1 / 27

Unsupervised learning Networks

ELE571 Digital Neural Networks. Unsupervised learning Networks. Associative Memory Networks. Associative Memory Networks. Associative Memory Networks recalls the original undistorted pattern from a distorted or partially-missing pattern. feedforward type (one-shot-recovery)

salena
Download Presentation

Unsupervised learning Networks

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. ELE571 Digital Neural Networks Unsupervised learning Networks • Associative Memory Networks

  2. Associative Memory Networks Associative Memory Networks recalls the original undistorted pattern from a distorted or partially-missing pattern. • feedforward type (one-shot-recovery) • feedback type e.g., the Hopfield network (iterative recovery)

  3. Associative Memory Model (feedfoward type) a nonlinear unit: e.g. threshold b b(k) W = b(k)m b(m) T a(m) = a(k) W could be (1) symmetric or not, (2) square or not

  4. Bidirectional Associative Memory

  5. a1 = [1 1 1 1 –1 1 1 1 1] a2 = [1 -1 1 -1 1 -1 1 –1 1] X=[ ] 1 1 1 1 –1 1 1 1 1 1 -1 1 -1 1 -1 1 –1 1 2 0 2 0 0 0 2 0 2 0 2 0 2 -2 2 0 2 0 2 0 2 0 0 0 2 0 2 0 2 0 2 -2 2 0 2 0 0 -2 0 -2 2 -2 0 -2 0 0 2 0 2 -2 2 0 2 0 2 0 2 0 0 0 2 0 2 0 2 0 2 -2 2 0 2 0 2 0 2 0 0 0 2 0 2 The weight matrix W =XTX=

  6. Associative Memory Model (feedback type) anew aold • W must be • normalized • W = WH W=X+X

  7. Each iteration in AMM(W) comprises two substeps: • (a) Projection of aold onto W-plane • anet = W aold • Remap the net-vector to closest symbol vector: • anew = T[anet] The two substeps in one iteration can be sumarized as one procedure: anew = arg minanew  ss|| anew - Waold ||

  8. initial vector Γρ x -plane symbol vector perfect attractor Linear projection onto x-plane = g-update in DML, shown as 2 steps in one AMM iteration Resymbolization nonlinear mapping = ^ s-update in DML, shown as ^ anet is the (least-square-error) projection of aold onto the (column) subspace of W.

  9. Common Assumptions on Signal for Associative Retrieval • Inherent properties of the signals (patterns) • to be retrieved: • Orthogonality • Higher-order statistics • Constant modulus property • FAE -Property • others

  10. gX=gH S = v S = ŝ Blind Recovery of MIMO System Channel Source Observation X S ε1 H g +

  11. h11 g1 h21 h1p +           g2 S h2p gq hq1 hqp Blind Recovery of MIMO System s1     si sp Goal: to find g, such that v gH, and v S = [ 0 .. 0 1 0 .. 0 ]S = sj

  12. Assumptions on MIMO System For Deterministic H, …… Signal Recoverability • H is PR (perfectly recoverable) if and only if H has full column rank, i.e. an inverse exists.

  13. H=[ ] H=[ ] 1 2 1 3 1 2 1 2 1 2 1 2 Examples for Flat MIMO non-recoverable recoverable

  14. H=[ ] 1 2 1 3 1 2 If perfectly recoverable, e.g. • Parallel Equalizer • ŝ = H+X = G X

  15. gX=gH S = v S = ŝ Example: Wireless MIMO System X S ε1 H g + Signal Constellation Ci Signal recovery via g:

  16. FAE()-Property: Finite-Alphabet Exclusiveness Given v s = ŝ, for ŝ to be always a valid symbol for any valid symbol vector s, if and only if v  [ 0 .. 0 ±1 0 .. 0 ]

  17. Theorem: FAE -Property Suppose that a v W = b. For the output b to be always a valid symbol sequence given whatever v, the necessary and sufficient condition is that v = E(k) . In other words, it is impossible to produce a valid but different output symbol vector.

  18. ], S = [ -1 +1 +1 +1 +1 +1 • • +1 +1 +1 -1 -1 +1 • • -1 -1 +1 -1 +1 +1 • • FAE()-Property: Finite-Alphabet Exclusiveness v S = [valid symbols] if and only if v  [ 0 .. 0 ±1 0 .. 0 ] If v =[ 0 .. 0 ±1 0 .. 0 ] If v ≠[ 0 .. 0 ±1 0 .. 0 ]

  19. Blind-BLAST gX=gH S = v S = ŝ “EM”: • The E step determines the best guess of the membership function zj . • The M step determines the best parameters, θn , which maximizes the likelihood function. ŝ = T[gX ] ŝ [Finite Alphabet] E-step M-step ĝ= ŝX+ š= T[ĝ X] E-step ğ= šX+ M-step š= T[ĝ X] =T[ ŝX+ X] =T[ ŝ W] Combined EM:

  20. :threshold Associative Memory Network ŝnew ŝnew W=X+X ŝnew ŝold=ŝ ŝnew= Sign[ŝoldW]

  21. Nearest symbol vector initial vector ŝ gx -plane ŝ’ = T[ŝW] = Linear projection onto gx - plane g= ŝold X+ = FA nonlinear mapping

  22. Definition: Perfect Attractor of AMM(W) • A symbol vector a* is a "perfect attractor" of AMM(W) if and only if • a* is symbol vector • a* = W a*

  23. Let v= [ f(1) ≠0 f(2) … f(p)] Let v’= [ f(1) ≠0 f(2) … f(p-1) 0] Let ûi= [ ui (1) ui(2) … ui(p) ]T 1 q 2 Let ǖi = v ûi Let ǖ’i = v’ ûi 1 p = 0 1 Thus f(p) =0

  24. MatLab Exercise Compare the two programs and determine the differences in performance. Why such a difference?

  25. p = zeros(1,100); for j=1:100; S = sign(randn(5,200)); A = randn(5,5)+eye(5); X = A*S + 0.01*randn(5,200); s = sign(randn(200,1)); W = X'*inv(X*X')*X; for i=1:20; sold = s; s = tanh(100*W*s); s = sign(s); end while norm(s-W*s)> 5.0, s = sign(randn(200,1)); for i=1:20; sold = s; s = tanh(100*W*s); s = sign(s); end end p(j) = max(abs(S*s)); end hist(p)

  26. p = zeros(1,100); for j=1:100; S = sign(randn(5,200)); A = randn(5,5)+eye(5); X = A*S + 0.01*randn(5,200); s = sign(randn(200,1)); W = X'*inv(X*X')*X; for i=1:20; sold = s; s = tanh(100*W*s); s = sign(s); end p(j) = max(abs(S*s)); end hist(p)

More Related