In which I fall back to testing temporal pooling on the simplest possible problem, that of fixed sequences presented in isolation from each other. I find that simple sequences do get a temporal pooling representation that is both sensitive and specific. Longer sequences are not fully covered due to the decay rate in temporal pooling scores; this needs rethinking. Sequences which start the same and later diverge give rise to pooled representations over them both together, as well as each unique sub-sequence.
In my recent post I tested a temporal pooling algorithm on an input stream consisting of fixed sequences randomly mixed together, with poor results. On Rob Freeman’s suggestion I went on to test it with the simpler problem of fixed sequences in isolation. I should have done that first, of course…
The problem uses the same fixed sequences as last time but presented randomly in series, separated by a small gap. As before this problem comes with an interactive demo.
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The sequence are well predicted after a few hundred time steps, and consequently temporal pooling emerges in the higher-level region.
As before, let’s try to measure this.
So I “warmed up” the system for 4000 time steps, keeping the following 2000 time steps. I filtered them down to where the given pattern occurs (excluding the first step it appears on, which is unpredictable), and selected the top 3 most frequently active temporal pooling cells. These are my candidates.
It is then a simple matter to compute sensitivity, specificity and precision for each candidate cell. The results are shown in the following table.
| pattern | candidate | in-steps | out-steps | active-in | active-out | sensitivity | specificity | precision |
|---|---|---|---|---|---|---|---|---|
| run-0-5 | 0 | 154 | 925 | 154 | 0 | 1.00 | 1.00 | 1.00 |
| run-0-5 | 1 | 154 | 925 | 154 | 0 | 1.00 | 1.00 | 1.00 |
| run-0-5 | 2 | 154 | 925 | 154 | 0 | 1.00 | 1.00 | 1.00 |
| rev-5-1 | 0 | 96 | 990 | 96 | 0 | 1.00 | 1.00 | 1.00 |
| rev-5-1 | 1 | 96 | 990 | 96 | 0 | 1.00 | 1.00 | 1.00 |
| rev-5-1 | 2 | 96 | 990 | 96 | 0 | 1.00 | 1.00 | 1.00 |
| run-6-10 | 0 | 132 | 945 | 132 | 104 | 1.00 | 0.89 | 0.56 |
| run-6-10 | 1 | 132 | 945 | 132 | 116 | 1.00 | 0.88 | 0.53 |
| run-6-10 | 2 | 132 | 945 | 132 | 116 | 1.00 | 0.88 | 0.53 |
| jump-6-12 | 0 | 116 | 965 | 116 | 132 | 1.00 | 0.86 | 0.47 |
| jump-6-12 | 1 | 116 | 965 | 116 | 132 | 1.00 | 0.86 | 0.47 |
| jump-6-12 | 2 | 116 | 965 | 116 | 128 | 1.00 | 0.87 | 0.48 |
| twos | 0 | 231 | 846 | 198 | 0 | 0.86 | 1.00 | 1.00 |
| twos | 1 | 231 | 846 | 198 | 0 | 0.86 | 1.00 | 1.00 |
| twos | 2 | 231 | 846 | 198 | 0 | 0.86 | 1.00 | 1.00 |
| saw-10-15 | 0 | 203 | 879 | 203 | 0 | 1.00 | 1.00 | 1.00 |
| saw-10-15 | 1 | 203 | 879 | 174 | 0 | 0.86 | 1.00 | 1.00 |
| saw-10-15 | 2 | 203 | 879 | 174 | 0 | 0.86 | 1.00 | 1.00 |
The results are encouraging. The first two patterns are pooled over
perfectly by several candidate cells. The next two—run-6-10 and
jump-6-12—show good sensitivity but lower specificity (precision
only 50%). The last two patterns—twos and saw-10-15—show good
specificity but reduced sensitivity at 86% (although one cell managed
to cover saw-10-15 completely).
Patterns run-6-10 and jump-6-12 share their first few elements
so are probably pooled over by the same cells (thus 50%-50%
precision). I hope there will be other cells which distinguish them by
pooling over the parts which make them unique.
The last two patterns are the longest sequences. The temporal pooling scores decay over time and so towards the end of the sequence they may start to be dominated by current feed-forward activation, even in the absence of bursting.
Let’s look over the same plot as last time, showing candidate cell
activations over each pattern instance. This display helps to diagnose
problems with sensitivity (coverage of the target sequence) so it only
makes sense to look at those with imperfect sensitivity, twos and
saw-10-15. Here we show cells in red for active and in black for
inactive.
The plot shows that indeed that some cells falter at the end of the sequence, while others start later, missing the first part but reaching to the end. The logical OR of these cells would be enough to fully cover the sequence.
I ran a simulation with another even higher-level region, and the statistics are essentially the same. The decay rate of temporal pooling scores is the same in the higher region so the same thing is happening. I think the mechanism needs reworking here.
Two patterns had problems with specificity, run-6-10 and
jump-6-12. Their values are as follows
run-6-10 |
[6 7 8 9 10] |
jump-6-12 |
[6 7 8 11 12] |
Since the first step of all sequences is unpredictable it is ignored for pooling. The remainder consists of 2 shared values and 2 unique values each. If we look down the list of candidate cells there are in fact some cells which cover 50% (2 steps) of each pattern with 100% specificity:
| pattern | candidate | in-steps | out-steps | active-in | active-out | sensitivity | specificity | precision |
|---|---|---|---|---|---|---|---|---|
| run-6-10 | 0 | 132 | 945 | 132 | 104 | 1.00 | 0.89 | 0.56 |
| run-6-10 | 1 | 132 | 945 | 132 | 116 | 1.00 | 0.88 | 0.53 |
| run-6-10 | … | … | … | … | … | … | … | … |
| run-6-10 | 13 | 132 | 945 | 66 | 0 | 0.50 | 1.00 | 1.00 |
| run-6-10 | 14 | 132 | 945 | 66 | 0 | 0.50 | 1.00 | 1.00 |
| jump-6-12 | 0 | 116 | 965 | 116 | 132 | 1.00 | 0.86 | 0.47 |
| jump-6-12 | 1 | 116 | 965 | 116 | 132 | 1.00 | 0.86 | 0.47 |
| jump-6-12 | … | … | … | … | … | … | … | … |
| jump-6-12 | 16 | 116 | 965 | 58 | 0 | 0.50 | 1.00 | 1.00 |
| jump-6-12 | 17 | 116 | 965 | 58 | 0 | 0.50 | 1.00 | 1.00 |
So that’s good—we have some cells representing “something starting
with [6 7 8]” and other cells specific to each which appear at their
point of divergence.
The mechanism I am using (which, by the way, was first articulated by Jake Bruce) produces a stream of temporal pooling cells over the life of a sequence. That has some nice-seeming properties and some potential problems.
If we see a sequence starting from the second element, it will activate some of these pooling cells but not the ones which are normally activated on the first element. It seems to me that these original pooling cells should be reactivated somehow from observing later parts of the sequence. I have raised this question before. One way this could work is by lateral activation between temporal pooling cells; as they stay active they continue to learn lateral connections to each other. Then any subset of them could complete the missing ones if enough lateral activation were enough to activate cells even without feed-forward input.
As always, I value your advice.
–Felix
The demo here was compiled from Comportex 0.0.3-SNAPSHOT with ComportexViz 0.0.3-SNAPSHOT.
The extra analysis code is here: temporal_pooling_experiments.clj.