[From Bill Powers (971207.0815 MST)]
Earliest sunset of the year today.
Rupert Young (971207 1400 UT) --
{Rupert]
I'd like to go back, if I may, to a query I had a while ago when my email
wasn't working.
So, keeping it simple with just two connections, if w1 and w2 are
the weights and s1 and s2 are the incoming perceptions the current
perception.
p = (w1 * s1) + (w2 * s2)
[Rick Marken (971029.0930)]
Yes. This is the current hypothesis about the nature of the perceptual
functions that produce sensation signals (level two perceptions).
The functions that produce signals representing more complex perceptual
variables (like transitions, relationships, principles, etc) would, of
course, be quite different.
[Rupert]
But in some sense would not the higher functions still be the same because
all that neurons can do is a weighted sum of inputs.
Not so. We use linear weightings in our models because we can't solve many
nonlinear equations, not because neurons can only compute linear sums. My
basic hypothesis is that neurons can (roughly) add, subtract, multiply, and
divide, and also compute squares, cubes, logs, square roots, and many other
kinds of continuous functions -- approximately. They can also compute
dynamic functions like time integrations and differentiations. They do this
by analog computation, not symbolic or digital computations. Of course
these functions are not very exact, but in an adaptive system they can
probably become as exact as needed to account for the accuracy of living
control processes.
Suppose there were two signals, x and y, representing positions along
orthogonal axes in space. A perceptual signal made of x^2 + y^2 would
represent the square of the distance of the point (x,y) from some zero
point. The output signal of this control system would have to contain some
kind of switching so it could preserve negative feedback for all positive
and negative values of x and y, but assuming that could be done, we would
have a control system that controlled a position to keep it on a circle
with a radius whose square is equal to the reference signal of the control
system. Disturbances tangent to the radius would not be resisted; those
along the radius would be resisted.
That's just an example out of thin air. The actual way in which we control
in radius and angle probably involves many control systems each working in
a slightly different direction, with a higher system selecting which subset
of control systems is to be used over various arcs of a circle. That would
be more consistent with what is known about perception of directions in
space -- those visual vectors that Georgeopolis writes about. But we can
imagine simpler arrangements that would in principle do the same thing, and
get an understanding of control processes that deals with the same
_effects_ while avoiding the immense difficulties in achieving them
computationally in the anatomically correct way. The HPCT diagram is really
just a user-friendly fiction, but as long as we all know it is we can still
work out some useful principles.
I think you can take it for granted that perceptual input functions can
compute just about any reasonable function of the input signals that you
could think up.
(Bill Powers (971029.1155 MST)
[Bill]
So, keeping it simple with just two connections, if w1 and w2 are the
weights and s1 and s2 are the incoming perceptions the current >perception.
[Rupert]
p = (w1 * s1) + (w2 * s2)
I have some questions about this but first I'd like to check if this looks
right so far. Is it ok ?
]Bill]
Yes. To anticipate what your problem with this might be, you should try two
control systems operating at the same level, with differently-weighted
input functions.
[Rupert]
Ok, I tried that and it works. Though the main thing I am trying to
understand is how is it that a control system is sensitive to one input
vector and not another.
The vector is _created_ by the weightings of the input function. It doesn't
exist in the two inputs s1 and s2. There is no objective vector that the
input function has to "recognize." This is hard to get used to if you've
heard only the conventional, naive-realist, version of perception, in which
the environment contains entities which the perceptual functions then have
to recognize and represent as signals.
In our little universe there are two variables, s1 and s2. They can vary
independently in any old way, depending on how external forces act on them.
But now suppose you create a control system that senses their sum: s1 + s2
(both weights are 1). The perceptual signal is compared with some desired
value, and the error signal is amplified to produce an output that affects
both s1 and s2. The result is that the _SUM_ is under control: we end up
with s1+s2 = r, the reference value that exists at the moment.
Before we introduced the control system, a plot of s1 against s2 would just
show some spaghetti-like trajectory of the point (s1,s2), as external
forces pushed the values of the two variables around in random ways. A sort
of space-filling curve. But after the control system was built and turned
on, the plot would look like this:
* | s1
* |
* |
* |
* |
------------------------------------ s2
> *
> *
> *
> *
> *
>
That would be for r = 0. For other values of r, the line would shift in a
direction at right angles to the line.
Suddenly there is a relationship between s1 and s2: their sum is constant,
at an adjustable value. All those external forces can still push the point
(s1,s2) back and forth along the line, but they can't push it off the line
any more. Try it with your simulation and see.
One dimension of this two-dimensional universe has been brought under
control, and in the process we have _created_ a vector in the direction of
the line (or rather, normal to it).
If we weight the two variables differently as we sum them, we get a
perception equal to w1*s1 + w2*s2. We will find that a new vector has been
created, with a different slope. Now the disturbances can move (s1,s2)
along the line in the different direction, but still not off the line.
What's happened is that we have put one degree of freedom of this little
universe under control, while leaving the point (s1,s2) free to change in
the uncontrolled degree of freedom. The degree of freedom that's now under
control is in the direction at right angles to the line.
What happens if we now build a second control system, sensing the same two
variables but giving them different weights? This will create another line
at an angle to the first line. If the second control system were the only
one, disturbances could now move (s1,s2) along that line, but not at right
angles to it (or at any other angle except along the line).
When _both_ control systems are operating, each one keeps (s1,s2) on its
own line, and the result is obvious: the only place (s1,s2) can go is to
the intersection of the two lines. Now disturbances can't move the point at
all. There are no more degrees of freedom left.
When the reference signals of the two control systems change, the
corresponding line moves at right angles to the direction of the line, so
the line stays parallel to a specific direction set by the input weights.
This means that (s1,s2) has to move so as to stay on the moving
intersection between the lines. So as you alter the two reference signals
you move the point around in two dimensions and all disturbances, in any
direction in 2-space, are resisted.
[Rupert]
What if the inputs don't lay on this vector, there would still be a signal
wouldn't there, but for the wrong perception ?
The two perceptions would just be whatever they are. If they aren't on
either of the two lines, both control systems would contain an error
signal. Each control system "sees" a universe in which only one weighted
sum of the two variables exists. That sum has whatever value it has. _All_
positions of the point (s1,s2) will produce some value of perceptual signal
in both systems, except (0,0). And even 0 is a value, so we can forget the
"except."
The other way way of looking at perception, which I think is what is giving
you trouble, is to think of a black box with many outputs and many inputs.
When the inputs are in one state, one of the outputs is activated. When
they're in a different state, a _different_ output is activated. So the
perceptual function is indicating _which input pattern is present_. This is
called identification or recognition. This is the sort of model that's used
by neural networkers, I think, and is the conceptual basis for conventional
models of perception. That's why everyone talks about "classification."
The output that is activated indicates the class to which the whole input
pattern belongs.
That kind of arrangement might exist at what I call the category level,
although as I described it it falsely implies that we can perceive only one
category at a time in a mutually-exclusive way. But at the lower levels,
where we exert continuous control in many degrees of freedom at the same
time, this kind of model of perception just won't work. We need to have
_all_ the perceptions in _all_ the degrees of freedom present at the same
time, continuously variable, so we can have many control systems working in
parallel.
People who haven't thought much about models of perception come into PCT in
a nice naive state, and when I describe how the model works they say, "Oh,
okay." So I don't take the time to go through all this. But people who have
already been studying models of perception have most probably been assuming
the other kind of model, in which "patterns" are "recognized." That's the
box with multiple inputs and outputs, with only one output at a time being
activated to signal the "right category." And of course if they've missed
my infrequent discussions of this difference in models, they can get very
confused trying to make the perception-creation model work like the
pattern-recognition model. I hope this discussion has alerted you to the
difference, and has helped with some of your problems.
Best,
Bill P.
