SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Figure 5: Three estimated Hessian elements at  $x_0 = (0,0)^T$  for a 6-layer ReLU Network  $f:\mathbb{R}^2\to \mathbb{R}$  trained on the Nested Interactions dataset. (a) SmoothHess (SH) is estimated with isotropic covariance  $\Sigma = \sigma^2 I$  using granularly sampled  $\sigma^2\in \{1e - 6,\dots ,10\}$ . At minute  $\log_{10}\sigma^2 < -4$  either hyper-local noisy behavior is captured, or smoothing is so negligible that the smoothed function is approximately piece-wise linear with a low-magnitude Hessian. For  $\sigma^2$  ranging from  $\log_{10}\sigma^2 = -4$  to  $\log_{10}\sigma^2 = 0$  we see SmoothHess reflects the interactions in the dataset: Starting at  $\log_{10}\sigma^2 = -4$  both  $x_{1}x_{1}$  and  $x_{1}x_{2}$  have an interaction  $\approx 1$  until the interaction between  $x_{1}x_{1}$  begins to dip to 0 around  $\log_{10}\sigma^2 = -1.5$ . Finally around  $\log_{10}\sigma^2 = -1$  the interaction  $x_{1}x_{2}$  begins to dip toward -5, until  $\log_{10}\sigma^2 = 0$  when  $\sigma^2$  is so large that samples outside the training data distribution are incorporated into SmoothHess estimation. (b) The Hessian of the SoftPlus smoothed function  $f_{\beta}$  (SP Hess) is computed using granularly sampled  $\beta \in \{1e - 1,\dots ,1e4\}$ . Here, as  $\beta$  is decreased, it is not apparent that the variety of interactions in the Nested Interactions dataset are captured, either in relative ordering or magnitude.
super-pixel (outlined in black) and each other super-pixel are visualized as a heatmap overlaid upon the image. In order to facilitate comparison across images and methods, we standardize the heatmap colorbar to range between the most negative and most positive interaction values on a per-image and method basis.
One interesting trend seen in each case is that there is a strong positive interaction between the chosen super-pixel and one other super-pixel which (a) is spatially nearby and (b) contains the class object of interest. For example, in the first row, the side-view mirror of the car positively interacts with the front wheel. In the second row, the tip of the frogs head can be seen to interact positively with the side of the head. In the third row, the upper and lower portions of the dogs front leg have a strong positive interaction.
Due to the subjectivity of this comparison, we include quantitative results above each image. Specifically, we indicate the  $\mathcal{P}_{MSE}$  each method achieves within an  $\varepsilon = 0.25$  ball around each image. Optimal smoothing parameters were chosen for each method for this task (see Table 3). It can be seen that SmoothHess achieves the lowest  $\mathcal{P}_{MSE}$  in each case by a wide margin. Thus, SmoothHess may be the preferable option if one wishes for a visualization which best reflects the network's behaviour in an  $\varepsilon = 0.25$  ball around the image.
$$
\mathcal {P} _ {M S E} = 2. 0 e - 4
$$
SmoothHess
$$
\mathcal {P} _ {M S E} = 1. 0 e - 4
$$
$$
\mathcal {P} _ {M S E} = 7. 0 e - 4
$$
SoftPlus Hessian
$$
\mathcal {P} _ {M S E} = 3. 0 e - 4
$$
$$
\mathcal {P} _ {M S E} = 1. 5 e - 3
$$
Swish Hessian
$$
\mathcal {P} _ {M S E} = 4. 0 e - 4
$$
$$
\mathcal {P} _ {M S E} = 8. 0 e - 4
$$
Figure 6: Visualization of interactions between super-pixels found for a ResNet18 on CIFAR10 by SmoothHess, SoftPlus Hessian and Swish Hessian. Results are shown for test images of a car, frog and dog in the first second and third rows respectively. Each image is visualised in column one. Images are segmented into 20-25 super-pixels using the SLIC algorithm [2]. Interactions are summed between each pair of features in each pair of super-pixels. We show interactions with one given super-pixel in each image, outlined in black. SmoothHess, SoftPlus Hessian and Swish Hessian interactions for this super-pixel are visualized as heatmaps overlaid upon the image in columns two, three and four, respectively. The heatmap colorbar is standardized to range between the minimum and maximum interactions on each image-method pair separately, to facilitate comparison. Quantitative  $\mathcal{P}_{MSE}$  results for  $\varepsilon = 0.25$  are shown above each method-image pairing, with SmoothHess achieving the lowest  $\mathcal{P}_{MSE}$  in all cases.