Re: Detecting linear transformations of a uniform distribution.

On Aug 22, 12:26 pm, Ray Koopman <koop...@xxxxxx> wrote:
On Aug 22, 6:39 am, LEJ Brouwer <intuitioni...@xxxxxxxxx> wrote:
Hi,

I was wondering whether anyone could suggest ways of solving a
problem I have been thinking about:

Suppose I have a very large set of samples taken from a multivariate
uniform distribution (e.g. with pdf f(x)=constant over some large
region of support, e.g. over a hypercube with range [-1000,1000]
along each of D axes/variables).

Now suppose a linear transformation is applied to the sample data
(e.g. by scaling each of the axes independently and perform a D-
dimensional rotation). Is there anyway to perform a local test on
the samples (e.g. in some ball of radius D around the origin) which
can detect the transformation that has been applied?

Assuming that a suitable method exists of determining the
transformation, how confident would we be of our estimate
as a function of sample density?

I hope the question is clear and I look forward to your responses.

Best wishes,

Sabbir.

Unless the ball is large enough to include the corners of the
transformed region, the density will be uniform throughout and
will contain no information about the transformation.

That's sufficient but not necessary. What is necessary is that the
radius of the ball be at least as large as the largest rescaling
factor. But even then you will be able to say only what the
rescaling factors were; you won't be able to say what the rotation
was.
.

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