Mutual information for symmetric rank-one matrix estimation: A proof of the replica formula

Setting and main results

A standard and natural setting is the case of additive white Gaussian noise (AWGN) of known variance $\Delta$,

2 Main result

Our central result is a proof of the expression for the asymptotic $n\!\to\!+\infty$ mutual information per variable via the so-called replica symmetric potential function $i_{\rm RS}(E;\Delta)$ defined as

Fix $\Delta\!>\!0$ and assume $P_{0}$ is a discrete distribution such that $i_{\rm RS}(E;\Delta)$ given by (2) has at most three stationary points. Then

The proof of the existence of the limit does not require the above hypothesis on $P_{0}$. Also, it was first shown in [Krzakala et al. (2016)] that for all $n$, $I({\textbf{S}};{\textbf{W}})/n\!\leq\!\min_{E\in[0,v]}i_{\rm RS}(E;\Delta)$, an inequality that we will use in the proof section. It is conceptually useful to define the following threshold:

Define $\Delta_{\rm Opt}$ as the first non-analyticity point of the asymptotic mutual information per variable as $\Delta$ increases, that is formally $\Delta_{\rm Opt}\!\vcentcolon=\!\sup\{\Delta|\lim_{n\to+\infty}I({\textbf{S}};{\textbf{W}})/n\ \text{is analytic in}\ ]0,\Delta[\}$.

It is natural to conjecture that the vector-MMSE is given by $\text{argmin}_{E\in[0,v]}i_{\rm RS}(E;\Delta)$ for all $\Delta\!\neq\!\Delta_{\rm RS}$, but our proof does not quite yield the full statement.

For $\Delta\!>\!0$ small enough, the fixed point equation corresponding to (4) has a unique solution for all noise values in $]0,\Delta[$. We define $\Delta_{\rm AMP}$ as the supremum of all such $\Delta$.

In the limit $n\!\to\!+\infty$, AMP initialized without any knowledge other than $P_{0}$ yields upon convergence the asymptotic matrix-MMSE as well as the asymptotic vector-MMSE iff $\Delta\!<\!\Delta_{\rm AMP}$ or $\Delta\!>\!\Delta_{\rm RS}$, namely $E^{\infty}\!=\!\text{argmin}_{E\in[0,v]}i_{\rm RS}(E;\Delta)$.

$\Delta_{\rm AMP}$ can be read off the replica potential (2): by differentiation of (2) one finds a fixed point equation that corresponds to (4). Thus $\Delta_{\rm AMP}$ is the smallest solution of $\partial i_{\rm RS}/\partial E\!=\!\partial^{2}i_{\rm RS}/\partial E^{2}\!=\!0$; in other words it is the “first” horizontal inflexion point that appears in $i_{\rm RS}(E;\Delta)$ when we increase $\Delta$.

3 Discussion

With our hypothesis on $P_{0}$ there are only three possible scenarios: $\Delta_{\rm AMP}\!<\!\Delta_{\rm RS}$ (one “first order” phase transition); $\Delta_{\rm AMP}\!=\!\Delta_{\rm RS}\!<\!+\infty$ (one “higher order” phase transition); $\Delta_{\rm AMP}\!=\!\Delta_{\rm RS}\!=\!+\infty$ (no phase transition). In the sequel we will have in mind the most interesting case, namely one first order phase transition, where we determine the gap between the algorithmic AMP and information theoretic performance. The cases of no phase transition or higher order phase transition, which present no algorithmic gap, are basically covered by the analysis of [Deshpande and Montanari (2014)] and follow as a special case from our proof. The only cases that would require more work are those where $P_{0}$ is such that (2) develops more than three stationary points and more than one phase transition is present.

We note for further use in the proof section that $E^{\infty}\!=\!E_{\rm good}(\Delta)$ for $\Delta\!<\!\Delta_{\rm AMP}$ and $E^{\infty}\!=\!E_{\rm bad}(\Delta)$ for $\Delta\!>\!\Delta_{\rm AMP}$. Definition 4 is equivalent to $\Delta_{\rm AMP}\!=\!\sup\{\Delta|E^{\infty}\!=\!E_{\rm good}(\Delta)\}$. Moreover we will also use that $i_{\rm RS}(E_{\rm good};\Delta)$ is analytic on $]0,\Delta^{\prime}[$, $i_{\rm RS}(E_{\rm bad};\Delta)$ is analytic on $]\Delta_{\rm AMP},+\infty[$, and the only non-analyticity point of $\min_{E\in[0,v]}i_{\rm RS}(E;\Delta)$ is at $\Delta_{\rm RS}$.

4 Relation to other works

Explicit single-letter characterization of the mutual information in the rank-one problem has attracted a lot of attention recently. Particular cases of (3) have been shown rigorously in a number of situations. A special case when $s_{i}\!=\!\pm 1\!\sim\!{\rm Ber}(1/2)$ already appeared in [Korada and Macris (2009)] where an equivalent spin glass model is analysed. Very recently, [Krzakala et al. (2016)] has generalized the results of [Korada and Macris (2009)] and, notably, obtained a generic matching upper bound. The same formula has been also rigorously computed following the study of AMP in [Deshpande and Montanari (2014)] for spike models (provided, however, that the signal was not too sparse) and in [Deshpande et al. (2015)] for strictly symmetric community detection.

For rank-one symmetric matrix estimation problems, AMP has been introduced by [Rangan and Fletcher (2012)], who also computed the state evolution formula to analyse its performance, generalizing techniques developed by [Bayati and Montanari (2011)] and [Javanmard and Montanari (2013)]. State evolution was further studied by [Deshpande and Montanari (2014)] and [Deshpande et al. (2015)]. In [Lesieur et al. (2015b, a)], the generalization to larger rank was also considered.

The general formula proposed by [Lesieur et al. (2015a)] for the conditional entropy and the MMSE on the basis of the heuristic cavity method from statistical physics was not demonstrated in full generality. Worst, all existing proofs could not reach the more interesting regime where a gap between the algorithmic and information theoretic perfomances appears, leaving a gap with the statistical physics conjectured formula (and rigorous upper bound from [Krzakala et al. (2016)]). Our result closes this conjecture and has interesting non-trivial implications on the computational complexity of these tasks.

Our proof technique combines recent rigorous results in coding theory along the study of capacity-achieving spatially coupled codes [Hassani et al. (2010); Kudekar et al. (2011); Yedla et al. (2014); Barbier et al. (2016)] with other progress, coming from developments in mathematical physics putting on a rigorous basis predictions of spin glass theory [Guerra (2005)]. From this point of view, the theorem proved in this paper is relevant in a broader context going beyond low-rank matrix estimation. Hundreds of papers have been published in statistics, machine learning or information theory using the non-rigorous statistical physics approach. We believe that our result helps setting a rigorous foundation of a broad line of work. While we focus on rank-one symmetric matrix estimation, our proof technique is readily extendable to more generic low-rank symmetric matrix or low-rank symmetric tensor estimation. We also believe that it can be extended to other problems of interest in machine learning and signal processing, such as generalized linear regression, features/dictionary learning, compressed sensing or multi-layer neural networks.

Two examples: Wigner spike model and community detection

In order to illustrate the consequences of our results we shall present two examples. In the first one we are given data distributed according to the spiked Wigner model where the vector s is a Bernoulli random vector, $S_{i}\!\sim\!{\rm Ber}(\rho)$. For large enough densities (i.e. $\rho\!>\!0.041(1)$), [Deshpande and Montanari (2014)] computed the matrix-MMSE and proved that AMP is a computationally efficient algorithm that asymptotically achieves the matrix-MMSE for any value of the noise $\Delta$. Our results allow to close the gap left open by [Deshpande and Montanari (2014)]: on one hand we now obtain rigorously the MMSE for $\rho\!\leq\!0.041(1)$, and on the other one, we observe that for such values of $\rho$, and as $\Delta$ decreases, there is a small region where two local minima coexist in $i_{\rm RS}(E;\Delta)$. In particular for $\Delta_{\rm AMP}\!<\!\Delta\!<\!\Delta_{\rm Opt}=\Delta_{\rm RS}$ the global minimum corresponding to the MMSE differs from the local one that traps AMP, and a computational gap appears (see figure 1). While the region where AMP is Bayes optimal is quite large, the region where is it not, however, is perhaps the most interesting one. While this is by no means evident, statistical physics analogies with physical phase transitions in nature suggest that this region should be hard for a very broad class of algorithms.

For small $\rho$ our results are consistent with the known optimal and algorithmic thresholds predicted in sparse PCA [Amini and Wainwright (2008); Berthet and Rigollet (2013)], that treats the case of sub-extensive \rho\!=\!\mathchoice{{\scriptstyle\mathcal{O}}}{{\scriptstyle\mathcal{O}}}{{\scriptscriptstyle\mathcal{O}}}{\scalebox{0.7}{\scriptscriptstyle\mathcal{O}}}(1) values. Another interesting line of work for such probabilistic models appeared in the context of random matrix theory (see [Baik et al. (2005)] and references therein) and predicts that a sharp phase transition occurs at a critical value of the noise $\Delta_{\rm spectral}\!=\!\rho^{2}$ below which an outlier eigenvalue (and its principal eigenvector) has a positive correlation with the hidden signal. For larger noise values the spectral distribution of the observation is indistinguishable from that of the pure random noise.

We now consider the problem of detecting two communities (groups) with different sizes $\rho n$ and $(1\!-\!\rho)n$, that generalizes the one considered in [Deshpande et al. (2015)]. One is given a graph where the probability to have a link between nodes in the first group is $p\!+\!\mu(1\!-\!\rho)/(\rho\sqrt{n})$, between those in the second group is $p\!+\!\mu\rho/(\sqrt{n}(1\!-\!\rho))$, while interconnections appear with probability $p\!-\!\mu/\sqrt{n}$. With this peculiar “balanced” setting, the nodes in each group have the same degree distribution with mean $pn$, making them harder to distinguish. According to the universality property described in section 1.1, this is equivalent to a model with AWGN of variance $\Delta\!=\!p(1\!-\!p)/\mu^{2}$ where each variable $s_{i}$ is chosen according to $P_{0}(s)\!=\!\rho\delta(s\!-\!\sqrt{(1\!-\!\rho)/\rho})\!+\!(1\!-\!\rho)\delta(s\!+\!\sqrt{\rho/(1\!-\!\rho)}).$ Our results for this problemNote that here since $E\!=\!v\!=\!1$ is an extremum of $i_{\rm RS}(E;\Delta)$, one must introduce a small bias in $P_{0}$ and let it then tend to zero at the end of the proofs. are summarized on the right hand side of figure 2. For $\rho\!>\rho_{c}\!=\!1/2\!-\!\sqrt{1/12}$ (black point), it is asymptotically information theoretically possible to get an estimation better than chance if and only if $\Delta\!<\!1$. When $\rho\!<\!\rho_{c}$, however, it becomes possible for much larger values of the noise. Interestingly, AMP and spectral methods have the same transition and can find a positive correlation with the hidden communities for $\Delta\!<\!1$, regardless of the value of $\rho$. Again, a region $[\Delta_{\rm AMP},\Delta_{\rm Opt}\!=\!\Delta_{\rm RS}]$ exists where a computational gap appears when $\rho\!<\!\rho_{c}$.

One can investigate the very low $\rho$ regime where we find that the information theoretic transition goes as $\Delta_{\rm Opt}(\rho\!\to\!0)\!=\!1/(4\rho|\log{\rho}|)$. Now if we assume that this result stays true even for \rho\!=\!\mathchoice{{\scriptstyle\mathcal{O}}}{{\scriptstyle\mathcal{O}}}{{\scriptscriptstyle\mathcal{O}}}{\scalebox{0.7}{\scriptscriptstyle\mathcal{O}}}(1) (which is a speculation at this point), we can choose $\mu\!\to\!(1\!-\!p)\rho\sqrt{n}$ such that the small group is a clique. Then the problem corresponds to a “balanced” version of the famous planted clique problem [d’Aspremont et al. (2007)]. We find that the AMP/spectral approach finds the hidden clique when it is larger than $\sqrt{np/(1\!-\!p)}$, while the information theoretic transition translates into size of the clique $4p\log(n)/(1\!-\!p)$. This is indeed reminiscent of the more classical planted clique problem at $p\!=\!1/2$ with its gap between $\log(n)$ (information theoretic), $\sqrt{n/e}$ (AMP [Deshpande and Montanari (2015)]) and $\sqrt{n}$ (spectral [d’Aspremont et al. (2007)]). Since in our balanced case the spectral and AMP limits match, this suggests that the small gain of AMP in the standard clique problem is simply due to the information provided by the distribution of local degrees in the two groups (which is absent in our balanced case). We believe this correspondence strengthens the claim that the AMP gap is actually a fundamental one.

Proofs

The crux of our proof rests on an auxiliary “spatially coupled system”. The hallmark of spatially coupled models is that one can tune them so that the gap between the algorithmic and information theoretical limits can be eliminated, while at the same time the mutual information is maintained unchanged for the coupled and original models. Roughly speaking, this means that it is possible to algorithmically compute the information theoretical limit of the original model because a suitable algorithm is optimal on the coupled system.

The spatially coupled construction used here is very similar to the one used for the coupled Curie-Weiss model [Hassani et al. (2010)]. We consider a ring of length $L\!+\!1$ ($L$ even) with blocks positioned at $\mu\!\in\!\{0,\dots,L\}$ and coupled to neighboring blocks $\{\mu\!-\!w,\dots,\mu\!+\!w\}$. The positions $\mu$ are taken modulo $L\!+\!1$ and $w\!\in\!\{0,\ldots,L/2\}$ is an integer equal to the size of the coupling window. The coupled model is

where the index $i_{\mu}\!\in\!\{1,\dots,n\}$ (resp. $j_{\nu}$) belongs to the block $\mu$ (resp. $\nu$) along the ring, $\mathbf{\Lambda}$ is an $(L\!+\!1)\!\times\!(L\!+\!1)$ matrix which describes the strength of the coupling between blocks, and $Z_{i_{\mu}j_{\nu}}\!\sim\!\mathcal{N}(0,1)$ are i.i.d. For the proof to work, the matrix elements have to be chosen appropriately. We assume that: $i)$ $\mathbf{\Lambda}$ is a doubly stochastic matrix; $ii)$ $\Lambda_{\mu\nu}$ depends on $|\mu\!-\!\nu|$; $iii)$ $\Lambda_{\mu\nu}$ is not vanishing for $|\mu\!-\!\nu|\leq w$ and vanishes for $|\mu\!-\!\nu|\!>\!w$; $iv)$ $\mathbf{\Lambda}$ is smooth in the sense $|\Lambda_{\mu\nu}\!-\!\Lambda_{\mu+1\nu}|\!=\!\mathcal{O}(w^{-2})$; $v)$ $\mathbf{\Lambda}$ has a non-negative Fourier transform. All these conditions can easily be met, the simplest example being a triangle of base $2w\!+\!1$ and height $1/(w\!+\!1)$. The construction of the coupled system is completed by introducing a seed in the ring: we assume perfect knowledge of the signal components $\{s_{i_{\mu}}\}$ for $\mu\!\in\!\mathcal{B}\!\vcentcolon=\!\{-w\!-\!1,\dots,w\!-\!1\}\mod L\!+\!1$. This seed is what allows to close the gap between the algorithmic and information theoretical limits and therefore plays a crucial role. Note it can also be viewed as an “opening” of the chain with pinned boundary conditions.

Our first crucial result states that the mutual information $I_{w,L}({\textbf{S}};{\textbf{W}})$ of the coupled and original systems are the same in a suitable asymptotic limit.

For any $w\!\in\!\{0,\ldots,L/2\}$ the following limits exist and are equal: $\lim_{L\to+\infty}\lim_{n\to+\infty}I_{w,L}({\textbf{S}};{\textbf{W}})/(n(L\!+\!1))\!=\!\lim_{n\to+\infty}I({\textbf{S}};{\textbf{W}})/n$.

An immediate corollary is that non-analyticity points (w.r.t $\Delta$) of the mutual informations are the same in the coupled and original models. In particular, defining $\Delta_{{\rm Opt,coup}}\!\vcentcolon=\!\sup\{\Delta\mid\lim_{L\to+\infty}\lim_{n\to+\infty}I_{w,L}({\textbf{S}};{\textbf{W}})/(n(L\!+\!1))\ \text{is analytic in}\ ]0,\Delta[\}$, we have $\Delta_{{\rm Opt,coup}}\!=\!\Delta_{{\rm Opt}}$.

where the mmse function is defined as in section 1.2. From the monotonicity of the mmse function we have $E_{\mu}^{t+1}\!\leq\!E_{\mu}^{t}$ for all $\mu\!\in\!\{0,\ldots,L\}$, a partial order which implies that $\lim_{t\to+\infty}{{\textbf{E}}}^{t}\!=\!{{\textbf{E}}}^{\infty}$ exists. This allows to define an algorithmic threshold: $\Delta_{{\rm AMP},w,L}\!\vcentcolon=\!\sup\{\Delta|E^{\infty}_{\mu}\!\leq\!E_{\rm good}(\Delta)\ \forall\ \mu\}$. We show (equality holds but is not directly needed)

Let $\Delta_{\rm AMP,coup}\!\vcentcolon=\!\liminf_{w\to+\infty}\liminf_{L\to+\infty}\Delta_{{\rm AMP},w,L}$. We have $\Delta_{\rm AMP,coup}\!\geq\!\Delta_{\rm RS}$.

Proof sketch of theorem 1 First we prove (3) for $\Delta\leq\Delta_{\rm Opt}$. It is known [Deshpande and Montanari (2014)] that the matrix-MSE of AMP when $n\!\to\!+\infty$ is equal to $v^{2}\!-\!(v\!-\!E^{t})^{2}$. This cannot improve the matrix-MMSE, hence

For $\Delta\!\leq\!\Delta_{\rm AMP}$ we have $E^{\infty}\!=\!E_{\rm good}(\Delta)$ which is the global minimum of (2) so the left hand side of (7) is equal to the derivative of $\min_{E\in[0,v]}i_{\rm RS}(E;\Delta)$ w.r.t $\Delta^{-1}$. Thus using a matrix version of the well known I-MMSE relation [Guo et al. (2005)] we get

Integrating this relation on $[0,\Delta]\!\subset\![0,\Delta_{\rm AMP}]$ and checking that $\min_{E\in[0,v]}i_{\rm RS}(E;0)\!=\!H(S)$ (the Shannon entropy of $P_{0}$) we obtain $\min_{E\in[0,v]}i_{\rm RS}(E;\Delta)\!\leq\!\liminf_{n\to+\infty}I({\textbf{S}};{\textbf{W}})/n$. But we know $I({\textbf{S}};{\textbf{W}})/n\!\leq\!\min_{E\in[0,v]}i_{\rm RS}(E;\Delta)$ [Krzakala et al. (2016)], thus we already get (3) for $\Delta\!\leq\!\Delta_{\rm AMP}$. We notice that $\Delta_{\rm AMP}\!\leq\!\Delta_{\rm Opt}$. While this might seem intuitively clear, it follows from $\Delta_{\rm RS}\!\geq\!\Delta_{\rm AMP}$ (by their definitions) which together with $\Delta_{\rm AMP}\!>\!\Delta_{\rm Opt}$ would imply from (3) that $\lim_{n\to+\infty}I({\textbf{S}};{\textbf{W}})/n$ is analytic at $\Delta_{\rm Opt}$, a contradiction. The next step is to extend (3) to the range $[\Delta_{\rm AMP},\Delta_{\rm Opt}]$. Suppose for a moment $\Delta_{\rm RS}\!\geq\!\Delta_{\rm Opt}$. Then both functions on each side of (3) are analytic on the whole range $]0,\Delta_{\rm Opt}[$ and since they are equal for $\Delta\!\leq\!\Delta_{\rm AMP}$, they must be equal on their whole analyticity range and by continuity, they must also be equal at $\Delta_{\rm Opt}$ (that the functions are continuous follows from independent arguments on the existence of the $n\!\to\!+\infty$ limit of concave functions). It remains to show that $\Delta_{\rm RS}\!\in\,]\Delta_{\rm AMP},\Delta_{\rm Opt}[$ is impossible. We proceed by contradiction, so suppose this is true. Then both functions on each side of (3) are analytic on $]0,\Delta_{\rm RS}[$ and since they are equal for $]0,\Delta_{\rm AMP}[\subset]0,\Delta_{\rm RS}[$ they must be equal on the whole range $]0,\Delta_{\rm RS}[$ and also at $\Delta_{\rm RS}$ by continuity. For $\Delta\!>\!\Delta_{\rm RS}$ the fixed point of state evolution is $E^{\infty}\!=\!E_{\rm bad}(\Delta)$ which is also the global minimum of $i_{\rm RS}(E;\Delta)$, hence (8) is verified. Integrating this inequality on $]\Delta_{\rm RS},\Delta[\subset]\Delta_{\rm RS},\Delta_{\rm Opt}[$ and using $I({\textbf{S}};{\textbf{W}})/n\!\leq\!\min_{E\in[0,v]}i_{\rm RS}(E;\Delta)$ again, we find that (3) holds for all $\Delta\!\in\![0,\Delta_{\rm Opt}]$. But this implies that $\min_{E\in[0,v]}i_{\rm RS}(E;\Delta)$ is analytic at $\Delta_{\rm RS}$, a contradiction.

We now prove (3) for $\Delta\!\geq\!\Delta_{\rm Opt}$. Note that the previous arguments showed that necessarily $\Delta_{\rm Opt}\!\leq\!\Delta_{\rm RS}$. Thus by lemmas 6 and 7 (and the sub-optimality of AMP as shown as before) we obtain $\Delta_{\rm RS}\leq\Delta_{\rm AMP,coup}\!\leq\!\Delta_{\rm Opt,coup}\!=\!\Delta_{\rm Opt}\!\leq\!\Delta_{\rm RS}$. This shows that $\Delta_{\rm Opt}\!=\!\Delta_{\rm RS}$ (this is the point where spatial coupling came in the game and we do not know of other means to prove such an equality). For $\Delta\!>\!\Delta_{\rm RS}$ we have $E^{\infty}\!=\!E_{\rm bad}(\Delta)$ which is the global minimum of $i_{\rm RS}(E;\Delta)$. Therefore we again have (8) in this range and the proof can be completed by using once more the integration argument, this time over the range $[\Delta_{\rm RS},\Delta]\!=\![\Delta_{\rm Opt},\Delta]$.

where $\langle\!-\!\rangle_{t}$ is the Gibbs average w.r.t the interpolated Hamiltonian, q is the vector of overlaps $q_{\mu}\!\vcentcolon=\!\sum_{i_{\mu}=1}^{n}s_{i_{\mu}}x_{i_{\mu}}/n$. If we can choose matrices such that $\mathbf{\Lambda}^{\prime}\!>\!\mathbf{\Lambda}$, the difference of quadratic forms in the Gibbs bracket is negative and we obtain an inequality in the large size limit. We use this scheme to interpolate between the fully decoupled system $w\!=\!0$ and the coupled one $1\!\leq\!w\!<\!L/2$ and then between $1\!\leq\!w<\!L/2$ and the fully connected system $w\!=\!L/2$. The $w\!=\!0$ system has $\Lambda_{\mu\nu}\!=\!\delta_{\mu\nu}$ with eigenvalues $(1,1,\dots,1)$. For the $1\!\leq\!w\!<\!L/2$ system, we take any stochastic translation invariant matrix with non-negative discrete Fourier transform (of its rows): such matrices have an eigenvalue equal to $1$ and all others in $[0,1[$ (the eigenvalues are precisely equal to the discrete Fourier transform). For $w\!=\!L/2$ we choose $\Lambda_{\mu\nu}\!=\!1/(L\!+\!1)$ which is a projector with eigenvalues $(0,0,\dots,1)$. With these choices we deduce that the free energies and mutual informations are ordered as $I_{w=0,L}+\mathcal{O}(1)\!\leq\!I_{w,L}+\mathcal{O}(1)\!\leq\!I_{w=L/2,L}+\mathcal{O}(1)$. To conclude the proof we divide by $n(L\!+\!1)$ and note that the limits of the leftmost and rightmost mutual informations are equal, provided the limit exists. Indeed the leftmost term equals $L$ times $I({\textbf{S}};{\textbf{W}})$ and the rightmost term is the same mutual information for a system of $n(L\!+\!1)$ variables. Existence of the limit follows by a subadditivity inequality which itself is proven by a similar interpolation [Guerra (2005)].

Proof sketch of lemma 7 Fix $\Delta\!<\!\Delta_{\rm RS}$. We show that, for $w$ large enough, the coupled state evolution recursion (6) must converge to a fixed point $E_{\mu}^{\infty}\!\leq\!E_{\rm good}(\Delta)$ for all $\mu$. The main intuition behind the proof is to use a “potential function” whose “energy” can be lowered by small perturbation of a fixed point that would go above $E_{\rm good}(\Delta)$ [Yedla et al. (2014); Barbier et al. (2016)]. The relevant potential function $i_{w,L}({{\textbf{E}}},\Delta)$ is in fact the replica potential of the coupled system, and equals up to a constant $(2w\!+\!1)Lv^{2}/4\Delta$

Acknowledgments

J.B and M.D acknowledge funding from the Swiss National Science Foundation (grant num. 200021-156672). Part of the research has received funding from the European Research Council under the European Union’s 7th Framework Programme (FP/2007-2013/ERC Grant Agreement 307087-SPARCS). This work was done in part while F.K and L.Z were visiting the Simons Institute for the Theory of Computing.

References