Unsupervised learning The Hebb rule – Neurons that fire together wire together. PCA

Unsupervised learning • The Hebb rule – Neurons that fire together wire together. • PCA • RF development with PCA

Classical Conditioning

Classical Conditioning and Hebb’s rule Ear A Nose B Tongue “When an axon in cell A is near enough to excite cell B and repeatedly and persistently takes part in firing it, some growth process or metabolic change takes place in one or both cells such that A’s efficacy in firing B is increased” D. O. Hebb (1949)

The generalized Hebb rule: where xiare the inputs and y the output is assumed linear: Results in 2D

Example of Hebb in 2D w (Note: here inputs have a mean of zero)

On the board: • Solve simple linear first order ODE • Fixed points and their stability for non linear ODE. • Describe potential and gradient descent

In the simplest case, the change in synaptic weight w is: where x are input vectors and y is the neural response. Assume for simplicity a linear neuron: So we get: Now take an average with respect to the distribution of inputs, get:

If a small change Δw occurs over a short time Δt then: (in matrix notation) If <x>=0 , Q is the covariance function. What is then the solution of this simple first order linear ODE ? (Show on board)

Mathematics of the generalized Hebb rule The change in synaptic weight w is: where x are input vectors and y is the neural response. Assume for simplicity a linear neuron: So we get:

Taking an average of the distribution of inputs And using and We obtain

In matrix form Where J is a matrix of ones, e is a vector in direction (1,1,1 … 1), and or Where

The equation therefore has the form If k1 is not zero, this has a fixed point, however it is usually not stable. If k1=0 then have:

The Hebb rule is unstable – how can it be stabilized while preserving its properties? The stabilized Hebb (Oja) rule. Where: Appoximateto first order in η: Now insert Get: Normalize

} y Therefore The Oja rule therefore has the form:

Average In matrix form:

Using this rule the weight vector converges to • the eigen-vector of Q with the highest eigen-value. It is often called a principal component or PCA rule. • The exact dynamics of the Oja rule have been solved by Wyatt and Elfaldel 1995 • Variants of networks that extract several principal components have been proposed (e.g: Sanger 1989)

Therefore a stabilized Hebb (Oja neuron) carries out Eigen-vector, or principal component analysis (PCA).

Using this rule the weight vector converges to the eigen-vector of Q with the highest eigen-value. It is often called a principal component or PCA rule. Another way to look at this: Where for the Oja rule: At the f.p: where So the f.p is an eigen-vector of Q. The condition means that w is normalized. Why? Could there be other choices for β?

Show that the Oja rule converges to the state |w^2|=1 • The Oja rule in matrix form: • Multiply by w, get • Bonus question for H.W: • The equivalence above • That the direction of the PC is the direction of maximal variance (HKP – pg 202)

Show that the f.p of the Oja rule is such that the largest eigen-vector with the largest eigen-value (PC) is stable while others are not (from HKP – pg 202). Start with: Assume w=ua+εub where ua and ub are eigen-vectors with eigen-values λa,b

Get: (show on board) Therefore: That is stable only when λa> λb for every b.

Unsupervised learning The Hebb rule – Neurons that fire together wire together. PCA