SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Proof of Corollary 3.2. We have  $M_{\mathbf{I}_{n_\ell}}(z) = 1 / (z - 1)$ , hence  $\kappa_{\ell} = \alpha_{\ell} - 1$ ,  $\mu_{\ell} = 1 - 1 / \alpha_{\ell}$ , and  $\kappa_{\min} = \alpha_0 / \alpha_{\min} - 1$ . Finally, for any fixed teacher vector satisfying  $\| \tilde{\mathbf{w}}_*\|^2 = n_0$ , we have  $\psi(z) = 1 / (z + 1)$  if  $\tilde{\Sigma}_0 = \mathbf{I}_{n_0}$ . Substituting these results into (23), we obtain (25).
This recovers the result obtained in our previous work [13], and in the single-layer case  $L = 1$  recovers results obtained by Rocks and Mehta [14, 15], and by Hastie et al. [5] (see Appendix D). In the slightly more general case of unstructured weights but structured features, we have
Corollary 3.3. If  $\tilde{\mathbf{S}}_{\ell} = \mathbf{I}_{n_{\ell}}$  for  $\ell = 1,\ldots ,L$ , but  $\tilde{\mathbf{S}}_0\neq \mathbf{I}_{n_0}$ , we have, for any target satisfying  $\| \tilde{\mathbf{w}}_*\|^2 = n_0$ ,
$$
\epsilon = \left\{ \begin{array}{l l} \left(\sum_ {\ell = 1} ^ {L} \frac {1}{\alpha_ {\ell} - 1}\right) \kappa_ {0} \psi (\kappa_ {0}) - \frac {\kappa_ {0} ^ {2}}{\mu_ {0}} \psi^ {\prime} (\kappa_ {0}) + \left(\frac {1 - \mu_ {0}}{\mu_ {0}} + \sum_ {\ell = 1} ^ {L} \frac {1}{\alpha_ {\ell} - 1}\right) \eta^ {2}, & \alpha_ {0}, \alpha_ {\min } > 1 \\ \frac {\kappa_ {\min } \psi (\kappa_ {\min })}{1 - \alpha_ {\min }} + \frac {\alpha_ {\min }}{1 - \alpha_ {\min }} \eta^ {2}, & \alpha_ {\min } <   1, \alpha_ {\min } <   \alpha_ {0} \\ \frac {\alpha_ {0}}{1 - \alpha_ {0}} \eta^ {2}, & \alpha_ {0} <   1, \alpha_ {0} <   \alpha_ {\min }. \end{array} \right. \tag {26}
$$
Proof of Corollary 3.3. (26) follows from substituting the results of Corollary 3.2 into (23).
In the special case  $L = 1$ , this recovers the result obtained using rigorous methods in contemporaneous work by Bach [40], posted to the arXiv one day after the first version of our work [43]. Here, as the data spectrum and target vector enter the generalization error in nearly the same way as in the case of linear regression, all of the intuitions developed in that case can be carried over [4-7, 16-18].
Another useful simplification can be obtained by further averaging over isotropically-distributed teachers  $\tilde{\mathbf{w}}_{*} \sim \mathcal{N}(\mathbf{0}, \mathbf{I}_{n_{0}})$ , which gives
Corollary 3.4. Let  $\bar{\epsilon} = \mathbb{E}_{\tilde{\mathbf{W}}_*} \sim \mathcal{N}(\mathbf{0}, \mathbf{I}_{n_0})[\epsilon]$ . Then, we have
$$
\bar {\epsilon} = \left\{ \begin{array}{l l} \left(1 + \sum_ {\ell = 1} ^ {L} \frac {1 - \mu_ {\ell}}{\mu_ {\ell}}\right) \frac {\kappa_ {0}}{\alpha_ {0}} + \left(\sum_ {\ell = 0} ^ {L} \frac {1 - \mu_ {\ell}}{\mu_ {\ell}}\right) \eta^ {2}, & \alpha_ {0}, \alpha_ {\min } > 1 \\ \frac {\alpha_ {\min } \kappa_ {\min } / \alpha_ {0}}{1 - \alpha_ {\min }} + \frac {\alpha_ {\min }}{1 - \alpha_ {\min }} \eta^ {2}, & \alpha_ {\min } <   1, \alpha_ {\min } <   \alpha_ {0} \\ \frac {\alpha_ {0}}{1 - \alpha_ {0}} \eta^ {2}, & \alpha_ {0} <   1, \alpha_ {0} <   \alpha_ {\min }. \end{array} \right. \tag {27}
$$
Proof of Corollary 3.4. Observing that  $\mathbb{E}_{\tilde{\mathbf{w}}_*}\psi (z) = -M_{\tilde{\Sigma}_0}(-z)$ , the claim follows from (23).
In the special case of a single layer of unstructured feature weights  $(L = 1, \hat{\Sigma}_1 = \mathbf{I}_{n_1})$ , this recovers the result of recent work by Maloney et al. [19], who used a planar diagram method to the generalization error of single-hidden-layer linear RFMs with unstructured weights (see Appendix D).
Another important simplifying case of Proposition 3.2 is the limit in which the hidden layer widths are large, in which the generalization error of the deep RFM reduces to that of a shallow model, as given by Corollary 3.1. More precisely, we have a large-width expansion given by:
Corollary 3.5. In the large-width regime  $\alpha_{1},\ldots ,\alpha_{L}\gg 1$ , assuming that the weight spectra have finite moments, the generalization error (23) expands as
$$
\epsilon = - \frac {\kappa_ {0} ^ {2}}{\mu_ {0}} \psi^ {\prime} (\kappa_ {0}) + \frac {1 - \mu_ {0}}{\mu_ {0}} \eta^ {2} + \left(\sum_ {\ell = 1} ^ {L} \frac {\mathbb {E} _ {\tilde {\sigma} _ {\ell}} [ \tilde {\sigma} _ {\ell} ^ {2} ]}{\mathbb {E} _ {\tilde {\sigma} _ {\ell}} [ \tilde {\sigma} _ {\ell} ] ^ {2}} \frac {1}{\alpha_ {\ell}}\right) (\kappa_ {0} \psi (\kappa_ {0}) + \eta^ {2}) + \mathcal {O} \left(\alpha_ {1} ^ {- 2}, \dots , \alpha_ {L} ^ {- 2}\right) \tag {28}
$$
in the regime  $\alpha_0 > 1$ ; if  $\alpha_0 < 1$  the generalization error does not depend on the hidden layer widths so long as they are greater than 1.
Proof of Corollary 3.5. See Appendix E.
4 How does weight structure affect generalization?
The first salient feature of the learning curves given by Proposition 3.2 is that the addition of weight structure does not alter the phase diagram of generalization, which is illustrated in Figure 1. There are three qualitatively distinct phases present, depending on the data density and minimum layer width: the overparameterized regime  $\alpha_0, \alpha_{\mathrm{min}} > 1$ , the bottlenecked regime  $\alpha_{\mathrm{min}} < 1, \alpha_{\mathrm{min}} < \alpha_0$ , and the overdetermined regime  $\alpha_0 < 1, \alpha_0 < \alpha_{\mathrm{min}}$ . This dependence on the narrowest hidden layer matches our previous work on models with unstructured weights [13] $^1$ , and can be observed in the solutions to the ridge regression problem for fixed data (Appendix C). As  $\alpha_{\ell} \downarrow 1, \kappa_{\ell} \downarrow 0$  and  $\mu_{\ell} \downarrow 0$ , and the generalization error diverges. Similarly, the generalization error diverges as  $\alpha_{\mathrm{min}} \uparrow 1$ , or  $\alpha_0 \uparrow 1$  in the presence of label noise. However, there are not multiple descents in these deep linear models, consistent with the qualitative picture of the effect of nonlinearity given by previous works [9, 10].
The second salient feature of Proposition 3.2 is that the matrices  $\tilde{\Sigma}_{\ell}$  enter the generalization error independently; there are no 'interaction' terms involving products of the correlation matrices for different layers. This decoupling is expected given that the features are Gaussian and independent across layers [30]. Moreover, under the rescaling  $\tilde{\Sigma}_{\ell}^{\prime} = \tau_{\ell}\tilde{\Sigma}_{\ell}$  for  $\tau_{\ell} > 0$ , we have  $\kappa_{\ell}^{\prime} = \tau_{\ell}\kappa_{\ell}$  and  $\mu_{\ell}^{\prime} = \mu_{\ell}$  (we show this explicitly in Appendix B). Therefore, (23) is sensitive only to the overall scale of  $\tilde{\Sigma}_{0}$ , not to the scales of  $\tilde{\Sigma}_{1},\ldots ,\tilde{\Sigma}_{L}$ . This scale-invariance can be observed directly from the ridgeless limit of the ridge regression estimator (7).
We can gain intuition for the effect of having  $\tilde{\Sigma}_{\ell} \neq \mathbf{I}_{n_{\ell}}$  for  $\ell \geq 1$  through the following argument:
Lemma 4.1. Under the conditions of Proposition 3.2, in the regime  $\alpha_0, \alpha_{\min} > 1$ , we have
$$
\epsilon \geq \left(\sum_ {\ell = 1} ^ {L} \frac {1}{\alpha_ {\ell} - 1}\right) \kappa_ {0} \psi (\kappa_ {0}) - \frac {\kappa_ {0} ^ {2}}{\mu_ {0}} \psi^ {\prime} (\kappa_ {0}) + \left(\frac {1 - \mu_ {0}}{\mu_ {0}} + \sum_ {\ell = 1} ^ {L} \frac {1}{\alpha_ {\ell} - 1}\right) \eta^ {2}. \tag {29}
$$
That is, the generalization error for a given  $\bar{\Sigma}_1,\dots ,\bar{\Sigma}_L$  is bounded from below by the generalization error for  $\bar{\Sigma}_{\ell} = \mathbf{I}_{n_{\ell}}$  for  $\ell = 1,\ldots ,L$
Proof of Lemma 4.1. In Appendix B, we show that  $\mu_{\ell} \leq 1 - 1 / \alpha_{\ell}$  for any weight spectrum, which implies that  $(1 - \mu_{\ell}) / \mu_{\ell} \geq 1 / (\alpha_{\ell} - 1)$ . Substituting these bounds in to the general expression for the generalization error in this regime from (23), the claim follows.
Therefore, having  $\tilde{\Sigma}_{\ell} \neq \mathbf{I}_{n_{\ell}}$  for  $\ell = 1, \ldots, L$  cannot improve generalization in the  $\alpha_0, \alpha_{\min} > 1$  regime. This is consistent with the large-width expansion in Corollary 3.5, where we can apply Jensen's inequality to bound the weight-dependence of the correction as  $\mathbb{E}_{\tilde{\sigma}_{\ell}}[\tilde{\sigma}_{\ell}^{2}] / \mathbb{E}_{\tilde{\sigma}_{\ell}}[\tilde{\sigma}_{\ell}]^{2} \geq 1$ , with equality only when the weights are unstructured. In other regimes,  $\tilde{\Sigma}_1, \dots, \tilde{\Sigma}_L$  do not affect the generalization error. In contrast, a similar argument shows that anisotropy in  $\tilde{\Sigma}_0$  can be beneficial in the target-averaged case, at least in the absence of label noise. We formalize this as:
Lemma 4.2. Under the conditions of Corollary 3.4, in the absence of label noise  $(\eta = 0)$ , we have
$$
\bar {\epsilon} \leq \left\{ \begin{array}{l l} \left(1 + \sum_ {\ell = 1} ^ {L} \frac {1 - \mu_ {\ell}}{\mu_ {\ell}}\right) \left(1 - \frac {1}{\alpha_ {0}}\right) \mathbb {E} \left[ \tilde {\sigma} _ {0} \right], & \alpha_ {0}, \alpha_ {\min } > 1 \\ \frac {\left(1 - \alpha_ {\min } / \alpha_ {0}\right)}{1 - \alpha_ {\min }} \mathbb {E} \left[ \tilde {\sigma} _ {0} \right], & \alpha_ {\min } <   1, \alpha_ {\min } <   \alpha_ {0} \\ 0, & \alpha_ {0} <   1, \alpha_ {0} <   \alpha_ {\min }. \end{array} \right. \tag {30}
$$
That is,  $\bar{\epsilon}$  for a given  $\bar{\Sigma}_0$  is bounded from above by the generalization error for a flat spectrum  $\hat{\Sigma}_0 = \mathbb{E}[\tilde{\sigma}_0]\mathbf{I}_{n_0}$ .
Previous works on deep RFMs have used several different parameterizations of the thermodynamic limit [3-17, 19-21]. We detail the conversion between these conventions in Appendix D.
Proof of Lemma 4.2. In Appendix B, we show that  $\kappa_0 \leq (\alpha_0 - 1)\mathbb{E}[\tilde{\sigma}_0]$ . As its defining equation (22) is of the same form as (20), the corresponding bound for  $\kappa_{\mathrm{min}}$  follows immediately:  $\kappa_{\mathrm{min}} \leq (\alpha_0 / \alpha_{\mathrm{min}} - 1)\mathbb{E}[\tilde{\sigma}_0]$ . Substituting these bounds into (27) with  $\eta = 0$ , the claim follows.
If  $\mathbb{E}[\tilde{\sigma}_0]$  is not finite, then this bound is entirely vacuous:  $\bar{\epsilon} \leq \infty$ . If we do not average over isotropically-distributed targets, then the effect of anisotropy in  $\tilde{\Sigma}_0$  is harder to analyze. Previous works have, however, analyzed the interaction of data structure with a fixed target in great detail for models with  $L = 0$  or  $L = 1$ , showing that targets that align with the top eigenvectors of  $\tilde{\Sigma}_0$  are easier to learn [5, 16, 17, 42, 44].
5 Power law spectra
We can gain further intuition for the effect of weight structure by considering an approximately solvable model for anisotropic spectra: power laws [16, 19, 28]. Power law data spectra have recently attracted considerable attention as a possible model for explaining the scaling laws of generalization observed in large language models [16, 19, 28, 32]. Maloney et al. [19] proposed a single-hidden-layer  $(L = 1)$  linear RFM with power-law-structured data and unstructured weights as a model for neural scaling laws. Does introducing power law structure into the weights affect the scaling laws predicted by deep linear RFMs? We have the following result:
Corollary 5.1. At finite size, define each covariance matrix  $\tilde{\Sigma}_{\ell}$  such that its  $j$ -th eigenvalue is  $\tilde{\sigma}_{\ell,j} = \tilde{\varsigma}_{\ell}(n_{\ell}/j)^{1+\omega_{\ell}}$  for some fixed scale factor  $\tilde{\varsigma}_{\ell} > 0$  and exponent  $\omega_{\ell} > 0$ . Then, the limiting target-averaged generalization error is approximately
$$
\bar {\epsilon} \simeq \left\{ \begin{array}{l l} \left(1 + \Omega_ {L} + \sum_ {\ell = 1} ^ {L} \frac {1}{\alpha_ {\ell} - 1}\right) \chi (\alpha_ {0}) + \left(\omega_ {0} + \Omega_ {L} + \sum_ {\ell = 0} ^ {L} \frac {1}{\alpha_ {\ell} - 1}\right) \eta^ {2}, & \alpha_ {0}, \alpha_ {\min } > 1 \\ \frac {\chi \left(\alpha_ {0} / \alpha_ {\min }\right)}{1 - \alpha_ {\min }} + \frac {\alpha_ {\min }}{1 - \alpha_ {\min }} \eta^ {2}, & \alpha_ {\min } <   1, \alpha_ {\min } <   \alpha_ {0} \\ \frac {\alpha_ {0}}{1 - \alpha_ {0}} \eta^ {2}, & \alpha_ {0} <   1, \alpha_ {0} <   \alpha_ {\min }, \end{array} \right. \tag {31}
$$
where  $\Omega_L = \sum_{\ell=1}^L \omega_\ell$  and for  $z > 1$  we have  $\chi(z) \simeq -M_{\tilde{\Sigma}_0^{-1}}(z)/z$  given by  $\chi(z) = \tilde{\varsigma}_0 \left\{ k(z^{\omega_0} - 1) + [2 + \omega_0(1 - k)] (1 - 1/z) \right\}$  for  $k = \mathrm{sinc}[\pi/(1 + \omega_0)]^{-(1 + \omega_0)}$ .
Proof of Corollary 5.1. Using the dictionary of notation in Appendix D, we can plug the approximate solutions for  $\kappa_{\ell}$  and  $\mu_{\ell}$  derived by Maloney et al. [19] into (27) to obtain (31).
Therefore, the power law exponents  $\omega_{1},\dots ,\omega_{L}$  of the weight covariances beyond the first layer, which enter only through their sum  $\Omega_L$  , do not affect the scaling laws of the generalization error with the dataset size and network widths. In particular, in the absence of label noise  $(\eta = 0)$  we can approximate the scaling of (31) in the regimes of large or small hidden layer width by
$$
\bar {\epsilon} \sim \left\{ \begin{array}{l l} \alpha_ {0} ^ {\omega_ {0}}, & \alpha_ {\min } > 1, \alpha_ {0} \gg 1, \\ \left(\alpha_ {0} / \alpha_ {\min }\right) ^ {\omega_ {0}}, & \alpha_ {\min } <   1, \alpha_ {0} / \alpha_ {\min } \gg 1, \end{array} \right. \tag {32}
$$
which recovers the results found by Maloney et al. [19] for  $L = 1$  with unstructured weights. This behavior, and the agreement of (31) with numerical experiments, is illustrated in Figure 2. Consistent with Lemma 4.1, generalization with power-law weight structure is never better than with unstructured weights, as can be seen by comparing (31) with (25).
6 Bayesian inference and the Gibbs estimator at large prior variance
Thus far, we have focused on ridge regression (6). Though this is the most commonly-considered estimator in studies of random feature models [3-17, 19-21], one might ask whether our qualitative findings—in particular, that feature weight structure beyond the first layer is generally harmful for generalization—carry over to other estimators. Our approach to Proposition 3.2 is easily extensible to the setting of zero-temperature Bayesian inference, which has recently attracted substantial interest [13, 27, 34, 37, 45-47], sparked by work from Li and Sompolinsky [34]. In this case, we take seriously the Gibbs distribution  $p(\mathbf{v}) \propto e^{-\beta L}$ , which in the ridge regression case was simply a convenient tool, and interpret it as the Bayes posterior for a Gaussian likelihood of variance  $1 / \beta$  and a Gaussian prior with covariance  $\Gamma_{L + 1} / (\beta \lambda)$ . It is in this context conventional to fix  $\lambda = 1 / \beta$ , such that the prior variance does not scale with  $\beta$ . We can then study the average of the generalization error
(a)
Figure 2: Generalization for power-law spectra. (a). Target-averaged generalization error  $\bar{\epsilon}$  as a function of training data density  $1 / \alpha_0$  for shallow models ( $L = 1$ ) of varying hidden layer width  $\alpha_{1} / \alpha_{0}$  in the absence of label noise ( $\eta = 0$ ). Here, the data and weight spectra have identical power law decay  $\omega_0 = \omega_1 = 1$ . (b). As in (a), but in the presence of label noise ( $\eta = 1 / 2$ ). (c). As in (b), but for fixed hidden layer width  $\alpha_{1} / \alpha_{0} = 4$ , fixed data exponent  $\omega_0 = 1$ , and varying weight exponents  $\omega_{1}$ . In all cases, solid lines show the predictions of (31), while dots with error bars show the mean and standard error over 100 realizations of numerical experiments with  $n_0 = 1000$ . See Appendix F for details of our numerical methods.
(13) under this posterior in the zero-temperature limit  $\beta \rightarrow \infty$ , which we refer to as the generalization error of the Gibbs estimator. We emphasize that this is not identical to the Bayesian minimum mean squared error (MMSE) estimator given by the posterior mean, which would coincide with the ridgeless estimator in the zero-temperature limit (see Appendix A).
For a deep RFM, this simply has the effect of adding a "thermal" variance term to the generalization error of the ridgeless estimator, which we describe in detail in Appendices A and C. We have:
Proposition 6.1. With the same setup as in Proposition 3.2, the generalization error of the Gibbs estimator for a RFM is
$$
\epsilon_ {\mathrm {B R F}} = \epsilon_ {\text {r i d g e l e s s}} + \left\{ \begin{array}{l l} \prod_ {\ell = 0} ^ {L} \frac {\kappa_ {\ell}}{\alpha_ {\ell}}, & \alpha_ {0}, \alpha_ {\min } > 1 \\ 0, & \text {o t h e r w i s e ,} \end{array} \right. \tag {33}
$$
where  $\epsilon_{\mathrm{ridgeless}}$  is given by Proposition 3.2, and  $\kappa_{\ell}$  is defined as in (20).
Proof of Proposition 6.1. We derive (33) alongside Proposition 3.2 in Appendix A.
The Gibbs estimator is sensitive to the scale of the random feature weight distributions through  $\kappa_{\ell}$ , while as noted above the ridgeless estimator is not sensitive to their overall scale. This direct dependence on  $\kappa_{\ell}$  means that the simple argument of Lemma 4.1 cannot be applied. Indeed, in the limit of large prior variance, where the thermal variance term dominates, structure can improve the performance of the Gibbs estimator. We make this result precise in the following lemma:
Lemma 6.1. In the setting of Proposition 6.1, consider Bayesian RFMs with weight covariances scaled as  $\tau_{\ell}\tilde{\Sigma}_{\ell}$  for  $\ell = 1,\dots ,L$  . Then, in the non-trivial regime  $\alpha_0,\alpha_{\mathrm{min}} > 1$  where the thermal variance is non-vanishing, we have
$$
\lim  _ {\tau_ {1}, \dots , \tau_ {L} \rightarrow \infty} \frac {\epsilon_ {\mathrm {B R F}}}{\prod_ {\ell = 1} ^ {L} \tau_ {\ell}} = \prod_ {\ell = 0} ^ {L} \frac {\kappa_ {\ell}}{\alpha_ {\ell}} \leq \frac {\kappa_ {0}}{\alpha_ {0}} \varsigma^ {2} \prod_ {\ell = 1} ^ {L} \left(1 - \frac {1}{\alpha_ {\ell}}\right), \tag {34}
$$
where the scalars  $\kappa_{\ell}$  are defined in terms of the un-scaled covariances  $\tilde{\Sigma}_{\ell}$  as in (20) and  $\varsigma^2 \equiv \prod_{\ell=1}^{L} \mathbb{E}_{\tilde{\sigma}_{\ell}}[\tilde{\sigma}_{\ell}]$ . Therefore, in the limit of large prior variance, including structure in the weight priors is generically advantageous for generalization. If  $\mathbb{E}_{\tilde{\sigma}_{\ell}}[\tilde{\sigma}_{\ell}]$  is not finite, then the bound is vacuous.
Proof of Lemma 6.1. The first part of (34) follows from (33) using the scaling properties of  $\kappa_{\ell}$ , while the bound follows from the bounds on  $\kappa_{\ell}$  derived as part of Lemma 4.2.
In contrast, weight structure is generally harmful for the Bayesian RFM in the limit of small prior variance, as its performance then coincides with the ridgeless RFM, as can be seen from the scaling
of  $\kappa_{\ell}$ . This example illustrates that there are cases in which, depending on the estimator used, weight structure in deeper layers can sometimes be helpful for generalization. However, whereas the ridgeless estimator is commonly used in practice, the Gibbs estimator is less standard, and the limit of large prior variance is certainly artificial. Therefore, we emphasize that we give this example to show that the behavior of the ridgeless estimator is not entirely general, not to show that weight structure can be helpful in practical settings.
We have computed learning curves for models with many layers of structured Gaussian random features learning a linear target function, showing that structure beyond the first layer is generally detrimental for generalization. This result is consistent with the intuition that in deep linear models learning a single target direction it is sufficient to modify the representation only at the first layer [13, 36]. It will be interesting to investigate whether this intuition carries over to nonlinear networks learning complex tasks, particularly including multi-index targets [35, 48]. Moreover, we have considered only linear, Gaussian models. As mentioned in the Introduction, past works have established Gaussian equivalence theorems for nonlinear RFMs with unstructured Gaussian feature weights. It will be important to investigate the effect of feature weight structure on Gaussian equivalence in future work, and determine whether our qualitative results carry over to nonlinear RFMs in the proportional limit.
Though our results are obtained using the replica trick, and we do not address the possibility of replica symmetry breaking, they should be rigorously justifiable given the convexity of the ridge regression problem [29, 33, 41]. We note that the replica approach makes it straightforward to handle models of any finite depth [30]. The relevant averages could of course be computed with alternative random matrix theory techniques, which could allow for a fully rigorous proof [5, 19-21]. Another more challenging setting to study with either the replica trick or rigorous techniques would be that in which one allows for correlations between weights in different layers. This setting could qualitatively capture aspects of feature learning in deep networks, which induces couplings across depth [45].
In closing, we note that RFMs with structured weights may also have relevance for biological neural networks. A recent study by Pandey et al. [39] considered RFMs with a single layer of random features  $(L = 1)$  with correlated rows  $(\Gamma_{1} \neq \mathbf{I}_{n_{0}})$ . In several biologically-inspired settings, they showed that introducing this structure could improve generalization, consistent with our results. More broadly, biological neural networks are imbued with rich priors [49]; investigating what insights deep structured models can afford for neuroscience will be an interesting subject for further study.
We thank Alexander Atanasov, Blake Bordelon, Benjamin S. Ruben, and James B. Simon for helpful discussions and comments on a draft of our manuscript. JAZ-V and CP were supported by NSF Award DMS-2134157 and NSF CAREER Award IIS-2239780. CP received additional support from a Sloan Research Fellowship. This work has been made possible in part by a gift from the Chan Zuckerberg Initiative Foundation to establish the Kempner Institute for the Study of Natural and Artificial Intelligence.