[EM] issue space revisited

[Forest Simmons]

I apologize for the delay in my next installment on this topic, but I want
to work out some good examples before I go too much further abstractly.

Meanwhile, if you want something to whet your appetite along these lines
look up the topic "tensor voting" on google.com

One site it will cite is

http://citeseer.nj.nec.com/gomes02variational.html

The abstract of the cited reference is

Abstract: We present a novel algorithm for recovering a smooth manifold of
unknown dimension and topology from a set of points known to belong to it.
Numerous applications in computer vision can be naturally interpreted as
instanciations of this fundamental problem. Recently, a non-iterative
discrete approach, tensor voting, has been introduced to solve this
problem and has been applied successfully to various applications.

The title of the cited paper is

A variational approach to recovering a manifold from sample points (2002).

The authors are

Josi Gomes and Aleksandra Mojsilovic.

Their method of reconstructing a manifold from sample information is much
more sophisticated than anything I had in mind for reconstruction of issue
space from the ballot information.

It's interesting that they use "tensor voting" as a tool in the
reconstruction process.

Note that the spaces they are interested in are associated with computer
vision, not the issue spaces associated with their voting method.

One never knows what nuggets of knowledge one might glean from such sites
that might give insight into some public election method.

How did I come across "tensor voting"?

While developing my method of reconstructing issue space, I had to modify
the Singular Value Decomposition of the "ballot matrix" in order to remove
the clone bias from the metric of the space. [By ballot matrix, I mean the
matrix whose columns are indexed by the candidates and whose rows are
indexed by the voters, and whose values are the ballot rankings or ratings
of the candidates by the voters.]

While adapting the SVD, I thought of a generalization of the SVD that
would be valuable for data compression in general.

I wondered if anybody had ever thought of that generalization before, so I
started searching the internet with key words that described the method.

Sure enough, somebody else was already using it under the name N-mode SVD.

N-mode SVD is to tensors as ordinary SVD is to matrices. In other words
ordinary SVD is just 2-mode SVD, since matrices are indexed with two
dimensions.

Any way, thats how the word "tensor" got into the mix, and I was
pleasantly surprised to see "tensor voting" come up in this connection.

More on applications of n-mode SVD:

Imagine that you are trying to compress a sequence of images for a DVD.
You have at least four natural dimensions to work with: the two
coordinates of the pixel on the screen, the time coordinate, and the RGB
intensity dimension. In this application a 4-mode SVD would be very
natural.  One of the more sophisticated applications by the authors of the
above paper requires 8-mode SVD because they are considering normal
vectors and other features, as well as three spatial coordinates.

My unique (as near as I can tell) idea for an application of n-mode SVD to
data processing is generalizing the "delay coordinate" idea, which leads
to n-mode SVD applications where n is on the order of log base two of the
number of sample points.

Forest