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Basis identification through convex optimization

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##### Abstract

Suppose one has a highly redundant spanning set for a

finite-dimensional Hilbert space, knows that a

subset of the spanning set is an orthonormal basis

for the space, and wants to identify that subset.

To identify such a subset, most standard models

typically require that the elements of the spanning set

are arranged in a particular order.

This dissertation develops a novel convex optimization problem,

inspired by compressed sensing, and conjectures that the

set of minimizers to this problem can be used to

identify bases in a given spanning set. In particular,

given an injective matrix X, consider the problem of

minimizing the sum of the absolute values of all entries

in the matrix product XY subject to Y being a left inverse of X.

This dissertation shows that for a given injective

matrix X, the set of such minimizers is a

nonempty, compact, convex set and conjectures that the

extreme points of this set can be used to find a subset

of the rows of X that is a basis for the domain of X.

An analysis of this conjecture is given with particular attention

given to the case when the rows of X are a concatenation

of two orthonormal bases. In this case, it is shown that if a

left inverse Z of X is a minimizer of the sum of the absolute values of

the entries of the matrix product XY subject

to Y being a left inverse of X, and Z identifies an orthonormal

basis in the rows of X, in a way made precise herein,

then Z is an extreme point of

the set of minimizers. Furthermore, conditions are developed that

ensure that such a left inverse Z exists. Last, some special

cases are developed where the above conjecture is shown to hold.