{1495} revision 9 modified: 01-22-2020 01:45 gmt

Why multifactor?

  • Take a simple MLP. Let xx be the layer activation. X 0X^0 is the input, X 1X^1 is the second layer (first hidden layer). These are vectors, indexed like x i ax^a_i .
  • Then X 1=WX 0X^1 = W X^0 or x j 1=ϕ(Σ i=1 Nw ijx i 0)x^1_j = \phi(\Sigma_{i=1}^N w_{ij} x^0_i) . ϕ\phi is the nonlinear activation function (ReLU, sigmoid, etc.)
  • In standard STDP the learning rule follows Δwf(x pre(t),x post(t)) \Delta w \propto f(x_{pre}(t), x_{post}(t)) or if layer number is aa Δw a+1f(x a(t),x a+1(t))\Delta w^{a+1} \propto f(x^a(t), x^{a+1}(t))
    • (but of course nobody thinks there 'numbers' on the 'layers' of the brain -- this is just referring to pre and post synaptic).
  • In an artificial neural network, Δw aEw ij aδ j ax i \Delta w^a \propto - \frac{\partial E}{\partial w_{ij}^a} \propto - \delta_{j}^a x_{i} (Intuitively: the weight change is proportional to the error propagated from higher layers times the input activity) where δ j a=(Σ k=1 Nw jkδ k a+1)ϕ \delta_{j}^a = (\Sigma_{k=1}^{N} w_{jk} \delta_k^{a+1}) \partial \phi where ϕ\partial \phi is the derivative of the nonlinear activation function, evaluated at a given activation.
  • f(i,j)[x,y,θ,ϕ] f(i, j) \rightarrow [x, y, \theta, \phi]
  • k=13.165 k = 13.165
  • x=round(i/k) x = round(i / k)
  • y=round(j/k) y = round(j / k)
  • θ=a(ikx)+b(ikx) 2 \theta = a (\frac{i}{k} - x) + b (\frac{i}{k} - x)^2
  • ϕ=a(jky)+b(jky) 2 \phi = a (\frac{j}{k} - y) + b (\frac{j}{k} - y)^2