Week 48: Autoencoders and summary of course

Overview of week 48

Lecture Monday, November 24

Readings and Videos

Lab sessions

Autoencoders: Overarching view

Autoencoders are artificial neural networks capable of learning efficient representations of the input data (these representations are called codings) without any supervision (i.e., the training set is unlabeled). These codings typically have a much lower dimensionality than the input data, making autoencoders useful for dimensionality reduction.

Autoencoders learn to encode the input data into a lower-dimensional representation, and then decode it back to the original data. The goal of autoencoders is to minimize the reconstruction error, which measures how well the output matches the input. Autoencoders can be seen as a way of learning the latent features or hidden structure of the data, and they can be used for data compression, denoising, anomaly detection, and generative modeling.

Powerful detectors

More importantly, autoencoders act as powerful feature detectors, and they can be used for unsupervised pretraining of deep neural networks.

Lastly, they are capable of randomly generating new data that looks very similar to the training data; this is called a generative model. For example, you could train an autoencoder on pictures of faces, and it would then be able to generate new faces. Surprisingly, autoencoders work by simply learning to copy their inputs to their outputs. This may sound like a trivial task, but we will see that constraining the network in various ways can make it rather difficult. For example, you can limit the size of the internal representation, or you can add noise to the inputs and train the network to recover the original inputs. These constraints prevent the autoencoder from trivially copying the inputs directly to the outputs, which forces it to learn efficient ways of representing the data. In short, the codings are byproducts of the autoencoder’s attempt to learn the identity function under some constraints.

First introduction of AEs

Autoencoders were first introduced by Rumelhart, Hinton, and Williams in 1986 with the goal of learning to reconstruct the input observations with the lowest error possible.

Why would one want to learn to reconstruct the input observations? If you have problems imagining what that means, think of having a dataset made of images. An autoencoder would be an algorithm that can give as output an image that is as similar as possible to the input one. You may be confused, as there is no apparent reason of doing so. To better understand why autoencoders are useful we need a more informative (although not yet unambiguous) definition.

Autoencoder structure

Autoencoders are neural networks where the outputs are its own inputs. They are split into an encoder part which maps the input \( \boldsymbol{x} \) via a function \( f(\boldsymbol{x},\boldsymbol{W}) \) (this is the encoder part) to a so-called code part (or intermediate part) with the result \( \boldsymbol{h} \)

where \( \boldsymbol{W} \) are the weights to be determined. The decoder parts maps, via its own parameters (weights given by the matrix \( \boldsymbol{V} \) and its own biases) to the final ouput

The goal is to minimize the construction error.

Schematic image of an Autoencoder

More on the structure

In most typical architectures, the encoder and the decoder are neural networks since they can be easily trained with existing software libraries such as TensorFlow or PyTorch with back propagation.

In general, the encoder can be written as a function \( g \) that will depend on some parameters

where \( \mathbf{h}_{i}\in\mathbb{R}^{q} \) (the latent feature representation) is the output of the encoder block where we evaluate it using the input \( \mathbf{x}_{i} \).

Decoder part

Note that we have \( g:\mathbb{R}^{n}\rightarrow\mathbb{R}^{q} \) The decoder and the output of the network \( \tilde{\mathbf{x}}_{i} \) can be written then as a second generic function of the latent features

where \( \tilde{\mathbf{x}}_{i}\mathbf{\in }\mathbb{R}^{n} \).

Training an autoencoder simply means finding the functions \( g(\cdot) \) and \( f(\cdot) \) that satisfy

Typical AEs

The standard setup is done via a standard feed forward neural network (FFNN), or what is called a Feed Forward Autoencoder.

A typical FFNN architecture has an odd number of layers and is symmetrical with respect to the middle layer.

Typically, the first layer has a number of neurons \( n_{1} = n \) which equals the size of the input observation \( \mathbf{x}_{\mathbf{i}} \).

As we move toward the center of the network, the number of neurons in each layer drops in some measure. The middle layer usually has the smallest number of neurons. The fact that the number of neurons in this layer is smaller than the size of the input, is often called the bottleneck.

Feed Forward Autoencoder

Mirroring

In almost all practical applications, the layers after the middle one are a mirrored version of the layers before the middle one. For example, an autoencoder with three layers could have the following numbers of neurons:

\( n_{1} = 10 \), \( n_{2} = 5 \) and then \( n_{3} = n_{1} = 10 \) where the input dimension is equal to ten.

All the layers up to and including the middle one, make what is called the encoder, and all the layers from and including the middle one (up to the output) make what is called the decoder.

If the FFNN training is successful, the result will be a good approximation of the input \( \tilde{\mathbf{x}}_{i}\approx\mathbf{x}_{i} \).

What is essential to notice is that the decoder can reconstruct the input by using only a much smaller number of features than the input observations initially have.

Output of middle layer

The output of the middle layer \( \mathbf{h}_{\mathbf{i}} \) are also called a learned representation of the input observation \( \mathbf{x}_{i} \).

The encoder can reduce the number of dimensions of the input observation and create a learned representation of the input that has a smaller dimension \( q < n \).

This learned representation is enough for the decoder to reconstruct the input accurately (if the autoencoder training was successful as intended).

Activation Function of the Output Layer

In autoencoders based on neural networks, the output layer's activation function plays a particularly important role. The most used functions are ReLU and Sigmoid.

ReLU

The ReLU activation function can assume all values in the range \( \left[0,\infty\right] \). As a remainder, its formula is

This choice is good when the input observations \(\mathbf{x}_{i}\) assume a wide range of positive values. If the input \( \mathbf{x}_{i} \) can assume negative values, the ReLU is, of course, a terrible choice, and the identity function is a much better choice. It is then common to replace to the ReLU with the so-called Leaky ReLu or just modified ReLU.

The ReLU activation function for the output layer is well suited for cases when the input observations \(\mathbf{x}_{i}\) assume a wide range of positive real values.

Sigmoid

The sigmoid function \( \sigma \) can assume all values in the range \( [0,1] \),

This activation function can only be used if the input observations \( \mathbf{x}_{i} \) are all in the range \( [0,1] \) or if you have normalized them to be in that range. Consider as an example the MNIST dataset. Each value of the input observation \( \mathbf{x}_{i} \) (one image) is the gray values of the pixels that can assume any value from 0 to 255. Normalizing the data by dividing the pixel values by 255 would make each observation (each image) have only pixel values between 0 and 1. In this case, the sigmoid would be a good choice for the output layer's activation function.

Cost/Loss Function

If an autoencoder is trying to solve a regression problem, the most common choice as a loss function is the Mean Square Error

Binary Cross-Entropy

If the activation function of the output layer of the AE is a sigmoid function, thus limiting neuron outputs to be between 0 and 1, and the input features are normalized to be between 0 and 1 we can use as loss function the binary cross-entropy. This cots/loss function is typically used in classification problems, but it works well for autoencoders. The formula for it is

Reconstruction Error

The reconstruction error (RE) is a metric that gives you an indication of how good (or bad) the autoencoder was able to reconstruct the input observation \( \mathbf{x}_{i} \). The most typical RE used is the MSE

Temporary summary on Autoencoders

What is an Autoencoder? An autoencoder (AE) is a neural network trained to reconstruct its input: \( \hat{x}=\mathrm{Dec}(\mathrm{Enc}(x)) \).

Training objective: minimize reconstruction loss, e.g. MSE \mathcal{L}(\theta,\phi)=\frac{1}{N}\sum_{i=1}^N \|x^{(i)}-g_\phi(f_\theta(x^{(i)}))\|^2_2.

Linear autoencoders and the PCA theorem

The covariance matrix is given as

Let us now assume that we can perform a series of orthogonal transformations where we employ some orthogonal matrices \( \boldsymbol{S} \). These matrices are defined as \( \boldsymbol{S}\in {\mathbb{R}}^{p\times p} \) and obey the orthogonality requirements \( \boldsymbol{S}\boldsymbol{S}^T=\boldsymbol{S}^T\boldsymbol{S}=\boldsymbol{I} \). The matrix can be written out in terms of the column vectors \( \boldsymbol{s}_i \) as \( \boldsymbol{S}=[\boldsymbol{s}_0,\boldsymbol{s}_1,\dots,\boldsymbol{s}_{p-1}] \) and \( \boldsymbol{s}_i \in {\mathbb{R}}^{p} \).

More details

Assume also that there is a transformation \( \boldsymbol{S}^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}=\boldsymbol{C}[\boldsymbol{y}] \) such that the new matrix \( \boldsymbol{C}[\boldsymbol{y}] \) is diagonal with elements \( [\lambda_0,\lambda_1,\lambda_2,\dots,\lambda_{p-1}] \).

That is we have

since the matrix \( \boldsymbol{S} \) is not a data dependent matrix. Multiplying with \( \boldsymbol{S} \) from the left we have

and since \( \boldsymbol{C}[\boldsymbol{y}] \) is diagonal we have for a given eigenvalue \( i \) of the covariance matrix that

The PCA Theorem

In the derivation of the PCA theorem we will assume that the eigenvalues are ordered in descending order, that is \( \lambda_0 > \lambda_1 > \dots > \lambda_{p-1} \).

The eigenvalues tell us then how much we need to stretch the corresponding eigenvectors. Dimensions with large eigenvalues have thus large variations (large variance) and define therefore useful dimensions. The data points are more spread out in the direction of these eigenvectors. Smaller eigenvalues mean on the other hand that the corresponding eigenvectors are shrunk accordingly and the data points are tightly bunched together and there is not much variation in these specific directions. Hopefully then we could leave it out dimensions where the eigenvalues are very small. If \( p \) is very large, we could then aim at reducing \( p \) to \( l < < p \) and handle only \( l \) features/predictors.

The Algorithm before theorem

Here's how we would proceed in setting up the algorithm for the PCA, see also discussion below here. Set up the datapoints for the design/feature matrix \( \boldsymbol{X} \) with \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \), with the predictors/features \( p \) referring to the column numbers and the entries \( n \) being the row elements.

Further steps

• Center the data by subtracting the mean value for each column. This leads to a new matrix \( \boldsymbol{X}\rightarrow \overline{\boldsymbol{X}} \).
• Compute then the covariance/correlation matrix \( \mathbb{E}[\overline{\boldsymbol{X}}^T\overline{\boldsymbol{X}}] \).
• Find the eigenpairs of \( \boldsymbol{C} \) with eigenvalues \( [\lambda_0,\lambda_1,\dots,\lambda_{p-1}] \) and eigenvectors \( [\boldsymbol{s}_0,\boldsymbol{s}_1,\dots,\boldsymbol{s}_{p-1}] \).
• Order the eigenvalue (and the eigenvectors accordingly) in order of decreasing eigenvalues.
• Keep only those \( l \) eigenvalues larger than a selected threshold value, discarding thus \( p-l \) features since we expect small variations in the data here.

Writing our own PCA code

We will use a simple example first with two-dimensional data drawn from a multivariate normal distribution with the following mean and covariance matrix (we have fixed these quantities but will play around with them below):

Note that the mean refers to each column of data. We will generate \( n = 10000 \) points \( X = \{ x_1, \ldots, x_N \} \) from this distribution, and store them in the \( 1000 \times 2 \) matrix \( \boldsymbol{X} \). This is our design matrix where we have forced the covariance and mean values to take specific values.

Implementing it

The following Python code aids in setting up the data and writing out the design matrix. Note that the function multivariate returns also the covariance discussed above and that it is defined by dividing by \( n-1 \) instead of \( n \).

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from IPython.display import display
n = 10000
mean = (-1, 2)
cov = [[4, 2], [2, 2]]
X = np.random.multivariate_normal(mean, cov, n)

Now we are going to implement the PCA algorithm. We will break it down into various substeps.

First Step

The first step of PCA is to compute the sample mean of the data and use it to center the data. Recall that the sample mean is

and the mean-centered data \( \bar{X} = \{ \bar{x}_1, \ldots, \bar{x}_n \} \) takes the form

When you are done with these steps, print out \( \mu_n \) to verify it is close to \( \mu \) and plot your mean centered data to verify it is centered at the origin! The following code elements perform these operations using pandas or using our own functionality for doing so. The latter, using numpy is rather simple through the mean() function.

df = pd.DataFrame(X)
# Pandas does the centering for us
df = df -df.mean()
# we center it ourselves
X_centered = X - X.mean(axis=0)

Scaling

Alternatively, we could use the functions we discussed earlier for scaling the data set. That is, we could have used the StandardScaler function in Scikit-Learn, a function which ensures that for each feature/predictor we study the mean value is zero and the variance is one (every column in the design/feature matrix). You would then not get the same results, since we divide by the variance. The diagonal covariance matrix elements will then be one, while the non-diagonal ones need to be divided by \( 2\sqrt{2} \) for our specific case.

Centered Data

Now we are going to use the mean centered data to compute the sample covariance of the data by using the following equation

where the data points \( x_i \in \mathbb{R}^p \) (here in this example \( p = 2 \)) are column vectors and \( x^T \) is the transpose of \( x \). We can write our own code or simply use either the functionaly of numpy or that of pandas, as follows

print(df.cov())
print(np.cov(X_centered.T))

Note that the way we define the covariance matrix here has a factor \( n-1 \) instead of \( n \). This is included in the cov() function by numpy and pandas. Our own code here is not very elegant and asks for obvious improvements. It is tailored to this specific \( 2\times 2 \) covariance matrix.

# extract the relevant columns from the centered design matrix of dim n x 2
x = X_centered[:,0]
y = X_centered[:,1]
Cov = np.zeros((2,2))
Cov[0,1] = np.sum(x.T@y)/(n-1.0)
Cov[0,0] = np.sum(x.T@x)/(n-1.0)
Cov[1,1] = np.sum(y.T@y)/(n-1.0)
Cov[1,0]= Cov[0,1]
print("Centered covariance using own code")
print(Cov)
plt.plot(x, y, 'x')
plt.axis('equal')
plt.show()

Exploring

Depending on the number of points \( n \), we will get results that are close to the covariance values defined above. The plot shows how the data are clustered around a line with slope close to one. Is this expected? Try to change the covariance and the mean values. For example, try to make the variance of the first element much larger than that of the second diagonal element. Try also to shrink the covariance (the non-diagonal elements) and see how the data points are distributed.

Diagonalize the sample covariance matrix to obtain the principal components

Now we are ready to solve for the principal components! To do so we diagonalize the sample covariance matrix \( \Sigma \). We can use the function np.linalg.eig to do so. It will return the eigenvalues and eigenvectors of \( \Sigma \). Once we have these we can perform the following tasks:

• We compute the percentage of the total variance captured by the first principal component
• We plot the mean centered data and lines along the first and second principal components
• Then we project the mean centered data onto the first and second principal components, and plot the projected data.
• Finally, we approximate the data as

where \( v_0 \) is the first principal component.

Collecting all Steps

Collecting all these steps we can write our own PCA function and compare this with the functionality included in Scikit-Learn.

The code here outlines some of the elements we could include in the analysis. Feel free to extend upon this in order to address the above questions.

# diagonalize and obtain eigenvalues, not necessarily sorted
EigValues, EigVectors = np.linalg.eig(Cov)
# sort eigenvectors and eigenvalues
#permute = EigValues.argsort()
#EigValues = EigValues[permute]
#EigVectors = EigVectors[:,permute]
print("Eigenvalues of Covariance matrix")
for i in range(2):
    print(EigValues[i])
FirstEigvector = EigVectors[:,0]
SecondEigvector = EigVectors[:,1]
print("First eigenvector")
print(FirstEigvector)
print("Second eigenvector")
print(SecondEigvector)
#thereafter we do a PCA with Scikit-learn
from sklearn.decomposition import PCA
pca = PCA(n_components = 2)
X2Dsl = pca.fit_transform(X)
print("Eigenvector of largest eigenvalue")
print(pca.components_.T[:, 0])

This code does not contain all the above elements, but it shows how we can use Scikit-Learn to extract the eigenvector which corresponds to the largest eigenvalue. Try to address the questions we pose before the above code. Try also to change the values of the covariance matrix by making one of the diagonal elements much larger than the other. What do you observe then?

Classical PCA Theorem

We assume now that we have a design matrix \( \boldsymbol{X} \) which has been centered as discussed above. For the sake of simplicity we skip the overline symbol. The matrix is defined in terms of the various column vectors \( [\boldsymbol{x}_0,\boldsymbol{x}_1,\dots, \boldsymbol{x}_{p-1}] \) each with dimension \( \boldsymbol{x}\in {\mathbb{R}}^{n} \).

The PCA theorem states that minimizing the above reconstruction error corresponds to setting \( \boldsymbol{W}=\boldsymbol{S} \), the orthogonal matrix which diagonalizes the empirical covariance(correlation) matrix. The optimal low-dimensional encoding of the data is then given by a set of vectors \( \boldsymbol{z}_i \) with at most \( l \) vectors, with \( l < < p \), defined by the orthogonal projection of the data onto the columns spanned by the eigenvectors of the covariance(correlations matrix).

The PCA Theorem

To show the PCA theorem let us start with the assumption that there is one vector \( \boldsymbol{s}_0 \) which corresponds to a solution which minimized the reconstruction error \( J \). This is an orthogonal vector. It means that we now approximate the reconstruction error in terms of \( \boldsymbol{w}_0 \) and \( \boldsymbol{z}_0 \) as

We are almost there, we have obtained a relation between minimizing the reconstruction error and the variance and the covariance matrix. Minimizing the error is equivalent to maximizing the variance of the projected data.

We could trivially maximize the variance of the projection (and thereby minimize the error in the reconstruction function) by letting the norm-2 of \( \boldsymbol{w}_0 \) go to infinity. However, this norm since we want the matrix \( \boldsymbol{W} \) to be an orthogonal matrix, is constrained by \( \vert\vert \boldsymbol{w}_0 \vert\vert_2^2=1 \). Imposing this condition via a Lagrange multiplier we can then in turn maximize

Taking the derivative with respect to \( \boldsymbol{w}_0 \) we obtain

meaning that

The direction that maximizes the variance (or minimizes the construction error) is an eigenvector of the covariance matrix! If we left multiply with \( \boldsymbol{w}_0^T \) we have the variance of the projected data is

If we want to maximize the variance (minimize the construction error) we simply pick the eigenvector of the covariance matrix with the largest eigenvalue. This establishes the link between the minimization of the reconstruction function \( J \) in terms of an orthogonal matrix and the maximization of the variance and thereby the covariance of our observations encoded in the design/feature matrix \( \boldsymbol{X} \).

The proof for the other eigenvectors \( \boldsymbol{w}_1,\boldsymbol{w}_2,\dots \) can be established by applying the above arguments and using the fact that our basis of eigenvectors is orthogonal, see Murphy chapter 12.2. The discussion in chapter 12.2 of Murphy's text has also a nice link with the Singular Value Decomposition theorem. For categorical data, see chapter 12.4 and discussion therein.

For more details, see for example Vidal, Ma and Sastry, chapter 2.

Geometric Interpretation and link with Singular Value Decomposition

For a detailed demonstration of the geometric interpretation, see Vidal, Ma and Sastry, section 2.1.2.

Principal Component Analysis (PCA) is by far the most popular dimensionality reduction algorithm. First it identifies the hyperplane that lies closest to the data, and then it projects the data onto it.

The following Python code uses NumPy’s svd() function to obtain all the principal components of the training set, then extracts the first two principal components. First we center the data using either pandas or our own code

import numpy as np
import pandas as pd
from IPython.display import display
np.random.seed(100)
# setting up a 10 x 5 vanilla matrix 
rows = 10
cols = 5
X = np.random.randn(rows,cols)
df = pd.DataFrame(X)
# Pandas does the centering for us
df = df -df.mean()
display(df)

# we center it ourselves
X_centered = X - X.mean(axis=0)
# Then check the difference between pandas and our own set up
print(X_centered-df)
#Now we do an SVD
U, s, V = np.linalg.svd(X_centered)
c1 = V.T[:, 0]
c2 = V.T[:, 1]
W2 = V.T[:, :2]
X2D = X_centered.dot(W2)
print(X2D)

PCA assumes that the dataset is centered around the origin. Scikit-Learn’s PCA classes take care of centering the data for you. However, if you implement PCA yourself (as in the preceding example), or if you use other libraries, don’t forget to center the data first.

Once you have identified all the principal components, you can reduce the dimensionality of the dataset down to \( d \) dimensions by projecting it onto the hyperplane defined by the first \( d \) principal components. Selecting this hyperplane ensures that the projection will preserve as much variance as possible.

W2 = V.T[:, :2]
X2D = X_centered.dot(W2)

PCA and scikit-learn

Scikit-Learn’s PCA class implements PCA using SVD decomposition just like we did before. The following code applies PCA to reduce the dimensionality of the dataset down to two dimensions (note that it automatically takes care of centering the data):

#thereafter we do a PCA with Scikit-learn
from sklearn.decomposition import PCA
pca = PCA(n_components = 2)
X2D = pca.fit_transform(X)
print(X2D)

After fitting the PCA transformer to the dataset, you can access the principal components using the components variable (note that it contains the PCs as horizontal vectors, so, for example, the first principal component is equal to

pca.components_.T[:, 0]

Another very useful piece of information is the explained variance ratio of each principal component, available via the \( explained\_variance\_ratio \) variable. It indicates the proportion of the dataset’s variance that lies along the axis of each principal component.

Example of Cancer Data

We can now repeat the above but applied to real data, in this case the Wisconsin breast cancer data. Here we compute performance scores on the training data using logistic regression.

import matplotlib.pyplot as plt
import numpy as np
from sklearn.model_selection import  train_test_split 
from sklearn.datasets import load_breast_cancer
from sklearn.linear_model import LogisticRegression
cancer = load_breast_cancer()

X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)

logreg = LogisticRegression()
logreg.fit(X_train, y_train)
print("Train set accuracy from Logistic Regression: {:.2f}".format(logreg.score(X_train,y_train)))
# We scale the data
from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
scaler.fit(X_train)
X_train_scaled = scaler.transform(X_train)
X_test_scaled = scaler.transform(X_test)
# Then perform again a log reg fit
logreg.fit(X_train_scaled, y_train)
print("Train set accuracy scaled data: {:.2f}".format(logreg.score(X_train_scaled,y_train)))
#thereafter we do a PCA with Scikit-learn
from sklearn.decomposition import PCA
pca = PCA(n_components = 2)
X2D_train = pca.fit_transform(X_train_scaled)
# and finally compute the log reg fit and the score on the training data	
logreg.fit(X2D_train,y_train)
print("Train set accuracy scaled and PCA data: {:.2f}".format(logreg.score(X2D_train,y_train)))

We see that our training data after the PCA decomposition has a performance similar to the non-scaled data.

Instead of arbitrarily choosing the number of dimensions to reduce down to, it is generally preferable to choose the number of dimensions that add up to a sufficiently large portion of the variance (e.g., 95%). Unless, of course, you are reducing dimensionality for data visualization — in that case you will generally want to reduce the dimensionality down to 2 or 3. The following code computes PCA without reducing dimensionality, then computes the minimum number of dimensions required to preserve 95% of the training set’s variance:

pca = PCA()
pca.fit(X_train)
cumsum = np.cumsum(pca.explained_variance_ratio_)
d = np.argmax(cumsum >= 0.95) + 1

You could then set \( n\_components=d \) and run PCA again. However, there is a much better option: instead of specifying the number of principal components you want to preserve, you can set \( n\_components \) to be a float between 0.0 and 1.0, indicating the ratio of variance you wish to preserve:

pca = PCA(n_components=0.95)
X_reduced = pca.fit_transform(X_train)

Incremental PCA

One problem with the preceding implementation of PCA is that it requires the whole training set to fit in memory in order for the SVD algorithm to run. Fortunately, Incremental PCA (IPCA) algorithms have been developed: you can split the training set into mini-batches and feed an IPCA algorithm one minibatch at a time. This is useful for large training sets, and also to apply PCA online (i.e., on the fly, as new instances arrive).

Randomized PCA

Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. This is a stochastic algorithm that quickly finds an approximation of the first d principal components. Its computational complexity is \( O(m \times d^2)+O(d^3) \), instead of \( O(m \times n^2) + O(n^3) \), so it is dramatically faster than the previous algorithms when \( d \) is much smaller than \( n \).

Kernel PCA

The kernel trick is a mathematical technique that implicitly maps instances into a very high-dimensional space (called the feature space), enabling nonlinear classification and regression with Support Vector Machines. Recall that a linear decision boundary in the high-dimensional feature space corresponds to a complex nonlinear decision boundary in the original space. It turns out that the same trick can be applied to PCA, making it possible to perform complex nonlinear projections for dimensionality reduction. This is called Kernel PCA (kPCA). It is often good at preserving clusters of instances after projection, or sometimes even unrolling datasets that lie close to a twisted manifold. For example, the following code uses Scikit-Learn’s KernelPCA class to perform kPCA with an

from sklearn.decomposition import KernelPCA
rbf_pca = KernelPCA(n_components = 2, kernel="rbf", gamma=0.04)
X_reduced = rbf_pca.fit_transform(X)

Other techniques

There are many other dimensionality reduction techniques, several of which are available in Scikit-Learn.

Here are some of the most popular:

• Multidimensional Scaling (MDS) reduces dimensionality while trying to preserve the distances between the instances.
• Isomap creates a graph by connecting each instance to its nearest neighbors, then reduces dimensionality while trying to preserve the geodesic distances between the instances.
• t-Distributed Stochastic Neighbor Embedding (t-SNE) reduces dimensionality while trying to keep similar instances close and dissimilar instances apart. It is mostly used for visualization, in particular to visualize clusters of instances in high-dimensional space (e.g., to visualize the MNIST images in 2D).
• Linear Discriminant Analysis (LDA) is actually a classification algorithm, but during training it learns the most discriminative axes between the classes, and these axes can then be used to define a hyperplane onto which to project the data. The benefit is that the projection will keep classes as far apart as possible, so LDA is a good technique to reduce dimensionality before running another classification algorithm such as a Support Vector Machine (SVM) classifier discussed in the SVM lectures.

Back to Autoencoders: Linear Autoencoders

PyTorch example

We will, again, use the MNIST database, which has \( 60000 \) training examples and a test set of 10000 handwritten numbers. The images have only one color channel and have a size of \( 28\times 28 \) pixels. We start by uploading the data set.

# import the Torch packages
# transforms are used to preprocess the images, e.g. crop, rotate, normalize, etc
import torch
from torchvision import datasets,transforms

# specify the data path in which you would like to store the downloaded files
# ToTensor() here is used to convert data type to tensor

train_dataset = datasets.MNIST(root='./mnist_data/', train=True, transform=transforms.ToTensor(), download=True)
test_dataset = datasets.MNIST(root='./mnist_data/', train=False, transform=transforms.ToTensor(), download=True)
print(train_dataset)
batchSize=128

#only after packed in DataLoader, can we feed the data into the neural network iteratively
train_loader = torch.utils.data.DataLoader(dataset=train_dataset, batch_size=batchSize, shuffle=True)
test_loader = torch.utils.data.DataLoader(dataset=test_dataset, batch_size=batchSize, shuffle=False)

We visualize the images here using the \( imshow \) function function and the \( make\_grid \) function from PyTorch to arrange and display them.

# package we used to manipulate matrix
import numpy as np
# package we used for image processing
from matplotlib import pyplot as plt
from torchvision.utils import make_grid

def imshow(img):
    npimg = img.numpy()
    #transpose: change array axis to correspond to the plt.imshow() function     
    plt.imshow(np.transpose(npimg, (1, 2, 0))) 
    plt.show()

# load the first 16 training samples from next iteration
# [:16,:,:,:] for the 4 dimension of examples, first dimension take first 16, other dimension take all data
# arrange the image in grid
examples, _ = next(iter(train_loader))
example_show=make_grid(examples[:16,:,:,:], 4)

# then display them
imshow(example_show)

Our autoencoder consists of two parts, see also the TensorFlow example above. The encoder and decoder parts are represented by two fully connected feed forward neural networks where we use the standard Sigmoid function. In the encoder part we reduce the dimensionality of the image from \( 28\times 28=784 \) pixels to first \( 16\times 16=256 \) pixels and then to 128 pixels. The 128 pixel representation is then used to define the representation of the input and the input to the decoder part. The latter attempts to reconstruct the images.

import torch.nn as nn
import torch.nn.functional as F
import torch.optim as optim

# Network Parameters
num_hidden_1 = 256  # 1st layer num features
num_hidden_2 = 128  # 2nd layer num features (the latent dim)
num_input = 784  # MNIST data input (img shape: 28*28)


# Building the encoder
class Autoencoder(nn.Module):
    def __init__(self, x_dim, h_dim1, h_dim2):
        super(Autoencoder, self).__init__()
        # encoder part
        self.fc1 = nn.Linear(x_dim, h_dim1)
        self.fc2 = nn.Linear(h_dim1, h_dim2)
        # decoder part
        self.fc3 = nn.Linear(h_dim2, h_dim1)
        self.fc4 = nn.Linear(h_dim1, x_dim)

    def encoder(self, x):
        x = torch.sigmoid(self.fc1(x))
        x = torch.sigmoid(self.fc2(x))
        return x

    def decoder(self, x):
        x = torch.sigmoid(self.fc3(x))
        x = torch.sigmoid(self.fc4(x))
        return x

    def forward(self, x):
        x = self.encoder(x)
        x = self.decoder(x)
        return x

# When initializing, it will run __init__() function as above
model = Autoencoder(num_input, num_hidden_1, num_hidden_2)

We define here the cost/loss function and the optimizer we employ (Adam here).

# define loss function and parameters
optimizer = optim.Adam(model.parameters())
epoch = 100
# MSE loss will calculate Mean Squared Error between the inputs 
loss_function = nn.MSELoss()

print('====Training start====')
for i in range(epoch):
    train_loss = 0
    for batch_idx, (data, _) in enumerate(train_loader):
        # prepare input data
        inputs = torch.reshape(data,(-1, 784)) # -1 can be any value.
        # set gradient to zero
        optimizer.zero_grad()
        # feed inputs into model
        recon_x = model(inputs)
        # calculating loss 
        loss = loss_function(recon_x, inputs)
        # calculate gradient of each parameter
        loss.backward()
        train_loss += loss.item()
        # update the weight based on the gradient calculated
        optimizer.step()
    if i%10==0:    
        print('====> Epoch: {} Average loss: {:.9f}'.format(i, train_loss ))
print('====Training finish====')

As we have trained the network, we will now reconstruct various test samples to see if the model can generalize to data which were not included in the training set.

# load 16 images from testset
inputs, _ = next(iter(test_loader))
inputs_example = make_grid(inputs[:16,:,:,:],4)
imshow(inputs_example)

#convert from image to tensor
#inputs=inputs.cuda()
inputs=torch.reshape(inputs,(-1,784))

# get the outputs from the trained model
outputs=model(inputs)

#convert from tensor to image
outputs=torch.reshape(outputs,(-1,1,28,28))
outputs=outputs.detach().cpu()

#show the output images
outputs_example = make_grid(outputs[:16,:,:,:],4)
imshow(outputs_example)

After training the auto-encoder, we can now use the model to reconstruct some images. In order to reconstruct different training images, the model has learned to recognize how the image looks like and describe it in the 128-dimensional latent space. In other words, the visual information of images is compressed and encoded in the 128-dimensional representations. As we assume that samples from the same categories should be more visually similar than those from different classes, the representations can then be used for image recognition, i.e., handwritten digit images recognition in our case.

One simple way to recognize images is to randomly select ten training samples from each class and annotate them with the corresponding label. Then given the test data, we can predict which classes they belong to by finding the most similar labelled training samples to them.

# get 100 image-label pairs from training set
x_train, y_train = next(iter(train_loader))

# 10 classes, 10 samples per class, 100 in total
candidates = np.random.choice(batchSize, 10*10)

# randomly select 100 samples
x_train = x_train[candidates]
y_train = y_train[candidates]

# display the selected samples and print their labels

imshow(make_grid(x_train[:100,:,:,:],10))
print(y_train.reshape(10, 10))

# get 100 image-label pairs from test set
x_test, y_test = next(iter(train_loader))
candidates_test = np.random.choice(batchSize, 10*10)

x_test = x_test[candidates_test]
y_test = y_test[candidates_test]

# display the selected samples and print their labels
imshow(make_grid(x_test[:100,:,:,:],10))

print(y_test.reshape(10, 10))

# compute the representations of training and test samples
#h_train=model.encoder(torch.reshape(x_train.cuda(),(-1,784)))
#h_test=model.encoder(torch.reshape(x_test.cuda(),(-1,784)))
h_train=model.encoder(torch.reshape(x_train,(-1,784)))
h_test=model.encoder(torch.reshape(x_test,(-1,784)))

# find the nearest training samples to each test instance, in terms of MSE
MSEs = np.mean(np.power(np.expand_dims(h_test.detach().cpu(), axis=1) - np.expand_dims(h_train.detach().cpu(), axis=0), 2), axis=2)
neighbours = MSEs.argmin(axis=1)
predicts = y_train[neighbours]

# print(np.stack([y_test, predicts], axis=1))
print('Recognition accuracy according to the learned representation is %.1f%%' % (100 * (y_test == predicts).numpy().astype(np.float32).mean()))