[EM] Candidate-Space Method

[Forest Simmons]

A brief progress report:

Let the entry in the ith row and jth column of a matrix be a zero or a one
depending on whether or not the ith voter approves of the jth candidate.

Call this matrix A.

The rows of this matrix represent the approval ballots of the voters, so
we could call the row space of the matrix "Voter Space."

Each column of this matrix gives a voter profile of one candidate, so we
might call the column space of the matrix "Candidate Space."

It is well known that the row and column spaces of a matrix are linearly
isomorphic, so that it is possible to move candidates from Candidate Space
into Voter Space via the isomorphism, and then use distances in the
Voter Space to estimate voter preferences.

Of course there are many possible isomorphisms and many possible metrics
for measuring distance.  Different choices will lead to different results.

Mike has raised the question of the most natural metric. I raise the
question of the most natural isomorphism.

I have some ideas along these lines which I will be able to share more
fully at a later date. Just a minimal sketch for those familiar with
eigenvectors:

Let B be the transpose of A.

The normalized eigenvectors of A*B and B*A form natural orthonormal bases
for the (respective) column and row spaces of A.

The transformation T that takes elements of the first basis onto
corresponding elements of the second basis is in some ways the most
natural isomorphism.

If we use the Euclidean metric, then this transformation is distance
preserving, i.e. it is an isometry.

To the physicist and statistician I say this: The eigenvectors give the
principal axes, and the moments about corresponding axes are equal in the
two spaces.

[The moments are the eigenvalues, and corresponding eigenvectors have
equal eigenvalues.]

To be continued with details, examples, etc. ......

Forest

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