"Dynamical Sampling in Paley-Wiener Spaces of Graphs"Bailey, VictorDynamical Sampling is, in a sense, a hypernym classifying the set of inverse problems arising from considering samples of a signal and its future states under the iterative action of a linear operator. One problem in this area is the following: let $f \in l^2(I)$ where $I=\{1, \ldots, d\}$. Suppose for $\Omega \subset I$ we know $\{{ A^j f(i)} : j= 0, \ldots l_i, i\in \Omega\}$ for some $A: l^2(I) \to l^2(I)$. What are conditions on $\Omega, A$, and $l_i$ that allow the stable reconstruction of $f$? Recently, the framework of Dynamical Sampling has been applied to combinatorial graphs. Setting our iteration matrix as the graph Laplacian, $M$, we have found conditions for $\Omega$ and $l_i$ (as given above) to admit stable reconstruction of Paley-Wiener spaces (denoted $PW_{\lambda_k}$ where $\lambda_k$ is some eigenvalue of $M$) of graphs. Also, given $\tilde{y}_i = A_if +\eta_i$, where $A_i = \mathcal{S}_\Omega M^{i-1}$ for $i \in \{1, \ldots, L\}$, $f \in PW_{\lambda_k}$, $\mathcal{S}_\Omega$ is the sampling operator for some $\Omega \subseteq \{1, \ldots, d\}$, and $\eta_i$ are i.i.d. random variables with a zero mean and a variance matrix $\sigma^{2}I$, we have obtained a bound for the expected error of approximately reconstructing $f$ via a least squares solution which has been shown to be an improvement on a previously known bound in the case of no subsampling. In this talk, I will discuss the aforementioned results as well as other recent developments in Dynamical Sampling. |
| http://univie.ac.at/projektservice-mathematik/e/talks/Bailey_2021-07_V Bailey ICCHA Paper.pdf |
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