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A Notational conventions
In this Appendix, we define the notational conventions used throughout the paper. We write  $\mathbb{R}_+^n = \{\mathbf{x} \in \mathbb{R}^n : x_1, \ldots, x_n \geq 0\}$  for the non-negative reals, and  $\mathbb{N} = \{0, 1, \ldots\}$  for the natural numbers. For a vector  $\pmb{\lambda} \in \mathbb{R}_+^n$ , we write
$$
\mathbf {x} \sim \operatorname {P o i s s o n} (\boldsymbol {\lambda}) \tag {A.1}
$$
if the probability mass function of  $\mathbf{x}\in \mathbb{N}^n$  is
$$
p (\mathbf {x}) = \prod_ {j = 1} ^ {n} \frac {\lambda_ {j} ^ {x _ {j}}}{x _ {j} !} e ^ {- \lambda_ {j}}. \tag {A.2}
$$
For scalars  $\alpha > 0$  and  $\lambda > 0$ , we write that a vector  $\mathbf{x} \in \mathbb{R}_+^n$  is distributed as
$$
\mathbf {x} \sim \operatorname {G a m m a} (\alpha , \lambda) \tag {A.3}
$$
if its probability density function is
$$
p (\mathbf {x}) = \prod_ {j = 1} ^ {n} \frac {x _ {j} ^ {\alpha - 1}}{\Gamma (\alpha)} e ^ {- \lambda x _ {j}}. \tag {A.4}
$$
Similarly, for vectors  $\alpha, \lambda \in \mathbb{R}_+^n$ , we write
$$
\mathbf {x} \sim \operatorname {G a m m a} (\boldsymbol {\alpha}, \boldsymbol {\lambda}) \tag {A.5}
$$
$$
p (\mathbf {x}) = \prod_ {j = 1} ^ {n} \frac {x _ {j} ^ {\alpha_ {j} - 1}}{\Gamma \left(\alpha_ {j}\right)} e ^ {- \lambda_ {j} x _ {j}}. \tag {A.6}
$$
For vectors  $\mathbf{x},\mathbf{y}\in \mathbb{R}_+^n$  , we write  $\mathbf{x}\odot \mathbf{y}$  for their Hadamard (elementwise) product
$$
(\mathbf {x} \odot \mathbf {y}) _ {i} = x _ {i} y _ {i} \tag {A.7}
$$
and  $\mathbf{x} \otimes \mathbf{y}$  for their elementwise ratio
$$
\left(\mathbf {x} \oslash \mathbf {y}\right) _ {i} = \frac {x _ {i}}{y _ {i}}. \tag {A.8}
$$
B Detailed derivation of circuit model
In this Appendix, we give a detailed derivation of the circuit models introduced in Sections 3 and 6 of the main text. Here, we focus on the setting of the sampling circuit; the circuit algorithm to compute the MAP introduced in Section 3 of the main text can be recovered at each step by dropping the additive noise terms from the dynamics.
We recall that our goal is to sample the posterior  $p(\mathbf{c} \mid \mathbf{s})$  over concentrations  $\mathbf{c}$  given OSN activity  $\mathbf{s}$ , for a Poisson likelihood
$$
\mathbf {s} \mid \mathbf {c} \sim \text {P o i s s o n} (\mathbf {r} _ {0} + \mathbf {A c}) \tag {B.1}
$$
and a Gamma prior
$$
\mathbf {c} \sim \operatorname {G a m m a} (\boldsymbol {\alpha}, \boldsymbol {\lambda}). \tag {B.2}
$$
Here, we use the conventions of Appendix A for Poisson and Gamma random vectors. Discarding normalization constants that do not depend on  $\mathbf{c}$ , the density of the posterior is then
$$
\begin{array}{l} p (\mathbf {c} \mid \mathbf {s}) \propto p (\mathbf {s} \mid \mathbf {c}) p (\mathbf {c}) (B.3) \\ \propto \left[ \prod_ {k = 1} ^ {n _ {\mathrm {O S N}}} \left(\mathbf {r} _ {0} + \mathbf {A c}\right) _ {k} ^ {s _ {k}} e ^ {- (\mathbf {r} _ {0} + \mathbf {A c}) _ {k}} \right] \left[ \prod_ {j = 1} ^ {n _ {\mathrm {o d o r}}} c _ {j} ^ {\alpha_ {j} - 1} e ^ {- \lambda_ {j} c _ {j}} \right]. (B.4) \\ \end{array}
$$
We will sample from this distribution using Langevin dynamics, without explicitly constraining the concentration estimate to be non-negative [21, 22]. This corresponds to the stochastic dynamics
$$
\dot {\mathbf {c}} (t) = \boldsymbol {\nabla} _ {\mathbf {c}} \log p (\mathbf {c} \mid \mathbf {s}) + \boldsymbol {\eta} (t), \tag {B.5}
$$
where  $\pmb{\eta}(t)$  is an  $n_{\mathrm{odor}}$ -dimensional zero-mean Gaussian noise process with  $\mathbf{E}[\eta_j(t)\eta_{j'}(t')] = 2\delta_{jj'}\delta(t - t')$ . The gradient of the log-posterior is
$$
\frac {\partial \log p (\mathbf {c} \mid \mathbf {s})}{\partial c _ {j}} = \sum_ {k = 1} ^ {n _ {\mathrm {O S N}}} \left(\frac {s _ {k}}{(\mathbf {r} _ {0} + \mathbf {A c}) _ {k}} - 1\right) A _ {k j} + \frac {\alpha_ {j} - 1}{c _ {j}} - \lambda_ {j}, \tag {B.6}
$$
or, in vector notation,
$$
\nabla_ {\mathbf {c}} \log p (\mathbf {c} \mid \mathbf {s}) = \mathbf {A} ^ {\top} [ \mathbf {s} \oslash (\mathbf {r} _ {0} + \mathbf {A c}) - \mathbf {1} ] + (\boldsymbol {\alpha} - \mathbf {1}) \oslash \mathbf {c} - \boldsymbol {\lambda}. \tag {B.7}
$$
The Langevin dynamics then become
$$
\dot {\mathbf {c}} (t) = \mathbf {A} ^ {\top} [ \mathbf {s} \oslash (\mathbf {r} _ {0} + \mathbf {A c}) - \mathbf {1} ] + (\boldsymbol {\alpha} - \mathbf {1}) \oslash \mathbf {c} - \boldsymbol {\lambda} + \boldsymbol {\eta} (t). \tag {B.8}
$$
This is closely related to the Langevin sampling algorithm introduced in equation (3.9) of Grabska-Barwińska et al. [21], but here we are not separately inferring odor concentration and odor presence. We note that, for  $\alpha_{j} > 1$ , the term  $(\alpha - 1) \otimes \mathbf{c}$  gives a force that diverges for small  $c_{j}$ .
However, in this setup single neurons encode single odors. We instead want to allow for a distributed code, where the population of neurons responsible for sampling may not directly encode single odors. To distribute the code, we follow previous work by Masset et al. [38] in using the "complete recipe" for stochastic gradient MCMC [37]. From the "complete recipe", we know that the dynamics
$$
\tau_ {\mathrm {g}} \dot {\mathbf {c}} (t) = \boldsymbol {\Gamma} \boldsymbol {\Gamma} ^ {\top} \boldsymbol {\nabla} _ {\mathbf {c}} \log p (\mathbf {c} | \mathbf {s}) + \boldsymbol {\Gamma} \boldsymbol {\xi} (t), \tag {B.9}
$$
for a (expectantly named) time constant  $\tau_{\mathrm{g}} > 0$  and any (potentially non-square) matrix  $\Gamma \in \mathbb{R}^{n_{\mathrm{odor}} \times n_{\mathrm{g}}}$  such that  $\Gamma \Gamma^{\top}$  is positive-definite, will have as their stationary distribution the posterior  $p(\mathbf{c} \mid \mathbf{s})$ . Here, we have replaced the  $n_{\mathrm{odor}}$ -dimensional noise process  $\boldsymbol{\eta}(t)$  with an  $n_{\mathrm{g}}$ -dimensional noise process  $\boldsymbol{\xi}(t)$  with zero mean and covariance
$$
\mathbb {E} \left[ \xi_ {j} (t) \xi_ {j ^ {\prime}} \left(t ^ {\prime}\right) \right] = 2 \tau_ {\mathrm {g}} \delta_ {j j ^ {\prime}} \delta \left(t - t ^ {\prime}\right). \tag {B.10}
$$
Then, we can write the concentration estimate as
$$
\mathbf {c} (t) = \boldsymbol {\Gamma} \mathbf {g} (t) \tag {B.11}
$$
where the activity of the neurons  $\mathbf{g} \in \mathbb{R}^{n_{\mathrm{g}}}$  follows the dynamics
$$
\tau_ {\mathrm {g}} \dot {\mathbf {g}} (t) = \boldsymbol {\Gamma} ^ {\top} \boldsymbol {\nabla} _ {\mathbf {c}} \log p (\mathbf {g} \mid \mathbf {s}) \Bigg | _ {\mathbf {c} = \boldsymbol {\Gamma} \mathbf {g}} + \boldsymbol {\xi} (t). \tag {B.12}
$$
Using the gradient of the log-posterior computed above, we then have
$$
\tau_ {\mathrm {g}} \dot {\mathbf {g}} (t) = (\mathbf {A} \boldsymbol {\Gamma}) ^ {\top} [ \mathbf {s} \oslash (\mathbf {r} _ {0} + \mathbf {A} \boldsymbol {\Gamma} \mathbf {g}) - \mathbf {1} ] + \boldsymbol {\Gamma} ^ {\top} [ (\boldsymbol {\alpha} - \mathbf {1}) \oslash (\boldsymbol {\Gamma} \mathbf {g}) ] - \boldsymbol {\Gamma} ^ {\top} \boldsymbol {\lambda} + \boldsymbol {\xi} (t). \tag {B.13}
$$
These dynamics include two divisive non-linearities, which can be complex to implement in biophysical models of single neurons [80]. Using the approach proposed in Chalk et al. [81], we can linearize the inference by introducing two additional cell types that have as their fixed points the elementwise divisions  $\mathbf{s} \oslash (\mathbf{r}_0 + \mathbf{A}\Gamma \mathbf{g})$  and  $(\alpha - 1) \oslash (\Gamma \mathbf{g})$ . We then introduce a population  $\mathbf{p}$  of  $n_{\mathrm{OSN}}$  neurons, with dynamics
$$
\tau_ {\mathrm {p}} \dot {\mathbf {p}} (t) = \mathbf {s} - \mathbf {p} \odot \left(\mathbf {r} _ {0} + \mathbf {A} \boldsymbol {\Gamma} \mathbf {g}\right), \tag {B.14}
$$
such that the fixed point is
$$
\mathbf {p} ^ {*} = \mathbf {s} \oslash (\mathbf {r} _ {0} + \mathbf {A} \boldsymbol {\Gamma} \mathbf {g}). \tag {B.15}
$$
We finally introduce a third population  $\mathbf{z}$  of  $n_{\mathrm{odor}}$  neurons, with dynamics
$$
\tau_ {z} \dot {\mathbf {z}} (t) = \boldsymbol {\alpha} - \mathbf {1} - \mathbf {z} \odot (\Gamma \mathbf {g}), \tag {B.16}
$$
$$
\mathbf {z} ^ {*} = (\boldsymbol {\alpha} - \mathbf {1}) \oslash (\boldsymbol {\Gamma} \mathbf {g}). \tag {B.17}
$$
To be more precise, the cell types  $\mathbf{p}$  and  $\mathbf{z}$  compute the desired elementwise divisions in the pseudosteady-state regime  $\tau_{\mathrm{p}},\tau_{\mathrm{z}}\downarrow 0$
Putting everything together, this gives the circuit dynamics
$$
\mathbf {c} (t) = \boldsymbol {\Gamma} \mathbf {g} (t) \tag {B.18}
$$
$$
\tau_ {\mathrm {g}} \dot {\mathbf {g}} (t) = (\mathbf {A} \boldsymbol {\Gamma}) ^ {\top} (\mathbf {p} - \mathbf {1}) + \boldsymbol {\Gamma} ^ {\top} (\mathbf {z} - \boldsymbol {\lambda}) + \boldsymbol {\xi} (t) \tag {B.19}
$$
$$
\tau_ {\mathrm {p}} \dot {\mathbf {p}} (t) = \mathbf {s} - \mathbf {p} \odot \left(\mathbf {r} _ {0} + \mathbf {A} \boldsymbol {\Gamma} \mathbf {g}\right) \tag {B.20}
$$
$$
\tau_ {z} \dot {\mathbf {z}} (t) = \boldsymbol {\alpha} - \mathbf {1} - \mathbf {z} \odot \mathbf {c}, \tag {B.21}
$$
where we recall that the covariance of the zero-mean Gaussian noise process  $\pmb{\xi}(t)$  is
$$
\mathbb {E} \left[ \xi_ {j} (t) \xi_ {j ^ {\prime}} \left(t ^ {\prime}\right) \right] = 2 \tau_ {\mathrm {g}} \delta_ {j j ^ {\prime}} \delta \left(t - t ^ {\prime}\right). \tag {B.22}
$$
We note that the two constant terms in the dynamics of  $\mathbf{g}$  can be grouped into an overall leak term  $-\Gamma^{\top}(\mathbf{A}^{\top}\mathbf{1} + \lambda)$ . As detailed in the main text, we interpret  $\mathbf{p}$  as  $\bar{\mathbf{M}}/\mathbf{T}$  cells,  $\mathbf{g}$  as granule cells, and  $\mathbf{z}$  as cortical feedback.
C Stability of the MAP fixed point
In this Appendix, we aim to get some understanding for how the introduction of the cell type  $\mathbf{p}$  to represent the elementwise division affects the inference, particularly when  $\tau_{\mathrm{p}}$  is comparable to  $\tau_{\mathrm{g}}$ . For simplicity, we specialize to the case of an exponential prior (i.e., we set  $\alpha = 1$ ), in which case the general circuit (3) simplifies to
$$
\mathbf {c} (t) = \boldsymbol {\Gamma} \mathbf {g} (t)
$$
$$
\tau_ {\mathrm {g}} \dot {\mathbf {g}} (t) = (\mathbf {A} \boldsymbol {\Gamma}) ^ {\top} (\mathbf {p} - \mathbf {1}) - \boldsymbol {\Gamma} ^ {\top} \boldsymbol {\lambda} \tag {C.1}
$$
$$
\tau_ {\mathbf {p}} \dot {\mathbf {p}} (t) = \mathbf {s} - \mathbf {p} \odot (\mathbf {r} _ {0} + \mathbf {A} \mathbf {\Gamma} \mathbf {g}).
$$
C.1 Analysis of a two-cell circuit
To build intuition, we first consider a circuit with  $n_{\mathrm{odor}} = n_{\mathrm{OSN}} = n_{\mathrm{g}} = 1$ :
$$
c (t) = \gamma g (t)
$$
$$
\tau_ {\mathrm {g}} \frac {d g}{d t} = a \gamma (p - 1) - \gamma \lambda \tag {C.2}
$$
$$
\tau_ {\mathrm {p}} \frac {d p}{d t} = s - p \left(r _ {0} + a \gamma g\right).
$$
In this case, it is useful to non-dimensionalize the system. Re-scale time as
$$
\tilde {t} = \frac {a \gamma}{\tau_ {\mathrm {g}}} t, \tag {C.3}
$$
and define the dimensionless parameters and input
$$
\tilde {\tau} = \frac {\tau_ {\mathrm {p}}}{\tau_ {\mathrm {g}}}, \tag {C.4}
$$
$$
\tilde {r} _ {0} = \frac {r _ {0}}{\alpha \gamma}, \tag {C.5}
$$
$$
\tilde {\lambda} = \frac {\lambda}{a}, \tag {C.6}
$$
$$
\tilde {s} = \frac {s}{\alpha \gamma}. \tag {C.7}
$$
Then, the dynamics are
$$
\frac {d g}{d \tilde {t}} = p - 1 - \tilde {\lambda} \tag {C.8}
$$
$$
\tilde {\tau} \frac {d p}{d \tilde {t}} = \tilde {s} - p \left(\tilde {r} _ {0} + g\right). \tag {C.9}
$$
These dynamics have a single fixed point at
$$
g _ {*} = \frac {\tilde {s}}{1 + \tilde {\lambda}} - \tilde {r} _ {0} \tag {C.10}
$$
$$
p _ {*} = 1 + \tilde {\lambda}. \tag {C.11}
$$
Linearizing about this fixed point, we have
$$
\frac {d}{d \tilde {t}} \binom {\delta g} {\delta g} = \mathbf {M} \binom {\delta g} {\delta g} \tag {C.12}
$$
$$
\mathbf {M} = \frac {1}{\tilde {\tau}} \left( \begin{array}{c c} 0 & \tilde {\tau} \\ - (1 + \tilde {\lambda}) & - (1 + \tilde {\lambda}) ^ {- 1} \tilde {s} \end{array} \right). \tag {C.13}
$$
The eigenvalues of  $\mathbf{M}$  are easily computed as
$$
\Lambda_ {\pm} (\mathbf {M}) = \frac {- \tilde {s} \pm \sqrt {\tilde {s} ^ {2} - 4 \tilde {\tau} (1 + \tilde {\lambda}) ^ {3}}}{2 \tilde {\tau} (1 + \tilde {\lambda})}. \tag {C.14}
$$
For any positive  $\tilde{\tau}$  and non-negative  $\tilde{\lambda}$ , we have
$$
\begin{array}{l} \operatorname {R e} \Lambda_ {\pm} (\mathbf {M}) = \frac {- \tilde {s} \pm \operatorname {R e} \sqrt {\tilde {s} ^ {2} - 4 \tilde {\tau} (1 + \tilde {\lambda}) ^ {3}}}{2 \tilde {\tau} (1 + \tilde {\lambda})} (C.15) \\ = \left\{ \begin{array}{l l} \frac {- \tilde {s} \pm \sqrt {\tilde {s} ^ {2} - 4 \tilde {\tau} (1 + \tilde {\lambda}) ^ {3}}}{2 \tilde {\tau} (1 + \tilde {\lambda})} & \tilde {s} ^ {2} - 4 \tilde {\tau} (1 + \tilde {\lambda}) ^ {3} > 0 \\ \frac {- \tilde {s}}{2 \tilde {\tau} (1 + \tilde {\lambda})} & \text {o t h e r w i s e ,} \end{array} \right. (C.16) \\ \end{array}
$$
hence it is easy to see that the real parts of both of these eigenvalues are strictly negative so long as  $\tilde{s} > 0$ , meaning that the system is stable. If  $\tilde{s} = 0$ —which should be exceedingly rare if the OSNs have a baseline rate—then
$$
\left. \Lambda_ {\pm} (\mathbf {M}) \right| _ {\tilde {s} = 0} = \pm 2 i \sqrt {\frac {1 + \tilde {\lambda}}{\tilde {\tau}}}, \tag {C.17}
$$
and there can be oscillations. Therefore, a linear stability analysis suggests that the MAP fixed point of this two-cell circuit should be stable even for large  $\tilde{\tau}$  in the presence of a non-zero input, though we expect the relaxation timescales to grow as  $\tilde{\tau}$  becomes larger.
C.2 Analysis of the full circuit
We now consider the full MAP circuit
$$
\tau_ {\mathrm {g}} \dot {\mathbf {g}} (t) = (\mathbf {A} \boldsymbol {\Gamma}) ^ {\top} (\mathbf {p} - \mathbf {1}) - \boldsymbol {\Gamma} ^ {\top} \boldsymbol {\lambda} \tag {C.18}
$$
$$
\tau_ {\mathbf {p}} \dot {\mathbf {p}} (t) = \mathbf {s} - \mathbf {p} \odot (\mathbf {r} _ {0} + \mathbf {A} \boldsymbol {\Gamma} \mathbf {g}).
$$
We assume that  $n_{\mathrm{OSN}} < n_{\mathrm{odor}} \leq n_{\mathrm{g}}$  and that  $\mathbf{A}\Gamma$  is of full row rank, i.e., it has rank  $n_{\mathrm{OSN}}$ . We recall that  $\mathbf{g} \in \mathbb{R}_{+}^{n_{\mathrm{g}}}$  and  $\mathbf{p} \in \mathbb{R}_{+}^{n_{\mathrm{g}}}$ . Therefore, the dynamics of  $\mathbf{g}$  span only an  $n_{\mathrm{odor}}$ -dimensional subspace, and, in particular, the  $\mathbf{p}$ -dependent term affects only an  $n_{\mathrm{OSN}}$ -dimensional subspace.
We start by observing that the dynamics of  $\mathbf{p}$  depend on  $\mathbf{g}$  only through
$$
\mathbf {q} \equiv \mathbf {A} \boldsymbol {\Gamma} \mathbf {g}. \tag {C.19}
$$
Importantly, if  $\mathbf{A}\Gamma$  is positivity-preserving, then  $\mathbf{q} \in \mathbb{R}_{+}^{n_{\mathrm{OSN}}}$ . Then, we have the closed dynamics
$$
\begin{array}{l} \tau_ {\mathrm {g}} \dot {\mathbf {q}} = (\mathbf {A} \boldsymbol {\Gamma} \mathbf {\Gamma} ^ {\top} \mathbf {A} ^ {\top}) (\mathbf {p} - \mathbf {1}) - \mathbf {A} \boldsymbol {\Gamma} \mathbf {\Gamma} ^ {\top} \boldsymbol {\lambda} \tag {C.20} \\ \tau_ {\mathbf {p}} \dot {\mathbf {p}} = \mathbf {s} - \mathbf {p} \odot (\mathbf {r} _ {0} + \mathbf {q}) \\ \end{array}
$$
for  $(\mathbf{q},\mathbf{p})\in \mathbb{R}_{+}^{2n_{\mathrm{OSN}}}$ . The fixed point of this system is of course determined by the conditions  $d(\mathbf{q},\mathbf{p}) / dt = \mathbf{0}$ , which gives
$$
\left(\mathbf {A} \boldsymbol {\Gamma} \boldsymbol {\Gamma} ^ {\top} \mathbf {A} ^ {\top}\right) \left(\mathbf {p} ^ {*} - \mathbf {1}\right) = \mathbf {A} \boldsymbol {\Gamma} \boldsymbol {\Gamma} ^ {\top} \boldsymbol {\lambda} \tag {C.21}
$$
$$
\mathbf {p} ^ {*} \odot \left(\mathbf {r} _ {0} + \mathbf {q} ^ {*}\right) = \mathbf {s} \tag {C.22}
$$
subject to the non-negativity constraints. By our assumptions on the rank of  $\mathbf{A}\Gamma$ , the symmetric matrix  $\mathbf{A}\Gamma \mathbf{T}^{\top} \mathbf{A}^{\top}$  is positive-definite and thus can be inverted to solve the first condition for  $\mathbf{p}^{*}$ :
$$
\mathbf {p} ^ {*} = \mathbf {1} + \left(\mathbf {A} \boldsymbol {\Gamma} \boldsymbol {\Gamma} ^ {\top} \mathbf {A} ^ {\top}\right) ^ {- 1} \mathbf {A} \boldsymbol {\Gamma} \boldsymbol {\Gamma} ^ {\top} \boldsymbol {\lambda}. \tag {C.23}
$$
For the second condition to be satisfied, we can see that the elements of  $\mathbf{p}^*$  must be strictly positive at the fixed point, which gives a self-consistency condition on  $\mathbf{A}$ ,  $\Gamma$ , and  $\lambda$ . Assuming that this holds, we then have
$$
\mathbf {q} ^ {*} = \mathbf {s} \oslash \mathbf {p} ^ {*} - \mathbf {r} _ {0}, \tag {C.24}
$$
which again gives a self-consistency condition as non-negativity is required.
Assuming these conditions hold, we can then linearize the dynamics about the fixed point. For convenience, we non-dimensionalize time through
$$
\tilde {t} = \frac {t}{\tau_ {\mathrm {g}}}, \tag {C.25}
$$
$$
\tilde {\tau} = \frac {\tau_ {\mathrm {P}}}{\tau_ {\mathrm {g}}}, \tag {C.26}
$$
which yields the linearized dynamics
$$
\frac {d}{d \tilde {t}} \binom {\delta \mathbf {q}} {\delta \mathbf {p}} = \mathbf {M} \binom {\delta \mathbf {q}} {\delta \mathbf {p}} \tag {C.27}
$$
$$
\begin{array}{l} \mathbf {M} = \left( \begin{array}{c c} \mathbf {0} & (\mathbf {A} \boldsymbol {\Gamma} \mathbf {\Gamma} ^ {\top} \mathbf {A} ^ {\top}) \\ - \tilde {\tau} ^ {- 1} \operatorname {d i a g} \left(\mathbf {p} ^ {*}\right) & - \tilde {\tau} ^ {- 1} \operatorname {d i a g} \left(\mathbf {r} _ {0} + \mathbf {q} ^ {*}\right) \end{array} \right) (C.28) \\ = \left( \begin{array}{c c} \mathbf {0} & (\mathbf {A} \boldsymbol {\Gamma} \mathbf {\Gamma} ^ {\top} \mathbf {A} ^ {\top}) \\ - \tilde {\tau} ^ {- 1} \operatorname {d i a g} \left(\mathbf {p} ^ {*}\right) & - \tilde {\tau} ^ {- 1} \operatorname {d i a g} \left(\mathbf {s} \oslash \mathbf {p} ^ {*}\right) \end{array} \right). (C.29) \\ \end{array}
$$
Using the fact that the diagonal matrices  $\mathrm{diag}(\mathbf{p}^*)$  and  $\mathrm{diag}(\mathbf{s} \oslash \mathbf{p}^*)$  commute, the characteristic polynomial of  $\mathbf{M}$  is
$$
\begin{array}{l} \det  (\mathbf {M} - \mu \mathbf {I} _ {2 n _ {\mathrm {O S N}}}) = \det  \left( \begin{array}{c c} \mu \mathbf {I} _ {n _ {\mathrm {O S N}}} & - \left(\mathbf {A} \boldsymbol {\Gamma} \mathbf {T} ^ {\top} \mathbf {A} ^ {\top}\right) \\ \tilde {\tau} ^ {- 1} \operatorname {d i a g} \left(\mathbf {p} ^ {*}\right) & \tilde {\tau} ^ {- 1} \operatorname {d i a g} \left(\mathbf {s} \oslash \mathbf {p} ^ {*}\right) + \mu \mathbf {I} _ {n _ {\mathrm {O S N}}} \end{array} \right) (C.30) \\ = \det [ \mu^ {2} \mathbf {I} _ {n _ {\mathrm {O S N}}} + \mu \tilde {\tau} ^ {- 1} \operatorname {d i a g} (\mathbf {s} \oslash \mathbf {p} ^ {*}) + \tilde {\tau} ^ {- 1} (\mathbf {A} \boldsymbol {\Gamma} \boldsymbol {\Gamma} ^ {\top} \mathbf {A} ^ {\top}) \operatorname {d i a g} (\mathbf {p} ^ {*}) ]. (C.31) \\ \end{array}
$$
One case that is particularly easy to solve is when the symmetric positive-definite matrix  $\mathbf{A}\mathbf{T}\mathbf{T}^{\top}\mathbf{A}^{\top}$  is in fact diagonal, with positive diagonal entries  $a_{j}$ . Then, we have
$$
\det  \left(\mathbf {M} - \mu \mathbf {I} _ {2 n _ {\mathrm {O S N}}}\right) = \prod_ {j = 1} ^ {n _ {\mathrm {O S N}}} \left[ \mu^ {2} + \frac {s _ {j}}{\tilde {\tau} p _ {j} ^ {*}} \mu + \frac {a _ {j} p _ {j} ^ {*}}{\tilde {\tau}} \right], \tag {C.32}
$$
meaning that the eigenvalues of  $\mathbf{M}$  are
$$
\mu_ {j, \pm} = \frac {1}{2} \left[ - \frac {s _ {j}}{\tilde {\tau} p _ {j} ^ {*}} \pm \sqrt {\left(\frac {s _ {j}}{\tilde {\tau} p _ {j} ^ {*}}\right) ^ {2} - 4 \frac {a _ {j} p _ {j} ^ {*}}{\tilde {\tau}}} \right], \tag {C.33}
$$
which under the given assumptions always have strictly positive real part for non-negative inputs.
Now, more generally, assume that  $\mathbf{M}$  is diagonalizable, and let  $\mathbf{m}$  be a un-normalized eigenvector of  $\mathbf{M}$  with eigenvalue  $\mu$ . Then, writing
$$
\mathbf {m} = \left( \begin{array}{l} \mathbf {u} \\ \mathbf {v} \end{array} \right) \tag {C.34}
$$
for  $\mathbf{u},\mathbf{v}\in \mathbb{C}^{n_{\mathrm{OSN}}}$  , the eigenvector condition
$$
\mathbf {M} \mathbf {m} = \mu \mathbf {m} \tag {C.35}
$$
implies that  $\mathbf{u}$  and  $\mathbf{v}$  satisfy
$$
\frac {1}{\tau_ {\mathrm {g}}} \left(\mathbf {A} \boldsymbol {\Gamma} \boldsymbol {\Gamma} ^ {\top} \mathbf {A} ^ {\top}\right) \mathbf {v} = \mu \mathbf {u} \tag {C.36}
$$
$$
- \frac {1}{\tau_ {\mathrm {p}}} \operatorname {d i a g} \left(\mathbf {p} ^ {*}\right) \mathbf {u} - \frac {1}{\tau_ {\mathrm {p}}} \operatorname {d i a g} \left(\mathbf {s} \oslash \mathbf {p} ^ {*}\right) \mathbf {v} = \mu \mathbf {v}. \tag {C.37}
$$
Assuming that  $\mu$  is non-zero, we can solve the first equation for  $\mathbf{u}$  and then substitute the result into the second to obtain a quadratic eigenproblem for  $\mu$  and  $\mathbf{v}$ :
$$
\mu^ {2} \mathbf {v} + \mu \frac {1}{\tau_ {\mathrm {p}}} \operatorname {d i a g} (\mathbf {s} \oslash \mathbf {p} ^ {*}) \mathbf {v} + \frac {1}{\tau_ {\mathrm {p}} \tau_ {\mathrm {g}}} \operatorname {d i a g} (\mathbf {p} ^ {*}) \left(\mathbf {A} \boldsymbol {\Gamma} \boldsymbol {\Gamma} ^ {\top} \mathbf {A} ^ {\top}\right) \mathbf {v} = 0. \tag {C.38}
$$
We will not attempt to solve this eigenproblem, but will instead attempt to extract information about possible values of  $\mu$  for a fixed  $\mathbf{v}$ . Suppose (without loss of generality given the assumption that it is nonzero, as otherwise we may divide by its norm) that  $\mathbf{v}$  is a unit vector. Then, acting with  $\mathbf{v}^{\dagger}$  from the left, we have
$$
\mu^ {2} + \frac {a}{\tau_ {\mathrm {p}}} \mu + \frac {b}{\tau_ {\mathrm {p}} \tau_ {\mathrm {g}}} = 0. \tag {C.39}
$$
where we define the coefficients
$$
a \equiv \mathbf {v} ^ {\dagger} \operatorname {d i a g} (\mathbf {s} \oslash \mathbf {p} ^ {*}) \mathbf {v} \tag {C.40}
$$
$$
b \equiv \mathbf {v} ^ {\dagger} \operatorname {d i a g} \left(\mathbf {p} ^ {*}\right) \left(\mathbf {A} \boldsymbol {\Gamma} \boldsymbol {\Gamma} ^ {\top} \mathbf {A} ^ {\top}\right) \mathbf {v}. \tag {C.41}
$$
So long as  $\mathbf{s}$  and  $\mathbf{p}^*$  are positive,  $a$  is real and positive. Let
$$
\tilde {\boldsymbol {\lambda}} \equiv (\mathbf {A} \boldsymbol {\Gamma} \boldsymbol {\Gamma} ^ {\top} \mathbf {A} ^ {\top}) ^ {- 1} \mathbf {A} \boldsymbol {\Gamma} \boldsymbol {\Gamma} ^ {\top} \boldsymbol {\lambda} \tag {C.42}
$$
$$
\mathbf {p} ^ {*} = \mathbf {1} + \tilde {\lambda}. \tag {C.43}
$$
By assumption, the elements of  $\tilde{\lambda}$  are strictly greater than  $-1$ . Then,