SlowGuess/NeurIPS_2023_OCR huggingface dataset:
explanations. Indeed, according to [11, 15], optimal transport plans are potential surrogates or even substitutes to causal counterfactuals. Optimal Transport plan provides for any sample  $\pmb{x} \in \mu$  a sample  $\gamma_{\pi}(\pmb{x}) \in \nu$ , the closest in average on the pairing process. [15] prove that these optimal transport plans can even coincide with causal counterfactuals when available. Relying on this definition of OT counterfactual, Prop. 1 demonstrates that gradients of the optimal OTNN solution provide almost surely the direction to the counterfactual  $\gamma_{\pi}(\pmb{x}) = \pmb{x} - t\nabla_{\pmb{x}}\pmb{f}^{*}(\pmb{x})$ . Even if  $t$  is only partially known, using  $t = \pmb{f}^{*}(\pmb{x})$ , we know that  $\pmb{x}_{\delta}$  is on the decision boundary (Corr. 1) and is both an adversarial attack and a counterfactual explanation and  $|t| \geq |\pmb{f}^{*}(\pmb{x})|$  is on the path to the other distribution.
Thus the learning process of OTNNs induces a strong constraint on the gradients of the neural network, aligning them to the optimal transport plan. We claim that is the reason why the simple Saliency Maps for OTNNs have very good properties: We will demonstrate in the Sec 5 that, for the Saliency Map explanations:  $(i)$  metrics scores are higher or comparable to other explanation methods (which is not the case for unconstrained networks), thus it has higher ranks;  $(ii)$  distance to other attribution methods such as Smoothgrad is imperceptible;  $(iii)$  scores obtained on metrics that can be compared between networks are higher than those obtained with unconstrained networks;  $(iv)$  alignment with human explanations is impressive.
We conduct experiments with networks learnt on FashionMNIST [73], and 22 binary labels of CelebA [44] datasets, Cat vs Dog (binary classification,  $224x224x3$  uncentered images), and ImageNet [17]. Note that labels in CelebA are very unbalanced (see Table 2 in Appendix A.2, with for instance less than  $5\%$  samples for Mustache or Wearing_Hat).
Architectures used for OTNNs and unconstrained networks are similar (same number of layers and neurons, a VGG for FashionMNIST and CelebA, a ResNet50 for Cat vs Dog andImagenet). We also train an alternative of ResNet50 OTNN with twice the number of parameters (50 M). Unconstrained networks use batchnorm and ReLU layers for activation, whereas OTNNs only use GroupSort2 [5, 56] activation. OTNNs are built using the DEEL.LIP<sup>1</sup> library, using Björck orthogonalization projection algorithm for linear layers. Note that several other approaches can be used for orthogonalization without altering the theoretical results; these might potentially enhance experimental outcome scores. The loss functions are cross-entropy for unconstrained networks (categorical for multiclass, and sigmoid for multilabel settings), and hKR  $\mathcal{L}_{\lambda,m}^{hKR}$  (and the multiclass variants see appendix A.2.3) for OTNNs. We train all networks with Adam optimizer [39]. Details on architectures and parameters are given in Appendix A.2.
<sup>1</sup>https://github.com/deel-ai/deel-lip distributed under MIT License (MIT)
Dataset
Fash. MNIST	CelebA	Cat vs Dog	Imagenet
OTNN	Uncst.	OTNN	Uncst.	OTNN	Uncst.	OTNN	Uncst.
Attribution	μFidelity-Uniform (↑ is better)
Saliency	0.156	-0.001	0.244	0.052	0.091	0.080	0.240	0.004
SmoothGrad	0.114	-0.001	0.248	0.018	0.012	-0.004	0.001	-0.002
Integ. Grad.	-0.005	-0.013	0.149	0.093	0.022	0.024	0.046	0.022
Grad. Input	-0.017	-0.009	0.168	0.074	0.013	0.009	0.009	0.000
GradCam	0.215	0.02	0.028	0.002	0.101	0.052	0.029	0.046
Attribution	μFidelity-Zero (↑ is better)
Saliency	0.246	0.034	0.325	0.082	0.121	0.079	0.147	0.049
SmoothGrad	0.332	0.052	0.324	0.091	0.011	-0.004	0.001	0.002
Integ. Grad.	0.543	0.134	0.400	0.125	0.037	0.027	0.057	0.023
Grad. Input	0.479	0.079	0.439	0.093	0.019	0.004	0.020	-0.001
GradCam	0.161	0.046	0.127	0.061	0.136	0.049	0.048	0.068
Attribution	Stability Spearman rank (↓ is better)
Saliency	0.59	0.91	0.51	0.77	0.58	0.69	0.60	0.74
SmoothGrad	0.55	0.82	0.52	0.95	0.64	0.82	0.62	0.82
Integ. Grad.	0.61	0.79	0.52	0.87	0.61	0.76	0.60	0.74
Distance Saliency smoothgrad (↓ is better)
Saliency	7.5e-03	7.0e-02	3.1e-4	1.4e-1	3.7e-8	4.1e-8	3.7e-8	4.3e-8
Table 1: Comparison of XAI metrics for different attributions methods and dataset for OTNN and unconstrained networks.
Classification performance: OTNN models achieve comparable results to unconstrained ones, confirming claims of [8]: they reach  $88.5\%$  average accuracy on FashionMNIST (Table 6), and  $81\%$  (resp.  $82\%$ ) average Sensitivity (resp. Specificity) over labels on CelebA (Table 6 in Appendix A.3). We use Sensitivity and Specificity for CelebA to take into consideration the unbalanced labels. OTNNs achieve  $96\%$  accuracy ( $98\%$  for the unconstrained version) on Cat vs Dog and  $67\%$  ( $75\%$  for the unconstrained version) on Imagenet. The ResNet50 OTNN with 50M parameters achieves  $70\%$  accuracy on Imagenet.
We present the results of quantitative evaluations of XAI metrics to compare the Saliency Map method with other explanation methods on OTNN, and more generally compare XAI explanations methods on these networks and their unconstrained counterparts. On CelebA, we only present the results for the label Mustache, but results for the other labels are similar. Parameters for explanation methods and metrics are given in Appendix A.4. We have chosen to present in Table 1 two SoTA XAI metrics that enable comparison between OTNNs and unconstrained networks.  $\mu$ Fidelity metric [9] is a well-known method that measures the correlation between important variables defined by the explanation method and the model score decreases when these variables are reset to a baseline state (or replaced by uniform noise). Another important property for explanations is their stability for nearby samples. In [74], the authors proposed Stability metrics based on the  $L_{2}$  distance. To better evaluate this stability and make it comparable for different models, we replace the  $L_{2}$  distance by  $1 - \rho$ ,  $\rho$  being the Spearman rank correlation. Other model-dependent metrics are described in the Appendix. Results from the 50M parameter ResNet OTNN are included in the human alignment study (Fig 4) to illustrate that enhancing the model's complexity can bolster both the accuracy and alignment. The following observations can be drawn from Table 1:
Saliency Map on OTNNs exhibit more fidelity and stability : We confirm and amplify the results in [23]. Table. 1 clearly states that for most of the explanation methods, the  $\mu$ Fidelity, zero or uniform, is significantly higher for OTNNs. And above all, Saliency Map score for OTNNs is always higher than any other attribution method score for unconstrained models. A similar observation holds for the Stability Spearman rank : OTNN scores are better whatever the attribution method.
Saliency Map method on OTNNs is equivalent to other attribution methods: We observe that the scores from the Saliency Maps and other methods are very similar for OTNN, with Saliency Maps consistently ranking among the top attribution methods. For the unconstrained case, Saliency Maps are occasionally outperformed by other attribution methods. Notably, for the ResNet architecture, attribution methods other than Saliency Maps and GradCAM yield more erratic results for
saaee
SmoonnGrad
(a) OTNN
Figure 3: Comparison of Saliency Map and SmoothGrad explanations for (a) OTNN and (b) unconstrained network for, from left to right, CelebA, Cat vs Dog and Imagenet datasets.
(b) Unconstrained
fidelity metrics. To highlight these results, we compare the  $L_{2}$  distance between Saliency Maps and SmoothGrad explanations, as suggested by [2, 23, 65, 26]. The explanation distances for OTNN are significantly lower than for the unconstrained ones and closely approach zero, indicating that for OTNN, averaging over a large set of noisy inputs—as in SmoothGrad—is unnecessary. This is illustrated in Fig.3.
OTNNs explanations are aligned with human ones: Adopting the method presented in [43], using the ClickMe dataset, we follow strictly their experimental methodology and use their code<sup>2</sup> to compute the human feature alignment of OTNN Saliency Maps and compare with the others models tested in [22]– more than 100 recent deep neural networks. In Figure 4, we demonstrate that OTNN models Saliency Maps, which also carries a theoretical interpretation as the direction of the transport plan, is more aligned with human attention than any other tested models and significantly surpasses the Pareto front discovered by [22]. The OTNN model is even more aligned than a ResNet50 model trained with a specific alignment objective, proposed by [22], and called Harmonized ResNet50. This finding is interesting as it indicates OTNNs are less prone to relying on spurious correlations [25] and better capture human visual strategies for object recognition. The implications of these results are crucial for both cognitive science and industrial applications. A model that more closely aligns with human attention and visual strategies can provide a more comprehensive understanding of how vision operates for humans, and also enhance the predictability, interpretability, and performance of object recognition models in industry settings. Furthermore, the drop in alignment observed in recent models highlights the necessity of considering the alignment of model visual strategies with human attention while developing object recognition models to reduce the reliance on spurious correlations and ensure that our models get things right for the right reasons.
Qualitative results: Using the learnt OTNN on FashionMNIST, CelebA (Mouth Slightly Open label), Cat vs Dog and Imagenet, Fig. 6,5 present the original image, average gradients  $\nabla_{x}f_{j}$  over the channels, and images in the direction of the transport plan (Prop. 1), other samples are given in Appendix A.5. We can see that most of the gradients are visually consistent, showing clearly what has to be changed in the input image to modify the class and act as a counterfactual explanation. This is less clear for the Imagenet examples. This could be due to the difficulty of defining a transport plan for each pair of the 1000 classes. However, feature visualizations in Figure 1 show that the internal representation of the classes is still more interpretable than the unconstraint one. More generally, we observe that the gradient gives clear information about how the classifier makes its decision. For instance, for the cat, it shows that the classifier does not need to encode perfectly the concept of a cat, but mainly to identify the color of the eyes and size of the nose.
6 Conclusions and broader impact
In this paper, we study the explainability properties of OTNN (Optimal Transport Neural Networks) that are 1-Lipschitz constrained neural networks trained with an optimal transport dual loss. We
$^{2}$ https://github.com/serre-lab/Harmonization
Figure 4: OTNN are aligned with Human attention. Our study shows that the Saliency Map of OTNN model is highly aligned with human attention.
Figure 5: Samples of counterfactuals for FashionMNIST dataset on different classes and targets: (left) source image, (center) gradient image, (right) counterfactual of the form  $\pmb{x} - t*\hat{f} (\pmb {x})\nabla_{\pmb{x}}\hat{f} (\pmb {x})$  for  $t > 1$
establish that the gradient of the optimal solution aligns with the transportation plan direction, and the closest decision boundary point (adversarial example) also lies in this gradient direction at a distance of the absolute value of the network output. Relying on a formal definition of Optimal Transport counterfactuals, we build a link between the OTNN gradients and counterfactual explanations. We thus show that OTNNs lose jointly targets the classification task and induces a gradient alignment to the transportation plan. These beneficial properties of the gradient substantially enhance the Saliency Map XAI method for OTNN s. The experiments show that the simple Saliency Map has top-rank scores on state-of-the-art XAI metrics, and largely outperforms any method applied to unconstrained networks. Besides, as far as we know, our models are the first large provable 1-Lipschitz neural models that match state-of-the-art results on large problems. And even if counterfactual explanations are less compelling onImagenet, probably due to the complexity of transport for large number of classes, we prove that OTNNs Saliency Maps are impressively aligned to human explanations.
Broader impact. This paper demonstrates the value of OTNNs for critical problems. OTNNs are certifiably robust and explainable with the simple Saliency Map method (highly aligned with human explanations) and have accuracy performances comparable to unconstrained networks. Though OTNNs take 3-6 times longer to train than unconstrained networks, they have similar computational costs during inference. We hope that this contribution will raise a great interest in these OTNN networks.
CeelebA
mouth opened
mouth closed
mouth closed
CeeleB
blurry
CeebeA
→
→
not blurry
rosy cheeks
Caaee
cabbage butterfly  $\rightarrow$  monarch butterfly
Figure 6: Samples of counterfactuals from different datasets: (left) source image, (center) gradient image, (right) counterfactual of the form  $\pmb{x} - t*\hat{f} (\pmb {x})\nabla_{x}\hat{f} (\pmb {x})$ , for  $t > 1$
cat  $\rightarrow$  dog
w/o rosy cheeks
great grey owl  $\rightarrow$  bald eagle
$\longrightarrow$
dog  $\rightarrow$  cat
rosy cheeks
This work has benefited from the AI Interdisciplinary Institute ANITI, which is funded by the French "Investing for the Future – PIA3" program under the Grant agreement ANR-19-P3IA-0004. The authors gratefully acknowledge the support of the DEEL project.<sup>3</sup>
[1] E. M. Achour, F. Malgouyres, and F. Mamalet. Existence, stability and scalability of orthogonal convolutional neural networks, 2021.
[2] J. Adebayo, J. Gilmer, M. Muelly, I. Goodfellow, M. Hardt, and B. Kim. Sanity checks for saliency maps. In Proceedings of the 32nd International Conference on Neural Information Processing Systems, NIPS'18, page 9525-9536, Red Hook, NY, USA, 2018. Curran Associates Inc.
<sup>3</sup>https://www.deel.ai/
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A.1 Additional definition and proofs
Let us first recall the optimal transport problem associated with the minimization of  $\mathcal{L}_{\lambda,m}^{hKR}$ :
$$
\inf  _ {f \in L i p _ {1} (\Omega)} \mathcal {L} _ {\lambda (f), m} ^ {h K R} = \inf  _ {\pi \in \Pi_ {\lambda} ^ {p} (\mu , \nu)} \int_ {\Omega \times \Omega} | \boldsymbol {x} - \boldsymbol {z} | d \pi + \pi_ {\boldsymbol {x}} (\Omega) + \pi_ {\boldsymbol {z}} (\Omega) - 1 \tag {3}
$$
Where  $\Pi_{\lambda}^{p}(\mu, \nu)$  is the set consisting of positive measures  $\pi \in \mathcal{M}_{+}(\Omega \times \Omega)$  which are absolutely continuous with respect to the joint measure  $d\mu \times d\nu$  and  $\frac{d\pi_{\mathrm{x}}}{d\mu} \in [p, p(m + \lambda)]$ ,  $\frac{d\pi_{\mathrm{z}}}{d\nu} \in [1 - p, (1 - p)(m + \lambda)]$ . We name  $\pi^{*}$  the optimal transport plan according to Eq.3 and  $f^{*}$  the associated potential function.
Proof of proposition 1: According to [56], we have
$$
\left| \left| \nabla_ {x} f ^ {*} (\boldsymbol {x}) \right| \right| = 1
$$
almost surely and
$$
\mathbb {P} _ {(\boldsymbol {x}, \boldsymbol {y}) \sim \pi^ {*}} (| f ^ {*} (\boldsymbol {x}) - f ^ {*} (\boldsymbol {y}) | = | | \boldsymbol {x} - \boldsymbol {y} | |) = 1
$$
Following the proof of proposition 1 in [29] and [3] we have :
Given  $\pmb{x}_{\alpha} = \alpha * x + (1 - \alpha)y, 0 \leq \alpha \leq 1$
$$
\mathbb {P} _ {(\boldsymbol {x}, y) \sim \pi^ {*}} \left(\nabla_ {x} f ^ {*} (\boldsymbol {x} _ {\alpha}) = \frac {\boldsymbol {x} _ {\alpha} - y}{| | \boldsymbol {x} _ {\alpha} - y | |}\right) = 1.
$$
So for for  $\alpha = 1$  we have
$$
\mathbb {P} _ {(\pmb {x}, y) \sim \pi^ {*}} \left(\nabla_ {\pmb {x}} f ^ {*} (\pmb {x}) = \frac {\pmb {x} - y}{| | \pmb {x} - y | |}\right) = 1
$$
$$
\mathbb {P} _ {(\boldsymbol {x}, y) \sim \pi^ {*}} (y = \boldsymbol {x} - \nabla_ {x} f ^ {*} (\boldsymbol {x}). | | \boldsymbol {x} - y | |) = 1
$$
This proves the proposition 1 by choosing  $t = ||\pmb {x} - y||$
Proof of proposition 2: Let  $\mu$  and  $\nu$  two distributions with disjoint support with minimal distance  $\epsilon$  and  $f^{*}$  an optimal solution minimizing the  $\mathcal{L}_{\lambda,m}^{hKR}$  with  $m < 2\epsilon$ . According to [56],  $f^{*}$  is  $100\%$  accurate. Since the classification is based on the sign of  $f$  we have:  $\forall x \in \mu, f^{*}(x) \geq 0$  and  $\forall y \in \nu, f^{*}(y) \leq 0$ . Given  $x \in \mu$  and  $y = tr_{\pi}(x) = x - t\nabla_x f^*(x)$  and  $y \in \nu$ . According to the previous proposition we have:
$$
\begin{array}{l} \left| f ^ {*} (\boldsymbol {x}) - f ^ {*} (y) \right| = \left| \left| \boldsymbol {x} - y \right| \right| \\ | f ^ {*} (\boldsymbol {x}) - f ^ {*} (y) | = \left\| \boldsymbol {x} - (\boldsymbol {x} - t \nabla_ {x} f ^ {*} (\boldsymbol {x})) \right\| \\ \left| f ^ {*} (\boldsymbol {x}) - f ^ {*} (y) \right| = \left| \left| t \nabla_ {x} f ^ {*} (\boldsymbol {x})\right) \right| | \\ \left| f ^ {*} (\boldsymbol {x}) - f ^ {*} (y) \right| = t. | | \nabla_ {x} f ^ {*} (\boldsymbol {x})) | | \quad (t \geq 0) \\ \left| f ^ {*} (\boldsymbol {x}) - f ^ {*} (y) \right| = t \quad \left(\nabla_ {x} f ^ {*} (\boldsymbol {x}) = 1\right) \\ f ^ {*} (\boldsymbol {x}) - f ^ {*} (y) = t \quad (f ^ {*} (\boldsymbol {x}) \geq 0, f ^ {*} (y) \leq 0) \\ f ^ {*} (y) = f ^ {*} (\boldsymbol {x}) - t \\ \end{array}
$$
since  $f^{*}(y) \leq 0$  we obtain:
$$
f ^ {*} (\boldsymbol {x}) \leq t
$$
Since  $f^{*}$  is continuous,  $\exists t' > 0$  such that  $\pmb{x}_{\delta} = \pmb{x} - t'\nabla_{x}f^{*}(\pmb{x})$  and  $f^{*}(\pmb{x}_{\delta}) = 0$ . We have:
$$
\begin{array}{l} \left| f ^ {*} (\boldsymbol {x}) - f ^ {*} \left(\boldsymbol {x} _ {\delta}\right) \right| \leq | | \boldsymbol {x} - \boldsymbol {x} _ {\delta} | | \\ f ^ {*} (\boldsymbol {x}) \leq | | \boldsymbol {x} - (\boldsymbol {x} - t ^ {\prime} \nabla_ {x} f ^ {*} (\boldsymbol {x})) | | \\ f ^ {*} (\boldsymbol {x}) \leq t ^ {\prime} \\ \end{array}
$$
$$
\begin{array}{l} \left| f ^ {*} \left(\boldsymbol {x} _ {\delta}\right) - f ^ {*} (y) \right| \leq \left\| \boldsymbol {x} _ {\delta} - y \right\| \\ - f ^ {*} (y) \leq | | (\boldsymbol {x} - t ^ {\prime} \nabla_ {x} f ^ {*} (\boldsymbol {x})) - (\boldsymbol {x} - t \nabla_ {x} f ^ {*} (\boldsymbol {x})) | | \\ - f ^ {*} (y) \leq t - t ^ {\prime} \\ - f ^ {*} (y) \leq | | \boldsymbol {x} - y | | - t ^ {\prime} \\ \end{array}
$$
Then, if  $f^{*}(\pmb{x}) < t'$  we have
$$
f ^ {*} (\boldsymbol {x}) - f ^ {*} (y) <   t ^ {\prime} + \left\| \boldsymbol {x} - y \right\| - t ^ {\prime}
$$
$$
f ^ {*} (\boldsymbol {x}) - f ^ {*} (y) <   \left\| \boldsymbol {x} - y \right\|
$$
which is a contradiction so  $f^{*}(\pmb{x}) = t'$  and
$$
\boldsymbol {x} _ {\delta} = \boldsymbol {x} - f ^ {*} (\boldsymbol {x}) \nabla_ {x} f ^ {*} (\boldsymbol {x})
$$
A.2 Parameters and architectures
FashionMNIST has 50,000 images for training and 10,000 for test of size  $28 \times 28 \times 1$ , with 10 classes.
CelebA contains 162,770 training samples, 19,962 samples for test of size  $218 \times 178 \times 3$ . We have used a subset of 22 labels: Attractive, Bald, Big_Nose, Black_Hair, Blond_Hair, Blurry, Brown_Hair, Eyeglasses, Gray_Hair, Heavy_Makeup, Male, Mouth_Slightly_Open, Mustache, Receding_Hairline, Rosy_Cheeks, Sideburns, Smiling, Wearing_Earrings, Wearing_Hat, Wearing_Lipstick, Wearing_Necktie, Young.
Note that labels in CelebA are very unbalanced (see Table 2, with less than  $5\%$  samples for Mustache or Wearing_Hat for instance). Thus we will use Sensibility and Specificity as metrics.
Table 2: CelebA label distribution: proportion of positive samples in training set (testing set) [bold: very unbalanced labels]
Attractive	Bald	Big_Nose	Black_Hair	Blond_Hair
0.51 (0.50)	0.02 (0.02)	0.24 (0.21)	0.24 (0.27)	0.15 (0.13)
Blurry	Brown_Hair	Eyeglasses	Gray_Hair	Heavy_Makeup
0.05 (0.05)	0.20 (0.18)	0.06 (0.06)	0.04 (0.03)	0.38 (0.40)
Male	Mouth_Slightly_Open	Mustache	Receding_Hairline	Rosy_Cheeks
0.42 (0.39)	0.48 (0.50)	0.04 (0.04)	0.08 (0.08)	0.06 (0.07)
Sideburns	Smiling	Wearing_Earrings	Wearing_Hat	Wearing_Lipstick
0.06 (0.05)	0.48 (0.50)	0.19 (0.21)	0.05 (0.04)	0.47 (0.52)
Wearing_Necktie	Young
0.12 (0.14)	0.78 (0.76)
Cat vs Dog contains 17400 training samples, 5800 test samples of various size.
Imagenet contains 1M training samples, 100 000 samples for test of various size.
preprocessing: For FashionMNIST Images are normalized between [0, 1] with no augmentation. For CelebA dataset, data augmentation is used with random crop, horizontal flip, random brightness, and random contrast. For imagenet and cat vs dog we use the standart preprocessing of resnet (with no normalization in [0, 1])
A.2.2 Architectures
As indicated in the paper, linear layers for OTNN and unconstrained networks are equivalent (same number of layers and neurons), but unconstrained networks use batchnorm and ReLU layer for activation, whereas OTNN only use GroupSort2 [5, 56] activation. OTNN are built using DEEL.LIP library.