invariants

[From Bill Powers (960101.1115)]

Yes, Happy New Year to all. When I was a kid, science fiction stories
about the far future were set in 1980. Here we are in the future beyond
the future. It's not as different as I once hoped it would be, but we're
working on it.

···

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Rick Marken (951231.1900) --

     ... anyone who understood what was going on in the basic
     compensatory tracking task could have come up with these analyses.
     But you do have be able see the generality of the principle
     illustrated by these experiments; in particular you have to be able
     to see that the perception of distance between cursor and target in
     the compensatory tracking task is the same, in principle, as the
     perception of the grammatical structure or phonemic content of an
     utterance.

Exactly so. The person who understands these basic relationships tries
them out FIRST as an explanation of behavior. A lot of our discussions
arise from those who try some other explanation first, and then try to
fit PCT into the left-over spaces.
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Shannon Williams (960101) --

     We are not visualizing this "pattern generator" the same way. I
     visualize something that changes its patterns every time that it is
     used. The parts of the pattern that correctly generates an output
     which will achieve control of the input should remain steady. But
     it will not be invariant at all.

An invariant is not a constant, as I have come to understand it. I think
it's worthwhile discussing invariants, to see the difference in how we
would handle patterns in a control model, as opposed to an open-loop
pattern generator model.

Consider the equation

y = a*x1 + b*x2

This could represent a simple perceptual function with two inputs
(weighted according to a and b) and one output, y. There are ways in
which x1 and x2 can vary which result in y varying, and other ways they
can vary that result in y remaining constant.

If y is constant, the points defined by x1 and x2 will lie on a straight
line with a slope defined by a and b (a straight line because I chose a
linear function of two variables). All pairs of values of x1 and x2 that
lie on this line produce the same output y (because we have said that y
is constant), so we can say that the value of y is _invariant_ with
respect to this particular way of changing x1 and x2. The perceptual
function will treat all pairs of inputs that lie on this line as the
same perception.

We can also change x1 and x2 in a way that moves the point x1,x2 _at
right angles_ to the line of invariance. This will produce different
values of y. Or, approaching the same result from another direction, if
we change the value of y to a new constant value, we will have a new
straight line parallel to the original line, but different from it. So
if the point x1,x2 moves from anywhere on the first line to anywhere on
the second line, y will change from its first invariant value to a new
invariant value. The kind of invariance remains the same: y is still
invariant with respect to combined changes in x1 and x2 that move the
point parallel to the line of invariance -- in the same spatial
direction.

         > y = 6 *
         > *
         > * *
         > # * *
     x2 | # * y = 1 *
         > * * *
         > * * *
         * # * y = -5 * #
         > * * #
         > * # * #
         > * # * #
         > * * # #
         * *
         > *
          ----------------------------------------------------
                                   x1

                     Fig. 1: Lines of invariance.

The starred lines in Fig. 1 show the ways x1 and x2 can change together
that leave y constant (I haven't specified a and b; they're whatever
produces these lines). The #'s show a general point moving in x1-x2
space as the values of x1 and x2 change independently. Starting with the
upper-left # and tracing the movement down , to the right, and then
finally up and to the right, you can see that the value of y will start
at something greater than 6, decine to 1 and then to -5, and finally
become constant with y a little less than -5 as the #'s begin moving up
and to the right parallel to the line at y = -5. Notice that the value
of y does not reflect any movements of the # points parallel to the
lines of invariance. It reflects only changes in directions at right
angles to those lines.

Note that the directions of the lines of invariance are determined by
the constants a and b. Those constants are properties of the input
function and determine which lines of invariance will exist.

Suppose we're controlling the perceptual signal y at a value y0. The
error, y0 - y, is amplified and send to two output functions, which
affect the values x1 and x2 in appropriate directions. The result will
not be to bring x1 and x2 to any particular values, but to bring y to
the value y0. This means that disturbances which act in the x1-x2 plane
parallel to the lines of invariance in the figure above will NOT be
resisted. Why not? Because they will leave y invariant: the control
system will detect no change in the error. If the reference value is y0
= 1 unit, disturbances could move the # marks parallel to the line y = 1
above without causing any change in the system's action.

So this control system does not control x1 and x2 in the general x1-x2
plane. It controls only in a direction at right angles to the lines of
invariance.

Suppose that x1 is the magnitude of one formant and x2 is the magnitude
of another. We can assume that the same phoneme will be heard if some
function of x1 and x2 remains constant. This will most probably involve
curves, not lines, of invariance. The perceptual function is probably
nonlinear in x1 and x2. But it is still true that any combination of
formant magnitudes that leaves the output of the function anywhere on
one curve of invariance will lead to the same perception, the same
phoneme. The value of the perceptual signal will be the same anywhere
along the curve.

By changing the reference level for the perceptual signal of the
phoneme-control system, a higher system can change the phoneme that is
heard. That is, it can move the two formant magnitudes x1 and x2 so the
point they define lies near a new curve of invariance. A listener who is
perceiving through a similar input function will hear the phoneme
change, too. However, a sound spectrometer will not necessarily show the
same combination of formant magnitudes each time the same phoneme is
generated. The reason is that there is no control parallel to the curves
of invariance -- nor does there have to be any control. The listener and
the speaker will hear the same new phoneme, even though the formants may
change in combinations that also have an irrelevant component. Note that
you can have a change from one phoneme to another over a wide range of
changes in the two formants. One of the formants might increase,
decrease, or remain the same in magnitude, depending on what the other
one does. Just look at Fig. 1, and pick any two points on different
lines of invariance, with each line representing a different phoneme
(along a continuum, as from eeee-ih-eh ... oh-ooo). Think of a trombone,
not a trumpet.

This principle can be extended to input functions having any number of
inputs, and for any kinds of nonlinear functions (it even works with
nultiple-valued functions). The principle will remain the same. There
will be lines, or curves, or surfaces, or hypersurfaces of invariance,
parallel to which disturbances will not be resisted. Changing the
reference signal will result in moving the output of the perceptual
function to a new surface, etc.. An observer using the same perceptual
function will see a change of the controlled variable from one value to
another, even though an objective measure of the actual inputs to the
perceptual function shows that the values do not repeat from one
instance of the same perception to the next.

A control system, therefore, is not like an open-loop pattern generator.
It does not simply convert inputs to outputs. It varies its outputs to
make the value of some function of its inputs match a reference signal,
and in doing so it may actually produce many different values of the
inputs to the perceptual function while matching the same reference
condition. The perception it controls can, in general, be recognized by
an observer only if the observer passes the same physical inputs through
a highly similar input function. A sound spectrometer can't reveal the
invariances.
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     This reference *must* evolve. If it does not evolve then it would
     mean that our perception of the physical world is not subject to
     our goals. In other words, the sounds we hear as infants would
     sound the same to us as adults.

The perceptual functions and output functions must evolve; a reference
signal is just a signal with a particular magnitude. I haven't dealt
with change of organization. I think it's best to start by defining what
has to evolve; then we can see what aspects of the system need to be
modified during evolution.
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Best to all,

Bill P.