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The standard justification for manifold learning is that the map from the latent to observed spaces is nonlinear. For example, here is how another StackExchange user justified Isomap over PCA:

Here we are looking for 1-dimensional structure in 2D. The points lie along an S-shaped curve. PCA tries to describe the data with a linear 1-dimensional manifold, which is simply a line; of course a line fits these data quite bad. Isomap is looking for a nonlinear (i.e. curved!) 1-dimensional manifold, and should be able to discover the underlying S-shaped curve.

PCA vs Isomap

However, in my experience, either PCA does comparably well to a nonlinear model or the nonlinear model also fails. For example, consider this result:

PCA vs Isomap on sine function map

A simple latent variable changes over time. There are three maps into observation space. Two are noise; one is a sine wave (see Code 1 below). Clearly, a large value in observation space does not correspond to a large $x$ value in latent space. Here is the data colored by index:

enter image description here

In this case, PCA does as well as Isomap. My first question: Why does PCA do well here? Isn't the map nonlinear?


You might say this problem is too simple. Here's a more complicated example. Let's introduce two nonlinearities: a nonlinear latent space and a nonlinear maps. Here, the latent variable is shaped like an "S". And the maps are GP distributed, meaning if there are $J$ maps, each $f_j(x) \sim \mathcal{N}(0, K_x)$, where $K_x$ is the covariance matrix based on the kernel function (see Code 2 below). Again, PCA does well. In fact, the GPLVM whose data generating process is being matched exactly appears to not deviate much from its PCA initialization:

PCA vs Isomap vs GPLVM on GP-distributed maps

So again I ask: What's going on here? Why am I not breaking PCA?


Finally, the only way I can break PCA and still get something a bit structured from a manifold learner is if I literally "embed" the latent variable into a higher dimensional space (see Code 3 below):

PCA vs Isomap vs GPLVM on embedded data

To summarize, I have a few questions that I assume are related to a shared misunderstanding:

  1. Why does PCA do well on a simple nonlinear map (a sine function)? Isn't the modeling assumption that such maps are linear?

  2. Why does PCA do as well as GPLVM on a doubly nonlinear problem? What's especially surprising is that I used the data generating process for a GPLVM.

  3. Why does the third case finally break PCA? What's different about this problem?

I appreciate this is a broad question, but I hope that someone with more understanding of the issues can help synthesize and refine it.

EDIT:

PCA on a latent variable that is not linearly separable and with nonlinear maps:

circles


Code

1. Linear latent variable, nonlinear map

import matplotlib.pyplot as plt
import numpy as np
from   sklearn.decomposition import PCA
from   sklearn.manifold import Isomap


def gen_data():
    n_features = 3
    n_samples  = 500
    time       = np.arange(1, n_samples+1)
    # Latent variable is a straight line.
    lat_var    = 3 * time[:, np.newaxis]
    data = np.empty((n_samples, n_features))
    # But mapping functions are nonlinear or nose.
    data[:, 0] = np.sin(lat_var).squeeze()
    data[:, 1] = np.random.normal(0, 1, size=n_samples)
    data[:, 2] = np.random.normal(0, 1, size=n_samples)
    return data, lat_var, time


data, lat_var, time = gen_data()

lat_var_pca = PCA(n_components=1).fit_transform(data)
lat_var_iso = Isomap(n_components=1).fit_transform(data)

fig, (ax1, ax2, ax3) = plt.subplots(1, 3)
fig.set_size_inches(20, 5)

ax1.set_title('True')
ax1.scatter(time, lat_var, c=time)
ax2.set_title('PCA')
ax2.scatter(time, lat_var_pca, c=time)
ax3.set_title('Isomap')
ax3.scatter(time, lat_var_iso, c=time)

plt.tight_layout()
plt.show()

2. Nonlinear latent variable, GP-distributed maps

from   GPy.models import GPLVM
import matplotlib.pyplot as plt
import numpy as np
from   sklearn.decomposition import PCA
from   sklearn.datasets import make_s_curve
from   sklearn.manifold import Isomap
from   sklearn.metrics.pairwise import rbf_kernel


def gen_data():
    n_features = 10
    n_samples  = 500

    # Latent variable is 2D S-curve.
    lat_var, time = make_s_curve(n_samples)
    lat_var = np.delete(lat_var, obj=1, axis=1)
    lat_var /= lat_var.std(axis=0)

    # And maps are GP-distributed.
    mean = np.zeros(n_samples)
    cov  = rbf_kernel(lat_var)
    data = np.random.multivariate_normal(mean, cov, size=n_features).T

    return data, lat_var, time


data, lat_var, time = gen_data()

lat_var_pca = PCA(n_components=2).fit_transform(data)
lat_var_iso = Isomap(n_components=2).fit_transform(data)
gp = GPLVM(data, input_dim=2)
gp.optimize()
lat_var_gp = gp.X

fig, (ax1, ax2, ax3, ax4) = plt.subplots(1, 4)
fig.set_size_inches(20, 5)

ax1.set_title('True')
ax1.scatter(lat_var[:, 0], lat_var[:, 1], c=time)
ax2.set_title('PCA')
ax2.scatter(lat_var_pca[:, 0], lat_var_pca[:, 1], c=time)
ax3.set_title('Isomap')
ax3.scatter(lat_var_iso[:, 0], lat_var_iso[:, 1], c=time)
ax4.set_title('GPLVM')
ax4.scatter(lat_var_gp[:, 0], lat_var_gp[:, 1], c=time)

plt.tight_layout()
plt.show()

3. Nonlinear latent variable embedded into higher dimensional space

from   GPy.models import GPLVM
import matplotlib.pyplot as plt
import numpy as np
from   sklearn.datasets import make_s_curve
from   sklearn.decomposition import PCA
from   sklearn.manifold import Isomap


def gen_data():
    n_features = 10
    n_samples = 500

    # Latent variable is 2D S-curve.
    lat_var, time = make_s_curve(n_samples)
    lat_var = np.delete(lat_var, obj=1, axis=1)
    lat_var /= lat_var.std(axis=0)

    # And maps are GP-distributed.
    data = np.random.normal(0, 1, size=(n_samples, n_features))
    data[:, 0] = lat_var[:, 0]
    data[:, 1] = lat_var[:, 1]

    return data, lat_var, time


data, lat_var, time = gen_data()

lat_var_pca = PCA(n_components=2).fit_transform(data)
lat_var_iso = Isomap(n_components=2).fit_transform(data)
gp = GPLVM(data, input_dim=2)
gp.optimize()
lat_var_gp = gp.X

fig, (ax1, ax2, ax3, ax4) = plt.subplots(1, 4)
fig.set_size_inches(20, 5)

ax1.set_title('True')
ax1.scatter(lat_var[:, 0], lat_var[:, 1], c=time)
ax2.set_title('PCA')
ax2.scatter(lat_var_pca[:, 0], lat_var_pca[:, 1], c=time)
ax3.set_title('Isomap')
ax3.scatter(lat_var_iso[:, 0], lat_var_iso[:, 1], c=time)
ax4.set_title('GPLVM')
ax4.scatter(lat_var_gp[:, 0], lat_var_gp[:, 1], c=time)

plt.tight_layout()
plt.show()

4. Latent variable that is not linearly separable with GP-distributed maps

from   GPy.models import GPLVM
import matplotlib.pyplot as plt
import numpy as np
from   sklearn.decomposition import PCA
from   sklearn.datasets import make_circles
from   sklearn.manifold import Isomap
from   sklearn.metrics.pairwise import rbf_kernel


def gen_data():
    n_features = 20
    n_samples  = 500
    lat_var, time = make_circles(n_samples)
    mean = np.zeros(n_samples)
    cov  = rbf_kernel(lat_var)
    data = np.random.multivariate_normal(mean, cov, size=n_features).T
    return data, lat_var, time


data, lat_var, time = gen_data()

lat_var_pca = PCA(n_components=2).fit_transform(data)
lat_var_iso = Isomap(n_components=2).fit_transform(data)
gp = GPLVM(data, input_dim=2)
gp.optimize()
lat_var_gp = gp.X

fig, (ax1, ax2, ax3, ax4) = plt.subplots(1, 4)
fig.set_size_inches(20, 5)

ax1.set_title('True')
ax1.scatter(lat_var[:, 0], lat_var[:, 1], c=time)
ax2.set_title('PCA')
ax2.scatter(lat_var_pca[:, 0], lat_var_pca[:, 1], c=time)
ax3.set_title('Isomap')
ax3.scatter(lat_var_iso[:, 0], lat_var_iso[:, 1], c=time)
ax4.set_title('GPLVM')
ax4.scatter(lat_var_gp[:, 0], lat_var_gp[:, 1], c=time)

plt.tight_layout()
plt.show()
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  • $\begingroup$ I don't think I have any chance of giving a full answer in the near future. PCA is able to break non-linear processes down into a series of smaller steps approximated by linear processes. One example is random walk in one dimension, where PCA returns progressively higher frequency sine-like waves. $\endgroup$ – ReneBt May 15 at 9:03
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    $\begingroup$ this question is so messed up! it's impossible to understand what you are doing without reading the code, and your use of graphs is inconsistent. there is no possible answer to such a question, because it is not coherent in its points and it doesn't really make sense overall. $\endgroup$ – carlo May 17 at 13:06
  • $\begingroup$ I agree with @carlo - I don't think you're comparing what you think you're comparing, or maybe you don't have a concrete definition of what you want to compare in the first place. What do you mean by 'performing well' here? Capturing the basis vectors of the underlying manifold? Minimizing some (specific!) measure of reconstruction error, either of the original data structure or predefined similarity measure? Your code also, at least with the current explanations, seems to exhibit a lack of understanding of the default behaviors and the mathematical theory of the underlying algorithms. $\endgroup$ – Don Walpola May 17 at 13:23
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    $\begingroup$ @carlo, what's inconsistent in my use of graphs? With the exception of the diagram from another SO post and the graph that was added to address a proposed answer, all of the graphs have the same x- and y-axes. $\endgroup$ – gwg May 17 at 14:22
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    $\begingroup$ I don't think I'll get an answer to my question, perhaps because it is ultimately a few questions about different nonlinear models, but this has been a useful discussion. Thanks. $\endgroup$ – gwg May 17 at 15:13
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The reason you are not breaking PCA is because your data is still "simple" and have strong "linear properties".

In your first example, the line example, we can summarize data as follows: the regression target will be larger, with respect x and y, i.e., in original feature space, the upper right corner.

In your second example, the S shaped example, we can summarize data as: the regression target will be larger, when x is small and y is small, i.e., in original feature space, the lower left corner.

The following example will break linear PCA. Because the is no linear relationship/features we can found to classify different classes. (Similar to the pearson correlation coefficient will be close to 0 for such data.)

enter image description here

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  • $\begingroup$ Please see my edit which now includes this example. PCA appears to do well on this problem. In fact, it does better than a GPLVM despite the nonlinear maps being GP-distributed. $\endgroup$ – gwg May 10 at 0:21
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    $\begingroup$ @gwg, I do not agree PCA was doing well in this case. The ultimate goal is trying to separate different classes, so, after PCA we want to see the classes are hopefully separable. The PCA results does not show that. $\endgroup$ – Haitao Du May 10 at 7:05
  • $\begingroup$ I see your point about PCA not generating a truly separable latent space, but I don't think I follow your argument about the nonlinear maps. Even in the first case, the sine function takes large values in observation space that are completely independent of the value in latent space (I've added an additional plot there). And in the nested circles example, there are 20 such nonlinear maps. I don't know how even non-nested circles are recoverable. $\endgroup$ – gwg May 10 at 14:22

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