Bernal S D_2010.pdf - University of Plymouth

4.3. LEARNING
4.3.2 S1-C1 CRTs
The belief propagalton operations between SI and CI need to approximate the max operation
implemented in the original HMAX model. To do this wc propose increasing the number of
states of the CI layer so that for each SI orienlalion there are Kcigmup states coding different
spatial arrangements of SI units. In this way all the CI states corresponding to the same SI
orienlalion can be grouped together and treated as a single state during the generation of the
outpui A mcs-sage. The operation to compute the output A message implements the sum over
all the stales of each CI group. In other words. CI nodes provide a dislribulion over SI features
and locations, which after marginalizing (summing) over tlie locations during the gcneraiion of
ihc outpui A message, provides an approximation to the max operation.
However, ihe number of different possible spatial arrangements of SI units converging on a
CI unit i.s given by the number of ^-combinations of the n-eiemenl set equal to the binomial
coefficient, ("). where n — ANa • ANc\ • 2 bands and it is the number of active units (assuming
binary values). For example, for n = 8 • 8 • 2 = 128 and k = .12, the number of possible spatial
arrangements is ('^2} ^ \(i^^. This is just a lower bound on Ihereal number of combinations, as
we would need to sum over all the diffcrenl values of k, and ihe weight values at each location
are not necessarily binary, but range from 0 to I. Creating a distribution for each CI node
containing/Tc] =^51 -Kngroup — 4- lO" slates, is obviously intractable.
Tor this reason, the value Kc\grouii is limited lo include only the most common arrangements
of S1 units for each orientation. Figure 4.7 portrdys a toy example of this method where the
numberofSI units is n —.3'3-9. the number of SI stales is A'51 = 4 (orientations), the number
of features per group is li(:\g„,up — 3 and the resulting number of C1 stales is Kci —4-3= 12.
The equation to calculate A(C1), which combines the bottom-up evidence, and Ac](S2), which
sends the bottom-up evidence lo S2, arc shown in the diagram.
Noie that, as illusiraied in the toy example, the weights are learned for each lixed CI state-i a.s
a function of the n afferent SI units and the j SI slates per node. This yields a weight matrix
(shown ontheboliom left) for each of the CM f states. However, theCITsof a Bayesian network
are defined as a function of the child and the parent stales, j and / respectively, for each rtxed
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