The Hebb rule Neurons that fire together wire together.

Transcription

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

2 Classical Conditioning and Hebbʼs 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)

3 The generalized Hebb rule: where x i are the inputs, η a learning rate, and y the output is assumed linear: y Results in 2D w1 w2 w3 w4 w5 x 1 x 2 x 3 x 4 x 5

4 Example of Hebb in 2D show simulation w + data points o synaptic weights (Note: here inputs have a mean of zero)

5 On the board: Solve simple linear first order ODE Fixed points and their stability for non linear ODE.

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

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

8 Example in 2D - 4 input vectors: x 1 = 0.8 x 2 = 1.0 x 3 = 1.0 x 4 = show on board calculation of Q Q = What are the eigen-values and eigen-vectors of Q? Qz = λz two solutions z 1 = λ 1 =1.62 z 2 = λ 2 = 0.02

9 This holds in general for: If: Qz i = λ i z i dw dt = ηqw w(t) = N a e ηλ j t z j j j=1 Then: where a j are set by the initial conditions. What does this mean compare to simulation plot.

10 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:

11 Taking an average of the distribution of inputs and using E[x i x j ] = x i x j = Q ij and E[x i ] = x i = µ We obtain

12 In matrix form E[Δw ] = η Q x 0 µj [ ]w ek 1 [ ] w y 0 (µ x 0 ) = η Q k 2 J Where J is a matrix of ones, e is a unit vector in direction (1,1,1 1), and or E[Δw ] = ηq ' w ek 1 Where Q ' = Q k 2 J

13 The equation therefore has the form If k 1 is not zero, this has a fixed point, however it is usually not stable. If k 1 =0 then have:

14 The Hebb rule is unstable how can it be stabilized while preserving its properties? The stabilized Hebb (Oja) rule. w ' i (t +1) = w ' i(t) + Δw i Where w(t) =1 (w + Δw) 2 Normalize Appoximate to first order in Δw: (show on board) Now insert Get:

15 Therefore } y The Oja rule therefore has the form:

16 In matrix form: Average

17 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)

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

19 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: So the f.p is an eigen-vector of Q. where λ = y 2 The condition λ = y 2 means that w is normalized. Why? -10 pts bonus question Could there be other choices for β?

20 Show that the Oja rule converges to the state w^2 =1 The Oja rule in matrix form: What is d w 2 dt = d(wt w) dt = 2w T dw dt d w 2 dt = 2ηy 2 (1 w 2 ) Do this in detail, and show stability bonus HW question (15 pt)

21 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=u a +εu b where u a and u b are eigenvectors with eigen-values λ a,b * HKP = Hertz, Krough, Palmer 1990.

22 Get: d(u a + εu b ) dt Therefore: (show on board) = η( λ a u a + ελ b u b λ a u a ελ a u b ελ b u ) a + O(ε 2 ) That is stable only when λ a > λ b for every b.

23 Finding multiple principal components the Sanger algorithm. Subtract projection onto accounted for subspace (Gram-Schmidt) Standard Oja Normalization

24 Homework 1: 1) For the input vectors: x 1 = x 2 = x 3 = x = x = x 6 = Calculate the correlation matrix, the covariance matrix and the eigen-vectors (30 pts). Draw the points and the direction of the eigen-vectors 2) Simulating Hebb synapses: 2a) Implement a simple Hebb neuron with random 2D input, tilted at an angle, θ=30 o with variances 1 and 3 and mean 0. Show the synaptic weight evolution. (200 patterns at least) 2b) Calculate the correlation matrix of the input data. Find the eigen-values, eigen-vectors of this matrix. Compare to 1a. 2c) Repeat 1a for an Oja neuron, compare to 1b. (70 pts) + bonus questions above (another 25 pt)

25 What did we learn up to here?

26 Visual Pathway Visual Cortex Receptive fields are: Binocular Orientation Selective Area 17 LGN Receptive fields are: Monocular Radially Symmetric Retina light electrical signals

27 Left Right Left Right Response (spikes/sec) Tuning curves

28 Orientation Selectivity Response (spikes/sec) Adult Response (spikes/sec) Adult angle Eye-opening Eye-opening angle

29 Response (spikes/sec) Left Right Right Left angle angle % of cells Rittenhouse et. al. group group

30 First use Hebb/PCA with toy examples then used with more realistic examples

31 Aim get selective neurons using a Hebb/PCA rule Simple example: r r r r

32 Why? The eigen-value equation has the form: Q can be rewritten in the equivalent form: And a possible solution can be written as the sum:

33 Inserting, and by orthogonality get: So for l=0, λ=2, and for l=1, λ=q, for l>1 there is no solution. So either w(r)= const with λ=2 or with λ=q.

34 Orientation selectivity from a natural environment: The Images:

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36 Natural Images, Noise, and Learning Retina image retinal activity present patches update weights LGN Patches from retinal activity image Cortex Patches from noise

37 Preprocessed images: (fig 5.9) 4 different independent examples Symmetrized means that our originals images were duplicated Rotated to 30,60,90 degrees and added to the data set to make it more rotationally symmetric

38 Response (spikes/sec) Left Right Right Left angle angle % of cells group Rittenhouse et. al group

39 Binocularity simple examples. Q = < x x > < x x > l l l r < x x > < x x > r l r r Q is a 2-eye correlation function. What is the solution of the eigen-value equation: Qw = λw w 1 = w 2 = 1 2 λ 1 = a + b λ 2 = a b +1 1

40 In a simpler case This implies Q ll =Q rr, that is eyes are equivalent. And the cross eye correlation is a scaled version of the one eye correlation. If: Qw = λw then: with w 1 = 1 2 m w 2 = 1 m 2 +m m

41 Positive correlations (η=0.2) Hebb with lower saturation at 0 Negative correlations (η=-0.2)

42 Lets now assume that Q is as above for the 1D selectivity example. Q 2 = Q ηq ηq Q Oscillating Correlated q(1+ η) Anti-correlated q(1 η) Constant 2(1+ η) 2(1 η)

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44 With 2D space included

45 2 partially overlapping eyes using natural images

46 Orientation selectivity and Ocular Dominance Left Eye Left Synapses Right Eye Right Synapses Left Right No. of Cells PCA Bin 5

47 What did we learned today?