Logistic regression

• This the perceptron with a sigmoid activation
  • It actually computes the probability that the input belongs to class 1
  • Decision boundaries may be obtained by comparing the probability to a threshold
  • These boundaries will be lines (hyperplanes in higher dimensions)
  • The sigmoid perceptron is a linear classifier

Estimating the model

• Given: Training data: $\left(X_{1}, y_{1}\right),\left(X_{2}, y_{2}\right), \ldots,\left(X_{N}, y_{N}\right)$
• $X$ are vectors, $y$ are binary (0/1) class values
• Total probability of data

$\begin{array}{l} P\left(\left(X_{1}, y_{1}\right),\left(X_{2}, y_{2}\right), \ldots,\left(X_{N}, y_{N}\right)\right)= \prod_{i} P\left(X_{i}, y_{i}\right) \\\\ =\prod_{i} P\left(y_{i} \mid X_{i}\right) P\left(X_{i}\right)=\prod_{i} \frac{1}{1+e^{-y_{i}\left(w_{0}+w^{T} X_{i}\right)}} P\left(X_{i}\right) \end{array}$

• Likelihood

$P(\text {Training data})=\prod_{i} \frac{1}{1+e^{-y_{i}\left(w_{0}+w^{T} X_{i}\right)}} P\left(X_{i}\right)$

• Log likelihood

$\begin{array}{l} \log P(\text {Training data})= \sum_{i} \log P\left(X_{i}\right)-\sum_{i} \log \left(1+e^{-y_{i}\left(w_{0}+w^{T} X_{i}\right)}\right) \end{array}$

• Maximum Likelihood Estimate

$w_{0}, w_{1}=\underset{w_{0}, w_{1}}{\operatorname{argmax}} \log P(\text {Training data})$

• Equals (note argmin rather than argmax)

$w_{0}, w_{1}=\underset{w_{0}, w}{\operatorname{argmin}} \sum_{i} \log \left(1+e^{-y_{i}\left(w_{0}+w^{T} X_{i}\right)}\right)$

• Identical to minimizing the KL divergence between the desired output and actual output $\frac{1}{1+e^{-\left(w_{0}+w^{T} X_{i}\right)}}$

MLP

Separable case

• The rest of the network may be viewed as a transformation that transforms data from non-linear classes to linearly separable features
  • We can now attach any linear classifier above it for perfect classification
  • Need not be a perceptron
  • Could even train an SVM on top of the features!
• For insufficient structures, the network may attempt to transform the inputs to linearly separable features
  • Will fail to separate exactly, but will try to minimize error
• The network until the second-to-last layer is a non-linear function $f(X)$ that converts the input space $X$ of into the feature space where the classes are maximally linearly separable

Lower layers

• Manifold hypothesis: For separable classes, the classes are linearly separable on a non-linear manifold
• Layers sequentially “straighten” the data manifold
• The “feature extraction” layer transforms the data such that the posterior probability may now be modelled by a logistic

Weight as a template

• In high dimensional space, all vectors are more or less the same length
  • Which means all $x$ are in this surface of sphere
• The perceptron fires if the input is within a specified angle of the weight
  • Represents a convex region on the surface of the sphere!
  • The network is a Boolean function over these regions
• Neuron fires if the input vector is close enough to the weight vector
  • If the input pattern matches the weight pattern closely enough
• The perceptron is a correlation filter!

Autoencoder

• The lowest layers of a network detect significant features in the signal
• The signal could be (partially) reconstructed using these features
  • Will retain all the significant components of the signal

Simplest autoencoder

• This is just PCA!
• The autoencoder finds the direction of maximum energy
• Simply varying the hidden representation will result in an output that lies along the major axis

Terminology

• Encoder
  • The “Analysis” net which computes the hidden representation
• Decoder
  • The “Synthesis” which recomposes the data from the hidden representation

Nonlinearity

• When the hidden layer has a linear activation the decoder represents the best linear manifold to fit the data
  • Varying the hidden value will move along this linear manifold
• When the hidden layer has non-linear activation, the net performs nonlinear PCA
  • The decoder represents the best non-linear manifold to fit the data
  • Varying the hidden value will move along this non-linear manifold

• The model is specific to the training data
  • Varying the hidden layer value only generates data along the learned manifold
  • Any input will result in an output along the learned manifold
  • But may not generalize beyond the manifold
    • Input unseen data may behave beyond intuitive manner, no constrain!
    • The decoder can only generate data on the manifold that the training data lie on
• This also makes it an excellent “generator” of the distribution of the training data

Dictionary-based techniques

• The decoder represents a source-specific generative dictionary
  • Exciting it will produce typical data from the source!

Signal separation

• Separation: Identify the combination of entries from both dictionaries that compose the mixed signal

• Given mixed signal and source dictionaries, find excitation that best recreates mixed signal
  • Simple backpropagation
• Intermediate results are separated signals