# Best rank-1 approximations without orthogonal invariance for the 1-norm

#### Abstract

Data measured in the real-world is often composed of both a true signal, such as an image or experimental response, and a perturbation, such as noise or weak secondary effects. Low-rank matrix approximation is one commonly used technique to extract the true signal from the data. Given a matrix representation of the data, this method seeks the nearest low-rank matrix where the distance is measured using a matrix norm. The classic Eckart-Young-Mirsky theorem tells us how to use the Singular Value Decomposition (SVD) to compute a best low-rank approximation of a matrix for any orthogonally invariant norm. This leaves as an open question how to compute a best low-rank approximation for norms that are not orthogonally invariant, like the 1-norm. In this thesis, we present how to calculate the best rank-1 approximations for 2-by-n and n-by-2 matrices in the 1-norm. We consider both the operator induced 1-norm (maximum column 1-norm) and the Frobenius 1-norm (sum of absolute values over the matrix). We present some thoughts on how to extend the arguments to larger matrices.

#### Degree

M.S.E.C.E.

#### Advisors

Gleich, Purdue University.

#### Subject Area

Applied Mathematics|Mathematics|Electrical engineering

Off-Campus Purdue Users:

To access this dissertation, please **log in to our
proxy server**.