Let's say I have a signal $s$ that is known to be an additive combination of two other signals:
$$<br /> s = s_1 + s_2<br />$$
And let's say we have a good model for signal 1 as a function of some known input variable $u$ that is parameterized by some vector $\theta$:
$$<br /> s_1 \approx \phi_{\theta}(u) <br />$$
Let's say we know that the first signal can be expressed through a sparse representation in the parameter space $\theta$ (the correct theta can be expressed as a linear combination of all of the thetas where very few coefficients are nonzero) but we know very little about $s_2$.

How can I decompose $s_1$ and $s_2$ ?