# Perceptual learning & invariants

[From Bill Powers (920612.1300) --

I'm sending this reply to a direct post to the net (and to John) because
the subject will be of general interest... here's John's post:

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Date: Thu, 11 Jun 92 21:29:54 CDT
From: gabriel@eid.anl.gov (John Gabriel)
Subject: Re: Perceptual Learning

Yes, the symmetry theory is one of my special interests, so thanks. What
Martin is saying is that the perceptual mechanism knows how things look
after being turned over, rotated or whatever from experience with various
objects. Circular means "same after rotation by any angle about normal to
plane of object". My background says a little more than Bill P. or Martin
perceive - that "well understood" experiences are all invariants, and that
symmetry or invariance is an important part of learning - particularly time
invariance, i.e. repeated experience from which PCT references are
constructed. In fact my other serious research interest at the moment is a
proof that any computer program (i.e. reproducible computation) is
completely defined by one input/output pair of values, and the set of all
the invariants of the program. Obvious in a way, but not observed until my
recent paper to be published in Beijing. The mathematics of PROPER proof is
formidable. If we take the view that any reproducible (understood?)
experience can be modelled by a computer simulation, this is a perhaps a

Please forward to CSG NET if you feel this would be a useful contribution.
::John
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John Gabriel (920611) via Bill Cunningham (920612.1345) --

Hello, John!

If a perceptual function sees something as "the same" when it's turned
over, I don't think it reports it as "the same." It simply goes on
reporting it without any change. OTHER perceptual functions might see
changes: apparent size, velocity, width-to-height ratio, and so on. If a
perceptual function truly perceives a shape as invariant with respect to
orientation, then as the shape is reoriented the perceptual signal from
that function simply remains constant -- the function doesn't see any
change.

But if a perceptual signal doesn't change, where is the need to control it?
This brings up a rather odd feature of perceptual invariants. An invariant
would remain constant under transformations that leave it unchanged. But
this implies that there are other transformations that will change it. For
example, if you have a square made of four toothpicks held together with
clay blobs at the corners, you'll see it as square in any orientation.
However, if you push in one corner, or someone does, the amount of
squareness perceived will decrease. You can now control the amount of
squareness perceived by pushing on other corners or pulling the disturbed
one out again. So CONTROL of an "invariant" depends on applying
transformations that DON'T preserve the invariance!

We tend to think of invariants (in terms of categories) as things that
either exist or don't. In terms of perception, however, they simply define
some canonical KIND of perception; they don't imply that this perception is
unchangeable. Every perceptual function that emits a signal that is a
function of multiple lower-level signals defines an invariant, in that
there are ways of changing the lower-level signals that will leave the
value of the function unchanged. Every weighted sum, for example, defines
an invariant. If p2a = A*p1a + B*p1b, then for every way that p1a can
change, there is a way that p1b can change that will keep p2a the same. But
if p1b DOESN't change in that particular way, then p2a WILL change. If p2a
is to be controllable by varying p1a and p1b, then the controlling system
must NOT vary p1a and p1b so that A*p1a + B*p1b = constant. If it did,
there would be no effect on p2a. So I repeat, control relies on causing
transformations that DON't preserve the invariance. You can't "control" a
circle by rotating it about an axis perpendicular to its center.

I said most of this in BCP, pp. 123, 125-6, and elsewhere, but such things
tend to get lost.

Some of the confusion about invariances (and control of perception in
general) probably comes from mixing control of a PARTICULAR perception with
control that entails CHANGING FROM ONE KIND OF PERCEPTION TO ANOTHER. The
latter case is quite different, and implies higher-level switching from one
lower-level system to another as the means of control. That probably
happens, but it's different from controlling the state of a single kind of
something I've noticed in general. I suspect that you and I see these
matters in pretty much the same way.

Once a symmetry has led to developing a particular perceptual function
through learning to keep a perception constant, the next thing is to learn
different states of the same perception -- that is, you perceive a circle,
and then learn that you can squash it and reproduce that appearance, too,
through operations that don't preserve radial symmetry. This isn't the same
as perceiving an ellipse; it means perceiving a circle that's in a
different state from the one you learned first. You don't see an ellipse,
you see a squashed circle. The same control system that controls for
circularity can also control for certain kinds of noncircularity if it's
given a reference signal different from the one that's matched by a perfect
circle. When presented with inputs that don't exemplify the "canonical"
symmetry, a perceptual function doesn't just switch off. It reports less of
the perception. And that amount of the perception, too, can become a
reference signal.

<If we take the view that any reproducible (understood?) experience can

be modelled by a computer simulation, this is a perhaps a useful result