Relative stability toward diffeomorphisms indicates performance in deep nets

√ Rf , suggesting that obtaining a smallRf is important to achieve good performance . We study how Rf depends on the size of the training set and compare it to a simple model of invariant learning . 1 Introduction . Deep learning algorithms LeCun et al . ( 2015 ) are now remarkably successful at a wide range of tasks Amodei et al . ( 2016 ) ; Huval et al . ( 2015 ) ; Mnih et al . ( 2013 ) ; Shi et al . ( 2016 ) ; Silver et al . ( 2017 ) . Yet , understanding how they can classify data in large dimensions remains a challenge . In particular , the curse of dimensionality associated with the geometry of space in large dimension prohibits learning in a generic setting Luxburg and Bousquet ( 2004 ) . If high-dimensional data can be learnt , then they must be highly structured . A popular idea is that during training , hidden layers of neurons learn a representation Le ( 2013 ) that is insensitive to aspects of the data unrelated to the task , effectively reducing the input dimension and making the problem tractable Ansuini et al . ( 2019 ) ; Recanatesi et al . ( 2019 ) ; Shwartz-Ziv and Tishby ( 2017 ) . Several quantities have been introduced to study this effect empirically . It includes ( i ) the mutual information between the hidden and visible layers of neurons Saxe et al . ( 2019 ) ; Shwartz-Ziv and Tishby ( 2017 ) , ( ii ) the intrinsic dimension of the neural representation of the data Ansuini et al . ( 2019 ) ; Recanatesi et al . ( 2019 ) and ( iii ) the projection of the label of the data on the main features of the network Kopitkov and Indelman ( 2020 ) ; Oymak et al . ( 2019 ) ; Paccolat et al . ( 2021a ) , the latter being defined from the top eigenvectors of the Gram matrix of the neural tangent kernel ( NTK ) Jacot et al . ( 2018 ) . All these measures support that the neuronal representation of the data indeed becomes well-suited to the task . Yet , they are agnostic to the nature of what varies in the data that need not being represented by hidden neurons , and thus do not specify what it is . Recently , there has been a considerable effort to understand the benefits of learning features for onehidden-layer fully connected nets . Learning features can occur and improve performance when the 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . true function is highly anisotropic , in the sense that it depends only on a linear subspace of the input space Bach ( 2017 ) ; Chizat and Bach ( 2020 ) ; Ghorbani et al . ( 2019 , 2020 ) ; Paccolat et al . ( 2021a ) ; Refinetti et al . ( 2021 ) ; Yehudai and Shamir ( 2019 ) . For image classification , such an anisotropy would occur for example if pixels on the edge of the image are unrelated to the task . Yet , fully-connected nets ( unlike CNNs ) acting on images tend to perform best in training regimes where features are not learnt Geiger et al . ( 2021 , 2020 ) ; Lee et al . ( 2020 ) , suggesting that such a linear invariance in the data is not central to the success of deep nets . Instead , it has been proposed that images can be classified in high dimensions because classes are invariant to smooth deformations or diffeomorphisms of small magnitude Bruna and Mallat ( 2013 ) ; Mallat ( 2016 ) . Specifically , Mallat and Bruna could handcraft convolution networks , the scattering transforms , that perform well and are stable to smooth transformations , in the sense that ‖f ( x ) − f ( τx ) ‖ is small if the norm of the diffeomorphism τ is small too . They hypothesized that during training deep nets learn to become stable and thus less sensitive to these deformations , thus improving performance . More recent works generalize this approach to more common CNNs and discuss stability at initialization Bietti and Mairal ( 2019a , b ) . Interestingly , enforcing such a stability can improve performance Kayhan and Gemert ( 2020 ) . Answering if deep nets become more stable to smooth deformations when trained and quantifying how it affects performance remains a challenge . Recent empirical results revealed that small shifts of images can change the output a lot Azulay and Weiss ( 2018 ) ; Dieleman et al . ( 2016 ) ; Zhang ( 2019 ) , in apparent contradiction with that hypothesis . Yet in these works , image transformations ( i ) led to images whose statistics were very different from that of the training set or ( ii ) were cropping the image , thus are not diffeophormisms . In Ruderman et al . ( 2018 ) , a class of diffeomorphisms ( low-pass filter in spatial frequencies ) was introduced to show that stability toward them can improve during training , especially in architectures where pooling layers are absent . Yet , these studies do not address how stability affects performance , and how it depends on the size of the training set . To quantify these properties and to find robust empirical behaviors across architectures , we will argue that the evolution of stability toward smooth deformations needs to be compared relatively to that of any deformation , which turns out to vary significantly during training . Note that in the context of adversarial robustness , attacks that are geometric transformations of small norm that change the label have been studied Alaifari et al . ( 2018 ) ; Alcorn et al . ( 2019 ) ; Athalye et al . ( 2018 ) ; Engstrom et al . ( 2019 ) ; Fawzi and Frossard ( 2015 ) ; Kanbak et al . ( 2018 ) ; Xiao et al . ( 2018 ) . These works differ for the literature above and from out study below in the sense that they consider worst-case perturbations instead of typical ones . 1.1 Our Contributions . ◦ We introduce a maximum entropy distribution of diffeomorphisms , that allow us to generate typical diffeomorphisms of controlled norm . Their amplitude is governed by a `` temperature '' parameter T . ◦ We define the relative stability to diffeomorphisms index Rf that characterizes the square magnitude of the variation of the output function f with respect to the input when it is transformed along a diffeomorphism , relatively to that of a random transformation of the same amplitude . It is averaged on the test set as well as on the ensemble of diffeomorphisms considered . ◦ We find that at initialization , Rf is close to unity for various data sets and architectures , indicating that initially the output is as sensitive to smooth deformations as it is to random perturbations of the image . ◦ Our central result is that after training , Rf correlates very strongly with the test error t : during training , Rf is reduced by several decades in current State Of The Art ( SOTA ) architectures on four benchmark datasets including MNIST Lecun et al . ( 1998 ) , FashionMNIST Xiao et al . ( 2017 ) , CIFAR-10 Krizhevsky ( 2009 ) and ImageNet Deng et al . ( 2009 ) . For more primitive architectures ( whose test error is higher ) such as fully connected nets or simple CNNs , Rf remains of order unity . For CIFAR10 we study 15 known architectures and find empirically that t ≈ 0.2 √ Rf . ◦ Rf decreases with the size of the training set P . We compare it to an inverse power 1/P expected in simple models of invariant learning Paccolat et al . ( 2021a ) . The library implementing diffeomorphisms on images is available online at github.com/pcslepfl/diffeomorphism . The code for training neural nets can be found at github.com/leonardopetrini/diffeo-sota and the corresponding pre-trained models at doi.org/10.5281/zenodo.5589870 . 2 Maximum-entropy model of diffeomorphisms . 2.1 Definition of maximum entropy model . We consider the case where the input vector x is an image . It can be thought as a function x ( s ) describing intensity in position s = ( u , v ) ∈ [ 0 , 1 ] 2 , where u and v are the horizontal and vertical coordinates . To simplify notations we consider a single channel , in which case x ( s ) is a scalar ( but our analysis holds for colored images as well ) . We denote by τx the image deformed by τ , i.e . [ τx ] ( s ) = x ( s − τ ( s ) ) . τ ( s ) is a vector field of components ( τu ( s ) , τv ( s ) ) . The deformation amplitude is measured by the norm ‖∇τ‖2 = ∫ [ 0,1 ] 2 ( ( ∇τu ) 2 + ( ∇τv ) 2 ) dudv . ( 1 ) To test the stability of deep nets toward diffeomorphisms , we seek to build typical diffeomorphisms of controlled norm ‖∇τ‖ . We thus consider the distribution over diffeomorphisms that maximizes the entropy with a norm constraint . It can be solved by introducing a Lagrange multiplier T and by decomposing these fields on their Fourier components , see e.g . Kardar ( 2007 ) or Appendix A . In this canonical ensemble , one finds that τu and τv are independent with identical statistics . For the picture frame not to be deformed , we impose fixed boundary conditions : τ = 0 if u = 0 , 1 or v = 0 , 1 . One then obtains : τu = ∑ i , j∈N+ Cij sin ( iπu ) sin ( jπv ) ( 2 ) where the Cij are Gaussian variables of zero mean and variance 〈C2ij〉 = T/ ( i2 + j2 ) . If the picture is made of n× n pixels , the result is identical except that the sum runs on 0 < i , j ≤ n. For large n , the norm then reads ‖∇τ‖2 = ( π2/2 ) n2T , and is dominated by high spatial frequency modes . It is useful to add another parameter c to cut-off the effect of high spatial frequencies , which can be simply done by constraining the sum in Eq.2 to i2 + j2 ≤ c2 , one then has ‖∇τ‖2 = ( π3/8 ) c2T . Once τ is generated , pixels are displaced to random positions . A new pixelated image can then be obtained using standard interpolation methods . We use two interpolations , Gaussian and bi-linear1 , as described in Appendix C. As we shall see below , this choice does not affect our result as long as the diffeomorphism induced a displacement of order of the pixel size , or larger . Examples are shown in Fig.1 as a function of T and c . 2.2 Phase diagram of acceptable diffeomorphisms . Diffeomorphisms are bijective , which is not the case for our transformations if T is too large . When this condition breaks down , a single domain of the picture can break into several pieces , as apparent in Fig.1 . It can be expressed as a condition on ∇τ that must be satisfied in every point in space Lowe ( 2004 ) , as recalled in Appendix B . This is satisfied locally with high probability if ‖τ‖2 1 , corresponding to T ( 8/π3 ) /c2 . In Appendix , we extract empirically a curve of similar form in the ( T , c ) plane at which a diffeomorphism is obtained with probability at least 1/2 . For much smaller T , diffeomorphisms are obtained almost surely . Finally , for diffeomorphisms to have noticeable consequences , their associated displacement must be of the order of magnitude of the pixel size . Defining δ2 as the average square norm of the pixel displacement at the center of the image in the unit of pixel size , it is straightforward to obtain from Eq.2 that asymptotically for large c ( cf . Appendix B for the derivation ) , δ2 = π 4 n2T ln ( c ) . ( 3 ) The line δ = 1/2 is indicated in Fig.1 , using empirical measurements that add pre-asymptotic terms to Eq.3 . Overall , the green region corresponds to transformations that ( i ) are diffeomorphisms with high probability and ( ii ) produce significant displacements at least of the order of the pixel size . 1Throughout the paper , if not specified otherwise , bi-linear interpolation is employed .

Summary: The authors consider the relationship between the performance of modern deep neural network architectures trained on standard benchmark natural image datasets and their relationship to stability with respect to smooth deformations of the input image. These issues have been studied previously in the context of (natural) adversarial examples and in theoretically-constructed (i.e. non-trainable) networks -- the motivation and contribution here is to examine them through the lens of trained networks as well. To study this issue empirically, the authors introduce a maximum-entropy distribution on diffeomorphisms, which (I believe) amounts to constructing the deformation vector field as a (gaussian) random combination of periodic Fourier basis functions with frequency-dependent weights; they undertake an empirical investigation of the properties of this distribution in order to find parameter regimes where it produces realistic small-magnitude diffeomorphisms of the image plane (rather than distorting it beyond recognition, or doing nothing). They then use this tool to construct a computational "relative diffeomorphism stability" metric, which is the average stability of a predictor with respect to such diffeomorphisms divided by the predictor's average stability with respect to scale-appropriate gaussian noise perturbations, called $R_f$. With $R_f$, several findings are reported: across numerous modern architectures (ResNet18; EfficientNet) and datasets (CIFAR10; ImageNet), randomly-initialized networks have $R_f$ near unity and it decreases after training; $R_f$ decreases with the size of the training set (measured by subsampling); and test set performance correlates well with $R_f$ after training in the context of the CIFAR10 dataset. The authors present a toy theoretical model to give a theoretical corroboration of the $R_f$ parameter's behavior for a shallow network, and provide some ablation studies in the context of presence or not of data augmentation in the training process and the type of interpolation used to apply the diffeomorphisms.