one can model the neural-network layer (6-5) with the simpler vector assignment

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Vector v = f(WW * xx)

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The vector expression WW * xx is equivalent to W * x + bias Note that the true input x is still available to the program, so that a vector expression can be assigned to x Figures 6-4b and 6-11 show simple applications

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Vector Models of Neural Networks

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We obtain a normalized neuron-layer pattern v1 with

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Vector v1 = abs(v) | Vector v1 = v/vnorm DOT vnorm = v1 * 1

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(6-6)

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(taxicab normalization, see also Section 3-7) or

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DOT vnormsq = v * v Vector v1 = v/vnorm | vnorm = sqrt(vnormsq)

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(6-7)

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(Euclidean normalization),4 so that the activations v1[i] or their squares v12[i] add up to 1 Usually the un-normalized vector v is not needed, and v1 in Eq (6-6) or (6-7) can simply be replaced with v Normalized activations are necessarily bounded The output activations v[i] of a softmax neuron layer defined by

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Vector v = exp(c * W * x) DOT vsum = v * 1 | | -- (c > 1) Vector v = v/vsum

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(6-8)

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are normalized and positive Each v[i] is enhanced or reduced depending on how large it is This contrast enhancement becomes more pronounced as the parameter c increases If no two output activations are equal, the largest v[i] approaches 1 as the parameter c increases, and all other v[i] go to 0 Such a softmax layer is a useful continuous approximation of a normalized maximum-selecting layer defined for the case of all nonnegative v[i] by

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Vector v^ = W * x | Vector v = swtch(v)

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(6-9a)

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(see also Section 3-8b) Another contrast-enhancement technique is thresholding, as in

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Vector v = swtch(c * W * x thresh) (c > 0)

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(6-9b)

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where thresh is a positive threshold value 6-4 Multilayer Networks Assume that the experiment protocol has declared neuron-activation vectors x, v, z, and connection-weight matrices W1, W2, with

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ARRAY x[nx], v[nv], z[nz], , W1[nv,nx], W2[nz,nv],

(6-10)

To save divisions, which are usually slower than multiplications, one can program

DOT vnormsq = v * v | vnormo1 = 1/sqrt(vnormsq) | Vector v1 = v*vnormo1

Neural-network Simulation

Then a DYNAMIC program segment can model a multilayer neural network by simply combining network-layer assignments, as in

Vector v = tanh(W1 * x) Vector z = W2 * v

(6-11)

The input pattern x feeds the v layer, the v layer feeds the z layer, and so on (Fig 6-1b) 6-5 Exercising a Neural-network Model

(a) Computing Successive Neuron Layer Outputs

Neuron-layer definitions such as Eqs (6-4) to (6-9) are normally sampleddata assignments that execute at the sampling times t0, t0 + COMINT, t0 + 2 COMINT, , t0 + TMAX = t0 + (NN 1)COMINT defined by the experiment protocol (Section 1-6) If an input pattern x = x(t) is programmed with a vector assignment such as

Vector x = A * sin(omega * t) + a * ran()

(6-12)

then subsequent network-layer assignments such as Eq (6-11) will generate the neuron-layer outputs v(t), z(t), for successive sampling times t One can now display or list selected neuron activations, say, v[19], as functions of the simulation time t t0 and TMAX default to 1 and NN 1 if the DYNAMIC program segment does not contain differential equations If t0 and TMAX are not specified, then t simply steps through t = 1, 2, , t0 + TMAX = NN 1

(b) Using Pattern-row Matrices

Instead of introducing the input pattern as a function of t as in Eq (6-12), one can define nx-dimensional input patterns x as selected rows of an N nx pattern-row matrix5 P declared and filled in the experiment-protocol script (Section 6-10a) After a DYNAMIC program segment specifies the value of the system variable iRow > 0, vector assignments such as

Vector x = P#

Vector x = (q alpha) * cos(P#) + c

(6-13)

Pattern-row matrices simplify computer programs because almost all computer languages store matrices row-by-row in memory Pattern vectors, though, are usually represented as column vectors, and most textbooks [15,16] define a pattern matrix as the nx N matrix XT whose columns are our N nx-dimensional pattern vectors

Vector Models of Neural Networks

automatically substitute the vector in the (iRow mod N)th row of P for P# DESIRE returns an error message if iRow < 1 In particular, the DYNAMIC-segment assignment