Non-linear hypothesis, neurons and the brain, model representation, and multi-class classification.

1. Motivations

1a. Non-linear Hypothesis

  • You can add more features
    • But it will be slow to process
  • If you have an image with 50 x 50 pixels (greyscale, not RGB)
    • n = 50 x 50 = 2500
    • quadratic features = (2500 x 2500) / 2
  • Neural networks are much better for a complex nonlinear hypothesis

1b. Neurons and the Brain

  • Origins
    • Algorithms that try to mimic the brain
  • Was very widely used in the 80s and early 90’s
    • Popularity diminished in the late 90’s
  • Recent resurgence
    • State-of-the-art techniques for many applications
  • The “one learning algorithm” hypothesis
    • Auditory cortex handles hearing
      • Re-wire to learn to see
    • Somatosensory cortex handles feeling
      • Re-wire to learn to see
    • Plug in data and the brain will learn accordingly
  • Examples of learning

2. Neural Networks

2a. Model Representation I

  • Neuron in the brain
    • Many neurons in our brain
    • Dendrite: receive input
    • Axon: produce output
      • When it sends a message through the Axon to another neuron
      • It sends to another neuron’s Dendrite
  • Neuron model: logistic unit
    • Yellow circle: body of neuron
    • Input wires: dendrites
    • Output wire: axon
  • Neural Network
    • 3 Layers
      • 1 Layer: input layer
      • 2 Layer: hidden layer
        • Unable to observe values
        • Anything other than input or output layer
      • 3 Layer: output layer
      • We calculate each of the layer-2 activations based on the input values with the bias term (which is equal to 1)
        • i.e. x0 to x3
        • We then calculate the final hypothesis (i.e. the single node in layer 3) using exactly the same logic, except in input is not x values, but the activation values from the preceding layer
      • The activation value on each hidden unit (e.g. a12 ) is equal to the sigmoid function applied to the linear combination of inputs
        • Three input units
        • Ɵ(1) is the matrix of parameters governing the mapping of the input units to hidden units
          • Ɵ(1) here is a [3 x 4] dimensional matrix
      • Three hidden units
        • Then Ɵ(2) is the matrix of parameters governing the mapping of the hidden layer to the output layer
          • Ɵ(2) here is a [1 x 4] dimensional matrix (i.e. a row vector)
      • Every input/activation goes to every node in following layer
        • Which means each “layer transition” uses a matrix of parameters with the following significance
          • j (first of two subscript numbers)= ranges from 1 to the number of units in layer l+1
          • i (second of two subscript numbers) = ranges from 0 to the number of units in layer l
          • l is the layer you’re moving FROM
  • Notation

2a. Model Representation II

  • Here we’ll look at how to carry out the computation efficiently through a vectorized implementation. We’ll also consider why neural networks are good and how we can use them to learn complex non-linear things
  • Forward propagation: vectorized implementation
    • g applies sigmoid-function element-wise to z
    • This process of calculating H(x) is called forward propagation
      • Worked out from the first layer
      • Starts off with activations of input unit
      • Propagate forward and calculate the activation of each layer sequentially
  • Similar to logistic regression if you leave out the first layer
    • Only second and third layer
    • Third layer resembles a logistic regression node
    • The features in layer 2 are calculated/learned, not original features
    • Neural network, learns its own features
      • The features a’s are learned from x’s
      • It learns its own features to feed into logistic regression
      • Better hypothesis than if we were constrained with just x1, x2, x3
      • We can have whatever features we want to feed to the final logistic regression function
      • Implemention in Octave for a2
        • a2 = sigmoid (Theta1 * x);
  • Other network architectures
    • Layer 2 and 3 are hidden layers

2. Neural Network Application

2a. Examples and Intuitions I

  • XOR/XNOR
    • XOR: or
    • XNOR: not or
  • AND function
    • Outputs 1 only if x1 and x2 are 1
    • Draw a table to determine if OR or AND
  • NAND function
    • NOT AND
  • OR function

2b. Examples and Intuitions II

  • NOT function
  • XNOR function
    • NOT XOR
    • NOT an exclusive or
      • Hence we would want
        • AND
        • Neither

2c. Multi-class Classification

  • Example: identify 4 classes
    • You would want a 4 x 1 vector for h_theta(X)
    • 4 logistic regression classifiers in the output layer
    • There will be 4 output
    • y would be a 4 x 1 vector instead of an integer