↩︎

The spiked Wigner ensemble is a prototypical model for high-dimensional inference. We study the spectral properties of an inhomogeneous rank-one spiked Wigner model in which the variance of each entry of the noise matrix is itself a random variable. In the high-dimensional limit, we derive exact equations for the spectral edges, the outlier eigenvalue, and the distribution of the components of the outlier eigenvector. These equations determine the BBP transition line that separates the gapped phase, where the signal is detectable, from the gapless phase. In the gapped regime, the distribution of the outlier eigenvector provides a natural estimator of the spike. We solve the equations for a noise matrix whose variances are generated from a truncated power-law distribution. In this case, the BBP transition line is non-monotonic, showing that an inhomogeneous noise can enhance signal detectability.

1 Introduction↩︎

Spiked matrix ensembles are paradigmatic models in high-dimensional statistics [1]. First introduced in the context of probabilistic principal component analysis [2], they provide fundamental insights into a broad class of inference problems, including graph clustering [3], synchronization on compact groups [4], and community detection [3]. In signal processing, spiked matrix ensembles model an observation matrix through a signal-plus-noise decomposition, providing an ideal setting for studying the inference of a rank-one signal corrupted by additive noise. The central question is then to determine under which conditions the signal can be detected and recovered in the high-dimensional limit [1].

Random matrix theory plays a central role in this class of problems [5]–[9], since the observation matrix takes the form of a low-rank perturbation of a large random matrix. A prototypical model in this context is the spiked Wigner ensemble [1], [5], [10]–[12], in which the entries of the noise matrix are independent and identically distributed random variables. As the signal strength increases, this model undergoes a well-known spectral transition [5], [6]: above a critical threshold, a single eigenvalue detaches from the bulk of the noise spectrum [13], and the eigenvector associated with this outlier eigenvalue becomes correlated with the spike vector, thereby providing a spectral estimator of the signal [14]. This threshold phenomenon is commonly referred to as the Baik-Ben Arous-Péché (BBP) transition [7].

The performance of spectral inference methods depends strongly on the assumptions made on the observation matrix. In particular, the effect of structural properties of the rank-one signal has been widely studied. Sparsity, for instance, has been incorporated into the signal to account for sample incompleteness [15]–[17], while correlations between signal components have been introduced through generative models [18]. From the viewpoint of random matrix theory, however, modifying the structure of the rank-one perturbation mainly affects the outlier eigenvalue and its associated eigenvector, while the bulk of the spectrum remains controlled by the noise matrix [6]. It is therefore natural to ask how structural features of the noise matrix shape the spectral properties of the ensemble. In the case of sparse noise matrices, for example, structural fluctuations may generate extremely large eigenvalues [19], [20], producing a tail in the spectrum that can suppress the gapped regime associated with the BBP transition and limit the applicability of spectral inference methods. Furthermore, applications in bioinformatics [21] and recommendation systems [22] suggest that the noise matrix may also contain a nonrandom component arising from structural features not captured by the signal alone. These observations highlight the importance of incorporating structure directly into the noise matrix.

In fact, recent works [9], [23]–[25] have introduced structured versions of the spiked Wigner model in which the standard assumptions of independence and identical distribution are relaxed. One line of work [9], [23] focuses on relaxing independence by considering rotationally invariant noise matrices with arbitrary potentials, thus introducing correlations among the matrix entries. Another direction, developed in [24], relaxes the assumption of identical distribution by considering inhomogeneous noise matrices, including a specific model with variances arranged in a block structure. This setting is closely related to multi-species spin-glass models [26], [27] and inference models with spatial coupling [28], [29]. By mapping the block-structured ensemble to an effective homogeneous form, the authors have determined the signal-to-noise threshold for the BBP transition in terms of the variance profile. A distinct class of structured models is considered in [25], where the noise matrix is sparse and its entries encode the edges of a random graph.

Motivated by these recent developments, we study an inhomogeneous spiked Wigner model in which the entries of the noise matrix are not identically distributed. In contrast to reference [24], where the variance profile is deterministic, we consider an ensemble in which the variances of the matrix elements are themselves random variables. This extension allows us to investigate how disorder in the variance profile modifies the spectral properties of the model and, in particular, how it affects the existence of a gapped phase and the detectability of the signal by spectral methods. Beyond its interest as a structured spiked Wigner ensemble, this random-matrix model naturally arises as the high-connectivity limit of the adjacency matrix of random graphs with arbitrary degree distributions [30]–[32], providing an analytically tractable setting in which the heterogeneous structure of random graphs is encoded directly in the noise matrix.

We derive exact equations for the spectral properties of the observation matrix and use them to determine when spectral methods can detect and estimate the signal in the high-dimensional limit. Specifically, we obtain equations for the spectral edges of the bulk and for the outlier eigenvalue, whose solutions yield the BBP critical line as a function of the variance profile, extending the class of spiked matrix ensembles for which sharp detectability thresholds can be established. When an outlier eigenvalue is present, the associated eigenvector provides a natural estimator of the spike vector. While the location of outlier eigenvalues has been extensively studied in this context [5], [6], [10]–[12], much less is known about the eigenvectors associated with extreme eigenvalues. Previous works have mainly focused on the overlap between the outlier eigenvector and the signal in spiked Wigner [14] and Wishart [33]–[35] ensembles. Here, we derive an analytic expression for the full distribution of the components of the outlier eigenvector as a function of the variance profile and the spike prior, which yields a simple prescription to estimate the signal and compute the associated mean-squared error. We illustrate our theory in a model where the individual variances follow a truncated power-law distribution. Interestingly, the BBP critical line of this model exhibits a non-monotonic behaviour as a function of the model parameters, showing that variance fluctuations can enhance detectability. Our theoretical findings are in excellent agreement with numerical diagonalizations of finite-dimensional observation matrices.

The remainder of the paper is organized as follows. In section 2, we define the inhomogeneous spiked Wigner model, review the main spectral properties of the homogeneous case, and discuss the relation between our model and previous works. In section 3, we derive the main equations for the spectral properties of the observation matrix for arbitrary spike priors and variance profiles. Section 4 presents the results obtained by solving these equations for a truncated power-law distribution of the variables defining the variance profile. Finally, section 5 summarizes the main results and presents our conclusions, while the appendix explains the derivation of the distributional equations that form the core of our analytic results.

2 The inhomogeneous spiked Wigner model↩︎

We address the problem of estimating a signal vector from observations of a matrix with an inhomogeneous noise structure. The data is modeled by a real symmetric random matrix \(\boldsymbol{A}_n \in\mathbb{R}^{n\times n}\) of the form \[\label{model} \boldsymbol{A}_n=\frac{1}{\sqrt{n}}\left(\boldsymbol{S}_{n}^{\odot 1/2} \odot \boldsymbol{W}_n \right) + \frac{\gamma}{n}\boldsymbol{X}_n\boldsymbol{X}_n^T,\tag{1}\] where \(\odot\) is the Hadamard product and \((\dots)^T\) denotes the transpose. The signal \(\boldsymbol{X}_n \in\mathbb{R}^n\) is formed by i.i.d. components drawn from a distribution \(P_{X}(x)\) with mean \(\langle X \rangle > 0\) and second moment \(\langle X^2 \rangle\). The parameter \(\gamma \geq 0\) is the signal-to-noise ratio that quantifies the signal strength. The noise component in Eq. (1 ) is defined in terms of two symmetric matrices: \(\boldsymbol{W}_n \in\mathbb{R}^{n\times n}\) and \(\boldsymbol{S}_n \in\mathbb{R}^{n\times n}\). The entries of \(\boldsymbol{W}_n\) are i.i.d. random variables sampled from a distribution with zero mean and unitary variance. The elements of \(\boldsymbol{S}_n\) are given by \(S_{ij}=S_iS_j\), where \(\{ S_i \}_{i=1}^n\) are i.i.d. variables drawn from a probability density \(P_S(s)\) with a finite support \(\Omega_S\) on the non-negative real line. Since \(S_i S_j/n\) is the variance of the \(ij\)-th entry of the noise matrix, we refer to \(P_S(s)\) as the variance profile.

The matrix \(\boldsymbol{A}_n\) models the observations of a rank-one signal \(\frac{\gamma}{n}\boldsymbol{X}_n \boldsymbol{X}_{n}^T\) corrupted by an additive noise matrix \(\frac{1}{\sqrt{n}}(\boldsymbol{S}_{n}^{\odot 1/2} \odot \boldsymbol{W}_n)\). The scaling with the matrix dimension \(n\) in Eq. (1 ) ensures that the eigenvalues of \(\boldsymbol{A}_n\) remain of \(\mathcal{O}(n^0)\) as \(n \rightarrow \infty\). The fluctuations of \(\{ S_i \}_{i=1}^n\) control the inhomogeneity of the noise. In the limit \(n \rightarrow \infty\), the spectrum of \(\boldsymbol{A}_n\) is generally dominated by the continuous part of the spectrum, which does not contain meaningful information about the rank-one signal matrix. The questions we will address in this work concern the role played by the variance profile \(P_S(s)\) or the inhomogeneous structure of the noise on the detection and recovery of \(\boldsymbol{X}_n\) [36]:

• Detection: can the spectrum of the observation matrix \(\boldsymbol{A}_n\) be distinguished from the spectrum of the pure noise component? How does the variance profile \(P_S(s)\) influence the detection of the signal?

• Recovery: once there is a clear distinction between signal and noise in the spectrum of \(\boldsymbol{A}_n\), can we compute an eigenvector estimator for the spike vector? How does the variance profile \(P_S(s)\) affect the quality of the signal recovery?

We will derive results for the spectral properties of \(\boldsymbol{A}_n\) in light of the above two aspects of the problem.

2.1 Spectral properties of interest↩︎

Let \(\{ \lambda_i(\boldsymbol{A}_n) \}_{i=1}^n\) be the eigenvalues of the observation matrix \(\boldsymbol{A}_n\). The primary quantity of interest is the empirical spectral density \[\label{esd-def} \rho(\lambda)=\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n \delta\left[ \lambda-\lambda_i(\boldsymbol{A}_n) \right].\tag{2}\] The variance profile \(P_S(s)\) has a bounded support, implying that \(\rho(\lambda)\) is nonzero on a finite interval \(\Omega_{\lambda}= (-\lambda_{*},\lambda_{*})\). The quantity \(\lambda_* > 0\) defines the spectral edge of \(\rho(\lambda)\). In the limit \(n \rightarrow \infty\), the spectra of rank-one perturbations of large random matrices typically contain a continuous part, forming the so-called bulk of eigenvalues, and eventually a single outlier eigenvalue \(\lambda_{\rm out}\) [5], [6], [37], separated from the bulk by a gap \(\lambda_{\rm out} - \lambda_{*} >0\). When the largest eigenvalue of the observation matrix \(\boldsymbol{A}_n\) is equal to \(\lambda_{\rm out}\), detection of the vector signal \(\boldsymbol{X}_n\) is possible, and \(\boldsymbol{X}_n\) can be estimated from the components of the outlier eigenvector [14]. An eigenvector \(\boldsymbol{R}_n(\lambda) \in \mathbb{R}^n\) associated to an eigenvalue \(\lambda(\boldsymbol{A}_n)\) satisfies \(\boldsymbol{A}_n \boldsymbol{R}_n(\lambda) = \lambda(\boldsymbol{A}_n) \boldsymbol{R}_n(\lambda)\), while the empirical distribution of the eigenvector components \(\{ R_{n,i}(\lambda) \}_{i=1}^n\) is defined as \[P_R(r|\lambda) = \lim_{n \rightarrow \infty} \frac{1}{n} \sum_{i=1}^n \delta\left[ r - R_{n,i}(\lambda) \right].\] Our goal is to derive equations for the spectral density \(\rho(\lambda)\), the spectral edge \(\lambda_*\), the outlier \(\lambda_{\rm out}\), and the eigenvector distributions \(P_R(r|\lambda_{\rm out})\) and \(P_R(r|\pm \lambda_{*})\), for arbitrary \(P_X\) and \(P_S\), which will allow us to fully characterize the detection and recovery of \(\boldsymbol{X}_n\) through spectral methods.

2.2 Relation to previous work↩︎

Equation (1 ) generalizes the homogeneous spiked Wigner model [5] to the case in which the entries of the noise matrix are not identically distributed. The homogeneous model is recovered by setting \(P_S(s)=\delta(s-\sigma)\), with \(\sigma > 0\). In this case, the elements of the noise matrix in Eq. (1 ) have constant variance \(\sigma^2/n\), and the spectral density follows the Wigner semicircle law [13] \[\label{wigner} \rho_{\rm sc}(\lambda)= \frac{1}{2\pi \sigma^2} \sqrt{4\sigma^2-\lambda^2},\tag{3}\] supported on the interval \(\Omega_{\lambda} = (-2\sigma,2\sigma)\).

The homogeneous spiked Wigner model undergoes a BBP transition at the critical line [38] \[\label{snr-wigner} \gamma = \sigma/\langle X^2\rangle,\tag{4}\] where \(\langle X^2\rangle\) is the second moment of \(P_X\). When \(\gamma > \sigma/\langle X^2\rangle\), the signal is strong enough for the spectrum of \(\boldsymbol{A}_n\) to be distinguishable from that of the noise matrix. In this regime, the spectrum contains an outlier [38] \[\label{out-wigner} \lambda_{{\rm out}} = \gamma \langle X^2\rangle+\frac{\sigma^2}{\gamma\langle X^2\rangle}\tag{5}\] separated from the bulk by a gap \(\lambda_{\rm out}-\lambda_* > 0\), with \(\lambda_{*} = 2 \sigma\). In this gapped phase, signal detection is possible, since the outlier eigenpair carries information about \(\boldsymbol{X}_n\). More specifically, the eigenvector \(\boldsymbol{R}_n(\lambda_{\rm out})\) associated to \(\lambda_{\rm out}\) is positively correlated with \(\boldsymbol{X}_n\). When \(\gamma \leq \sigma/\langle X^2\rangle\), there is no outlier and the spectrum of \(\boldsymbol{A}_n\) is gapless. In this regime, detection of \(\boldsymbol{X}_n\) by spectral methods is not possible, since the correlation between \(\boldsymbol{X}_n\) and the eigenvector at the spectral edge vanishes. A schematic illustration of the spectrum of the homogeneous spiked Wigner model is shown in figure 1.

Recently, random-matrix models with structured noise have attracted considerable attention [9], [23], [24]. In particular, reference [24] considers, for instance, a noise matrix whose variances \(\{ S_{ij} \}_{i,j=1}^N\) are organized in a finite number of blocks, so that entries within the same block are identically distributed. By applying a suitable rescaling of Eq. (1 ), the authors map the structured noise ensemble back to a homogeneous form, which allows them to determine the critical signal-to-noise ratio of the BBP transition in terms of the variance profile. While Eq. (1 ) is closely related to the model of [24], we consider a different class of variance matrices \(\boldsymbol{S}_n\). Rather than assuming a deterministic variance profile, we set \(S_{ij}=S_iS_j\), where \(\{ S_i \}_{i=1}^N\) are i.i.d. random variables drawn from \(P_S\). The distribution \(P_S\) shapes the structure of the noise matrix and thus controls the inhomogeneity of the model. As we show below, this factorized form of \(S_{ij}\) allows us to make significant progress in characterizing the spectral properties of Eq. (1 ) in the limit \(n \to \infty\). In particular, we derive exact equations for the spectral density of the observation matrix, the spectral edges, the BBP transition line, the outlier eigenvalue, and the distribution of the corresponding eigenvector, for arbitrary distributions \(P_X\) and \(P_S\). These equations provide a detailed understanding of how inhomogeneous noise affects spectral inference methods.

3 Analytic results for the spectral properties↩︎

In this section, we present the main analytic results for the spectral properties of \(\boldsymbol{A}_n\) that are relevant to characterize the detection and recovery of the rank-one signal. First, we provide equations that determine the continuous spectral density and its support. We also derive an equation for the average outlier eigenvalue and the full distribution of its associated eigenvector components, which yield a natural estimate of the rank-one signal in the gapped phase. Finally, we obtain an analytic expression for the mean squared error between the outlier eigenvector and the rank-one signal. This quantity characterizes the quality of the eigenvector estimator. All results in this section hold for arbitrary distributions \(P_X\) and \(P_S\).

The spectral properties of random matrices are encoded in the \(n \times n\) resolvent matrix \[\boldsymbol{G}_n(z)=(z\boldsymbol{I}_n-\boldsymbol{A}_n)^{-1}, \label{res}\tag{6}\] where \(\boldsymbol{I}_n\) is the \(n \times n\) identity matrix and \(z=\lambda-i\epsilon\), with \(\epsilon > 0\) and \(\lambda \in \mathbb{R}\). The spectral density in Eq. (2 ) can be written in terms of the diagonal elements of \(\boldsymbol{G}_n(z)\) as [39] \[\label{ESD-sum} \rho(\lambda) = \lim_{\epsilon \to 0^+}\lim_{n\to\infty} \frac{1}{n\pi} \sum_{i=1}^n {\rm Im} \, G_{n,ii}(\lambda-i\epsilon).\tag{7}\] In the limit \(\epsilon \rightarrow 0^{+}\), the diagonal components \(\{ G_{n,ii}(z) \}_{i=1}^n\) have non-zero imaginary parts inside the bulk of eigenvalues (\(\lambda \in \Omega_{\lambda}\)), being purely real outside the bulk (\(\lambda \notin \Omega_{\lambda}\)). In addition, the resolvent elements satisfy \(\lim_{|z|\to\infty} G_{n,ii}(z) = 0\). As shown in a series of works [40]–[43], the diagonal entries of \(\boldsymbol{G}_n(z)\) also give access to the statistics of the eigenvector components outside the bulk. In fact, by ordering the eigenvalues of \(\boldsymbol{A}_n\) as \(\lambda_1(\boldsymbol{A}_n) \geq \lambda_2(\boldsymbol{A}_n) \geq \dots \geq \lambda_n(\boldsymbol{A}_n)\), the \(i\)-th component of the leading eigenvector \(\boldsymbol{R}_{n}(\lambda_1)\) follows from the identity \[\label{r-def} \left(\sum_{k=1}^n R_{n,k}(\lambda_1) \right) R_{n,i}(\lambda_1) = -i\lim_{\epsilon \to 0^+} \epsilon\left[ \boldsymbol{G}_n(\lambda_1 - i\epsilon) \boldsymbol{1}_n \right]_i,\tag{8}\] where \(\boldsymbol{1}_n=(1,\cdots,1)^T\). Explicit derivations of Eq. (8 ) appear in [40], [41]. The above identity enables to compute the eigenvector distribution \(P_R(r|\lambda)\) for \(\lambda=\lambda_{*}\) and \(\lambda = \lambda_{\rm out}\) [43]. Thus, we have recast the problem of obtaining the spectral properties of \(\boldsymbol{A}_n\) into the computation of the resolvent matrix \(\boldsymbol{G}_n(z)\).

3.1 Inside the bulk: the empirical spectral density↩︎

The first step is to determine the elements \(\{ G_{n,ii}(z) \}_{i=1}^{n}\), which yield the spectral density through Eq. (2 ). By using the cavity method for dense random matrices [44], we show in appendix 6 that, in the limit \(n \rightarrow \infty\), the diagonal elements of the resolvent satisfy the distributional equation \[G_{ii}(z) \stackrel{\rm d}{=} \frac{1}{z - q(z) S_i}, \label{juhrt}\tag{9}\] with \(q(z) = \left\langle S \mkern1mu G(z) \right\rangle\). Here and elsewhere, the brackets \(\langle F(Y_{1},\dots, Y_{M} ) \rangle\) represent the ensemble average of an arbitrary function \(F(Y_{1},\dots, Y_{M} )\) of \(M\) random variables \(Y_{1},\dots, Y_{M}\) with their corresponding joint distribution. Although \(G_{ii}\) and \(S_i\) are not independent, it is straightforward to obtain from Eq. (9 ) a self-consistent equation for \(q(z)\), \[\label{q-final} q(z) = \int_{\Omega_S} \frac{ds \,s P_S(s)}{z-q(z)s}.\tag{10}\] The solutions of Eqs. (9 ) and (10 ) give the joint distribution of the real and imaginary parts of the diagonal elements of the resolvent in the complex plane [32]. From Eqs. (7 ) and (9 ), we get the following expression for the spectral density \[\label{ESD-final} \rho(\lambda)=\frac{1}{\pi}\lim_{\epsilon \to 0^+} {\rm Im} \left[ \int_{\Omega_S} \frac{ds \, P_S(s)}{z-q(z) s} \right].\tag{11}\] Equations (10 ) and (11 ) have been originally derived in references [30], [32] by taking the high-connectivity limit of the spectral density of random graphs with arbitrary degree distributions.

We finish this part by showing how the analytic expressions recover those for the homogeneous spiked Wigner ensemble. Setting \(P_S(s) = \delta(s - \sigma)\) in Eq. (10 ), we obtain the quadratic equation \(\sigma q^2(z) - z q(z) + \sigma =0\), whose solutions read \[q_{\pm}(z) = \frac{1}{2 \sigma} \left( z \pm \sqrt{ z^2 - 4 \sigma^2 } \right). \label{ytrsd}\tag{12}\] For \(|\lambda| < 2 \sigma\), the root \(q_{+}(z)\) leads to the Wigner semicircle law for \(\rho(\lambda)\) with support \(\Omega_{\lambda} = (-2 \sigma,2 \sigma)\).

3.2 Outside the bulk: spectral edge, outlier and the eigenvector distributions↩︎

The calculation of the spectral edge \(\lambda_*\) of \(\rho(\lambda)\) is crucial for the inference problem under study. From Eq. (10 ), we can derive an equation for \({\rm Im} q(z)\), \[{\rm Im} q(z)\left( \int_{\Omega_S} \frac{ds \, s^2 P_S(s)}{[\lambda- s \mkern1mu {\rm Re} q(z)]^2+[s \mkern1mu {\rm Im} q (z)]^2}-1 \right)=0. \label{tewds}\tag{13}\] For \(\lambda\) outside the support \(\Omega_{\lambda} = (-\lambda_{*}, \lambda_{*})\) of \(\rho(\lambda)\), \({\rm Im} G_{ii}(z)\) converges to zero as \(\epsilon \rightarrow 0^{+}\), implying that \(\lim_{\epsilon \rightarrow 0^{+}} {\rm Im}q(z)=0\). Thus, the above equation is trivially satisfied for \(|\lambda| \geq |\lambda_{*}|\). On the other hand, \(\lim_{\epsilon \rightarrow 0^{+}} {\rm Im}q(z)\) is nonzero for \(\lambda \in \Omega_{\lambda}\). As \(|\lambda|\) approaches \(|\lambda_{*}|\) from below, \(\lim_{\epsilon \rightarrow 0^{+}} {\rm Im}q(z)\) becomes very small, and Eq. (13 ) is satisfied provided \[\label{edge} \lim_{\epsilon \to 0^+}\int_{\Omega_S} \frac{ds \,s^2 P_S(s)}{\left[\pm \lambda_* - s \mkern1mu {\rm Re} q (\pm \lambda_*-i\epsilon ) \right]^2} = 1.\tag{14}\] By solving Eq. (10 ) and computing \(\lim_{\epsilon \to 0^+} {\rm Re} \mkern1mu q (z)\) as a function of \(\lambda\), the above condition determines the spectral edges \(\pm \lambda_*\). Note that the spectral density and its support are independent of the statistical properties of the spike \(\boldsymbol{X}_n\), which means that a rank-one perturbation of \(\mathcal{O}(n^{-1})\) does not affect the bulk spectrum of \(\boldsymbol{A}_n\).

Let us now determine the outlier eigenvalue \(\lambda_{\rm out}\) and the distribution \(P_R(r|\lambda)\) of the eigenvector components for \(\lambda \notin \Omega_{\lambda}\). Building on the cavity approach as discussed in [40], we derive in appendix 6 the following distributional equation for the eigenvector components \(R_i(\lambda)\) in the limit \(n \rightarrow \infty\), \[\label{evecs} R_i(\lambda) \stackrel{\rm d}{=} \left[ \sqrt{\left\langle S R^2(\lambda) \right\rangle} \sqrt{S}_i U_i + \gamma \left\langle X R (\lambda) \right\rangle X_i \right] G_{ii}(\lambda),\tag{15}\] where \(U_i\) is a Gaussian random variable with mean zero and variance one, while \(G_{ii}(\lambda)\) is computed from Eq. (9 ). This equation is strictly valid outside the bulk. The solutions of Eq. (15 ) allow us to study the statistics of the eigenvector components in terms of the model parameters, forming an interesting starting point to compute the moments of \(R_i(\lambda)\) and the outlier eigenvalue [40], [41], [43]. From Eq. (15 ), the first and second moments of \(R_i(\lambda)\) fulfill the equations \[\begin{align} \tag{16} \langle R (\lambda) \rangle & = & \gamma \langle X R(\lambda) \rangle \langle X\rangle \langle G(\lambda) \rangle, \\ \tag{17} \langle R^{2} (\lambda) \rangle & = & \langle S R^{2}(\lambda) \rangle \langle S G^{2}(\lambda) \rangle + \gamma^{2} \langle X R(\lambda) \rangle^{2} \langle X^2\rangle\langle G^{2}(\lambda) \rangle, \end{align}\] which depend on the correlations \(\langle X R(\lambda) \rangle\) and \(\langle S R^{2}(\lambda) \rangle\). Thus, we derive from Eq. (15 ) the additional expressions \[\begin{align} \langle X R(\lambda) \rangle &=& \gamma \langle X^2 \rangle \langle G(\lambda) \rangle \langle X R(\lambda) \rangle, \tag{18} \\ \langle X R(\lambda) \rangle^2 &=& \frac{\left[ 1-\langle S^2 G^{2}(\lambda) \rangle \right]}{\gamma^2 \langle X^2 \rangle \langle S G^{2}(\lambda) \rangle} \langle S R^{2}(\lambda) \rangle. \tag{19} \end{align}\] The solutions of the fixed-point Eqs. (16 19 ) yield the outlier \(\lambda_{\rm out}\), the spectral edge \(\lambda_{*}\), and the first two moments of the corresponding eigenvector components.

First, let us consider the trivial solution \(\langle X R(\lambda)\rangle = 0\) of Eq. (18 ), which represents the absence of correlation between the eigenvector and the spike. By setting \(\langle X R(\lambda) \rangle = 0\) in Eq. (19 ), we find that the right hand side of this equation is zero for values of \(\lambda\) such that \(\langle S^2 G^{2}(\lambda)\rangle = 1\). This condition coincides with Eq. (14 ), which determines the spectral edges. Hence, the solution \(\langle X R(\lambda) \rangle = 0\) appears strictly at \(\lambda = \pm \lambda_*\), and the first two moments of the eigenvector components at the spectral edges are given by \(\langle R(\pm \lambda_*) \rangle =0\) and \(\langle R^{2}(\pm \lambda_*) \rangle = \langle S R^{2}(\pm \lambda_*) \rangle\langle S G^2(\pm \lambda_*) \rangle\).

Next, we consider the case in which the eigenvector components have a nonzero correlation \(\langle X R(\lambda) \rangle\) with the spike. The linear Eq. (18 ) admits a nontrivial solution, \(\langle X R(\lambda) \rangle \neq 0\), when \(\lambda=\lambda_{\rm out} > 0\) satisfies \[\label{outlier} \lim_{\epsilon \to 0^+}\int_{\Omega_S} \frac{ds P_S(s)}{\left[ \lambda_{\rm out} - s {\rm Re} q(\lambda_{\rm out} - i \epsilon ) \right]} = \frac{1}{ \gamma\langle X^2\rangle }.\tag{20}\] If the solution of this equation detaches from the bulk (\(\lambda_{\rm out} > \lambda_{*}\)), the spectrum of \(\boldsymbol{A}_n\) exhibits an outlier at \(\lambda=\lambda_{\rm out}\). In this regime, the signal \(\boldsymbol{X}_n\) is detectable, since the spectrum of \(\boldsymbol{A}_n\) can be distinguished from the spectrum of the noise matrix in Eq. (1 ). The transition between the gapped phase, where the spectral gap \(\lambda_{\rm out} - \lambda_{*}\) is finite, and the gapless phase, where the spectrum of \(\boldsymbol{A}_n\) has no outlier, is known as the BBP transition [7]. Here, we can compute the BBP transition line by identifying the set of parameters for which \(\lambda_{\rm out} = \lambda_{*}\) solves simultaneously Eqs. (14 ) and (20 ). From Eqs. (16 ) and (17 ), we derive the first two moments of the eigenvector components associated to the outlier \[\begin{align} \frac{\langle R (\lambda_{\rm out}) \rangle}{\langle X R(\lambda_{\rm out}) \rangle } & = & \gamma \langle X\rangle \langle G(\lambda_{\rm out}) \rangle, \tag{21} \\ \frac{\langle R^{2}(\lambda_{\rm out})\rangle }{ \langle X R(\lambda_{\rm out}) \rangle^2 } &=& \frac{\gamma^2 \langle X^2\rangle \langle S \mkern1mu G^{2}(\lambda_{\rm out}) \rangle}{\left[ 1-\langle S^2 G^{2}(\lambda_{\rm out}) \rangle \right]} \langle S \mkern1mu G^{2}(\lambda_{\rm out}) \rangle + \gamma^2 \langle X^2\rangle \langle G^{2} (\lambda_{\rm out})\rangle . \tag{22} \end{align}\] Since the eigenvectors are defined up to an overall normalization, one can fix either \(\langle R (\lambda_{\rm out}) \rangle\) or \(\langle R^{2}(\lambda_{\rm out})\rangle\), which then fully determines the first and second moments through Eqs. (21 ) and (22 ).

The components of the outlier eigenvector provide a natural estimator for \(\boldsymbol{X}_n\) in the gapped phase. From the distributional Eq. (15 ), one can derive an explicit expression for the distribution of the eigenvector components corresponding to eigenvalues outside the bulk, \[P_R(r | \lambda) = \int_{\mathbb{R}} d x P_X(x) \int_{\Omega_{S}} d s P_S(s) \frac{1}{\sqrt{2 \pi \sigma_{R}^{2}(s|\lambda) }} \exp{\left( - \frac{1}{2 \sigma_{R}^{2}(s|\lambda) } \left[ r - \mu(s,x | \lambda) \right]^2 \right) }, \label{ured}\tag{23}\] where \[\mu(s,x|\lambda) = \frac{\gamma \langle X R(\lambda) \rangle x }{\lambda - s \langle S G(\lambda) \rangle }\] and \[\sigma_{R}^{2}(s|\lambda) = \frac{\langle S R^{2}(\lambda) \rangle s }{\left[ \lambda - s \langle S G(\lambda) \rangle \right]^2},\] with \(\lambda = \pm \lambda_{*}\) or \(\lambda=\lambda_{\rm out}\). For \(\lambda \notin \Omega_{\lambda}\), the distribution of the eigenvector components is thus a weighted superposition of Gaussian distributions with mean and variance governed by the spike prior \(P_X\) and the variance profile \(P_S\). Note that \(P_R(r|\lambda)\) is fully specified once the correlation \(\langle X R(\lambda) \rangle\) is fixed, reflecting the freedom in the normalization of the eigenvectors. Equation (23 ) provides a natural prescription for estimating the rank-one signal in the gapped phase (\(\lambda = \lambda_{\rm out}\)). We can access the quality of this estimate by computing the mean squared error \[\label{mse} {\rm MSE}(\boldsymbol{X}|\boldsymbol{R}) = \lim_{n \rightarrow \infty} \frac{1}{n} \sum_{i=1}^n \left[ X_{i,n} - R_{i,n}(\lambda_{\rm out}) \right]^{2} = \langle X^2 \rangle + \langle R^{2}(\lambda_{\rm out}) \rangle - 2 \langle X R(\lambda_{\rm out}) \rangle.\tag{24}\] Since \(\langle X R(\pm \lambda_*) \rangle = 0\) and \(\mu(s,x|\pm \lambda_*) = 0\), \(P_R(r | \pm \lambda_*)\) is independent of the spike distribution \(P_X\) at the spectral edges. Consequently, the eigenvectors at \(\lambda = \pm \lambda_*\) contain no information about the spike \(\boldsymbol{X}_n\).

We now recover well-known results for the homogeneous spiked Wigner model from our general equations. Setting \(P_S(s) = \delta( s - \sigma)\) and choosing the root \({\rm Re} q_{+}(\lambda) = \lambda/2 \sigma\) in Eq. (12 ), the solution of Eq. (14 ) gives the spectral edge \(\lambda_{*}=2 \sigma\) of the Wigner semicircle law. To obtain the outlier from Eq. (20 ), we instead select the root \({\rm Re} q_{-}(\lambda) = \frac{1}{2 \sigma} (\lambda - \sqrt{\lambda^2 - 4 \sigma^2} )\) in Eq. (12 ), which satisfies \(\lim_{\lambda \rightarrow \infty} {\rm Re} q_{-}(\lambda) = 0\) and therefore provides the physically relevant solution for \(q(\lambda)\) outside the bulk. Substituting \(P_S(s) = \delta( s - \sigma)\) and \({\rm Re} q_{-}(\lambda)\) into Eq. (20 ), we recover the outlier eigenvalue of the homogeneous spiked Wigner model, Eq. (5 ). From the condition \(\lambda_* = \lambda_{\rm out}\), we find the quadratic equation \(\gamma^2 \langle X^2 \rangle^2 - 2 \sigma \gamma \langle X^2 \rangle + \sigma^2 = 0\) for \(\gamma \langle X^2 \rangle\), whose solution yields the BBP transition line \(\gamma \langle X^2 \rangle = \sigma\), Eq. (4 ). The distribution of the eigenvector components of the homogeneous Wigner model follows by setting \(P_S(s) = \delta(s - \sigma)\) in Eq. (23 ). For \(\lambda = \pm \lambda_*\), the eigenvector components follow the Gaussian distribution \[P_R(r | \pm \lambda_*) = \frac{1}{\sqrt{2 \pi \langle R^{2}(\pm \lambda_*) \rangle }} \exp{\left[ - \frac{r^2}{2 \langle R^{2}(\pm \lambda_*) \rangle } \right]}, \label{ured1}\tag{25}\] with \(\lambda_{*}=2 \sigma\). For \(\lambda = \lambda_{\rm out}\) (see Eq. (5 )), we obtain \[P_R(r | \lambda_{\rm out}) = \int_{\mathbb{R}} d x P_X(x) \frac{1}{\sqrt{2 \pi \sigma_{R}^{2}(\lambda_{\rm out}) }} \exp{\left( - \frac{1}{2 \sigma_{R}^{2}(\lambda_{\rm out}) } \left[ r - \mu(x|\lambda_{\rm out}) \right]^2 \right) }, \label{ured2}\tag{26}\] where \[\mu(x|\lambda_{\rm out}) = \frac{\langle X R(\lambda_{\rm out}) \rangle x }{\langle X^2\rangle} \quad {\rm and} \quad \sigma_{R}^{2}(\lambda_{\rm out}) = \frac{\sigma^2 \langle R^{2}(\lambda_{\rm out}) \rangle }{(\gamma\langle X^2\rangle)^2}.\]

Finally, we determine the empirical distribution \(P_G(g|\lambda)\) of the resolvent entries \(G_{ii}(\lambda)\) for \(|\lambda| \geq |\lambda_*|\). Although this object is not directly relevant for the inference problem considered here, \(P_G(g|\lambda)\) plays an important role in random-matrix theory and the study of localization phenomena [32], [39], [45], [46], as it determines the local density of states. For \(|\lambda| \geq |\lambda_*|\), the distribution \(P_G(g|\lambda)\) is supported on the real line \(g \in \mathbb{R}\). From the nonlinear relation between \(G_{ii}\) and \(S_i\), given by Eq. (9 ), one can perform a change of variables and obtain, for arbitrary \(P_S\), the following expression \[\label{out-res-dist} P_{G}(g|\lambda) = \frac{1}{|{\rm Re} \mkern1mu q(\lambda)| g^2} P_S\left[ \frac{\lambda}{q(\lambda)} - \frac{1}{q(\lambda)g}\right] ,\tag{27}\] where the support of \(P_G(g|\lambda)\) is given by \(\Omega_G=\left[\frac{1}{\lambda-q(\lambda)s_{-}},\frac{1}{\lambda-q(\lambda) s_{+}}\right]\), with \(\Omega_S \in [s_{-},s_{+}]\).

4 Numerical results↩︎

In the previous section, we have shown how to determine the spectral properties of the observation matrix \(\boldsymbol{A}_n\) in the limit \(n\to\infty\). The equations presented in section 3 are valid for an arbitrary spike distribution \(P_X\) and variance profile \(P_S\) with finite support \(\Omega_S\). In this section, we examine the effects of noise inhomogeneity on the detection and recovery of the spike for specific choices of \(P_X\) and \(P_S\).

For the variance profile, we consider the following truncated power-law distribution \[\label{ps} P_S(s) = \left\{ \begin{array}{ll} \mathcal{A}^{-1} s^{\alpha}, & {\rm if} \,\, s \in \left[1-\frac{\Delta}{2},1+\frac{\Delta}{2}\right], \\ 0, & {\rm otherwise}, \end{array} \right.\tag{28}\] where \(\mathcal{A}\) is the normalization constant \[\mathcal{A} = \frac{1}{\alpha+1} \left[\left(1+\frac{\Delta}{2}\right)^{\alpha+1}-\left(1-\frac{\Delta}{2}\right)^{\alpha+1}\right].\] The parameter \(0 < \Delta \leq 2\) controls the width of the support \(\Omega_S=\left[1-\frac{\Delta}{2},1+\frac{\Delta}{2}\right]\), while the exponent \(-1 < \alpha < \infty\) governs the shape of \(P_S\). The mode of \(P_S(s)\) is located at \(s=1+\Delta/2\) or \(s=1-\Delta/2\) provided \(\alpha > 0\) or \(\alpha < 0\), respectively. For \(\alpha=0\), Eq. (28 ) reduces to the uniform distribution. The first and second moments of \(P_S\) are given by \[\label{moms-s} \langle S^{m} \rangle = \frac{\alpha+1}{\alpha+m+1} \frac{\left[\left(1+\frac{\Delta}{2}\right)^{\alpha+m+1}-\left(1-\frac{\Delta}{2}\right)^{\alpha+m+1} \right]}{\left[\left(1+\frac{\Delta}{2}\right)^{\alpha+1}-\left(1-\frac{\Delta}{2}\right)^{\alpha+1} \right]},\tag{29}\] with \(m=1,2\). The above expression is useful to understand the behaviour of \(P_S\) in certain limiting cases:

• As \(\alpha\to\infty\), both \(\langle S \rangle^2,\langle S^2\rangle \to \left(1+\frac{\Delta}{2}\right)^2\), thus \(P_S(s)\to\delta[s-(1+\frac{\Delta}{2})]\), and we recover the homogeneous spiked Wigner ensemble with standard deviation \(\sigma=1+\frac{\Delta}{2}\).

• As \(\alpha\to-1\) and \(\Delta \to 2\), both \(\langle S \rangle^2,\langle S^2\rangle \to 0\), and hence \(P_S(s)\to\delta(s)\). In this case, the noise component in Eq. (1 ) is zero.

For \(\Delta =0\), we recover the homogeneous spiked Wigner ensemble with \(\sigma=1\), since \(P_S(s) = \delta(s-1)\). Regarding the spike distribution \(P_X\), below we present results for a constant spike, where \(P_X(x) = \delta(x-1)\), and for spike components drawn from a Gaussian distribution.

4.1 Empirical spectral density↩︎

The spectral density \(\rho(\lambda)\) of \(\boldsymbol{A}_n\) is obtained from the solutions of Eqs. (10 ) and (11 ). Figure 2 illustrates the behaviour of \(\rho(\lambda)\) for the truncated power-law distribution \(P_S(s)\) defined in Eq. (28 ). As \(\alpha \to \infty\), \(P_{S}(s)\) develops a peak at the upper edge of its support, and \(\rho(\lambda)\) converges to the Wigner semicircle law \(\rho_{\rm sc}(\lambda)\) with \(\sigma=1+\frac{\Delta}{2}\) (see Eq. (3 )). In contrast, as \(\alpha\to-1\), the weight \(P_{S}(s)\) near the lower edge of the support \(\Omega_S\) increases, leading to an overall reduction in the variance of the noise contribution in Eq. (1 ). Accordingly, the support of \(\rho(\lambda)\) shrinks as \(\alpha\to-1\), vanishing in the particular case of \(\Delta=2\) (figure 2\((c)\)). Figure 2 also shows that results from numerical diagonalizations of finite-dimensional matrices \(\boldsymbol{A}_n\) are in excellent agreement with the solutions of Eqs. (10 ) and (11 ), thus confirming our theoretical predictions for \(n \rightarrow \infty\).

Figure 2\((c)\) shows that \(\rho(\lambda)\) may exhibit a divergence at \(\lambda=0\) when \(\Delta=2\). This singular behaviour originates from strong fluctuations in the imaginary parts of the diagonal elements of the resolvent [32], [45], [46]. Defining \(Y_{n,i}= {\rm Im} G_{n,ii}(0) \geq 0\), one can analytically determine, in the limit \(n \rightarrow \infty\), the empirical distribution \(P_Y(y)\) of \(\{ Y_{n,i} \}_{i=1}^n\). For \(P_S(s)\) given by Eq. (28 ), one finds that \({\rm Im} q(0) = 1\), and the distribution \(P_Y(y)\) reads \[\label{ldos} P_Y(y) = \left\{ \begin{array}{ll} \mathcal{A}^{-1} y^{-(\alpha+2)} , & {\rm if} \,\, y \in \Omega_Y, \\ 0, & {\rm otherwise}, \end{array} \right.\tag{30}\] where \(\Omega_Y = \left[\left(1+\frac{\Delta}{2}\right)^{-1},\left(1-\frac{\Delta}{2}\right)^{-1} \right]\) denotes the support of \(P_Y(y)\). According to Eq. (7 ), the spectral density \(\rho(\lambda)\) at \(\lambda=0\) is proportional to the first moment \(\langle Y \rangle = \int_{\Omega_{Y}} dy \, y \, P_Y(y)\). For \(\Delta=2\), this gives \[\langle Y \rangle = \frac{1}{\mathcal{A}}\int_{1/2}^{\infty} dy\, y^{-(\alpha+1)}.\] The above expression diverges for \(\alpha \leq 0\), in full agreement with the results in figure 2\((c)\). This singularity therefore stems from strong fluctuations in the diagonal elements of the resolvent, which are induced by arbitrarily small variances of the individual entries in the noise component of Eq. (1 ). As shown in previous studies [32], [45], this heterogeneity-induced divergence within the bulk of the spectrum is associated with localization of the corresponding eigenvectors on sites with small values of \(S_i\).

4.2 Spectral edge and outlier eigenvalue↩︎

The spectral edge \(\lambda_{*} > 0\) follows from the solutions of Eqs. (10 ) and (14 ). In figure 3\((a)\), we present results for different combinations of the parameters \(\alpha\) and \(\Delta\) that characterize the truncated power-law distribution \(P_S\) (see Eq. (28 )). As \(\Delta \to 0\), fluctuations in the variance matrix \(\boldsymbol{S}_n\) are suppressed and \(\lambda\) converges to the spectral edge \(\lambda_{*} = 2\) of the homogeneous spiked Wigner model. For \(\alpha \geq 0\), \(\lambda_{*}\) increases monotonically with \(\Delta\). In the limit \(\alpha \to \infty\), we recover the spectral edge \(\lambda_{*} = 2 (1 + \Delta/2)\) of the homogeneous spiked Wigner model with \(\sigma = 1 + \Delta/2\), since \(P_S(s) = \delta[ s - (1 + \Delta/2 ) ]\). For moderate negative values of \(\alpha\), when the mode of \(P_S(s)\) is located at the lower edge \(1 - \Delta/2\), \(\lambda_{*}\) becomes a non-monotonic function of \(\Delta\). Setting \(\Delta = 2\) and taking the limit \(\alpha \to -1\) yields \(P_S(s) = \delta(s)\), which results in \(\lambda_{*} \to 0\).

The spectral gap \(\lambda_{\rm out}-\lambda_{*}\) follows from the solutions of Eqs. (14 ) and (20 ). In figure 3\((b)\), we compare the solutions of these equations with numerical results from the diagonalization of finite-dimensional matrices \(\boldsymbol{A}_n\) for a deterministic spike prior \(P_X(x)=\delta(x-1)\), confirming our theoretical predictions in the limit \(n \to \infty\). When \(\lambda_{\rm out} - \lambda_{*} > 0\), the spectrum of \(\boldsymbol{A}_n\) is gapped and the signal \(\boldsymbol{X}_n\) is detectable through the outlier eigenpair. When \(\lambda_{\rm out} - \lambda_{*} \leq 0\), the spectrum is gapless and the spike vector cannot be detected by spectral methods. The vanishing of the spectral gap identifies the BBP transition line [7]. Figure 3\((b)\) shows that, for \(\alpha > 0\) and moderate values of the signal strength \(\gamma\), the spectral gap vanishes monotonically as the width \(\Delta\) of the support \(\Omega_S\) increases.

4.3 Eigenvectors outside the bulk↩︎

The distribution \(P_R(r|\lambda)\) of the eigenvector components outside the bulk follows directly from Eq. (23 ). In figure 4, we illustrate the behavior of \(P_R(r|\lambda)\) for a deterministic spike prior \(P_X(x)=\delta(x-1)\) and a truncated power-law distribution \(P_S(s)\) with maximum width \(\Delta=2\) and different shape parameters \(\alpha\). For each pair \((\Delta,\alpha)\), the spectrum of \(\boldsymbol{A}_n\) undergoes a BBP transition at a critical signal-to-noise ratio \(\gamma=\gamma_c\). When the signal is sufficiently strong (\(\gamma > \gamma_c\)), the spectrum contains an outlier \(\lambda_{\rm out}\), and figure 4\((a)\) shows the corresponding eigenvector distribution \(P_R(r|\lambda_{\rm out})\) for the choice \(\langle R \rangle = \langle X \rangle\). For \(\gamma \leq \gamma_c\), no outlier is present, and Eq. (23 ) yields the eigenvector distribution \(P_R(r|\pm \lambda_{*})\) at the spectral edges, shown in figure 4\((b)\) for the choice \(\langle R^2\rangle=1\). The eigenvector distributions are non-Gaussian, in contrast to the Gaussian behavior of the homogeneous spiked Wigner ensemble (see Eqs. (25 ) and (26 )). Figure 4 further shows that, in both phases (gapped and gapless), the distribution \(P_R(r|\lambda)\) concentrates around \(\langle R\rangle\) as \(\alpha\) decreases. This occurs because the fluctuations of \(R_i(\lambda)\) around the spike component \(X_i\) are controlled by \(\sqrt{S_i}\), which become smaller as \(\alpha \to -1\). Finally, numerical results for the distribution of the eigenvector components, obtained from the diagonalization of \(\boldsymbol{A}_n\) for finite \(n\), show excellent agreement with Eq. (23 ) in both regimes displayed in figure 4.

Figure 4\((a)\) shows that the eigenvector components associated with \(\lambda_{\rm out}\) are moderately spread around the mean \(\langle X \rangle = 1\) of the spike components. Thus, the outlier eigenvector \(\boldsymbol{R}_n(\lambda_{\rm out})\) provides an estimate of the rank-one signal \(\boldsymbol{X}_n\) in the gapped regime. The quality of this estimate can be measured by the mean squared error \({\rm MSE}(\boldsymbol{X}|\boldsymbol{R})\) between \(\boldsymbol{X}_n\) and \(\boldsymbol{R}_n(\lambda_{\rm out})\), given by Eq. (24 ) in the limit \(n \to \infty\). Figure 5 shows the behavior of the \({\rm MSE}\) as a function of \(\gamma\) for both deterministic and random spikes, with \(X_{n,i} = 1\) and \(X_{n,i} \sim \mathcal{N}(1,1)\), respectively. The \({\rm MSE}\) vanishes as \(\gamma \to \infty\), whereas it diverges as the BBP threshold \(\gamma = \gamma_c\) is approached. This divergence confirms that the eigenvector at the spectral edge does not carry information about the spike.

4.4 The BBP transition↩︎

The critical signal-to-noise ratio \(\gamma_c\) that identifies the BBP transition is obtained by solving simultaneously Eqs. (14 ) and (20 ). Figures 6\((a)\) and 6\((b)\) show \(\gamma_c(\alpha,\Delta)\) as a function of, respectively, \(\Delta\) and \(\alpha\), the parameters that characterize the variance profile \(P_S(s)\) defined in Eq. (28 ). For \(\gamma > \gamma_c(\alpha,\Delta)\), the spectrum of the observation matrix \(\boldsymbol{A}_n\) contains an outlier separated from the bulk by a finite gap, and the signal can be estimated from the associated eigenvector. For \(\gamma \leq \gamma_c(\alpha,\Delta)\), the spectrum of \(\boldsymbol{A}_n\) is gapless and the signal is not detectable by spectral methods.

As shown in figure 6, introducing randomness in the individual variances of the noise matrix makes signal detection more difficult for \(\alpha \geq 0\), since \(\gamma_c\) grows monotonically with \(\Delta\). In contrast, for \(\alpha < 0\), when the mode of \(P_S(s)\) lies at the lower edge \(1-\Delta/2\), \(\gamma_c\) displays a pronounced non-monotonic dependence on \(\Delta\). In particular, when \(\alpha \simeq -1\), \(\gamma_c\) decreases rapidly as \(\Delta \rightarrow 2\), showing that fluctuations in the variances of the noise matrix can also enhance signal detection. Figure 6 also shows that \(\gamma_c \rightarrow 1 + \Delta/2\) in the limit \(\alpha \rightarrow \infty\), recovering the BBP threshold of the homogeneous spiked Wigner model with \(\sigma = 1 + \Delta/2\) (see Eq. (4 )).

Lastly, we discuss the BBP critical line in the \((\Delta,\alpha)\) plane. In figure 7, we present results for the critical value \(\alpha_c(\Delta,\gamma)\) below which the spectrum of \(\boldsymbol{A}_n\) is gapped and the signal is detectable. Figure 7\((b)\) shows that \(\alpha_c(\Delta,\gamma)\) displays a non-monotonic dependence on \(\Delta\) for negative \(\alpha\). In particular, for low values of the signal strength \(\gamma\), a moderate increase of the width parameter \(\Delta\) completely suppresses the gapped phase. However, upon further increasing \(\Delta\), the gapped phase re-emerges near \(\Delta = 2\), due to the overall reduction of the noise caused by the many near-zero variances of the noise matrix entries in Eq. (1 ). For positive \(\alpha\), where the mode of \(P_S\) lies at the right edge of its support, \(\alpha_c(\Delta,\gamma)\) decreases monotonically with \(\Delta\).

5 Summary and conclusions↩︎

The spiked Wigner ensemble [5] is one of the simplest models for studying the inference of a low-rank signal corrupted by additive noise. In its standard homogeneous form, the entries of the noise matrix are independent and identically distributed random variables. In this work, we have investigated how inhomogeneities in the noise matrix shape the spectral properties of the spiked Wigner model. To this end, we have considered an analytically tractable extension in which the noise component is given by the Hadamard product of a Wigner matrix and a variance matrix with factorized entries of the form \(S_i S_j\), where the variables \(\{ S_i \}_{i=1}^n\) are drawn from an arbitrary distribution that defines the variance profile. This construction provides a simple way to introduce nontrivial noise inhomogeneity. It also arises naturally as the high-connectivity limit of sparse random matrices encoding the structure of random graphs [30], [32]. In the high-dimensional limit, we have derived exact equations for the spectral properties of the observation matrix relevant to inference: the empirical spectral density and its support, the outlier eigenvalue, the BBP transition line, and the full distribution of the eigenvector components associated with the outlier.

When the spectrum of the observation matrix contains an outlier eigenvalue detached from the bulk, the corresponding eigenvector can be used to estimate the rank-one signal. One of our main results is an analytic expression for the full distribution of the outlier eigenvector components. This distribution is a weighted superposition of Gaussian distributions, with mean and variance determined by both the signal prior and the variance profile. The eigenvector distribution thus provides a direct way to estimate the signal in the gapped regime, without requiring more sophisticated algorithms. At the BBP transition, the outlier merges with the spectral bulk and the estimator based on the leading eigenvector fails, since the eigenvectors at the spectral edges carry no information about the spike. We showed this explicitly by deriving the full distribution of the eigenvector components at the spectral edges, which have zero mean and are independent of the spike prior. These analytic results for the eigenvector distributions are also relevant from the viewpoint of random matrix theory, since existing works characterize the leading eigenvector of dense spiked random matrices mainly through its projection onto the spike vector [14], [47].

In the present model, both the outlier and the support of the continuous spectrum are determined by the variance profile of the noise matrix. We solved our general equations for the case in which the variables defining this profile are sampled from a truncated power-law distribution, controlled by a width parameter and a shape parameter. The resulting phase diagrams reveal a rich non-monotonic behaviour of the BBP transition line as a function of the noise-inhomogeneity parameters. In particular, this model exhibits a regime in which the critical signal-to-noise ratio decreases as the support of the variance profile broadens, showing that noise inhomogeneity can, in some cases, favour signal detection.

From a methodological perspective, the present work combines the cavity method for dense random matrices [44] with an analytic approach to the statistics of the eigenvector components outside the bulk, originally developed for sparse non-Hermitian random matrices [40], [41], [43]. We have shown that this approach can also be applied to symmetric matrices, with the simplification that no further regularization of the resolvent is required in this case [48]. The general character of the present method allows one to study leading eigenvector statistics and spectral inference in other symmetric spiked random-matrix models, including inhomogeneous versions of the spiked Wishart ensemble [7] and models with sparse noise [25]. The present framework can also be extended to study spectral inference in spiked non-Hermitian ensembles. Overall, our results show that the structure of the noise matrix plays an important role in high-dimensional inference, sometimes suppressing signal detection, and in other cases enhancing it.

Acknowledgements↩︎

5.0.0.1 Funding information

LSF acknowledges a fellowship from CNPq. FLM acknowledges financial support from CNPq (Grants No 402487/2023-0 and No 310255/2025-2) and from ICTP through the Associates Program (2023-2028).

References↩︎