corners of a cube, then there's no linear projection that preserves all the clusters


    

In [303]:

    

from tpe import TPE


    

In [304]:

    

tpe = TPE()


    

In [305]:

    

X_tpe = tpe.fit_transform(X)


    

In [306]:

    

plt.scatter(X_tpe[:,0],X_tpe[:,1],c=Y,linewidths=0,alpha=0.5)


    

    
Out[306]:





<matplotlib.collections.PathCollection at 0x11094b278>






    

In [307]:

    

one_nn_class_baseline(X_tpe,Y)


    

    
Out[307]:





0.99875000000000003

In [308]:

    

from sklearn.manifold import Isomap
iso = Isomap()
X_iso = iso.fit_transform(X)
plt.scatter(X_iso[:,0],X_iso[:,1],c=Y,linewidths=0,alpha=0.5)


    

    
Out[308]:





<matplotlib.collections.PathCollection at 0x12442d6a0>






    

In [309]:

    

one_nn_class_baseline(X_iso,Y)


    

    
Out[309]:





0.43375000000000002

In [310]:

    

from sklearn.manifold import LocallyLinearEmbedding
lle = LocallyLinearEmbedding()
X_lle = lle.fit_transform(X)
plt.scatter(X_lle[:,0],X_lle[:,1],c=Y,linewidths=0,alpha=0.5)


    

    
Out[310]:





<matplotlib.collections.PathCollection at 0x1167746d8>






    

In [311]:

    

from sklearn.manifold import TSNE
tsne = TSNE()
X_tsne = tsne.fit_transform(X)
plt.scatter(X_tsne[:,0],X_tsne[:,1],c=Y,linewidths=0,alpha=0.5)


    

    
Out[311]:





<matplotlib.collections.PathCollection at 0x114f75a90>






    

In [ ]:

    




    

In [312]:

    

one_nn_class_baseline(X_tsne,Y)


    

    
Out[312]:





0.99875000000000003

In [313]:

    

from sklearn.decomposition import KernelPCA


    

In [314]:

    

kpca = KernelPCA(n_components=2,kernel='rbf',gamma=0.05)
X_kpca = kpca.fit_transform(X)


    

In [315]:

    

plt.scatter(X_kpca[:,0],X_kpca[:,1],c=Y,linewidths=0,alpha=0.5)


    

    
Out[315]:





<matplotlib.collections.PathCollection at 0x11bf60550>






    

In [316]:

    

one_nn_class_baseline(X_kpca,Y)


    

    
Out[316]:





0.995

In [317]:

    

X_l,Y_l = generate_hypercube(8,n_tot=2000)
kpca = KernelPCA(n_components=2,kernel='rbf',gamma=0.06)
X_kpca = kpca.fit_transform(X_l)
plt.scatter(X_kpca[:,0],X_kpca[:,1],c=Y_l,linewidths=0,alpha=0.5)


    

    
Out[317]:





<matplotlib.collections.PathCollection at 0x11a49f080>






    

In [ ]:

    

# what if the clusters are not isotropic


    

In [427]:

    

from scipy.cluster import hierarchy


    

In [438]:

    

dim=6
radius=1
corners = [(np.array(i)*radius+radius)/2 for i in itertools.product([-1,1],repeat=dim)]


    

In [453]:

    

hierarchy.dendrogram(hierarchy.to_tree(corners));


    

    
Out[453]:





{'icoord': [[5.0, 5.0, 15.0, 15.0],
  [25.0, 25.0, 35.0, 35.0],
  [10.0, 10.0, 30.0, 30.0],
  [45.0, 45.0, 55.0, 55.0],
  [65.0, 65.0, 75.0, 75.0],
  [50.0, 50.0, 70.0, 70.0],
  [20.0, 20.0, 60.0, 60.0],
  [85.0, 85.0, 95.0, 95.0],
  [105.0, 105.0, 115.0, 115.0],
  [90.0, 90.0, 110.0, 110.0],
  [125.0, 125.0, 135.0, 135.0],
  [145.0, 145.0, 155.0, 155.0],
  [130.0, 130.0, 150.0, 150.0],
  [100.0, 100.0, 140.0, 140.0],
  [40.0, 40.0, 120.0, 120.0],
  [165.0, 165.0, 175.0, 175.0],
  [185.0, 185.0, 195.0, 195.0],
  [170.0, 170.0, 190.0, 190.0],
  [205.0, 205.0, 215.0, 215.0],
  [225.0, 225.0, 235.0, 235.0],
  [210.0, 210.0, 230.0, 230.0],
  [180.0, 180.0, 220.0, 220.0],
  [245.0, 245.0, 255.0, 255.0],
  [265.0, 265.0, 275.0, 275.0],
  [250.0, 250.0, 270.0, 270.0],
  [285.0, 285.0, 295.0, 295.0],
  [305.0, 305.0, 315.0, 315.0],
  [290.0, 290.0, 310.0, 310.0],
  [260.0, 260.0, 300.0, 300.0],
  [200.0, 200.0, 280.0, 280.0],
  [80.0, 80.0, 240.0, 240.0],
  [325.0, 325.0, 335.0, 335.0],
  [345.0, 345.0, 355.0, 355.0],
  [330.0, 330.0, 350.0, 350.0],
  [365.0, 365.0, 375.0, 375.0],
  [385.0, 385.0, 395.0, 395.0],
  [370.0, 370.0, 390.0, 390.0],
  [340.0, 340.0, 380.0, 380.0],
  [405.0, 405.0, 415.0, 415.0],
  [425.0, 425.0, 435.0, 435.0],
  [410.0, 410.0, 430.0, 430.0],
  [445.0, 445.0, 455.0, 455.0],
  [465.0, 465.0, 475.0, 475.0],
  [450.0, 450.0, 470.0, 470.0],
  [420.0, 420.0, 460.0, 460.0],
  [360.0, 360.0, 440.0, 440.0],
  [485.0, 485.0, 495.0, 495.0],
  [505.0, 505.0, 515.0, 515.0],
  [490.0, 490.0, 510.0, 510.0],
  [525.0, 525.0, 535.0, 535.0],
  [545.0, 545.0, 555.0, 555.0],
  [530.0, 530.0, 550.0, 550.0],
  [500.0, 500.0, 540.0, 540.0],
  [565.0, 565.0, 575.0, 575.0],
  [585.0, 585.0, 595.0, 595.0],
  [570.0, 570.0, 590.0, 590.0],
  [605.0, 605.0, 615.0, 615.0],
  [625.0, 625.0, 635.0, 635.0],
  [610.0, 610.0, 630.0, 630.0],
  [580.0, 580.0, 620.0, 620.0],
  [520.0, 520.0, 600.0, 600.0],
  [400.0, 400.0, 560.0, 560.0],
  [160.0, 160.0, 480.0, 480.0]],
 'dcoord': [[0.0, 1.0, 1.0, 0.0],
  [0.0, 1.0, 1.0, 0.0],
  [1.0, 1.4142135623730951, 1.4142135623730951, 1.0],
  [0.0, 1.0, 1.0, 0.0],
  [0.0, 1.0, 1.0, 0.0],
  [1.0, 1.4142135623730951, 1.4142135623730951, 1.0],
  [1.4142135623730951,
   1.7320508075688772,
   1.7320508075688772,
   1.4142135623730951],
  [0.0, 1.0, 1.0, 0.0],
  [0.0, 1.0, 1.0, 0.0],
  [1.0, 1.4142135623730951, 1.4142135623730951, 1.0],
  [0.0, 1.0, 1.0, 0.0],
  [0.0, 1.0, 1.0, 0.0],
  [1.0, 1.4142135623730951, 1.4142135623730951, 1.0],
  [1.4142135623730951,
   1.7320508075688772,
   1.7320508075688772,
   1.4142135623730951],
  [1.7320508075688772, 2.0, 2.0, 1.7320508075688772],
  [0.0, 1.0, 1.0, 0.0],
  [0.0, 1.0, 1.0, 0.0],
  [1.0, 1.4142135623730951, 1.4142135623730951, 1.0],
  [0.0, 1.0, 1.0, 0.0],
  [0.0, 1.0, 1.0, 0.0],
  [1.0, 1.4142135623730951, 1.4142135623730951, 1.0],
  [1.4142135623730951,
   1.7320508075688772,
   1.7320508075688772,
   1.4142135623730951],
  [0.0, 1.0, 1.0, 0.0],
  [0.0, 1.0, 1.0, 0.0],
  [1.0, 1.4142135623730951, 1.4142135623730951, 1.0],
  [0.0, 1.0, 1.0, 0.0],
  [0.0, 1.0, 1.0, 0.0],
  [1.0, 1.4142135623730951, 1.4142135623730951, 1.0],
  [1.4142135623730951,
   1.7320508075688772,
   1.7320508075688772,
   1.4142135623730951],
  [1.7320508075688772, 2.0, 2.0, 1.7320508075688772],
  [2.0, 2.2360679774997898, 2.2360679774997898, 2.0],
  [0.0, 1.0, 1.0, 0.0],
  [0.0, 1.0, 1.0, 0.0],
  [1.0, 1.4142135623730951, 1.4142135623730951, 1.0],
  [0.0, 1.0, 1.0, 0.0],
  [0.0, 1.0, 1.0, 0.0],
  [1.0, 1.4142135623730951, 1.4142135623730951, 1.0],
  [1.4142135623730951,
   1.7320508075688772,
   1.7320508075688772,
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In [454]:

    

Z = hierarchy.complete(corners)


    

In [456]:

    

l = hierarchy.to_tree(Z)


    

In [515]:

    

from sklearn.datasets import make_swiss_roll


    

In [514]:

    

tpe = TPE()


    

In [516]:

    

X,Y = make_swiss_roll(1000)


    

In [517]:

    

X_ = tpe.fit_transform(X)


    

In [518]:

    

plt.scatter(X_[:,0],X_[:,1],c=Y,linewidths=0,alpha=0.5)


    

    
Out[518]:





<matplotlib.collections.PathCollection at 0x136c04198>






    

In [519]:

    

tpe = TPE('ward')
X_ = tpe.fit_transform(X)
plt.scatter(X_[:,0],X_[:,1],c=Y,linewidths=0,alpha=0.5)


    

    
Out[519]:





<matplotlib.collections.PathCollection at 0x13702e8d0>






    

In [520]:

    

tsne = TSNE()
X_ = tsne.fit_transform(X)
plt.scatter(X_[:,0],X_[:,1],c=Y,linewidths=0,alpha=0.5)


    

    
Out[520]:





<matplotlib.collections.PathCollection at 0x11b51a358>






    

In [521]:

    

from mpl_toolkits.mplot3d import Axes3D
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.scatter(X[:,0],X[:,1],X[:,2],c=Y,linewidths=0,alpha=0.5)


    

    
Out[521]:





<mpl_toolkits.mplot3d.art3d.Path3DCollection at 0x1381b25f8>






    

In [ ]: