1 / 21

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

howard
Télécharger la présentation

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

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

  2. Classical Conditioning

  3. 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)

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

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

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

  7. 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:

  8. 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)

  9. 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:

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

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

  12. 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:

  13. 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

  14. } y Therefore The Oja rule therefore has the form:

  15. Average In matrix form:

  16. 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)

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

  18. 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?

  19. 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)

  20. 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

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

More Related