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