Representation

#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 β€œstraighten” the data manifold
  • The β€œfeature 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 β€œAnalysis” net which computes the hidden representation
  • Decoder
    • The β€œSynthesis” 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 β€œgenerator” 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
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