SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Farzad Pourkamali & Nicolas Macris
School of Computer and Communication Sciences,
Ecole Polytechnique Federale de Lausanne
{farzad.pourkamali,nicolas.macris}@epfl.ch
We consider a statistical model for matrix factorization in a regime where the rank of the two hidden matrix factors grows linearly with their dimension and their product is corrupted by additive noise. Despite various approaches, statistical and algorithmic limits of such problems have remained elusive. We study a Bayesian setting with the assumptions that (a) one of the matrix factors is symmetric, (b) both factors as well as the additive noise have rotational invariant priors, (c) the priors are known to the statistician. We derive analytical formulas for Rotation Invariant Estimators to reconstruct the two matrix factors, and conjecture that these are optimal in the large-dimension limit, in the sense that they minimize the average mean-square-error. We provide numerical checks which confirm the optimality conjecture when confronted to Oracle Estimators which are optimal by definition, but involve the ground-truth. Our derivation relies on a combination of tools, namely random matrix theory transforms, spherical integral formulas, and the replica method from statistical mechanics.
Matrix factorization (MF) is the problem of reconstructing two matrices  $X$  and  $Y$  from the noisy observations of their product. Applications in signal processing and machine learning abound, such as for example dimensionality reduction [1, 2], sparse coding [3-5], representation learning [6], robust principal components analysis [7, 8], blind source separation [9], or matrix completion [10, 11].
In this work we approach the problem from a Bayesian perspective and assume that an observation or data matrix  $S = \sqrt{\kappa} X Y + W$  is given to a statistician who knows the prior distributions of  $X$  and  $Y$  as well as the prior of the additive noise matrix  $W$  and the signal-to-noise ratio  $\kappa > 0$ . The task of the statistician is to construct estimators  $\Xi_{X}(\cdot), \Xi_{Y}(\cdot)$  for the matrix factors  $X, Y$ , that ideally, minimize the average mean-square-error (MSE)  $\mathbb{E}\|X - \Xi_{X}(S)\|_{\mathrm{F}}^{2}$  and  $\mathbb{E}\|Y - \Xi_{Y}(S)\|_{\mathrm{F}}^{2}$  ( $\|. \|_{\mathrm{F}}$  the Frobenius norm and  $\mathbb{E}$  the expectation w.r.t  $X, Y, W$ ). We consider priors which are rotation invariant for all three matrices  $X, Y, W$  and for  $X$  we furthermore impose that it is square and symmetric. These matrix ensembles are defined precisely in section 2.1, but the reader can keep in mind the examples of Wigner or Wishart matrices for  $X$ , and general Gaussian  $Y$  and  $W$  with i.i.d elements. We look at the asymptotic regime where all matrix dimensions and ranks tend to infinity at the same speed. We remark that the usual "rotation ambiguity" occurring in MF is not present because we impose that at least one of the two matrix factors is symmetric. We also remark that MF is different (and more difficult) than matrix denoising which would consist in constructing an estimator  $\Xi_{XY}(S)$  for the signal as a whole by minimizing  $\mathbb{E}\|X Y - \Xi_{XY}(S)\|_{\mathrm{F}}^{2}$ .
The rotation invariance of the model implies that the estimators minimizing the MSE belong to the class of rotation invariant estimators (RIE). RIEs are matrix estimators which have the same singular vectors (or eigenvectors) as the observation or data matrix. These estimators have been
proposed for matrix denoising problems (see references [12-16] for covariance estimation, [17] for cross-covariance estimation, and [18], [19, 20] for extensions to rectangular matrices). For the present MF model, we derive optimal estimators (minimizing the MSE) that belong to the RIE class and can be computed explicitly in the large dimensional limit from the observation matrix and the knowledge of the priors. We propose:
1. an explicit RIE to estimate  $\mathbf{X}$ , which requires the knowledge of the priors of both  $\mathbf{X}, \mathbf{Y}$  and of the noise  $\mathbf{W}$ . Moreover, under the assumption that  $\mathbf{X}$  is positive-semi-definite, a sub-optimal RIE can be derived which does not require any prior on  $\mathbf{X}$ .
2. an explicit RIE to estimate  $\mathbf{Y}$ , which requires the knowledge of the priors of the noise  $\mathbf{W}$  and  $\mathbf{X}$  only (the prior of  $\mathbf{Y}$  is not required).
3. combined with the singular value decomposition (SVD) of the observation matrix, our explicit RIEs provide a spectral algorithm to reconstruct both factors  $\mathbf{X}$  and  $\mathbf{Y}$ .
The derivation of the proposed estimators relies on the replica method from statistical mechanics combined with techniques from random matrix theory and finite-rank spherical integrals [21, 22]. Although the replica method is not rigorous and involves concentration assumptions, the derivation is entirely analytical and suggests that the estimators are optimal in the limit of large dimensions. This is corroborated by numerical calculations comparing our explicit RIEs with Oracle Estimators which are optimal by definition and involve the ground-truth matrices.
1.1 Related literature and discussion
When the matrices  $X$  and  $Y$  are assumed to have low-rank compared to their dimension, the mathematical theory of MF has enjoyed much progress under various settings (Bayesian, spectral, algorithmic) and fundamental information theoretical and algorithmic limits have been rigorously derived. The behavior of eigenvalues/singular values and eigenvector/singular vectors of finite-rank perturbations of a Gaussian matrix is studied in [23-25] which leads to spectral estimators when the noise matrix is Gaussian distributed. For the case of factorized prior, and Gaussian noise, closed form expressions have been established for the asymptotic Bayes-optimal estimation error [26-30], and iterative algorithms based on approximate message passing has been proposed [31, 32]. The low-rank matrix denoising problem has been addressed in various other settings, such as structured noise matrix [33, 34], mismatched estimation problem [35-38], and estimation in the regime with diverging aspect-ratio of matrices [39].
In extensive-rank regimes, when the rank grows like the matrix dimensions, despite various attempts there is no solid theory of MF. One approach is based on Approximate Message Passing (AMP) methods developed in [40-42]. Despite acceptable performance in practical settings [43], as pointed out in [44] the AMP algorithms developed in these works are (theoretically) sub-optimal. Other approaches rooted in statistical physics have been considered in [44-47] but have not led to explicit reconstructions of matrix factors or algorithms. A practical probabilistic approach to MF problem is based on variational Bayesian approximations [48-50], in which one tries to approximate the posterior distribution with proper distribution. In [51] it is shown that under Gaussian priors, the solution to the MF problem is a reweighted SVD of the observation matrix. We point out here that these estimators can be seen as a RIE and therefore there seems to be a rather close relation between the RIE studied here and the variational Bayesian approach. This also suggests that adapting RIEs to real data is an interesting direction for future research. Finally, let us also mention optimization approaches where one constructs estimators by following a gradient flow (or gradient descent) trajectory of a training loss of the type  $\|S - XY\|_{\mathrm{F}}^2 + \mathrm{reg}$ . term (see [52], [53] for analysis in rotation invariant models). Benchmarking these various other algorithmic approaches against our explicit RIEs (conjectured to be optimal) is outside the scope of this work and is left for future work.
Constraints such as sparsity or non-negativity of the matrix entries which have important applications [54] are not covered by our theory. Despite this drawback, we believe that the proposed estimators are important both for theoretical and practical purposes. Even in non-rotation invariant problems our explicit RIEs may serve as sub-optimal estimators, and as we show in an example they can be used as a "warmed-up" spectral initialization for more efficient algorithms (see for example [31, 55] for related ideas in other contexts). The methodology developed here may open up the way to further analysis in inference and learning problems perhaps also in the context of neural networks where extensive rank weight matrices must be estimated.
1.2 Organization and notations
In section 2, we introduce the precise MF model, general class of RIEs, and the Oracle estimators. In section 3, we present the explicit RIEs (and algorithm) to estimate  $\mathbf{X}$  and  $\mathbf{Y}$ . We provide the numerical examples and calculations in section 4. In section 5, we sketch the derivation of RIE for  $\mathbf{X}$ , while the one for  $\mathbf{Y}$  is similar and deferred to the appendices.
The following notations are used throughout. For a vector  $\gamma \in \mathbb{R}^N$  we denote by  $\Gamma \in \mathbb{R}^{N\times M}$  a matrix constructed as  $\Gamma = \left[ \begin{array}{cc}\Gamma_N & 0_{N\times (M - N)} \end{array} \right]$  with  $\Gamma_N\in \mathbb{R}^{N\times N}$  a diagonal matrix with diagonal  $\gamma$ . The same notations will also be used for the vector  $\sigma$  and the corresponding matrix  $\Sigma$  and . For a sequence of non-symmetric matrices  $A$  of growing size, we denote the limiting empirical singular value distribution (ESD) by  $\mu_A$ , and the limiting empirical eigenvalue distribution of  $AA^{\intercal}$  by  $\rho_A$ . For a sequence of symmetric matrices  $B$  of growing size, we denote the limiting empirical eigenvalue distribution by  $\rho_B$ , and the limiting eigenvalue distribution of  $B^2$  by  $\rho_{B^2}$ .
2 Matrix factorization model and rotation invariant estimators
2.1 Matrix factorization model
Let  $\pmb{X} = \pmb{X}^{\intercal} \in \mathbb{R}^{N \times N}$  a symmetric matrix distributed according to a rotationally invariant prior  $P_{X}(\pmb{X})$ , i.e., for any orthogonal matrix  $\pmb{O} \in \mathbb{R}^{N \times N}$  we have  $P_{X}(\pmb{O}\pmb{X}\pmb{O}^{\intercal}) = P_{X}(\pmb{X})$ . Let also  $\pmb{Y} \in \mathbb{R}^{N \times M}$  be distributed according to a bi-rotationally invariant prior  $P_{Y}(\pmb{Y})$ , i.e. for any orthogonal matrices  $\pmb{U} \in \mathbb{R}^{N \times N}, \pmb{V} \in \mathbb{R}^{M \times M}$  we have  $P_{Y}(\pmb{U}\pmb{Y}\pmb{V}^{\intercal}) = P_{Y}(\pmb{Y})$ . We observe the data matrix  $S \in \mathbb{R}^{N \times M}$ ,
$$
\boldsymbol {S} = \sqrt {\kappa} \boldsymbol {X} \boldsymbol {Y} + \boldsymbol {W} \tag {1}
$$
where  $\mathbf{W} \in \mathbb{R}^{N \times M}$  is also bi-rotationally invariant distributed, and  $\kappa \in \mathbb{R}_+$  is proportional to the signal-to-noise-ratio (SNR). The goal is to recover both factors  $\mathbf{X}$  and  $\mathbf{Y}$  from the data matrix  $\mathbf{S}$ . For definiteness, we consider the regime  $M \geq N$  with aspect ratio  $N / M \to \alpha \in (0,1]$  as  $N \to \infty$ . The case of  $\alpha > 1$  can be analyzed in the same manner and is presented in appendix F. Furthermore, we assume that the entries of  $\mathbf{X}, \mathbf{Y}$  and  $\mathbf{W}$  are of the order  $O(1 / \sqrt{N})$ . This scaling is such that the eigenvalues of  $\mathbf{X}$  and singular values of  $\mathbf{Y}, \mathbf{W}$  and  $\mathbf{S}$  are of the order  $O(1)$  as  $N \to \infty$ .
Assumption 1. The empirical eigenvalue distribution of  $X$  converge weakly to measure  $\rho_{X}$ , and the ESD of  $Y$ ,  $W$  converge weakly to measures  $\mu_{Y}, \mu_{W}$  with bounded support on the real line. Moreover, these measures are known to the statistician. He can deduce (in principle) these measures from the priors on  $X, Y, W$ .
Remark 1. In a general formulation of matrix factorization the hidden matrices have dimensions  $\mathbf{X} \in \mathbb{R}^{N \times H}$ ,  $\mathbf{Y} \in \mathbb{R}^{H \times M}$ , and in the Bayesian framework with bi-rotational invariant priors for both factors, the optimal estimators are trivially the zero matrix. Indeed, from bi-rotational invariance we have  $P_X(-\mathbf{X}) = P_X(\mathbf{X})$ ,  $P_Y(-\mathbf{Y}) = P_Y(\mathbf{Y})$ , which implies that the Bayesian estimate is zero. Here, by imposing that  $\mathbf{X} \in \mathbb{R}^{N \times N}$  is symmetric and  $P_X(\mathbf{O}\mathbf{X}\mathbf{O}^\top) = P_X(\mathbf{X})$ , we can break this symmetry and find non-trivial estimators. This is due to the fact that the map  $\mathbf{X} \to -\mathbf{X}$  cannot be realized as a (real) orthogonal transformation, so  $P_X(-\mathbf{X}) = P_X(\mathbf{X})$  does not hold in general (various examples are given in section 4 and appendices). Of course, if the prior is even, e.g. Wigner ensemble, again the Bayesian posterior estimate is trivially zero for both factors. As we will see our RIEs are consistent with these observations.
2.2 Rotation invariant estimators
To recover matrices  $X, Y$  from  $S$ , we consider two denoising problems. One is recovering  $X$  by treating both  $Y, W$  as "noise" matrices, and the other is estimating  $Y$  by treating  $X, W$  as "noise". As will become clear the procedure is not iterative, and the two denoising problems are solved independently and simultaneously. In the following, for each of these two problems, we introduce two rotation invariant classes of estimators and discuss their optimum Oracle estimators. We then provide an explicit construction and algorithm for RIEs which we conjecture have the optimum performance of Oracle estimators in the large  $N$  limit.
2.2.1 RIE class for  $X$
Consider the SVD of  $\pmb{S} = \pmb{U}_S\pmb{\Gamma}\pmb{V}_S^\intercal$ , where  $\pmb{U}_S \in \mathbb{R}^{N \times N}$ ,  $\pmb{V}_S \in \mathbb{R}^{M \times M}$  are orthogonal, and  $\pmb{\Gamma} \in \mathbb{R}^{N \times M}$  is a diagonal matrix with singular values of  $\pmb{S}$  on its diagonal,  $(\gamma_i)_{1 \leq i \leq N}$ . A rotational invariant estimator for  $\pmb{X}$  is denoted  $\Xi_X(\pmb{S})$ , and is constructed as:
$$
\boldsymbol {\Xi} _ {X} (\boldsymbol {S}) = \boldsymbol {U} _ {S} \operatorname {d i a g} \left(\xi_ {x 1}, \dots , \xi_ {x N}\right) \boldsymbol {U} _ {S} ^ {\intercal} \tag {2}
$$
where  $\xi_{x_1},\ldots ,\xi_{x_N}$  are the eigenvalues of the estimator.
First, we derive an Oracle estimator by minimizing the squared error  $\frac{1}{N}\left\| \boldsymbol{X} - \boldsymbol{\Xi}_X(\boldsymbol{S})\right\|_{\mathrm{F}}^2$  for a given instance, over the RIE class or equivalently over the choice of the eigenvalues  $(\xi_{x_i})_{1\leq i\leq N}$ . Let the eigen-decomposition of  $\boldsymbol{X}$  be  $\boldsymbol{X} = \sum_{i=1}^{N} \lambda_i \boldsymbol{x}_i \boldsymbol{x}_i^\intercal$  with  $\boldsymbol{x}_i \in \mathbb{R}^N$  eigenvectors of  $\boldsymbol{X}$ . The error can be expanded as:
$$
\frac {1}{N} \left\| \boldsymbol {X} - \boldsymbol {\Xi} _ {X} (\boldsymbol {S}) \right\| _ {\mathrm {F}} ^ {2} = \frac {1}{N} \sum_ {i = 1} ^ {N} \lambda_ {i} ^ {2} + \frac {1}{N} \sum_ {i = 1} ^ {N} \xi_ {x i} ^ {2} - \frac {2}{N} \sum_ {i = 1} ^ {N} \xi_ {x i} \sum_ {j = 1} ^ {N} \lambda_ {j} \left(\boldsymbol {u} _ {i} ^ {\mathsf {T}} \boldsymbol {x} _ {j}\right) ^ {2}
$$
where  $\pmb{u}_i$ 's are columns of  $U_S$ . Minimizing over  $\xi_{xi}$ 's, we find the optimum among the RIE class:
$$
\boldsymbol {\Xi} _ {X} ^ {*} (\boldsymbol {S}) = \sum_ {i = 1} ^ {N} \xi_ {x i} ^ {*} \boldsymbol {u} _ {i} \boldsymbol {u} _ {i} ^ {\intercal}, \quad \xi_ {x i} ^ {*} = \sum_ {j = 1} ^ {N} \lambda_ {j} \left(\boldsymbol {u} _ {i} ^ {\intercal} \boldsymbol {x} _ {j}\right) ^ {2} = \boldsymbol {u} _ {i} ^ {\intercal} \boldsymbol {X} \boldsymbol {u} _ {i} \tag {3}
$$
Expression (3) defines the Oracle estimator which requires the knowledge of signal matrix  $\mathbf{X}$ . Surprisingly, in the large  $N$  limit, the optimal eigenvalues  $\left(\xi_{x_i}^*\right)_{1 \leq i \leq N}$  can be computed from the observation matrix and knowledge of the measures  $\rho_X, \mu_Y, \mu_W$ . In the next section, we show that this leads to an explicitly computable (or algorithmic) RIE, which we conjecture to be optimal as  $N \to \infty$ , in the sense that its performance matches the one of the Oracle estimator.
Now we remark that the Oracle estimator is not only optimal within the rotation invariant class but is also Bayesian optimal. From the Bayesian estimation point of view, one wishes to minimize the average mean squared error (MSE)  $\mathrm{MSE}_{\hat{\mathbf{X}}} \equiv \frac{1}{N} \mathbb{E}\| \mathbf{X} - \hat{\mathbf{X}}(\mathbf{S})\|_{\mathrm{F}}^{2}$ , where the expectation is over  $\mathbf{X}, \mathbf{Y}, \mathbf{W}$ , and  $\hat{\mathbf{X}}(\mathbf{S})$  is an estimator of  $\mathbf{X}$ . The MSE is minimized for  $\hat{\mathbf{X}}^{*}(\mathbf{S}) = \mathbb{E}[X|\mathbf{S}]$  which is the posterior mean. Therefore, the posterior mean estimator has the minimum MSE (MMSE) among all possible estimators, in particular  $\mathrm{MSE}_{\hat{\mathbf{X}}^{*}} \leq \mathrm{MSE}_{\Xi_{\mathrm{X}}^{*}}$  for any  $N$ . In appendix A.1, we show that, for rotational invariant priors, the posterior mean estimator is inside the RIE class. Thus, since  $\Xi_{\mathrm{X}}^{*}(\mathbf{S})$  is optimum among the RIE class  $\mathrm{MSE}_{\Xi_{\mathrm{X}}^{*}} \leq \mathrm{MSE}_{\hat{\mathbf{X}}^{*}}$ . Therefore, we conclude that the Oracle estimator (3) is Bayesian optimal in the sense that  $\mathrm{MSE}_{\Xi_{\mathrm{X}}^{*}} = \mathrm{MSE}_{\hat{\mathbf{X}}^{*}} = \mathrm{MMSE}$ .
2.2.2 RIE class for  $Y$
Estimators for  $\mathbf{Y}$  from the rotation invariant class are denoted  $\Xi_Y(S)$ , and are constructed as:
$$
\boldsymbol {\Xi} _ {Y} (\boldsymbol {S}) = \boldsymbol {U} _ {S} \left[ \operatorname {d i a g} \left(\xi_ {y _ {1}}, \dots , \xi_ {y _ {N}}\right) \mid \mathbf {0} _ {N \times (M - N)} \right] \boldsymbol {V} _ {S} ^ {\intercal} \tag {4}
$$
where  $\xi_{y_1},\ldots ,\xi_{y_N}$  are the singular values of the estimator.
Let the SVD of  $\mathbf{Y}$  be  $\mathbf{Y} = \sum_{i=1}^{N} \sigma_i \mathbf{y}_i^{(l)} \mathbf{y}_i^{(r)\top}$  with  $\mathbf{y}_i^{(l)} \in \mathbb{R}^N$ ,  $\mathbf{y}_i^{(r)} \in \mathbb{R}^M$  the left and right singular vectors of  $\mathbf{Y}$ . To derive an Oracle estimator, we proceed as above. Expanding the error, we have:
$$
\frac {1}{N} \left\| \boldsymbol {Y} - \boldsymbol {\Xi} _ {Y} (\boldsymbol {S}) \right\| _ {\mathrm {F}} ^ {2} = \frac {1}{N} \sum_ {i = 1} ^ {N} \sigma_ {i} ^ {2} + \frac {1}{N} \sum_ {i = 1} ^ {N} \xi_ {y _ {i}} ^ {2} - \frac {2}{N} \sum_ {i = 1} ^ {N} \xi_ {y _ {i}} \sum_ {j = 1} ^ {N} \sigma_ {j} \left(\boldsymbol {u} _ {i} ^ {\intercal} \boldsymbol {y} _ {j} ^ {(l)}\right) \left(\boldsymbol {v} _ {i} ^ {\intercal} \boldsymbol {y} _ {j} ^ {(r)}\right)
$$
where  $\pmb{v}_i$ 's are columns of  $\pmb{V}_S$ . Minimizing over  $\xi_{y_i}$ 's, we find the optimum among the RIE class:
$$
\boldsymbol {\Xi} _ {Y} ^ {*} (\boldsymbol {S}) = \sum_ {i = 1} ^ {N} \xi_ {y _ {i}} ^ {*} \boldsymbol {u} _ {i} \boldsymbol {v} _ {i} ^ {\intercal}, \quad \xi_ {y _ {i}} ^ {*} = \sum_ {j = 1} ^ {N} \sigma_ {j} \left(\boldsymbol {u} _ {i} ^ {\intercal} \boldsymbol {y} _ {j} ^ {(l)}\right) \left(\boldsymbol {v} _ {i} ^ {\intercal} \boldsymbol {y} _ {j} ^ {(r)}\right) = \boldsymbol {u} _ {i} ^ {\intercal} \boldsymbol {Y} \boldsymbol {v} _ {i} \tag {5}
$$
Expression (5) defines the Oracle estimator which requires the knowledge of signal matrix  $\mathbf{Y}$ . Like for the case of  $\mathbf{X}$ , in the large  $N$  limit we can derive the optimal singular values  $\left(\xi_{y_i}^*\right)_{1\leq i\leq N}$  in terms
of the singular values of observation matrix and knowledge of the measures  $\rho_{X},\mu_{W}$ . This leads to an explicitly computable (or algorithmic) RIE, which is conjectured to be optimal as  $N\to \infty$ , in the sense that it has the same performance as the Oracle estimator. Note that unlike the estimator for  $X$  we do not need the knowledge of  $\mu_Y$ .
In appendix A.2, we show that for bi-rotationally invariant priors the posterior mean estimator  $\hat{\pmb{Y}}^{*}(\pmb {S}) = \mathbb{E}[\pmb {Y}|\pmb {S}]$  belongs to the RIE class, which (by similar arguments to the case of  $\pmb{X}$ ) implies that the Oracle estimator (5) is Bayesian optimal.
3 Algorithmic RIEs for the matrix factors
In this section, we present our explicit RIEs for  $X$ ,  $Y$  and the corresponding algorithm. We conjecture that their performance matches the one of Oracles estimators in the large  $N$  limit and they are therefore Bayesian optimal in this limit. Let us first give a brief reminder on useful transforms in random matrix theory.
3.1 Preliminaries on transforms in random matrix theory
For a probability density function  $\rho (x)$  on  $\mathbb{R}$ , the Stieltjes (or Cauchy) transform is defined as
$$
\mathcal {G} _ {\rho} (z) = \int_ {\mathbb {R}} \frac {1}{z - x} \rho (x) d x \quad \text {f o r} z \in \mathbb {C} \backslash \operatorname {s u p p} (\rho)
$$
By Plemelj formulae we have for  $x\in \mathbb{R}$
$$
\lim  _ {\epsilon \rightarrow 0 ^ {+}} \mathcal {G} _ {\rho} (x - \mathrm {i} \epsilon) = \pi \mathsf {H} [ \rho ] (x) + \pi \mathrm {i} \rho (x) \tag {6}
$$
with  $\mathsf{H}[\rho](x) = \mathrm{p.v.}\frac{1}{\pi}\int_{\mathbb{R}}\frac{\rho(t)}{x - t} dt$  the Hilbert transform of  $\rho$  (here p.v. stands for "principal value"). Denoting the inverse of  $\mathcal{G}_{\rho}(z)$  by  $\mathcal{G}_{\rho}^{-1}(z)$ , the  $R$ -transform of  $\rho$  is defined as [56]:
$$
\mathcal {R} _ {\rho} (z) = \mathcal {G} _ {\rho} ^ {- 1} (z) - \frac {1}{z}
$$
For a probability density function  $\mu$  with support contained in  $[-K, K]$  with  $K > 0$ , we define a generating function of (even) moments  $\mathcal{M}_{\mu}:[0, K^{-2}] \to \mathbb{R}_+$  as  $\mathcal{M}_{\mu}(z) = \int \frac{1}{1 - t^2z}\mu(t)dt - 1$ . For  $\alpha \in (0,1]$ , define  $T^{(\alpha)}(z) = (\alpha z + 1)(z + 1)$ , and  $\mathcal{H}_{\mu}^{(\alpha)}(z) = zT^{(\alpha)}\bigl(\mathcal{M}_{\mu}(z)\bigr)$ . The rectangular  $R$ -transform with aspect ratio  $\alpha$  is defined as [57]:
$$
\mathcal {C} _ {\mu} ^ {(\alpha)} (z) = T ^ {(\alpha) - 1} \left(\frac {z}{\mathcal {H} _ {\mu} ^ {(\alpha) - 1} (z)}\right)
$$
3.2 Explicit RIE for  $X$
The RIE for  $X$  is constructed as  $\widehat{\Xi_X^*}(S) = \sum_{i=1}^{N} \widehat{\xi}_{xi}^* \pmb{u}_i \pmb{u}_i^\top$  with eigenvalues  $\left(\widehat{\xi}_{xi}^*\right)_{1 \leq i \leq N}$ :
$$
\hat {\xi} _ {x _ {i}} ^ {*} = \frac {1}{2 \kappa \pi \bar {\mu} _ {S} (\gamma_ {i})} \operatorname {I m} \lim  _ {z \rightarrow \gamma_ {i} - \mathrm {i} 0 ^ {+}} \left\{\frac {1}{\zeta_ {3}} \left[ \mathcal {G} _ {\rho_ {X}} \left(\sqrt {\frac {z - \zeta_ {1}}{\kappa \zeta_ {3}}}\right) + \mathcal {G} _ {\rho_ {X}} \left(- \sqrt {\frac {z - \zeta_ {1}}{\kappa \zeta_ {3}}}\right)\right]\right\} \tag {7}
$$
where  $\gamma_{i}$  is the  $i$ -th singular value of  $S$ ,  $\bar{\mu}_{S}$  is the symmetrized limiting ESD of  $S$ , and
$$
\zeta_ {1} = \frac {1}{\mathcal {G} _ {\tilde {\mu} S} (z)} \mathcal {C} _ {\mu W} ^ {(\alpha)} \left(\mathcal {G} _ {\tilde {\mu} S} (z) \left[ \alpha \mathcal {G} _ {\tilde {\mu} S} (z) + \frac {1 - \alpha}{z} \right]\right) \tag {8}
$$
and  $\zeta_3$  satisfies:
$$
\left. \left(z - \zeta_ {1}\right) \mathcal {G} _ {\bar {\mu} _ {S}} (z) - 1 = \mathcal {C} _ {\mu_ {Y}} ^ {(\alpha)} \left(\frac {1}{\bar {\zeta} _ {3}} \left[ \alpha \mathcal {G} _ {\bar {\mu} _ {S}} (z) + \frac {1 - \alpha}{z} \right] \left[ \left(z - \zeta_ {1}\right) \mathcal {G} _ {\bar {\mu} _ {S}} (z) - 1 \right]\right) \right. \tag {9}
$$
Remark 2. If  $\rho_{X}$  is a symmetric measure,  $\rho_{X}(x) = \rho_{X}(-x)$ , then  $\mathcal{G}_{\rho_X}(-z) = -\mathcal{G}_{\rho_X}(z)$ . This implies that the optimal eigenvalues  $\left(\widehat{\xi}_{xi}^{*}\right)_{1\leq i\leq N}$  in (7) are all zero, and  $\widehat{\Xi}_X^* (\pmb {S}) = \pmb {0}$ , see figure 4.