Prelude: On Reduction and High Dimensional Pancakes#
If we’re going to be dealing with very large matrices as representing “language”, we’re going to need to build an intuition for dealing with them.
Remember large matrices are generally thought of as geometrical objects of high dimension.
import matplotlib.pyplot as plt
import numpy as np
plt.xkcd()
<contextlib.ExitStack at 0x7f1b7438b730>
High Dimensional Space is … Spikey?#
f,ax=plt.subplots(ncols=2,figsize=(10,4))
dims=np.arange(2,11)
sm=(np.sqrt(dims)-1)
ax[0].axhline(2, ls=':', c='grey')
ax[0].axhline(4, ls='--', c='grey')
ax[0].plot(dims, 2*sm,
label='Dia. of small sphere', c='#1e90ff')
ax[0].legend()
ax[0].set_xlabel('dimensions')
from scipy.special import gamma
vols=np.pi**(dims/2)/gamma(1+dims/2)
ax[1].plot(dims, vols,
label='Vol. of spheres', c='grey', ls=':')
ax[1].plot(dims, (sm**dims)*vols,
label='Vol. of small sphere', c='#1e90ff')
ax[1].set_ylim(0,10)
ax[1].legend()
<matplotlib.legend.Legend at 0x7f1b34856e60>
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(the volume of the hypercube is way up there)
A big take-away: there’s a lot of “volume” in high dimensions. Unimaginable amounts of “space.”
Often, though, the processes and definitions of things we want to “observe” (like the unit-spheres, or spheres touching those…) do not occupy that space in intuitive ways as dimensionality increases. The relative sizes and distances between things get very stretched out… almost diluted.
It’s easy to lose stuff in all that space.
Dimensionality Reduction#
Do we really need all those dimensions?
Do you need 3 dimensions to represent a racetrack? (track is technically banked, but…)
Do you need 2 dimensions to represent a 100m sprint? (runners technically standing apart, but…)
Text analysis tends to involve huge matrices, i.e. a lot of dimensions. But the “interesting stuff” is almost certainly happening in a lot fewer dimensions, if you slice it right.
What is Singular Value Decomposition?#
Decompose a matrix \(M\) into two orthonormal ones and a diagonal one:
\[M = U\Sigma V^*\]
This can be understood geometrically as saying
every linear transformation can be broken down into:
rotation \(\rightarrow\) scaling \(\rightarrow\) rotation
(credit: Georg-Johann, August 2010)
Matrix-as-Data? Principal Components Analysis (PCA)#
Low-Rank Approximation as “Compression”#
What if we did this pancake procedure…in “reverse”?
We have the “optimal hyperellipse”
throw away the dimensions that are least variant
In high dimensions, as we’ve seen, these should be radically “thinner” than our “most” variant directions.
Practically pancakes already!
from sklearn.decomposition import TruncatedSVD
from sklearn.datasets import load_sample_image
gs = load_sample_image('flower.jpg')[...,:3]@np.array([0.2125,0.7154,0.0721])
plt.imshow(gs, cmap='Greys_r')
plt.axis('off')
gs.shape
(427, 640)
def compress(n, ax=None):
svd = TruncatedSVD(n_components=n)
svd.fit(gs)
if ax is not None:
ax.imshow(svd.transform(gs)@svd.components_,
cmap='Greys_r')
ax.axis('off')
return svd
1-D Approximation of the image (each data point is a “row” of pixels)
from mpl_toolkits.axes_grid1 import make_axes_locatable
f,ax=plt.subplots(figsize=(10,10))
svd = compress(1, ax=ax)
# create new axes on the right and on the top of the current axes
divider = make_axes_locatable(ax)
# below height and pad are in inches
ax_loads = divider.append_axes("top", 1.1, pad=0.1)
ax_data = divider.append_axes("left", 1.1, pad=0.1)
# make some labels invisible
ax_loads.xaxis.set_tick_params(labelbottom=False)
ax_data.yaxis.set_tick_params(labelright=False)
loads = svd.components_
ax_loads.plot(loads[0], label='feature\nloads')
ax_loads.set_ylabel('feature\nloads')
ax_data.plot(svd.transform(gs), range(len(gs)), ls=':', label='projected\ndata')
ax_data.set_xlabel('projected data');
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2-D scatterplot projection of the “rows” (each point approximates an entire row of pixels in 2D)
f,ax =plt.subplots(ncols=2, figsize=(14,5))
pancake = compress(2, ax=ax[0]).transform(gs)
pc12 = ax[1].scatter(pancake[:,1], pancake[:,0],
c=np.arange(len(pancake)), cmap='cubehelix')
ax[1].set(ylabel='pc1', xlabel='pc2', title='2D projection of pixel rows')
plt.tight_layout()
f.colorbar(pc12, label='row #')
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<matplotlib.colorbar.Colorbar at 0x7f1b14f278b0>
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This is wild. Our intuition is really hard to build here, but the start and end rows are roughly similar. The middle “grows” then “shrinks”, reflected in the push and pull of green and pink, while the noise of petals is captured by the “fuzziness” of the points there.
Still, why are similar rows so far apart? Stuff seems weirdly spread out, and weirdly shaped.
How many dimensions?#
mow much should we “pancake”?
svd=compress(30);
plt.bar(x=range(30),height=svd.explained_variance_ratio_)
plt.title('Scree plot')
plt.ylabel('Explained Variance Ratio')
plt.xlabel('n. dimensions')
plt.axvline(3.5, ls='--', c='xkcd:rust', label='...ish?')
plt.legend();
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findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
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findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
findfont: Font family 'xkcd Script' not found.
findfont: Font family 'Humor Sans' not found.
findfont: Font family 'Comic Neue' not found.
findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
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findfont: Font family 'Humor Sans' not found.
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findfont: Font family 'Comic Sans MS' not found.
findfont: Font family 'xkcd' not found.
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findfont: Font family 'xkcd' not found.
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But how does it look?
def img_approx(mat, which):
num=len(which)
f,ax=plt.subplots(ncols=3, nrows=num//3,figsize=(16,5*(num//3)))
for n,idx in enumerate(which):
compress(idx, ax=ax.flatten()[n])
ax.flatten()[n].text(0.1, 0.8, f'{idx}', transform=ax[n].transAxes)
img_approx(gs, [5, 15, 30])
#
findfont: Font family 'xkcd' not found.
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Even though the variance has been captured, it’s hard to see “what it is” without detail. Our brains are not treating each row of pixels as a distince observation, anymore than we read a document of english text in isolation!
Individual Singular Vectors as “Slices”#
from scipy import linalg
def img_topics(mat, which):
U, s, Vh = linalg.svd(mat, full_matrices=False)
def each_topic(n, ax):
topic = U[:,n, np.newaxis]@(s[n]*Vh[np.newaxis,n,:])
ax.imshow(topic, cmap='Greys_r')
ax.axis('off')
f,ax=plt.subplots(ncols=len(which), figsize=(16,4))
for n,idx in enumerate(which):
ax[n].text(0.1, 0.8, f'{idx}', transform=ax[n].transAxes)
each_topic(idx, ax=ax[n])
img_topics(gs, range(3))
img_topics(gs, range(3,6))
img_topics(gs, range(6,9))
findfont: Font family 'xkcd' not found.
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NOTE
It doesn’t make sense to say each of the above are “clusters”, does it? REMEBER THIS.
Components of a matrix decomposition are not the same as clusters data.
Every “component” has a value for the space of every data point (not some cluster of them)!
While each is a group of pixels that do something together, and that “something” is orthogonal to all the other somethings (i.e. they do unique “stuff” in Euclidean space)
Non-Linear “Pancake” Flavors#
Manifold Learning#
Ok, let’s break the assumption that a linear projection gives us our reduction.
Most of the time, we lose the “global” interpretation of our “new” axes, but we want to gain local linearity. This way, we gain local relationships back in our lower dimensional relation. This object that maintains “local” distance relationships is a lower-diensional manifold. Sort’ve like squishing data into a… “floppy” pancake?
multi-dimensional scaling
t-distributed Stochastic Neighbor Embedding
uniform manifold approximation and projection1
from sklearn.decomposition import FastICA
from sklearn.manifold import TSNE
manifold = TSNE()
pancake = manifold.fit_transform(gs)
f,(img,ax)=plt.subplots(1,2, figsize=(12,5))
pc12 = ax.scatter(pancake[:,1], pancake[:,0],
c=np.arange(len(pancake)), cmap='cubehelix')
# plt.scatter(svd.transform(gs)[:,0], svd.transform(gs)[:,1])
ax.set(ylabel='pc1', xlabel='pc2', title='2D tSNE projection of pixel rows')
f.colorbar(pc12, label='row #')
img.imshow(gs, cmap='Greys_r')
img.axis('off')
plt.tight_layout()
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As hoped, each row is approximately touching it’s neighbor, except where there’s an actual jump in colors.
Notice global stuff is much less pronounced, instead giving us a hint at “what’s going on”, locally! See how the rows start and end at different positions, despite being very similar, objectively? This is because they are heading in different directions (growing white, vs shrinking white).
Independence & Blind Source Separation#
What if, instead of trying to find additive components that maximize variance (accounted for in each subsequent orthogonal dimension), we instead maximize the mutual independence of some number of components?
Classic example:
On a stage, you have some number (K) of microphones, and some number (N) of instruments
Can you use the unique recordings of mixed instrument audio to “unmix” the recordings? Yes!
In our case, each “microphone” is a data point, and each “instrument” is a component vector.
(source: scikit-learn.org)
Takeaway#
Dimensionality reduction is very powerful, and it feels like it. BUT…
there are serious implications when you choose a specific type of reduction… really depends on what you want to accomplish
Building an intuition takes time, but is crucial to proper usage and making sure you ask yourself (and others) good questions, constantly!
But where’s the text??
As mentioned, we are now focusing primarily on vector-based models for context-driven models of language. This means we need to be comfy with what big matrices actually are, the curse of dimensionality, and how different assumed goals deal with that.
And trust me, this will come very much in handy for the next section on clustering & topic modeling
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Often the best-performing, see interactive explanation