Original: Extended Abstract Track

Original: Data-driven emergence of convolutional structure in neural networks
Revised: Data-driven emergence of convolutional structure in neural networks

Original: Editors: List of editors’ names
Revised: Editors: Sophia Sanborn, Christian Shewmake Simone Azeglio, Arianna Di Bernardo Nina Miolane

Original: Abstract
Revised: Abstract

Original: Exploiting data invariances is crucial for eﬃcient learning in both artiﬁcial and biological neural circuits, but can neural networks learn apposite representations from scratch?
Revised: Exploiting data invariances is crucial for eﬃcient learning in both artiﬁcial and biological neural circuits, but can neural networks learn apposite representations from scratch?

Original: Convolutional neural networks, for example, were designed to exploit translation symmetry, yet learning convolutions directly from data has so far proven elusive.
Revised: Convolutional neural networks, for example, were designed to exploit translation symmetry, yet learning convolutions directly from data has so far proven elusive.

Original: Here, we show how initially fully-connected neural networks solving a discrimination task can learn a convolutional structure directly from their inputs, resulting in localised, space-tiling receptive ﬁelds that match the ﬁlters of a convolutional network trained on the same task.
Revised: Here, we show how initially fully-connected neural networks solving a discrimination task can learn a convolutional structure directly from their inputs, resulting in localised, space-tiling receptive ﬁelds that match the ﬁlters of a convolutional network trained on the same task.

Original: By carefully designing data models for the visual scene, we show that the emergence of this pattern is triggered by the non-Gaussian, higher-order local structure of the inputs, which has long been recognised as the hallmark of natural images.
Revised: By carefully designing data models for the visual scene, we show that the emergence of this pattern is triggered by the non-Gaussian, higher-order local structure of the inputs, which has long been recognised as the hallmark of natural images.

Original: We provide an analytical and numerical characterisation of the pattern-formation mechanism responsible for this phenomenon in a simple model and ﬁnd an unexpected link between receptive ﬁeld formation and tensor decomposition of higher-order input correlations.
Revised: We provide an analytical and numerical characterisation of the pattern-formation mechanism responsible for this phenomenon in a simple model and ﬁnd an unexpected link between receptive ﬁeld formation and tensor decomposition of higher-order input correlations.

Original: Introduction
Revised: Introduction

Original: Exploiting invariances in the inputs is crucial for constructing eﬃcient representations and accurate predictions in neural circuits.
Revised: Exploiting invariances in the inputs is crucial for constructing eﬃcient representations and accurate predictions in neural circuits.

Original: In neuroscience, translation invariance is at the heart of models of the visual system (DiCarlo et al., 2012; Yamins et al., 2014; Kar and DiCarlo, 2021; Spoerer et al., 2017), while in machine learning, convolutional neural networks are designed to exploit translation invariance (LeCun et al., 1990; Scherer et al., 2010).
Revised: In neuroscience, translation invariance is at the heart of models of the visual system (DiCarlo et al., 2012; Yamins et al., 2014; Kar and DiCarlo, 2021; Spoerer et al., 2017), while in machine learning, convolutional neural networks are designed to exploit translation invariance (LeCun et al., 1990; Scherer et al., 2010).

Original: While the two hallmarks of convolutions, namely localised receptive ﬁelds that tile the input space, can be implemented with fully-connected neural networks, learning convolutions directly from inputs in a fully-connected network has so far proven elusive (Urban et al., 2017; d ' Ascoli et al., 2019) without elaborate pruning (Pellegrini and Biroli, 2021) or regularisation strategies (Neyshabur, 2020).
Revised: While the two hallmarks of convolutions, namely localised receptive ﬁelds that tile the input space, can be implemented with fully-connected neural networks, learning convolutions directly from inputs in a fully-connected network has so far proven elusive (Urban et al., 2017; d ' Ascoli et al., 2019) without elaborate pruning (Pellegrini and Biroli, 2021) or regularisation strategies (Neyshabur, 2020).

Original: Whether convolutions can be learnt from scratch has thus been a central problem in neuroscience and machine learning since the seminal work by
Revised: Whether convolutions can be learnt from scratch has thus been a central problem in neuroscience and machine learning since the seminal work by

Original: Olshausen and Field (1996) on unsupervised learning.
Revised: Olshausen and Field (1996) on unsupervised learning.

Original: Here, we show how initially fully-connected neural networks solving a discrimination task can learn a convolutional structure directly from their inputs, resulting in localised,space-tiling receptive ﬁelds.
Revised: Here, we show how initially fully-connected neural networks solving a discrimination task can learn a convolutional structure directly from their inputs, resulting in localised,

Original: By carefully designing data models for the visual scene, we show that this phenomenon relies on the non-Gaussian, higher-order local structure of the inputs, which has long been recognized as the hallmark of natural images (Bell and

Original: Sejnowski, 1996).
Revised: [Figure]

Original: We characterise receptive ﬁeld formation analytically and numerically, Extended Abstract Track
Revised: architecture Input Output

Original: [Figure]
Revised: [Figure]

Original:  oscillatory 
Revised: localised RF oscillatory RF long

Original: [Figure]
Revised: [Figure]

Original:  low
Revised: range correlations same covariance per class vs. low

Original: [Figure]
Revised: [Figure]

Original: high
Revised: freq. RF high-freq.

Original: [Figure]

Original: RF
Revised: RF

Original: A
Revised: A

Original: B
Revised: B

Original: C weights w
Revised: C weights w

Original: D
Revised: D

Original: E
Revised: E

Original: F
Revised: F

Original: G time time
Revised: G time time

Original: I P R
Revised: I P R

Revised: I P R space-tiling receptive ﬁelds.

Revised: By carefully designing data models for the visual scene, we show that this phenomenon relies on the non-Gaussian, higher-order local structure of the inputs, which has long been recognized as the hallmark of natural images (Bell and

Revised: Sejnowski, 1996).

Original: I P R revealing an unexpected link with tensor decomposition of higher-order input cumulants.
Revised: We characterise receptive ﬁeld formation analytically and numerically, revealing an unexpected link with tensor decomposition of higher-order input cumulants.

Original: The receptive ﬁelds learnt by the fully-connected networks match the ﬁlters found by training a convolutional network on the same task.
Revised: The receptive ﬁelds learnt by the fully-connected networks match the ﬁlters found by training a convolutional network on the same task.

Original: These results provide a new perspective on the development of low-level feature detectors in various sensory modalities and pave the way for the study of higher-level invariances in cortical processing.
Revised: These results provide a new perspective on the development of low-level feature detectors in various sensory modalities, and pave the way for the study of higher-level invariances in cortical processing.

Original: Results 
Revised: Results A high-dimensional dataset with tunable higher-order

Original: A high-dimensional dataset with tunable

Original: higher-order moments
Revised: moments

Original: We train twolayer networks (ﬁg. 1A) on a minimal model of natural images capable of capturing higherorder spatial correlations.
Revised: We train twolayer networks (ﬁg. 1A) on a minimal model of natural images capable of capturing higherorder spatial correlations.

Original: We generated two-dimensional inputs inputs x = ( x ij ) by ﬁrst drawing a random vector z =
Revised: We generated two-dimensional inputs inputs x = ( x ij ) by ﬁrst drawing a random vector z =

Original: ( z ij ) from a centered Gaussian distribution with a covariance that renders the input distribution translation invariant along both dimensions.
Revised: ( z ij ) from a centered Gaussian distribution with a covariance that renders the input distribution translation invariant along both dimensions.

Original: Each pixel in the synthetic image x ij is then computed as
Revised: Each pixel in the synthetic image x ij is then computed as

Original: [Equation]
Revised: [Equation]

Original: Extended Abstract Track

Original: where ψ ( · ) is a symmetric, saturating non-linear function such as the error function, g > 0 is a gain factor, and the normalisation constant Z ( g ) ensures that pixels have unit variance for all values of g (see appendix A for details).
Revised: where ψ ( · ) is a symmetric, saturating non-linear function such as the error function, g > 0 is a gain factor, and the normalisation constant Z ( g ) ensures that pixels have unit variance for all values of g (see appendix A for details).

Original: Intuitively, the gain factor controls the sharpness in the images: a large gain factor results in images with sharp edges and important nonGaussian statistics (ﬁg. 1B), while images with a small gain factor are close to Gaussians in distribution.
Revised: Intuitively, the gain factor controls the sharpness in the images: a large gain factor results in images with sharp edges and important nonGaussian statistics (ﬁg. 1B), while images with a small gain factor are close to Gaussians in distribution.

Original: The task consists in discriminating inputs with short ( ξ − ) vs long ( ξ + ) correlation length.
Revised: The task consists in discriminating inputs with short ( ξ − ) vs long ( ξ + ) correlation length.

Original: Crucially, inputs have sharp edges, which is a visual indication of higherorder spatial correlations which cannot be captured by a simpler Gaussian model.
Revised: Crucially, inputs have sharp edges, which is a visual indication of higherorder spatial correlations which cannot be captured by a simpler Gaussian model.

Original: Indeed, as we show in ﬁg.
Revised: Indeed, as we show in ﬁg.

Original: 1E, samples from a Gaussian distribution with the same covariance as the inputs appear blurry in comparison.
Revised: 1E, samples from a Gaussian distribution with the same covariance as the inputs appear blurry in comparison.

Original: Learning convolutions directly from stimuli
Revised: Learning convolutions directly from stimuli

Original: We trained two-layer neural networks on this task using vanilla stochastic gradient descent, achieving test accuracy > 90%.
Revised: We trained two-layer neural networks on this task using vanilla stochastic gradient descent, achieving test accuracy > 90%.

Original: We plot the weight vector, or receptive ﬁeld (RF), of several hidden neurons of the trained networks in ﬁg. 1C.
Revised: We plot the weight vector, or receptive ﬁeld (RF), of several hidden neurons of the trained networks in ﬁg. 1C.

Original: The RF of half of the neurons are localised : they only have a few synaptic weights whose magnitude is signiﬁcantly larger than zero in a small region of input space.
Revised: The RF of half of the neurons are localised : they only have a few synaptic weights whose magnitude is signiﬁcantly larger than zero in a small region of input space.

Original: Neurons that detect short-range correlations develop diﬀerent representations, instead converging to highly oscillatory patterns.
Revised: Neurons that detect short-range correlations develop diﬀerent representations, instead converging to highly oscillatory patterns.

Original: We can quantify the localisation of receptive ﬁelds by computing the Inverse Participation Ratio (IPR) of their weight vector w = ( w i ), IPR( w )
Revised: We can quantify the localisation of receptive ﬁelds by computing the Inverse Participation Ratio (IPR) of their weight vector w = ( w i ), IPR( w )

Original: =
Revised: =

Original: (cid:16)(cid:80)
Revised: (cid:16)(cid:80)

Original: Di =1 w 

Revised: Di =1 w

Original: i (cid:17) / (cid:16)(cid:80)
Revised: i (cid:17) / (cid:16)(cid:80)

Original: Di =1 w 

Revised: Di =1 w

Original: i (cid:17) 2 .
Revised: i (cid:17) 2 .

Original: The IPR quantiﬁes the amount of non-zero components of a vector and is commonly used in quantum mechanics and random matrix theory (Metz et al., 2010).
Revised: The IPR quantiﬁes the amount of non-zero components of a vector and is commonly used in quantum mechanics and random matrix theory (Metz et al., 2010).

Original: We plot the IPR of all neurons during training in ﬁg. 1D.
Revised: We plot the IPR of all neurons during training in ﬁg. 1D.

Original: Localised neurons develop a large IPR over the course of training, while the IPR of neurons with oscillatory receptive ﬁelds remains very small.
Revised: Localised neurons develop a large IPR over the course of training, while the IPR of neurons with oscillatory receptive ﬁelds remains very small.

Original: Crucially, we found that the RFs of this fully-connected network are spread over the entire input range (ﬁg. S1A) and that they match the ﬁlters of a two-layer convolutional network trained on the same task (ﬁg. S1).
Revised: Crucially, we found that the RFs of this fully-connected network are spread over the entire input range (ﬁg. S1A) and that they match the ﬁlters of a two-layer convolutional network trained on the same task (ﬁg. S1).

Original: Learning convolutions requires higher-order input cumulants
Revised: Learning convolutions requires higher-order input cumulants

Original: As a control, we trained the same networks on a task where inputs for each class are Gaussian with the same covariance as the original data (ﬁg. 1B).
Revised: As a control, we trained the same networks on a task where inputs for each class are Gaussian with the same covariance as the original data (ﬁg. 1B).

Original: These inputs are still translation-invariant, but lack the non-trivial higher-order statistics.
Revised: These inputs are still translation-invariant, but lack the non-trivial higher-order statistics.

Original: Networks trained on these control inputs do not form localised receptive ﬁelds (ﬁg. 1F), instead converging to oscillatory patterns.
Revised: Networks trained on these control inputs do not form localised receptive ﬁelds (ﬁg. 1F), instead converging to oscillatory patterns.

Original: The kurtosis of all neurons stays also close to zero throughout learning (ﬁg. 1G).
Revised: The kurtosis of all neurons stays also close to zero throughout learning (ﬁg. 1G).

Original: Taken together, these results show that both translation invariance and non-trivial higher-order statistics are needed to learn localised receptive ﬁelds from scratch.
Revised: Taken together, these results show that both translation invariance and non-trivial higher-order statistics are needed to learn localised receptive ﬁelds from scratch.

Original: Existing theories of learning in neural networks fail to capture the formation of receptive ﬁelds
Revised: Existing theories of learning in neural networks fail to capture the formation of receptive ﬁelds

Original: The dynamics of (deep) linear networks depends only on the inputinput and the input-label covariance matrices Saxe et al.
Revised: The dynamics of (deep) linear networks depends only on the inputinput and the input-label covariance matrices Saxe et al.

Original: and can therefore not capture the formation of receptive ﬁelds, which is driven by non-Gaussian ﬂuctuations in the inputs.
Revised: and can therefore not capture the formation of receptive ﬁelds, which is driven by non-Gaussian ﬂuctuations in the inputs.

Original: Similarly, an analysis of the learning dynamics using the Gaussian Equivalence
Revised: Similarly, an analysis of the learning dynamics using the Gaussian Equivalence

Original: Theorem (Goldt et al., 2020; Hu and Lu, 2020; Goldt et al., 2021; Mei and Montanari, 2021) breaks down precisely when localised receptive ﬁelds form, highlighting the non-Gaussian nature of their formation.
Revised: Theorem (Goldt et al., 2020; Hu and Lu, 2020; Goldt et al., 2021; Mei and Montanari, 2021) breaks down precisely when localised receptive ﬁelds form, highlighting the non-Gaussian nature of their formation.

Original: We discuss this issue in more detail in appendix B.
Revised: We discuss this issue in more detail in appendix B.

Original: Connecting receptive ﬁelds to data geometry
Revised: Connecting receptive ﬁelds to data geometry

Original: An analysis of the gradient ﬂow dynamics of a simpliﬁed model revealed an interesting connection between receptive ﬁelds and data geometry.
Revised: An analysis of the gradient ﬂow dynamics of a simpliﬁed model revealed an interesting connection between receptive ﬁelds and data geometry.

Original: We studied the learning dynamics of a single neuron with a polynomial activation function.
Revised: We studied the learning dynamics of a single neuron with a polynomial activation function.

Original: The gradient ﬂow dynamics of this neuron depends only on the covariance matrices C µij =
Revised: The gradient ﬂow dynamics of this neuron depends only on the covariance matrices C µij =

Original: E (cid:104) x µi x µj (cid:105) and the fourth-order moments T µijk(cid:96) =
Revised: E (cid:104) x µi x µj (cid:105) and the fourth-order moments T µijk(cid:96) =

Original: E (cid:104) x µi x µj x µk x µ(cid:96) (cid:105) of each input class µ :
Revised: E (cid:104) x µi x µj x µk x µ(cid:96) (cid:105) of each input class µ .

Original: [Equation]

Original: Our analysis shows that the 4th-order cumulant ∆ T µ – obtained by subtracting the Gaussian contribution from T µ – is crucial for the formation of the RF.
Revised: Our analysis shows that the 4th-order cumulant ∆ T µ – obtained by subtracting the Gaussian contribution from T µ – is crucial for the formation of the RF.

Original: A tensor like ∆ T µ can be decomposed into its leading CP factors (Kolda and Bader, 2009), akin to the eigendecomposition of a matrix,
Revised: A tensor like ∆ T µ can be decomposed into its leading CP factors (Kolda and Bader, 2009), akin to the eigendecomposition of a matrix

Original: [Equation]
Revised: [Equation]

Original: where r is the rank of the decomposition (see ﬁg.
Revised: where r is the rank of the decomposition (see ﬁg.

Original: 2A for an illustration of a third-order tensor).
Revised: 2A for an illustration of a third-order tensor).

Original: When training on strongly non-Gaussian inputs with ∆ T µ (cid:54) = 0, the single neuron develops a localised receptive ﬁeld which mirrors the localisation of the leading CP factors of ∆ T µ (blue and grey lines in ﬁg. 2B).
Revised: When training on strongly non-Gaussian inputs with ∆ T µ (cid:54) = 0, the single neuron develops a localised receptive ﬁeld which mirrors the localisation of the leading CP factors of ∆ T µ (blue and grey lines in ﬁg. 2B).

Original: The CP factors also tile the input space.
Revised: The CP factors also tile the input space.

Original: If instead inputs are only weakly non-Gaussian, the CP factors, and hence also the weight, oscillate (ﬁg. 2C).
Revised: If instead inputs are only weakly non-Gaussian, the CP factors, and hence also the weight, oscillate (ﬁg. 2C).

Original: Extended Abstract Track

Original: Appendix A. Data models
Revised: Appendix A. Data models

Original: We conduct the experiments reported in ﬁg. 1 on a data set that consists of inputs x that can be one- or two-dimensional, divided in M distinct classes.
Revised: We conduct the experiments reported in ﬁg. 1 on a data set that consists of inputs x that can be one- or two-dimensional, divided in M distinct classes.

Original: Here, we illustrate the diﬀerent types of inputs in one dimension.
Revised: Here, we illustrate the diﬀerent types of inputs in one dimension.

Original: A data vector of the non-linear Gaussian process (NLGP) is given by x µ =
Revised: A data vector of the non-linear Gaussian process (NLGP) is given by x µ =

Original: Z − 1 ( g ) ψ ( g z µ ), where z µ is a zero-mean Gaussian vector of length L and covariance matrix[Equation] A.
Revised: Z − 1 ( g ) ψ ( g z µ ), where z µ is a zero-mean Gaussian vector of length L and covariance matrix

Original: [Equation]
Revised: [Equation]

Original: with i, j = 1 , 2 , . . .
Revised: with i, j = 1 , 2 , . . .

Original: , L .
Revised: , L .

Original: The covariance thus only depends on the distance between sites i and j , given by | i − j | .
Revised: The covariance thus only depends on the distance between sites i and j , given by | i − j | .

Original: The normalisation factor Z ( g ) is chosen such that Var ( x ) = 1.
Revised: The normalisation factor Z ( g ) is chosen such that Var ( x ) = 1.

Original: Throughout this work, we took ψ to be a symmetric saturating function ψ ( z ) = erf ( z / √ 2 ), for which Z ( g ) 2 = 2 / π arcsin (cid:0) g 2 / (1 + g 2 ) (cid:1) .
Revised: Throughout this work, we took ψ to be a symmetric saturating function ψ ( z ) = erf ( z / √ 2 ), for which Z ( g ) 2 = 2 / π arcsin (cid:0) g 2 / (1 + g 2 ) (cid:1) .

Original: We also enforce periodic boundary conditions.
Revised: We also enforce periodic boundary conditions.

Original: We create the Gaussian clone (GP) by drawing inputs from a Gaussian distribution with mean zero and the same covariance as the corresponding NLGP in each class.
Revised: We create the Gaussian clone (GP) by drawing inputs from a Gaussian distribution with mean zero and the same covariance as the corresponding NLGP in each class.

Original: The covariance of the NLGP can be evaluated analytically for ψ ( z ) =
Revised: The covariance of the NLGP can be evaluated analytically for ψ ( z ) =

Original: erf ( z / √ 2 ) and reads
Revised: erf ( z / √ 2 ) and reads

Original: [Equation]
Revised: [Equation]

Original: where we have used that fact that C ii = 1.
Revised: where we have used that fact that C ii = 1.

Original: The experiments on Gaussian processes (GP) are thus not performed on the Gaussian variables z ; they are performed on Gaussian random variables with covariance given in eq.
Revised: The experiments on Gaussian processes (GP) are thus not performed on the Gaussian variables z ; they are performed on Gaussian random variables with covariance given in eq.

Original: (A.2).
Revised: (A.2).

Original: In this way, we exclude the possibility that the change in the two-point correlation function from applying the non-linearity ψ is responsible for the emergence of receptive ﬁelds.
Revised: In this way, we exclude the possibility that the change in the two-point correlation function from applying the non-linearity ψ is responsible for the emergence of receptive ﬁelds.

Original: For 1-dimensional inputs, the fact that the covariances of the NLGP and the GP depend only on the distances between pixels | i − j | implies that they are circulant matrices (Horn and Johnson, 2012).
Revised: For 1-dimensional inputs, the fact that the covariances of the NLGP and the GP depend only on the distances between pixels | i − j | implies that they are circulant matrices (Horn and Johnson, 2012).

Original: These matrices display a number of useful properties: they can be diagonalised using discrete Fourier Transform (DFT), and thus any two circulant matrices of the same size can be jointly diagonalised and commute with each other.
Revised: These matrices display a number of useful properties: they can be diagonalised using discrete Fourier Transform (DFT), and thus any two circulant matrices of the same size can be jointly diagonalised and commute with each other.

Original: We use this fact in the analysis of the reduced model to diagonalise the dynamics of the synaptic weights.
Revised: We use this fact in the analysis of the reduced model to diagonalise the dynamics of the synaptic weights.

Original: We obtain the covariance for 2-dimensional inputs by taking the Kronecker product of the one-dimensional covariance matrix with itself.
Revised: We obtain the covariance for 2-dimensional inputs by taking the Kronecker product of the one-dimensional covariance matrix with itself.

Original: For any dimension, we indicate the total input size by D .
Revised: For any dimension, we indicate the total input size by D .

Original: Appendix B. The limits of Gaussian equivalence in describing the formation of receptive ﬁelds
Revised: Appendix B. The limits of Gaussian equivalence in describing the formation of receptive ﬁelds

Original: How can we capture the formation of receptive ﬁelds theoretically?
Revised: How can we capture the formation of receptive ﬁelds theoretically?

Original: There exist precise theories for learning in neural networks with linear activation functions Baldi and Hornik (1989); Le Cun et al.
Revised: There exist precise theories for learning in neural networks with linear activation functions Baldi and Hornik (1989); Le Cun et al.

Original: ; Krogh and Hertz (1992); Saxe et al.
Revised: ; Krogh and Hertz (1992); Saxe et al.

Original: (2014, 2019a);
Revised: (2014, 2019a);

Original: Advani et al.
Revised: Advani et al.

Original: However, the dynamics of even a deep linear network with several layers will only depend on the input-input and the input-label covariance matrices, i.e. the ﬁrst two moments of the data Saxe et al.
Revised: However, the dynamics of even a deep linear network with several layers will only depend on the input-input and the input-label covariance matrices, i.e. the ﬁrst two moments of the data Saxe et al.

Original: This formalism thus cannot capture the formation of receptive ﬁelds, which is driven by non-Gaussian ﬂuctuations in the inputs.
Revised: This formalism thus cannot capture the formation of receptive ﬁelds, which is driven by non-Gaussian ﬂuctuations in the inputs.

Original: An exact theory describing the learning dynamics is available for non-linear two-layer neural networks with large input size D → ∞ and a few neurons K ∼ O (1) in the hidden layer Saad and Solla (1995); Biehl and Schwarze (1995).
Revised: An exact theory describing the learning dynamics is available for non-linear two-layer neural networks with large input size D → ∞ and a few neurons K ∼ O (1) in the hidden layer Saad and Solla (1995); Biehl and Schwarze (1995).

Original: In this limit, one can derive a set of ordinary diﬀerential equations that predict the evolution of the (prediction mean-squared) test error pmse of a network, when training on Gaussian mixture classiﬁcation Reﬁnetti et al.
Revised: In this limit, one can derive a set of ordinary diﬀerential equations that predict the evolution of the (prediction mean-squared) test error pmse of a network, when training on Gaussian mixture classiﬁcation Reﬁnetti et al.

Original: In ﬁg. S2, we show the pmse of a network with K = 8 neurons trained on the Gaussian control task (blue lines) and verify that this theory yields matching predictions (blue crosses).
Revised: In ﬁg. S2, we show the pmse of a network with K = 8 neurons trained on the Gaussian control task (blue lines) and verify that this theory yields matching predictions (blue crosses).

Original: This type of analysis has recently been extended from mixtures of Gaussians to more complex input distributions thanks to the phenomenon of “Gaussian equivalence”, whereby the performance of a network trained on non-Gaussian inputs is still well captured by an appropriately chosen Gaussian model for the data.
Revised: This type of analysis has recently been extended from mixtures of Gaussians to more complex input distributions thanks to the phenomenon of “Gaussian equivalence”, whereby the performance of a network trained on non-Gaussian inputs is still well captured by an appropriately chosen Gaussian model for the data.

Original: This Gaussian equivalence was used successfully to analyse random features Liao and Couillet (2018); Seddik et al.
Revised: This Gaussian equivalence was used successfully to analyse random features Liao and Couillet (2018); Seddik et al.

Original: ; Mei and Montanari (2021) and neural networks with one or two layers, even when inputs were drawn from pre-trained generative models Goldt et al.
Revised: ; Mei and Montanari (2021) and neural networks with one or two layers, even when inputs were drawn from pre-trained generative models Goldt et al.

Original: Hu and Lu (2020); Goldt et al.
Revised: Hu and Lu (2020); Goldt et al.

Original: ; Loureiro et al. (2021).
Revised: ; Loureiro et al. (2021).

Original: In ﬁg. S2, we plot the test error of a network trained on
Revised: In ﬁg. S2, we plot the test error of a network trained on

Original: NLGP data together with the theoretical prediction obtained from applying the Gaussian
Revised: NLGP data together with the theoretical prediction obtained from applying the Gaussian

Original: Equivalence Theorem Goldt et al.
Revised: Equivalence Theorem Goldt et al.

Original: (2021) (GET).
Revised: (2021) (GET).

Original: Initially, the theoretical predictions from the GET (orange crosses) agree with the test error measured in the simulation (orange line), but the theory breaks down around time
Revised: Initially, the theoretical predictions from the GET (orange crosses) agree with the test error measured in the simulation (orange line) but the theory breaks down around time

Original: ≈ 10 2 , when predictions start deviating from simulations.
Revised: ≈ 10 2 , when predictions start deviating from simulations.

Original: The breakdown of the Gaussian theory coincides with the localisation of the receptive ﬁelds, as measured by their IPR (green line in the inset of ﬁg. S2).
Revised: The breakdown of the Gaussian theory coincides with the localisation of the receptive ﬁelds, as measured by their IPR (green line in the inset of ﬁg. S2).

Original: The increased localisation of the weights also coincides with a change in the statistics of the pre-activations of the hidden neurons, λ ∼ (cid:80) i
Revised: The increased localisation of the weights also coincides with a change in the statistics of the pre-activations of the hidden neurons, λ ∼ (cid:80) i

Original: w i
Revised: w i

Original: x i : the excess kurtosis of λ (orange line) is initially close to zero,Extended Abstract Track meaning that λ is approximately Gaussian, but decreases as the weights localise, indicating a transition to a non-Gaussian distribution.
Revised: x i : the excess kurtosis of λ (orange line) is initially close to zero, meaning that λ is approximately Gaussian, but decreases as the weights localise, indicating a transition to a non-Gaussian distribution.

Original: We can ﬁnally see from ﬁg. S2 that the network is only inﬂuenced by the second-order ﬂuctuations in both the NLGP and the GP at the beginning of training, since the pmse for models trained on NLGP and GP initially coincide.
Revised: We can ﬁnally see from ﬁg. S2 that the network is only inﬂuenced by the second-order ﬂuctuations in both the NLGP and the GP at the beginning of training, since the pmse for models trained on NLGP and GP initially coincide.

Original: Likewise, a network trained on GP and evaluated on NLGP test data has the same test accuracy as the network trained directly on
Revised: Likewise, a network trained on GP and evaluated on NLGP test data has the same test accuracy as the network trained directly on

Original: NLGP in the early stages of learning (red line).
Revised: NLGP in the early stages of learning (red line).

Original: The higher-order moments of the NLGP inputs start inﬂuencing learning only at a later stage, when the IPR of the weight vectors increases and the Gaussian theory breaks down.
Revised: The higher-order moments of the NLGP inputs start inﬂuencing learning only at a later stage, when the IPR of the weight vectors increases and the Gaussian theory breaks down.

Original: This sequential learning of increasingly higher-order statistics of the inputs is reminiscent of how neural networks learn increasingly complex functions during training.
Revised: This sequential learning of increasingly higher-order statistics of the inputs is reminiscent of how neural networks learn increasingly complex functions during training.

Original: Simplicity biases of this kind have been analysed in simple models of neural networks Schwarze and Hertz (1992); Saad and Solla (1995); Engel and Van den Broeck (2001); Saxe et al.
Revised: Simplicity biases of this kind have been analysed in simple models of neural networks Schwarze and Hertz (1992); Saad and Solla (1995); Engel and Van den Broeck (2001); Saxe et al.

Original: (2019b); Rahaman et al.
Revised: (2019b); Rahaman et al.

Original: and have been demonstrated in modern convolutional networks Kalimeris et al.
Revised: and have been demonstrated in modern convolutional networks Kalimeris et al.

Original: The sequential learning of increasingly higher-order statistics and the ensuing breakdown of the GET to describe learning is a result of independent interest which we will investigate further in future work.
Revised: The sequential learning of increasingly higher-order statistics and the ensuing breakdown of the GET to describe learning is a result of independent interest which we will investigate further in future work.