A Puzzle of No Return?

Here is an interesting problem. Consider the sum of two real signals \(x(t)\) and \(y(t)\) of the same length, \(z(t) := x(t) + \lambda y(t)\), where \(\lambda\) is real and non-zero. Let us decompose \(z(t)\) by some pursuit (e.g., OMP) using an overcomplete dictionary of unit norm functions \(\mathcal{D} = \{d_\gamma\}_{\gamma \in \Gamma}\), thus producing the model $$z(t) - \sum_{k \in\Gamma_z} \alpha_k d_{\gamma_k}(t) = R^{|\Gamma_z|}z(t).$$ What are the conditions on the dictionary, and the signals \(x(t)\) and \(y(t)\), such that we can "recover" the separate models of \(x(t)\) and \(y(t)\) from that of \(z(t)\), i.e., $$x(t) - \sum_{k\in \Gamma_z^x \subset \Gamma_z} \alpha_k d_{\gamma_k}(t) = R^{|\Gamma_z^x|}x(t)$$ $$y(t) - \sum_{k\in \Gamma_z^y\subset \Gamma_z} \alpha_k d_{\gamma_k}(t) = R^{|\Gamma_z^y|}y(t)$$ where \(\Gamma_z^x \cup \Gamma_z^y = \Gamma_z\), \(\Gamma_z^x \cap \Gamma_z^y = \oslash\), and \(|| R^{|\Gamma_z|}z(t) ||^2 \ge || R^{|\Gamma_z^x|}x(t) ||^2 + \lambda^2|| R^{|\Gamma_z^y|}y(t)||^2\)?

Is that too much to ask?

CRISSP is a research group in ADMT at Aalborg University Copenhagen (AAU-KBH), Denmark.

Authors:

  Bob L. Sturm
  Sofia Dahl
  Stefania Serafin

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