Relative stability toward diffeomorphisms indicates performance in deep nets^*

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 (von 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 (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 one-hidden-layer fully connected (FC) nets. Learning features can occur and improve performance when the 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, Ghorbani et al 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, Geiger et al 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, Bietti and Mairal 2019b). Interestingly, enforcing such a stability can improve performance (Kayhan and van 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

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 (τ_u (s), τ_v (s)). The deformation amplitude is measured by the norm

$$\begin{equation}{\Vert}\nabla \tau {{\Vert}}^{2}={\int }_{{[0,1]}^{2}}({(\nabla {\tau }_{u})}^{2}+{(\nabla {\tau }_{v})}^{2})\mathrm{d}u\,\mathrm{d}v.\end{equation} \tag{ 1 }$$

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 τ_u and τ_v 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:

$$\begin{equation}{\tau }_{u}=\sum\limits _{i,j\in {\mathbb{N}}^{+}}{C}_{ij}\,\sin (i\pi u)\sin (j\pi v)\end{equation} \tag{ 2 }$$

where the C_ij are Gaussian variables of zero mean and variance $\langle {C}_{ij}^{2}\rangle =T/({i}^{2}+{j}^{2})$. 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) n² T, 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 equation (2) to i² + j² ⩽ c², one then has ||∇τ||² = (π³/8) c² T.

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-linear ¹ , 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 figure 1 as a function of T and c.

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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 figure 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 equation (2) that asymptotically for large c (cf appendix B for the derivation),

$$\begin{equation}{\delta }^{2}=\frac{\pi }{4}{n}^{2}T\,\mathrm{ln}(c).\end{equation} \tag{ 3 }$$

The line δ = 1/2 is indicated in figure 1, using empirical measurements that add pre-asymptotic terms to equation (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.

Relative stability to diffeomorphisms. To quantify how a deep net f learns to become less sensitive to diffeomorphisms than to generic data transformations, we define the relative stability to diffeomorphisms R_f as:

$$\begin{equation}{R}_{f}=\frac{\langle {\Vert}f(\tau x)-f(x){{\Vert}}^{2}{\rangle }_{x,\tau }}{\langle {\Vert}f(x+\eta )-f(x){{\Vert}}^{2}{\rangle }_{x,\eta }}.\end{equation} \tag{ 4 }$$

where the notation ⟨⟩_y can indicate alternatively the mean or the median with respect to the distribution of y. In the numerator, this operation is made over the test set and over the ensemble of diffeomorphisms of parameters (T, c) (on which R_f implicitly depends). In the denominator, the average is on the test set and on the vectors η sampled uniformly on the sphere of radius ||η|| = ⟨||τx − x||⟩_x,τ . An illustration of what R_f captures is shown in figure 2. In the main text, we consider median quantities, as they reflect better the typical values of distribution. In appendix E.3 we show that our results for mean quantities, for which our conclusions also apply.

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Dependence of R _f on the diffeomorphism magnitude. Ideally, R_f could be defined for infinitesimal transformations, as it would then characterize the magnitude of the gradient of f along smooth deformations of the images, normalized by the magnitude of the gradient in random directions. However, infinitesimal diffeomorphisms move the image much less than the pixel size, and their definition thus depends significantly on the interpolation method used. It is illustrated in the left panels of figure 3, showing the dependence of R_f in terms of the diffeomorphism magnitude (here characterised by the mean displacement magnitude at the center of the image δ) for several interpolation methods. We do see that R_f becomes independent of the interpolation when δ becomes of order unity. In what follows we thus focus on R_f (δ = 1), which we denote R_f .

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SOTA architectures become relatively stable to diffeomorphisms during training, but are not at initialization. The central panels of figure 3 show R_f at initialization (shaded), and after training (full) for two SOTA architectures on four benchmark data sets. The first key result is that, at initialization, these architectures are as sensitive to diffeomorphisms as they are to random transformations. Relative stability to diffeomorphisms at initialization (guaranteed theoretically in some cases (Bietti and Mairal 2019a, Bietti and Mairal 2019b)) thus does not appear to be indicative of successful architectures.

By contrast, for these SOTA architectures, relative stability toward diffeomorphisms builds up during training on all the data sets probed. It is a significant effect, with values of R_f after training generally found in the range R_f ∈ [10⁻², 10⁻¹].

Standard data augmentation techniques (translations, crops, and horizontal flips) are employed for training. However, the results we find only mildly depend on using such techniques, see figure 12 in the appendix.

Learning relative stability to diffeos requires large training sets. How many data are needed to learn relative stability toward diffeomorphisms? To answer this question, newly initialized networks are trained on different training-sets of size P. R_f is then measured for CIFAR10, as indicated in the right panels of figure 3. Neural nets need a certain number of training points (P ∼ 10³) in order to become relatively stable toward smooth deformations. Past that point, R_f monotonically decreases with P. In a range of P, this decrease is approximately compatible with the an inverse behavior R_f ∼ 1/P found in the simple model of section 6. Additional results for MNIST and FashionMNIST can be found in figure 13, appendix E.3.

Simple architectures do not become relatively stable to diffeomorphisms. To test the universality of these results, we focus on two simple architectures: (i) a four-hidden-layer FC network (FullConn-L4) where each hidden layer has 64 neurons and (ii) LeNet (LeCun et al 1989) that consists of two convolutional layers followed by local max-pooling and three fully-connected layers.

Measurements of R_f for these networks are shown in figure 4. For the FC net, R_f ≈ 1 at initialization (as observed for SOTA nets) but grows after training on the full data set, showing that FC nets do not learn to become relatively stable to smooth deformations. It is consistent with the modest evolution of R_f (P) with P, suggesting that huge training sets would be required to obtain R_f < 1. The situation is similar for the primitive CNN LeNet, which only becomes slightly insensitive (R_f ≈ 0.6) in a single data set (CIFAR10), and otherwise remains larger than unity.

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Layers' relative stability monotonically increases with depth. Up to this point, we measured the relative stability of the output function for any given architecture. We now study how relative stability builds up as the input data propagate through the hidden layers. In figure 14 of appendix E.3, we report R_f as a function of depth for both simple and deep nets. What we observe is ${R}_{{f}_{0}}\approx 1$ independently of depth at initialization, and monotonically decreases with depth after training. Overall, the gain in relative stability appears to be well-spread through the net, as is also found for stability alone (Ruderman et al 2018).

Thus, SOTA architectures appear to become relatively stable to diffeomorphisms after training, unlike primitive architectures. This observation suggests that high performance requires such a relative stability to build up. To test further this hypothesis, we select a set of architectures that have been relevant in the SOTA progress over the past decade; we systematically train them in order to compare R_f to their test error _t. Apart from FC nets, we consider the already cited LeNet (5 layers and $\approx 60k$ parameters); then AlexNet (Krizhevsky et al 2012) and VGG (Simonyan and Zisserman 2015), deeper (8–19 layers) and highly over-parametrized (10–20M (million) params.) versions of the latter. We introduce batch-normalization in VGGs and skip connections with ResNets. Finally, we go to EfficientNets, that have all the advancements introduced in previous models and achieve SOTA performance with a relatively small number of parameters (<10M); this is accomplished by designing an efficient small network and properly scaling it up. Further details about these architectures can be found in table 1, appendix E.2.

The results are shown in figure 5. The correlation between R_f and _t is remarkably high (corr. coeff. ³ : 0.97), suggesting that generating low relative sensitivity to diffeomorphisms R_f is important to obtain good performance. In appendix E.3 we also report how changing the train set size P affects the position of a network in the (_t, R_f ) plane, for the four architectures considered in the previous section (figure 18). We also show that our results are robust to changes of δ, c (figure 21) and data sets (figure 20).

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What architectures enable a low R_f value? The latter can be obtained with skip connections or not, and for quite different depths as indicated in figure 5. Also, the same architecture (EfficientNetB0) trained by transfer learning from ImageNet—instead of directly on CIFAR10—shows a large improvement both in performance and in diffeomorphisms invariance. Clearly, R_f is much better predicted by _t than by the specific features of the architecture indicated in figure 5.

The relative stability to diffeomorphisms R_f can be written as R_f = D_f /G_f where G_f characterizes the stability with respect to additive noise and D_f the stability toward diffeomorphisms:

$$\begin{equation}{G}_{f}=\frac{\langle {\Vert}f(x+\eta )-f(x){{\Vert}}^{2}{\rangle }_{x,\eta }}{\langle {\Vert}f(x)-f(z){{\Vert}}^{2}{\rangle }_{x,z}},\qquad \quad {D}_{f}=\frac{\langle {\Vert}f(\tau x)-f(x){{\Vert}}^{2}{\rangle }_{x,\tau }}{\langle {\Vert}f(x)-f(z){{\Vert}}^{2}{\rangle }_{x,z}}.\end{equation} \tag{ 5 }$$

Here, we chose to normalize these stabilities with the variation of f over the test set (to which both x and z belong), and η is a random noise whose magnitude is prescribed as above. Stability toward additive noise has been studied previously in FC architectures (Novak et al 2018) and for CNNs as a function of spatial frequency in Tsuzuku and Sato (2019), Yin et al (2019).

The decrease of R_f with growing training set size P could thus be due to an increase in the stability toward diffeomorphisms (i.e. D_f decreasing with P) or a decrease of stability toward noise (G_f increasing with P). To test these possibilities, we show in figure 6 G_f (P), D_f (P) and R_f (P) for MNIST, Fashion MNIST and CIFAR10 for two SOTA architectures. The central results are that (i) stability toward noise is always reduced for larger training sets. This observation is natural: when more data needs to be fitted, the function becomes rougher. (ii) Stability toward diffeomorphisms does not behave universally: it can increase with P or decrease depending on the architecture and the training set. Additionally, G_f and D_f alone show a much smaller correlation with performance than R_f —see figures 15–17 in appendix E.3.

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In this section, we discuss the simplest model of invariance in data where stability to transformation builds up, that can be compared with our observations of R_f above. Specifically, we consider the 'stripe' model (Paccolat et al 2021b), corresponding to a binary classification task for Gaussian-distributed data points x = (x_||, x_⊥) where the label function depends only on one direction in data space, namely y(x) = y(x_||). Layers of y = +1 and y = −1 regions alternate along the direction x_||, separated by parallel planes. Hence, the data present d − 1 invariant directions in input-space denoted by x_⊥ as illustrated in figure 7 (left).

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When this model is learnt by a one-hidden-layer FC net, the first layer of weights can be shown to align with the informative direction (Paccolat et al 2021a). The projection of these weights on the orthogonal space vanishes with the training set size P as $1/\sqrt{P}$, an effect induced by the sampling noise associated to finite training sets.

In this model, R_f can be defined as:

$$\begin{equation}{R}_{f}=\frac{\langle {\Vert}f({x}_{{\Vert}},{x}_{\perp }+\nu )-f({x}_{{\Vert}},{x}_{\perp }){{\Vert}}^{2}{\rangle }_{x,\nu }}{\langle {\Vert}f(x+\eta )-f(x){{\Vert}}^{2}{\rangle }_{x,\eta }},\end{equation} \tag{ 6 }$$

where we made explicit the dependence of f on the two linear subspaces. Here, the isotropic noise ν is added only in the invariant directions. Again, we impose ||η|| = ||ν||. R_f (P) is shown in figure 7 (right). We observe that R_f (P) ∼ P⁻¹, as expected from the weight alignment mentioned above.

Interestingly, figure 3 for CIFAR10 and SOTA architectures support that the 1/P behavior is compatible with the observations for some range of P. In appendix E.3, figure 13, we show analogous results for MNIST and Fashion-MNIST. We observe the 1/P power-law scaling for ResNets. It suggests that for these architectures, learning to become invariant to diffeomorphisms may be limited by a naive measure of sampling noise as well. By contrast for EfficientNets, in which the decrease in R_f is more limited, a 1/P behavior cannot be identified.

A common belief is that stability to random noise (small G_f ) and to diffeomorphisms (small D_f ) are desirable properties of neural nets. Its underlying assumption is that the true data label mildly depends on such transformations when they are small. Our observations suggest an alternative view:

• (a)  
  Figures 6 and 16: better predictors are more sensitive to small perturbations in input space.
• (b)  
  As a consequence, the notion that predictors are especially insensitive to diffeomorphisms is not captured by stability alone, but rather by the relative stability R_f = D_f /G_f .
• (c)  
  We propose the following interpretation of figure 5: to perform well, the predictor must build large gradients in input space near the decision boundary—leading to a large G_f overall. Networks that are relatively insensitive to diffeomorphisms (small R_f ) can discover with less data that strong gradients must be there and generalize them to larger regions of input space, improving performance and increasing G_f .

This last point can be illustrated in the simple model of section 6, see figure 7 (left) panel. Imagine two data points of different labels falling close to the, e.g. left true decision boundary. These two points can be far from each other if their orthogonal coordinates differ. Yet, if R_f = 0 (now defined in equation (6)), then the output does not depend on the orthogonal coordinates, and it will need to build a strong gradient—in input space—along the parallel coordinate to fit these two data. This strong gradient will exist throughout that entire decision boundary, improving performance but also increasing G_f . Instead, if R_f = 1, fitting these two data will not lead to a strong gradient, since they can be far from each other in input space. Beyond this intuition, in this model decreasing R_f can quantitatively be shown to increase performance, see Paccolat et al (2021b).

We have introduced a novel empirical framework to characterize how deep nets become invariant to diffeomorphisms. It is jointly based on a maximum-entropy distribution for diffeomorphisms, and on the realization that stability of these transformations relative to generic ones R_f strongly correlates to performance, instead of just the diffeomorphisms stability considered in the past.

The ensemble of smooth deformations we introduced may have interesting applications. It could serve as a complement to traditional data-augmentation techniques (whose effect on relative stability is discussed in figure 12 of the appendix). A similar idea is present in Hauberg et al (2016), Shen et al (2020) but our deformations have the advantage of being easier to sample and data agnostic. Moreover, the ensemble could be used to build adversarial attacks along smooth transformations, in the spirit of Alaifari et al (2018), Engstrom et al (2019), Kanbak et al (2018). It would be interesting to test if networks robust to such attacks are more stable in relative terms, and how such robustness affects their performance.

Finally, the tight correlation between relative stability R_f and test error _t suggests that if a predictor displays a given R_f , its performance may be bounded from below. The relationships we observe _t(R_f ) may then be indicative of this bound, which would be a fundamental property of a given data set. Can it be predicted in terms of simpler properties of the data? Introducing simplified models of data with controlled stability to diffeomorphisms beyond the toy model of section 6 would be useful to investigate this key question.

We thank Alberto Bietti, Joan Bruna, Francesco Cagnetta, Pascal Frossard, Jonas Paccolat, Antonio Sclocchi and Umberto M Tomasini for helpful discussions. This work was supported by a grant from the Simons Foundation (#454953 Matthieu Wyart).

Under the constraint on the borders, τ_u and τ_v can be expressed in a real Fourier basis as in equation (2). By injecting this form into ||∇τ||² we obtain:

$$\begin{equation}{\Vert}\nabla \tau {{\Vert}}^{2}=\frac{{\pi }^{2}}{4}\sum\limits _{i,j\in {\mathbb{N}}^{+}}({C}_{ij}^{2}+{D}_{ij}^{2})({i}^{2}+{j}^{2})\end{equation} \tag{ 7 }$$

where D_ij are the Fourier coefficients of τ_v . We aim at computing the probability distributions that maximize their entropy while keeping the expectation value of ||∇τ||² fixed. Since we have a sum of quadratic random variables, the equipartition theorem (Beale 1996) applies: the distributions are normal and every quadratic term contributes in average equally to ||∇τ||². Thus, the variance of the coefficients follows $\frac{T}{{i}^{2}+{j}^{2}}$ where the parameter T determines the magnitude of the diffeomorphism.

Average pixel displacement magnitude δ . We derive here the large-c asymptotic behavior of δ (equation (3)). This is defined as the average square norm of the displacement field, in pixel units:

$$\begin{align*}\hfill {\delta }^{2}& ={n}^{2}{\int }_{{[0,1]}^{2}}{\Vert}\tau (u,v){{\Vert}}^{2}\mathrm{d}u\,\mathrm{d}v\hfill \\ \hfill & =2T{n}^{2}\sum\limits _{{i}^{2}+{j}^{2}\leqslant {c}^{2}}\frac{1}{{i}^{2}+{j}^{2}}{\int }_{{[0,1]}^{2}}{\sin }^{2}(i\pi u){\sin }^{2}(j\pi v)\mathrm{d}u\,\mathrm{d}v\hfill \\ \hfill & =\frac{T{n}^{2}}{2}\sum\limits _{{i}^{2}+{j}^{2}\leqslant {c}^{2}}\frac{1}{{i}^{2}+{j}^{2}}\hfill \\ \hfill & \approx \frac{T{n}^{2}}{2}{\int }_{1\leqslant {x}^{2}+{y}^{2}\leqslant {c}^{2}}\frac{1}{{x}^{2}+{y}^{2}}\mathrm{d}x\,\mathrm{d}y\hfill \\ \hfill & =\frac{\pi T{n}^{2}}{4}{\int }_{1}^{c}\frac{1}{r}\,\mathrm{d}r\hfill \\ \hfill & =\frac{\pi }{4}{n}^{2}T\,\mathrm{log}\,c,\hfill \end{align*}$$

where we approximated the sum with an integral, in the third step. The asymptotic relations for ||∇τ|| that are reported in the main text are computed in a similar fashion. In figure 8, we check the agreement between asymptotic prediction and empirical measurements. If δ ≪ 1, our results strongly depend on the choice of interpolation method. To avoid it, we only consider conditions for which δ ⩾ 1/2, leading to

$$\begin{equation}T > \frac{1}{\pi {n}^{2}\,\mathrm{log}\,c}.\end{equation} \tag{ 8 }$$

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Condition for diffeomorphism in the (T, c) plane. For a given value of c, there exists a temperature scale beyond which the transformation is not injective anymore, affecting the topology of the image and creating spurious boundaries, see figures 9(a)–(c) for an illustration. Specifically, consider a curve passing by the point s in the deformed image. Its tangent direction is u at the point s. When going back to the original image (s' = s − τ(s)) the curve gets deformed and its tangent becomes

$$\begin{equation}{u}^{\prime }=u-(u\cdot \nabla )\tau (s).\end{equation} \tag{ 9 }$$

A smooth deformation is bijective iff all deformed curves remain curves which is equivalent to have non-zero tangents everywhere

$$\begin{equation}\forall \enspace s,u\ne 0\quad {\Vert}{u}^{\prime }{\Vert}\ne 0.\end{equation} \tag{ 10 }$$

Imposing ||u'|| ≠ 0 does not give us any constraint on τ. Therefore, we constraint τ a bit more and allow only displacement fields such that u ⋅ u' > 0, which is a sufficient condition for equation (10) to be satisfied—cf figure 9(d). By extremizing over u, this condition translates into

$$\begin{equation}\frac{1}{2}\left(\sqrt{{({\partial }_{x}{\tau }_{x}-{\partial }_{y}{\tau }_{y})}^{2}+{({\partial }_{x}{\tau }_{y}+{\partial }_{y}{\tau }_{x})}^{2}}-{\partial }_{x}{\tau }_{x}-{\partial }_{y}{\tau }_{y}\right)< 1\end{equation} \tag{ 11 }$$

or, equivalently,

$$\begin{equation}{\Xi}=\frac{1}{2}\left(\sqrt{{\Vert}\nabla \tau {{\Vert}}^{2}-2\,\mathrm{det}(\nabla \tau )}-\mathrm{Tr}(\nabla \tau )\right)< 1,\end{equation} \tag{ 12 }$$

were we identified by Ξ the lhs of the inequality. We find that the median of the maximum of Ξ over all the image (||Ξ(s)||_∞) can be approximated by (see figure 8(b)):

$$\begin{equation}\underset{s}{\mathrm{max}}{\Xi}\simeq \frac{c}{2}\sqrt{{\pi }^{3}T\,\mathrm{log}\,c}.\end{equation} \tag{ 13 }$$

The resulting constraint on T reads

$$\begin{equation}T< \frac{4}{{\pi }^{3}{c}^{2}\,\mathrm{log}\,c}.\end{equation} \tag{ 14 }$$

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When a deformation is applied to an image x, each of its pixels gets mapped, from the original pixels grid, to new positions generally outside of the grid itself—cf figures 9(a) and (b). A procedure (interpolation method) needs to be defined to project the deformed image back into the original grid.

For simplicity of notation, we describe interpolation methods considering the square [0, 1]² as the region in between four pixels—see an illustration in figure 10(a). We propose here two different ways to interpolate between pixels and then check that our measurements do not depend on the specific method considered.

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Bi-linear interpolation. The bi-linear interpolation consists, as the name suggests, of two steps of linear interpolation, one on the horizontal, and one on the vertical direction—figure 10(b). If we look at the square [0, 1]² and we apply a deformation τ such that (0, 0) ↦ (u, v), we have

$$\begin{equation}x(u,v)=x(0,0)(1-u)(1-v)+x(1,0)u(1-v)+x(0,1)(1-u)v+x(1,1)uv.\end{equation} \tag{ 15 }$$

Gaussian interpolation. In this case, a Gaussian function ⁴ is placed on top of each point in the grid—cf figure 10. The pixel intensity x can be evaluated at any point outside the grid by computing

$$\begin{equation}x(s)=\frac{{\sum }_{i}x({s}_{i})G(s-{s}_{i})}{{\sum }_{i}G(s-{s}_{i})}.\end{equation} \tag{ 16 }$$

In order to fix the standard deviation σ of G, we introduce the participation ratio n. Given Ψ_i = G(s, s_i )|_s=(0.5,0.5), we define

$$\begin{equation}n=\frac{{\left({\sum }_{i}{{\Psi}}_{i}^{2}\right)}^{2}}{{\sum }_{i}{{\Psi}}_{i}^{4}}.\end{equation} \tag{ 17 }$$

The participation ratio is a measure of how many pixels contribute to the value of a new pixel, which results from interpolation. We fix σ in such a way that the participation ratio for the Gaussian interpolation matches the one for the bi-linear (n = 4), when the new pixel is equidistant from the four pixels around. This gives σ = 0.4715.

Notice that this interpolation method is such that it applies a Gaussian smoothing of the image even if τ is the identity. Consequently, when computing observables for f with the Gaussian interpolation, we always compare f(τx) to $f(\tilde{x})$, where $\tilde{x}$ is the smoothed version of x, in such a way that $f({\tau }^{[T=0]}x)=f(\tilde{x})$.

Empirical results dependence on interpolation. Finally, we checked to which extent our results are affected by the specific choice of interpolation method. In particular, blue and red colors in figures 3 and 13 correspond to bi-linear and Gaussian interpolation, respectively. The interpolation method only affects the results in the small displacement limit (δ → 0).

Note. Throughout the paper, if not specified otherwise, bi-linear interpolation is employed.

We introduced in section 5 the stability toward additive noise:

$$\begin{equation}{G}_{f}=\frac{\langle {\Vert}f(x+\eta )-f(x){{\Vert}}^{2}{\rangle }_{x,\eta }}{\langle {\Vert}f(x)-f(z){{\Vert}}^{2}{\rangle }_{x,z}}.\end{equation} \tag{ 18 }$$

We study here the dependence of G_f on the noise magnitude ||η||. In the η → 0 limit, we expect the network function to behave as its first-order Taylor expansion, leading to G_f ∝ ||η||². Hence, for small noise, G_f gives an estimate of the average magnitude of the gradient of f in a random direction η.

Empirical results. Measurements of G_f on SOTA nets trained on benchmark data-sets are shown in figure 11. We observe that the effect of non-linearities start to be significant around ||η|| = 1. For large values of the noise—i.e. far away from data-points—the average gradient of f does not change with training.

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In this appendix, we provide details on the training procedure, on the different architectures employed and some additional experimental results.

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E.1. Image classification training set-up:

E.2. Networks architectures

E.3. Additional figures

We present here:

• Figure 13. R_f as a function of P for MNIST and FashionMNIST with the corresponding predicted slope, omitted in the main text.
• Figure 14. Relative diffeomorphisms stability R_f as a function of depth for simple and deep nets.
• Figures 15 and 16. Diffeomorphisms and inverse of the Gaussian stability D_f and 1/G_f vs test error for CIFAR10 and the set of architectures considered in section 4.
• Figure 17. D_f , 1/G_f and R_f when using the mean in place of the median for computing averages ⟨⋅⟩.
• Figure 18. Curves in the (_t, R_f ) plane when varying the training set size P for FullyConnL4, LeNet, ResNet18 and EfficientNetB0.
• Figures 19 and 22. Error estimates for the main quantities of interest—often omitted in the main text for the sake of figures' clarity.

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