path: root/marginal.tex

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\begin{document}

\title{
  Conditioning the complexity of random landscapes on marginal minima
}

\author{Jaron Kent-Dobias}
\affil{Istituto Nazionale di Fisica Nucleare, Sezione di Roma I}

\maketitle
\begin{abstract}
\end{abstract}

\tableofcontents

\section{Introduction}

Systems with rugged landscapes are important across many disciplines, from the
physics to glasses and spin-glasses to the statistical inference problems. The
behavior of these systems is best understood when equilibrium or optimal
solutions are studied and averages can be taken statically over all possible
configurations. However, such systems are also infamous for their tendency to
defy equilibrium and optimal expectations in practice, due to the presence of
dynamic transitions or crossovers that leave physical or algorithmic dynamics
stuck exploring only a subset of configurations.

In some simple models of such landscapes, it was recently found that marginal
minima are significant as the attractors of gradient descent dynamics
\cite{Folena_2020_Rethinking, Folena_2023_On}. This extends to more novel
algorithms, like message passing \cite{} \textbf{Find out if this is true}.
\textbf{Think of other examples.}
While it is still not known how to predict which marginal minima will be
attractors, this ubiquity of behavior suggests that cartography of marginal
minima is a useful step in bounding out-of-equilibrium dynamical behavior.

In the traditional methods for analyzing the geometric structure of rugged
landscapes, it is not necessarily straightforward to condition an analysis on
the marginality of minima. Using the method of a Legendre transformation of the
Parisi parameter corresponding to a set of real replicas, one can force the
result to be marginal by restricting the value of that parameter, but this
results in only the marginal minima at the energy level at which they are the
majority of stationary points \cite{Monasson_1995_Structural}. It is now
understood that out-of-equilibrium dynamics usually goes to marginal minima at
other energy levels \cite{Folena_2023_On}.

The alternative, used to great success in the spherical models, is to start by
making a detailing understanding of the Hessian matrix at stationary points.
Then, one can condition the analysis on whatever properties of the Hessian are
necessary to lead to marginal minima. This strategy is so successful in the
spherical models because it is very straightforward to implement: a natural
parameter in the analysis of these models linearly shifts the spectrum of the
Hessian, and so fixing this parameter by whatever means naturally allows one to
require that the Hessian spectrum have a pseudogap.
Unfortunately this strategy is less straightforward to generalize. Many models
of interest, especially in inference problems, have Hessian statistics that are
poorly understood.

Here, we introduce a generic method for conditioning the statistics of
stationary points on their marginality. The technique makes use of a novel way
to condition an integral over parameters to select only those that result in a
certain value of the smallest eigenvalue of a matrix that is a function of
those parameters. By requiring that the smallest eigenvalue of the Hessian at
stationary points be zero, we restrict to marginal minima, either those with a
pseudogap in their bulk spectrum or those with outlying eigenvectors. We
provide a heuristic to distinguish these two cases. We demonstrate the method
on the spherical models, where it is unnecessary but instructive, and on
extensions of the spherical models with non-\textsc{goe} Hessians where the technique is
more useful.

\section{Conditioning on the smallest eigenvalue}



An arbitrary function $g$ of the minimum eigenvalue of a matrix $A$ can be expressed as
\begin{equation}
  g(\lambda_\textrm{min}(A))
  =\lim_{\beta\to\infty}\int\frac{d\mathbf x\,\delta(N-\mathbf x^T\mathbf x)e^{-\beta\mathbf x^TA\mathbf x}}{\int d\mathbf x'\,\delta(N-\mathbf x'^T\mathbf x')e^{-\beta\mathbf x'^TA\mathbf x'}}g\left(\frac{\mathbf x^TA\mathbf x}N\right)
\end{equation}
Assuming
\begin{equation}
  \begin{aligned}
    \lim_{\beta\to\infty}\int\frac{d\mathbf x\,\delta(N-\mathbf x^T\mathbf x)e^{-\beta\mathbf x^TA\mathbf x}}{\int d\mathbf x'\,\delta(N-\mathbf x'^T\mathbf x')e^{-\beta\mathbf x'^TA\mathbf x'}}&g\left(\frac{\mathbf x^TA\mathbf x}N\right)
    =\int\frac{d\mathbf x\,\delta(N-\mathbf x^T\mathbf x)\mathbb 1_{\operatorname{ker}(A-\lambda_\mathrm{min}(A)I)}(\mathbf x)}{\int d\mathbf x'\,\delta(N-\mathbf x'^T\mathbf x')\mathbb 1_{\operatorname{ker}(A-\lambda_\mathrm{min}(A)I)}(\mathbf x')}g\left(\frac{\mathbf x^TA\mathbf x}N\right) \\
    &=g(\lambda_\mathrm{min}(A))
    \frac{\int d\mathbf x\,\delta(N-\mathbf x^T\mathbf x)\mathbb 1_{\operatorname{ker}(A-\lambda_\mathrm{min}(A)I)}(\mathbf x)}{\int d\mathbf x'\,\delta(N-\mathbf x'^T\mathbf x')\mathbb 1_{\operatorname{ker}(A-\lambda_\mathrm{min}(A)I)}(\mathbf x')}
    =g(\lambda_\mathrm{min}(A))
  \end{aligned}
\end{equation}
The first relation extends a technique first introduced in
\cite{Ikeda_2023_Bose-Einstein-like} and used in
\cite{Kent-Dobias_2024_Arrangement}. A Boltzmann distribution is introduced
over a spherical model whose Hamiltonian is quadratic with interaction matrix
given by $A$. In the limit of zero temperature, the measure will concentrate on
the ground states of the model, which correspond with the eigenspace of $A$
associated with its minimum eigenvalue $\lambda_\mathrm{min}$. The second
relation uses the fact that, once restricted to the sphere $\mathbf x^T\mathbf
x=N$ and the minimum eigenspace, $\mathbf x^TA\mathbf
x=N\lambda_\mathrm{min}(A)$.

The relationship is formal, but we can make use of the fact that the integral
expression with a Gibbs distribution can be manipulated with replica
techniques, averaged over, and in general treated with a physicist's toolkit.
In particular, we have specific interest in using
$g(\lambda_\mathrm{min}(A))=\delta(\lambda_\mathrm{min}(A))$, a Dirac
delta-function, which can be inserted into averages over ensembles of matrices
$A$ (or indeed more complicated averages) in order to condition that the
minimum eigenvalue is zero.

\subsection{Simple example: shifted GOE}

We demonstrate the efficacy of the technique by rederiving a well-known result:
the large-deviation function for pulling an eigenvalue from the bulk of the
\textsc{goe} spectrum.
Consider an ensemble of $N\times N$ matrices $A=B+\omega I$ for $B$ drawn from the \textsc{goe} ensemble with entries
whose variance is $\sigma^2/N$. We know that the bulk spectrum of $A$ is a
Wigner semicircle with radius $2\sigma$ shifted by a constant $\omega$.
Therefore, for $\omega=2\sigma$, the minimum eigenvalue will typically be zero,
while for $\omega>2\sigma$ the minimum eigenvalue would need to be a large
deviation from the typical spectrum and its likelihood will be exponentially
suppressed with $N$. For $\omega<2\sigma$, the bulk of the typical spectrum contains
zero and therefore a larger $N^2$ deviation, moving an extensive number of
eigenvalues, would be necessary. This final case cannot be quantified by this
method, but instead the nonexistence of a large deviation linear in $N$ appears
as the emergence of an imaginary part in the function.

As an example, we compute
\begin{equation} \label{eq:large.dev}
  e^{NG_\sigma(\omega)}=P_{\lambda_\mathrm{min}(B+\omega I)=0}=\overline{\lim_{\beta\to\infty}\int\frac{d\mathbf x\,\delta(N-\mathbf x^T\mathbf x)e^{-\beta\mathbf x^T(B+\omega I)\mathbf x}}{\int d\mathbf x'\,\delta(N-\mathbf x'^T\mathbf x')e^{-\beta\mathbf x'^T(B+\omega I)\mathbf x'}}\,\delta\big(\mathbf x^T(B+\omega I)\mathbf x\big)}
\end{equation}
where the overline is the average over $B$, and we have defined the large
deviation function $G_\sigma(\omega)$. Using replicas to treat the denominator ($x^{-1}=\lim_{n\to0}x^{n-1}$)
and transforming the $\delta$-function to its Fourier
representation, we have
\begin{equation}
  e^{NG_\sigma(\omega)}=\overline{\lim_{\beta\to\infty}\lim_{n\to0}\int d\lambda\prod_{a=1}^n\left[d\mathbf x_a\,\delta(N-\mathbf x_a^T\mathbf x_a)\right]
  \exp\left\{-\beta\sum_{a=1}^n\mathbf x_a^T(B+\omega I)\mathbf x_a+\lambda\mathbf x_1^T(B+\omega I)\mathbf x_1\right\}}
\end{equation}
having introduced the parameter $\lambda$ in the Fourier representation of the $\delta$-function.
The whole expression, so transformed, is a simple exponential integral linear in the matrix $B$.
Taking the average over $B$, we have
\begin{equation}
  e^{NG_\sigma(\omega)}
  =\lim_{\beta\to\infty}\lim_{n\to0}\int d\lambda\prod_{a=1}^n\left[d\mathbf x_a\,\delta(N-\mathbf x_a^T\mathbf x_a)\right]
  \exp\left\{-Nn\beta\omega+N\lambda\omega+\frac{\sigma^2}{N}\left[\beta^2\sum_{ab}^n(\mathbf x_a^T\mathbf x_b)^2
    -2\beta\lambda\sum_a^n(\mathbf x_a^T\mathbf x_1)^2
    +\lambda^2N^2
  \right]\right\}
\end{equation}
We make the Hubbard--Stratonovich transformation to the matrix field $Q_{ab}=\frac1N\mathbf x_a^T\mathbf x_b$. This gives
\begin{equation}
  e^{NG_\sigma(\omega)}
  =\lim_{\beta\to\infty}\lim_{n\to0}\int d\lambda\,dQ\,
  \exp N\left\{
    -n\beta\omega+\lambda\omega+\sigma^2\left[\beta^2\sum_{ab}^nQ_{ab}^2
      +-\beta\lambda\sum_a^nQ_{1a}^2
    +\lambda^2
  \right]+\frac12\log\det Q\right\}
\end{equation}
where $Q_{aa}=1$ because of the spherical constraint. We can evaluate this
integral using the saddle point method. We make a replica symmetric ansatz for
$Q$, because this is a 2-spin model, but with the first row singled out because
of its unique coupling with $\lambda$. This gives
\begin{equation}
  Q=\begin{bmatrix}
    1&\tilde q_0&\tilde q_0&\cdots&\tilde q_0\\
    \tilde q_0&1&q_0&\cdots&q_0\\
    \tilde q_0&q_0&1&\ddots&q_0\\
    \vdots&\vdots&\ddots&\ddots&\vdots\\
    \tilde q_0&q_0&q_0&\cdots&q_0
  \end{bmatrix}
\end{equation}
with $\sum_{ab}Q_{ab}^2=n+2(n-1)\tilde q_0^2+(n-1)(n-2)q_0^2$, $\sum_aQ_{1a}^2=1+(n-1)\tilde q_0^2$,
and
\begin{equation}
  \log\det Q=(n-2)\log(1-q_0)+\log(1+(n-2)q_0-(n-1)\tilde q_0^2)
\end{equation}
Inserting these expressions and taking the limit of $n$ to zero, we find
\begin{equation}
  e^{NG_\sigma(\omega)}=\lim_{\beta\to\infty}\int d\lambda\,dq_0\,d\tilde q_0\,e^{N\mathcal S_\beta(q_0,\tilde q_0,\lambda)}
\end{equation}
with the effective action
\begin{equation}
  \mathcal S_\beta(q_0,\tilde q_0,\lambda)=\lambda\omega+\sigma^2\left[
    2\beta^2(q_0^2-\tilde q_0^2)-2\beta\lambda(1-\tilde q_0^2)+\lambda^2
  \right]-\log(1-q_0)+\frac12\log(1-2q_0+\tilde q_0^2)
\end{equation}
We need to evaluate the integral above using the saddle point method, but in the limit of $\beta\to\infty$.
We expect the overlaps to concentrate on one as $\beta$ goes to infinity. We therefore take
\begin{align}
  q_0=1-y\beta^{-1}-z\beta^{-2}+O(\beta^{-3})
  &&
  \tilde q_0=1-\tilde y\beta^{-1}-\tilde z\beta^{-2}+O(\beta^{-3})
\end{align}
However, taking the limit with $y\neq\tilde y$ results in an expression for the
action that diverges with $\beta$. To cure this, we must take $\tilde y=y$. The result is
\begin{equation}
  \mathcal S_\infty(y,z,\tilde z,\lambda)
  =\lambda\omega+\sigma^2\big[
    \lambda^2-4(y+z-\tilde z)
  \big]+\frac12\log\left(1+2\frac{z-\tilde z}{y^2}\right)
\end{equation}
Extremizing this action over the new parameters $y$, $\Delta z=z-\tilde z$, and $\lambda$, we have
\begin{align}
  \lambda=-\frac1\sigma\sqrt{\frac{\omega^2}{(2\sigma)^2}-1}
  &&
  y=\frac1{2\sigma}\left(\frac{\omega}{2\sigma}-\sqrt{\frac{\omega^2}{(2\sigma)^2}-1}\right)
  &&
  \Delta z=\frac1{4\sigma^2}\left(1-\frac{\omega}{2\sigma}\left(\frac\omega{2\sigma}-\sqrt{\frac{\omega^2}{(2\sigma)^2}-1}\right)\right)
\end{align}
Inserting this solution into $\mathcal S_\infty$ we find
\begin{equation}
  G_\sigma(\omega)
  =\mathop{\textrm{extremum}}_{y,\Delta z,\lambda}\mathcal S_\infty(y,\Delta z,\lambda)
    =-\frac{\omega}{2\sigma}\sqrt{\frac{\omega^2}{(2\sigma)^2}-1}
  +\log\left[
    \frac{\omega}{2\sigma}+\sqrt{\frac{\omega^2}{(2\sigma)^2}-1}
  \right]
\end{equation}
This function is plotted in Fig.~\ref{fig:large.dev}. For $\omega<2\sigma$ $G_\sigma(\omega)$ has an
imaginary part, which makes any additional integral over $\omega$ highly
oscillatory. This indicates that the existence of a marginal minimum for this
parameter value corresponds with a large deviation that grows faster than $N$,
rather like $N^2$, since in this regime the bulk of the typical spectrum is
over zero and therefore extensively many eigenvalues have to have large
deviations in order for the smallest eigenvalue to be zero. For
$\omega\geq2\sigma$ this function gives the large deviation function for the
probability of seeing a zero eigenvalue given the shift $\omega$.
$\omega=2\sigma$ is the maximum of the function with a real value, and
corresponds to the intersection of the average spectrum with zero, i.e., a pseudogap.

\begin{figure}
  \centering
  \includegraphics{figs/large_deviation.pdf}
  \caption{
    The large deviation function $G_\sigma(\omega)$ defined in
    \eqref{eq:large.dev} as a function of the shift $\omega$ to the
    \textsc{goe} diagonal. As expected, $G_\sigma(2\sigma)=0$, while for
    $\omega>2\sigma$ it is negative and for $\omega<2\sigma$ it gains an
    imaginary part.
  } \label{fig:large.dev}
\end{figure}

Marginal spectra with a pseudogap and those with simple isolated eigenvalues
are qualitatively different, and more attention may be focused on the former.
Here, we see what appears to be a general heuristic for identifying the saddle
parameters for which the spectrum is psedogapped: the equivalent of this
large-deviation functions will lie on the singular boundary between a purely
real and complex value.

\subsection{Application to complexity in random landscapes}

The situation in the study of random landscapes is often as follows: an
ensemble of smooth functions $H:\mathbb R^N\to\mathbb R$ define random
landscapes, often with their configuration space subject to one or more
constraints of the form $g(\mathbf s)=0$ for $\mathbf s\in\mathbb R^N$. The
geometry of such landscapes is studied by their complexity, or the average
logarithm of the number of stationary points with certain properties, e.g., of
marginal minima at a given energy.

Such problems can be studied using the method of Lagrange multipliers, with one introduced for every constraint. If the configuration space is defined by $r$ constraints, then the problem is to extremize
\begin{equation}
  H(\mathbf s)+\sum_{i=1}^r\omega_ig_i(\mathbf s)
\end{equation}
with respect to $\mathbf s$ and $\pmb\omega=\{\omega_1,\ldots,\omega_r\}$. The corresponding gradient and Hessian for the problem are
\begin{align}
  \nabla H(\mathbf s,\pmb\omega)=\partial H(\mathbf s)+\sum_{i=1}^r\omega_i\partial g_i(\mathbf s)
  &&
  \operatorname{Hess}H(\mathbf s,\pmb\omega)=\partial\partial H(\mathbf s)+\sum_{i=1}^r\omega_i\partial\partial g_i(\mathbf s)
\end{align}
The number of stationary points in a landscape for a particular realization $H$ is found by integrating over the Kac--Rice measure
\begin{equation}
  d\mu_H(\mathbf s,\pmb\omega)=d\mathbf s\,d\pmb\omega\,\delta\big(\nabla H(\mathbf s,\pmb\omega)\big)\,\delta\big(\mathbf g(\mathbf s)\big)\,\big|\det\operatorname{Hess}H(\mathbf s,\pmb\omega)\big|
\end{equation}
with a $\delta$-function of the gradient and the constraints ensuring that we
count valid stationary points, and the Hessian entering in the determinant of
the Jacobian of the argument to the $\delta$-function. It is usually more
interesting to condition the count on interesting properties of the stationary
points, like the energy,
\begin{equation}
  d\mu_H(\mathbf s,\pmb\omega\mid E)=d\mu_H(\mathbf s,\pmb\omega)\,\delta\big(NE-H(\mathbf s)\big)
\end{equation}
In this paper we in particular want to exploit our method to condition
complexity on the marginality of stationary points. We therefore define the
number of marginal points in a particular instantiation $H$ as
\begin{equation}
  \begin{aligned}
    &\mathcal N_\text{marginal}(E)
    =\int d\mu_H(\mathbf s,\pmb\omega\mid E)\,\delta\big(N\lambda_\mathrm{min}(\operatorname{Hess}H(\mathbf s,\pmb\omega))\big) \\
    &=\lim_{\beta\to\infty}\int d\mu_H(\mathbf s,\pmb\omega\mid E)
    \frac{d\mathbf x\,\delta(N-\mathbf x^T\mathbf x)\delta(\mathbf x^T\partial\mathbf g(\mathbf s))e^{\beta\mathbf x^T\operatorname{Hess}H(\mathbf s,\pmb\omega)\mathbf x}}
    {\int d\mathbf x'\,\delta(N-\mathbf x'^T\mathbf x')\delta(\mathbf x'^T\partial\mathbf g(\mathbf s))e^{\beta\mathbf x'^T\operatorname{Hess}H(\mathbf s,\pmb\omega)\mathbf x'}}
    \delta\big(\mathbf x^T\operatorname{Hess}H(\mathbf s,\pmb\omega)\mathbf x\big)
  \end{aligned}
\end{equation}
where the $\delta$-functions
\begin{equation}
  \delta(\mathbf x^T\partial\mathbf g(\mathbf s))
  =\prod_{s=1}^r\delta(\mathbf x^T\partial g_i(\mathbf s))
\end{equation}
ensure that the integrals are constrained to the tangent space of the configuration manifold at the point $\mathbf s$. This likewise allows us to define the complexity of marginal points at energy $E$ as
\begin{equation}
  \Sigma_\text{marginal}(E)
  =\frac1N\overline{\log\mathcal N_\text{marginal}(E)}
\end{equation}
In practice, this can be computed by introducing replicas to treat the
logarithm ($\log x=\lim_{n\to0}\frac\partial{\partial n}x^n$) and replicating
again to treat each of the normalizations in the numerator. This leads to the expression
\begin{equation}
  \begin{aligned}
    \Sigma_\text{marginal}(E)
    &=\lim_{\beta\to\infty}\lim_{n\to0}\frac1N\frac\partial{\partial n}\int\prod_{a=1}^n\Bigg[d\mu_H(\mathbf s_a,\pmb\omega_a\mid E)\,\delta\big((\mathbf x_a^1)^T\operatorname{Hess}H(\mathbf s_a,\pmb\omega_a)\mathbf x_a^1\big)\\
      &\qquad\times\lim_{m_a\to0}
      \left(\prod_{b=1}^{m_a} d\mathbf x_a^b\,\delta(N-(\mathbf x_a^b)^T\mathbf x_a^b)\delta((\mathbf x_a^b)^T\partial\mathbf g(\mathbf s_a))e^{\beta(\mathbf x_a^b)^T\operatorname{Hess}H(\mathbf s_a,\pmb\omega_a)\mathbf x_a^b}\right)\Bigg]
  \end{aligned}
\end{equation}

\section{Examples in random landscapes}

\subsection{Application to the spherical models}

\begin{align}
  C_{ab}=\frac1N\mathbf s_a\cdot\mathbf s_b
  &&
  R_{ab}=-i\frac1N\mathbf s_a\cdot\hat{\mathbf s}_b
  &&
  D_{ab}=\frac1N\hat{\mathbf s}_a\cdot\hat{\mathbf s}_b
  \\
  A_{ab}^{cd}=\frac1N\mathbf x_a^c\cdot\mathbf x_b^d
  &&
  X^c_{ab}=\frac1N\mathbf s_a\cdot\mathbf x_b^c
  &&
  \hat X^c_{ab}=\frac1N\hat{\mathbf s}_a\cdot\mathbf x_b^c
\end{align}

\begin{equation}
  \begin{aligned}
    &\sum_{ab}^n\left[\beta\omega A_{aa}^{bb}+\hat x\omega A_{aa}^{11}+\beta^2f''(1)\sum_{cd}^m(A_{ab}^{cd})^2+\hat x^2f''(1)(A_{ab}^{11})^2+\beta\hat xf''(1)\sum_c^m A_{ab}^{1c}\right]\\
    &+\hat\beta^2f(C_{ab})+(2\hat\beta(R_{ab}-F_{ab})-D_{ab})f'(C_{ab})+(R_{ab}^2-F_{ab}^2)f''(C_{ab})
    +\log\det\begin{bmatrix}C&iR\\iR&D\end{bmatrix}-\log\det F
  \end{aligned}
\end{equation}

$X^a$ is $n\times m_a$, and $A^{ab}$ is $m_a\times m_b$.
\begin{equation}
  \begin{bmatrix}
    C&iR&X^1&\cdots&X^n \\
    iR&D&i\hat X^1&\cdots&i\hat X^m\\
    (X^1)^T&i(\hat X^1)^T&A^{11}&\cdots&A^{1n}\\
    \vdots&\vdots&\vdots&\ddots&\vdots\\
    (X^n)^T&i(\hat X^n)^T&A^{n1}&\cdots&A^{nn}
  \end{bmatrix}
\end{equation}
$X_{ab}^c$ will be nonzero if the lowest eigenvector of the hessian at the
point $\mathbf s_c$ are correlated with the direction of the point $\mathbf
s_a$. Since the eigenvector problem is always expected to be replica symmetric,
we expect no $b$-dependence of this matrix. $A^{aa}$ is the usual
replica-symmetric overlap matrix of the spherical two-spin problem. $A^{ab}$
describes overlaps between eigenvectors at different stationary points and should be a constant $m_a\times m_b$ matrix.

We will discuss at the end of this paper when these order parameters can be expected to be nonzero, but in this and most isotropic problems all of the $X$s, $\hat X$s, and $A^{ab}$ for $a\neq b$ are zero.

\begin{equation}
  \Sigma_\textrm{marginal}(E)
  =\operatorname{max}_\omega\big[\Sigma(E,\omega)+G_{\sqrt{f''(1)}}(\omega)\big]
\end{equation}
where the maximum over $\omega$ needs to lie at a real value.

\subsection{Twin spherical model}

$\Omega=S^{N-1}\times S^{N-1}$
\begin{equation}
  H(\pmb\sigma)=H_1(\pmb\sigma^{(1)})+H_2(\pmb\sigma^{(2)})+\epsilon\pmb\sigma^{(1)}\cdot\pmb\sigma^{(2)}
\end{equation}
\begin{equation}
  \overline{H_s(\pmb\sigma_1)H_s(\pmb\sigma_2)}
  =Nf_s\left(\frac{\pmb\sigma_1\cdot\pmb\sigma_2}N\right)
\end{equation}

\begin{equation}
  \mathcal S(C,R,D,W,\hat\beta,\omega)
  =\frac12\frac1n
    \sum_{ab}\left(
      \hat\beta^2f(C_{ab})+(2\hat\beta R_{ab}-D_{ab})f'(C_{ab})+(R_{ab}^2-W_{ab}^2)f''(C_{ab})
    \right)
\end{equation}

\begin{equation}
  \mathcal S(C^{11},R^{11},D^{11},W^{11},\hat\beta)+\mathcal S(C^{22},R^{22},D^{22},W^{22},\hat\beta)
  -\epsilon(r_{12}+r_{21})-\omega_1(r^{11}_d-w^{11}_d)-\omega_2(r^{22}_d-w^{22}_d)+\hat\beta E
  +\frac12\log\det\begin{bmatrix}C^{11}&iR^{11}\\iR^{11}&D^{11}\end{bmatrix}
  +\frac12\log\det\left(
  \begin{bmatrix}C^{22}-q_{12}^2C^{11}&iR^{22}\\iR^{22}&D^{22}\end{bmatrix}
  \right)
  -\log\det(W^{11}W^{22}+W^{12}W^{21})
\end{equation}

\begin{equation}
  \begin{aligned}
    &\sum_a^n\left[\hat q_a(Q^{11}_{aa}+Q^{22}_{aa}-1)-\beta(\omega_1Q^{11}_{aa}+\omega_2Q^{22}_{aa}+2\epsilon Q^{12}_{aa})\right]
    +\lambda(\omega_1Q^{11}_{11}+\omega_2Q^{22}_{11}+2\epsilon Q^{12}_{11}) \\
    &+\sum_{i=1,2}f_i''(1)\left[\beta^2\sum_{ab}^n(Q^{ii}_{ab})^2-2\beta\lambda\sum_a^n(Q^{ii}_{1a})^2+\lambda^2(Q^{ii}_{11})^2\right]
    +\frac12\log\det\begin{bmatrix}
      Q^{11}&Q^{12}\\
      Q^{12}&Q^{22}
    \end{bmatrix}
  \end{aligned}
\end{equation}
\begin{equation}
  \log\det\begin{bmatrix}
    Q^{11}&Q^{12}\\
    Q^{12}&Q^{22}
  \end{bmatrix}
  +\log\det(Q^{11}Q^{22}-Q^{12}Q^{12})
\end{equation}

\subsection{Multi-species spherical model}

We consider models whose configuration space consists of the product of $r$
spheres, each with its own dimension $N_s$, or
$\Omega=S^{N_1-1}\times\cdots\times S^{N_r-1}$. Coordinates on this space we
will typically denote
$\pmb\sigma=(\pmb\sigma^{(1)},\ldots,\pmb\sigma^{(r)})\in\Omega$, with
$\pmb\sigma^{(s)}\in S^{N_s-1}$ denoting the subcomponent restricted to a
specific subsphere. The model can be thought of as consisting of centered
random functions $H:\Omega\to\mathbb R$ with covariance
\begin{equation}
  \overline{
    H(\pmb\sigma_1)H(\pmb\sigma_2)
  }
  =f\left(
      \frac{\pmb\sigma^{(1)}_1\cdot\pmb\sigma^{(1)}_2}{N_1},
      \ldots,
      \frac{\pmb\sigma^{(r)}_1\cdot\pmb\sigma^{(r)}_2}{N_r}
    \right)
\end{equation}
where $f:[-1,1]^r\to\mathbb R$ is an $r$-component function of the overlaps that defines the model.

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