Frame completions for optimally robust reconstruction

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.

Cited by in corpus (12)

• Near-Optimal Sensor Placement for Linear Inverse Problems
• Sparse Sampling for Inverse Problems with Tensors
• A determinantal point process for column subset selection
• Optimal completions of a frame
• Optimal frame designs for multitasking devices with weight restrictions
• Constructing all self-adjoint matrices with prescribed spectrum and diagonal
• DASS: Distributed Adaptive Sparse Sensing
• Optimal dual frames and frame completions for majorization
• Frame completions with prescribed norms: local minimizers and applications
• Multiplicative Lidskii's inequalities and optimal perturbations of frames
• Convex potentials and optimal shift generated oblique duals in shift invariant spaces
• Optimal frame completions with prescribed norms for majorization