Frame completions for optimally robust reconstruction

TL;DR

This paper explores how to optimally add measurement vectors to a frame in Hilbert spaces to minimize average reconstruction error, enhancing robustness in information fusion tasks.

Contribution

It introduces preliminary methods for augmenting frames to reduce mean square error in vector reconstruction under noisy measurements.

Findings

Adding vectors can significantly reduce reconstruction error.

Optimal frame augmentation improves robustness against noise.

Preliminary results guide effective frame completion strategies.

Abstract

In information fusion, one is often confronted with the following problem: given a preexisting set of measurements about an unknown quantity, what new measurements should one collect in order to accomplish a given fusion task with optimal accuracy and efficiency. We illustrate just how difficult this problem can become by considering one of its more simple forms: when the unknown quantity is a vector in a Hilbert space, the task itself is vector reconstruction, and the measurements are linear functionals, that is, inner products of the unknown vector with given measurement vectors. Such reconstruction problems are the subject of frame theory. Here, we can measure the quality of a given frame by the average reconstruction error induced by noisy measurements; the mean square error is known to be the trace of the inverse of the frame operator. We discuss preliminary results which help indicate how to add new vectors to a given frame in order to reduce this mean square error as much as possible.