text:
\subsection{A generative model for  Signal Model \ref{sigmod2}}
\label{generativemodel}
To help understand the model better (and also to simulate it), we describe here one plausible generative model that satisfies its required assumptions. This assumes that every element gets added to the support at magnitude $r$ and keeps increasing until it reaches magnitude $M$. Similarly, every element that began decreasing keeps decreasing until it becomes zero and gets removed from the support\footnote{Another possible generative model is: select $S_a$ out of the $2S_a$ current elements with magnitude $jr$ and increase them, and decrease the other $S_a$ elements}. We keep the signs of the elements the same except when the element first gets added (at that time, can set the sign to $\pm 1$ with equal probability).

To specify the generative model, first define
\bd
Define
\ben
\item Increasing set, $\Inc_t := \Iset_t(1) \cup \Iset_t(2) \dots \cup  \Iset_t(d-1)$
\item Decreasing set, $\Dec_t := \Dset_t(1) \cup \Dset_t(2) \dots \cup  \Dset_t(d-1)$
\item Constant set, $\Con_t:= \{i : |x_{t,i}| = M \}$. Clearly $\Con_t = N_t \setminus (\Inc_t \cup \Dec_t)$.
\een
\ed

The generative model is as follows. At each $t>0$,
\ben

\item Update the magnitudes for elements of the previous increasing, decreasing and constant sets.
\bea
(x_{t})_{\Inc_{t-1}}  \se [ |(x_{t-1})_{\Inc_{t-1}}| + r ] \ \sgn((x_{t-1})_{\Inc_{t-1}})   \nn \\
(x_{t})_{\Dec_{t-1}} \se [|(x_{t-1})_{\Dec_{t-1}}| - r] \ \sgn((x_{t-1})_{\Dec_{t-1}}) \nn \\
(x_{t})_{\Con_{t-1}} \se (x_{t-1})_{\Con_{t-1}}
\eea
where $\sgn(z)$ is a vector containing the signs of each element of $z$.

\item Select the newly added set, $\Iset_t(1) \subseteq N_{t-1}^c$, of size $S_a$ uniformly at random. Similarly select the new set of decreasing elements, $\Dset_t(d-1) \subseteq \Con_{t-1}$ of size $S_a$ uniformly at random. Set their values as:
\bea
(x_{t})_{\Iset_t(1)} \se r \ \underline{s}    \nn \\
(x_{t})_{\Dset_t(d-1)} \se (d-1)r \ \sgn((x_{t-1})_{\Dset_{d-1}})
\eea
where $\underline{s}$ is an $S_a \times 1$ signs' vector in which each element is $+1$ or $-1$ with probability $1/2$.

\item Compute:
\bea
\Iset_t(d) \se  \{i \in \Inc_{t-1} : |(x_{t-1})_i| = (d-1)r \} \nn \\
\Dset_t(0) \se  \{i \in \Dec_{t-1} : |(x_{t-1})_i| = r \}
\eea
and update the increasing, decreasing and constant sets:
\bea

\Inc_t \se \Inc_{t-1} \cup \Iset_t(1) \setminus \Iset_t(d) \nn \\
\Dec_t \se \Dec_{t-1} \cup \Dset_t(d-1) \setminus \Dset_t(0) \nn \\
\Con_t \se \Con_{t-1} \cup  \Iset_t(d) \setminus \Dset_t(d-1) \nn \\
N_t \se \Inc_t \cup \Dec_t \cup \Con_t
\eea

\een