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| Question: Which property do these two objects have in common? |
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| information that could be helpful for answering the question. In-context example(s) |
| Question: Which property do these two objects have in common? |
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+ "html": "| AigormmmrT.CartoonA Data: Image x ∈ [0,1]exwxt with c channels and wt pixels, classifier Φ. Result: CartoonX explanation ε ∈ [0,1]w×t for decision Φ(x). Hyperparameters: Sparsity level X > O, number of steps N, number of noise samples L. Initialize mask s := [1,..,1]T ∈ [0,1]k on DWT coefficients h = [h1,.., hk] with x = f(h), where f is the discrete inverse wavelet transform; Compute predicted label j* := arg maxi Φ(x); fori←1toNdo |
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| Invert wavelet mask s back to pixel space as & := f(h s) ; Clip the explanation ε into [0,1]w×t to obtain ε. Visualize ε; |
+
+# A.2 EXPERIMENT DETAILS
+
+Throughout our experiments with CartoonX and Pixel RDE, we used a learning rate of $\epsilon = 0 . 0 0 1$ , a sample size of $L = 6 4$ for the adaptive Gaussian noise, and $N = 2 0 0 0$ steps. Several different sparsity levels were used. We recommend specifying the sparsity level in terms of the number of mask entries $k$ , i.e., choosing the product $\lambda k$ . Pixel RDE typically requires a smaller sparsity level than CartoonX. We chose $\bar { \lambda k } \in [ \bar { 2 0 } , 8 0 ]$ for CartoonX and $\bar { \lambda } k \in [ 3 , 2 \bar { 0 } ]$ for Pixel RDE. The obfuscation strategy for Pixel RDE was chosen as Gaussian adaptive noise with mean and standard deviation computed for all pixel values (see Section 4.1.2). In Appendix 8, we show that Gaussian adaptive noise produces much more interpretable explanations than using a zero baseline perturbation. We implemented the DWT for CartoonX with the Pytorch Wavelets package, which is compatible with PyTorch gradient computations, and chose the Daubechies wavelet system with $J = 5$ scales and zero-padding. For the Integrated Gradients method, we used 100 steps, and for the Smoothgrad method, we used 10 samples and a standard deviation of 0.1.
+
+# A.3 SENSITIVITY TO HYPERPARAMETERS
+
+We compare CartoonX’s sensitivity to its main hyperparameters, i.e., the sparsity level $\lambda$ , the perturbation distribution $\gamma _ { s }$ , and the distortion measure $\bar { d ( \Phi ( x ) , \Phi ( y ) ) }$ . For each experiment, we fix all but one of the three parameters.
+
+# A.3.1 SENSITVITY TO THE SPARSITY LEVEL $\lambda$
+
+Figure 7 plots CartoonX explanations and Pixel RDEs for increasing $\lambda$ —the hyperparameter determining the explanation’s sparsity in the respective representation system. We find that CartoonX is less sensitive than Pixel RDE to $\lambda$ . In practice, this means one can find a suitable $\lambda$ faster for CartoonX than for Pixel RDE.
+
+# A.3.2 SENSITVITY TO THE DISTRIBUTION $\gamma _ { s }$
+
+Figure 8 plots CartoonX explanations for two choices of $\gamma _ { s }$ : (1) Gaussian adaptive noise (see Section 4.1.2) and (2) constant zero perturbations (i.e. $v = 0$ with probability one under $\gamma _ { s }$ ). We observe that the Gaussian adaptive noise gives much more meaningful explanations than the simple zero baseline perturbations.
+
+
+Figure 7: We compare the sensitivity of CartoonX and Pixel RDE to the sparsity level $\lambda$ . The top row depicts CartoonX, and the bottom row depicts Pixel RDE, for increasing values of $\lambda$ . Note that for $\lambda = 0$ , Pixel RDE is entirely yellow because the mask is initialized as $s ^ { \check { = } } [ 1 \ldots 1 ] ^ { T }$ and $\lambda = 0$ provides no incentive to make s sparser. For the same reason, CatoonX is simply the greyscale image for $\lambda = 0$ .
+
+
+Figure 8: We compare the sensitivity of CartoonX to the perturbation distribution $\gamma _ { s }$ . The top image was classified as a fountain and the bottom image as a viaduct. The second column depicts CartoonX with $\gamma _ { s }$ as Gaussian adaptive noise, and the third column depicts CartoonX with $\gamma _ { s }$ as constant zero perturbations (zero baseline). We observe that Gaussian adaptive noise is much more interpretable than the zero baseline.
+
+# A.3.3 SENSITVITY TO THE DISTORTION MEASURE $d ( \Phi ( x ) , \Phi ( y ) )$
+
+Figure 9 plots CartoonX explanations for the following four choices of $d ( \Phi ( x ) , \Phi ( y ) )$ , where $x$ is the original input, $y$ is the RDE obfuscation, and $\Phi$ outputs post-softmax probabilities:
+
+1. $d ( \Phi ( x ) , \Phi ( y ) ) = ( \Phi _ { j ^ { * } } ( x ) - \Phi _ { j ^ { * } } ( y ) ) ^ { 2 }$ , where $j ^ { * } : = \arg \operatorname* { m a x } _ { j } \Phi _ { j } ( x )$ (squared $\ell _ { 2 }$ in label )
+
+2. $d ( \Phi ( x ) , \Phi ( y ) ) = ( \Phi _ { j ^ { * } } ( x ) - 1 ) ^ { 2 }$ , where $j ^ { * } : = \arg \operatorname* { m a x } _ { j } \Phi _ { j } ( x )$ (maximize label)
+3. $d ( \Phi ( x ) , \Phi ( y ) ) = \lVert \Phi ( x ) - \Phi ( y ) \rVert _ { 2 }$ ( $\ell _ { 2 }$ probabilities)
+4. $d ( \Phi ( x ) , \Phi ( y ) ) = K L ( \Phi ( y ) , \Phi ( x ) )$ (KL-Divergence)
+
+The explanations for $d ( \Phi ( x ) , \Phi ( y ) )$ as “squared $\ell _ { 2 }$ in label”, “maximize label”, and ${ } ^ { 6 6 } \ell _ { 2 }$ probabilities” look indistinguishable. For $\bar { d } ( \Phi ( x ) , \mathbf { \bar { \Phi } } ( y ) )$ as KL-Divergence, we see a slightly less smooth explanation, which may be due to the fact that the KL-Divergence is unbounded unlike the other measures of distortion.
+
+
+Figure 9: We compare the sensitivity of CartoonX to four measures of distortion $d ( \Phi ( x ) , \Phi ( y ) )$ . Each of the measures of distortion is marked at the top of each column. We observe almost no difference in the CartoonX explanations for the four distortion measures.
+
+# A.4 RELIABILITY AND EXPLANATION ARTIFACTS
+
+We argue experimentally why CartoonX is more reliable than Pixel RDE for image data. More precisely, we show that CartoonX is less susceptible to so-called explanation artifacts than Pixel RDE. An explanation artifact is an unwanted phenomenon that we observed for Pixel RDE: instead of marking the relevant entries in $x$ , the mask $s$ creates artificial edges that end up making up an artificial class prototype. Explanation artifacts are problematic because they highlight not actual substructures but artificial structures that trigger the classification. Examples for explanation artifacts in Pixel RDE are given in Figure 11.
+
+
+Figure 10: CartoonX and Pixel RDE are both performed on the image of the blue sky. However, both methods are adjusted here to find evidence for the output probabilities of the image of the airplane instead of the blue sky. Pixel RDE, unlike CartoonX, can create an artificial airplane as evidence for an airplane in the smooth blue sky.
+
+Pixel RDE can produce artificial edges in smooth regions for the following reason: When $s$ has a curve-like structure in some region, unselected points near $s$ are replaced with perturbations that tend to differ from the values of the curve-like structure in $s$ . Thus, the curve-like structure also appears in the obfuscation and can produce low distortion if the structure makes up a prototypical class feature (see, for example, the airplane in Figure 10).
+
+We suspect CartoonX is inherently less susceptible to explanation artifacts for the following reason: Natural images tend to be piece-wise smooth, and piece-wise smooth images have sparse high-frequency DWT coefficients that cluster about the edges Stephane (2009b) (see for example ´ Figure 4). For a DWT mask to create artificial edges, it has to select a curve-like structure in the high-frequency coefficients (low-frequency coefficients cannot create edges) and replace surrounding unselected values with different values. However, in CartoonX, perturbations of high-frequency coefficients are Gaussian with low variance centered close to zero (see adaptive Gaussian noise in Section 4.1.2), which are not very different from the values along the selected curve due to the sparsity of the coefficients.
+
+We illustrate our previous reasoning about explanation artifacts in a controlled example (see Figure 10). We take an image $x ^ { ( \mathrm { s k y } ) }$ of a blue sky that is very smooth and an image $x ^ { \mathrm { ( p l a n e ) } }$ of a airplane. The goal is to show that Pixel RDE, unlike CartoonX, can create artificial evidence for the class airplane on the image of the smooth blue sky. We perform CartoonX and Pixel RDE on the blue sky image with the distortion function
+
+$$
+\forall y \in \mathbb { R } ^ { n } : \ d ( \Phi ( x ^ { ( \mathrm { s k y } ) } ) , \Phi ( y ) ) = 1 0 ^ { 6 } \| \Phi ( x ^ { ( \mathrm { p l a n e } ) } ) - \Phi ( y ) \| _ { 2 } ,
+$$
+
+and a sparsity level of $\lambda = 8 0 0 0 0$ . As expected, we observe that Pixel RDE, unlike CartoonX, can create an artificial plane in the smooth blue sky(see Figure 10).
+
+A.5 EXPLAINING THROUGH IMAGENET HISTORY: FROM ALEXNET TO RESNETXT50
+
+In the deep learning community, it is well-known that AlexNet (Krizhevsky et al., 2012) provided a major breakthrough in deep learning, improving the top-5 error on ImageNet from $2 5 \%$ to $16 \%$ . Since then, deep learning based ImageNet classifiers have continued to drastically improve on ImageNet—achieving less than $6 \%$ top-5 error in 2016. In Figure 12, we compare CartoonX for four ImageNet classifiers with increasing performance, starting with AlexNet (top-1 accuracy $5 6 . 5 5 \%$ , AlexNet), VGG16 (top-1 accuracy $7 1 . 5 9 \%$ , Simonyan & Zisserman (2015)), InceptionV3 (top-1 accuracy $7 7 . 2 9 \%$ , Szegedy et al. (2016)), and ResNeXt50 (top-1 accuracy $7 7 . 6 2 \%$ , Xie et al. (2017).) Throughout the experiment, the CartoonX hyperparameters for a given image are not changed for any of the four classifiers.
+
+# A.6 EXPLAINING MISCLASSIFICATIONS WITH CARTOONX
+
+In Figure 13, 14, 15, and 16, we provide further examples where CartoonX provides insightful explanations for misclassified images.
+
+# A.7 CARTOONX COMPARED ON RANDOM IMAGENET SAMPLES
+
+Figure 17, 18, 19, and 20 compares CartoonX to Pixel RDE (Macdonald et al., 2019), Integrated Gradients (Sundararajan et al., 2017), Smoothgrad (Smilkov et al., 2017), Guided Backprop (Springenberg et al., 2015), and (Bach et al., 2015) on random Imagenet samples classified by VGG16.
+
+# A.8 CARTOONX FAILURES
+
+We also show failures of CartoonX in Figure 21. These are examples of explanations that are not interpretable and seem to fail at explaining the model prediction. Notably, most failure examples are also not particularly well explained by other state-of-the-art methods. It is challenging to state the underlying reason for the CatoonX failures with certainty (there is always the possibility that the neural net bases its decision on non-interpretable grounds). We intentionally also showed uninterpretable CartoonX explanations that were not too sparse (all or almost black explanations) since one can typically fix these explanations by decreasing $\lambda$ .
+
+
+Figure 11: Explanation artifacts in Pixel RDE. We observe that Pixel RDE tends to create edges that are not a subset of the edges in the original input image. These edges can make prototypical artifact patterns such as wrinkles in the cloak (first row), coral tentacles (second row), or chain mail (third row).
+
+
+Figure 12: We compare CatoonX explanations for classifications by AlexNet (Krizhevsky et al., 2012), VGG16 (Simonyan & Zisserman, 2015), InceptionV3 (Szegedy et al., 2016), and ResNeXt50 (Xie et al., 2017). Green labels mark correct classifications and red labels mark wrong classifactions.
+
+
+Figure 13: Explaining misclassifications with CartoonX on Imagenet and MobileNetV3-Small.
+
+
+Figure 14: Explaining misclassifications with CartoonX on Imagenet and MobileNetV3-Small.
+
+
+Figure 15: Explaining misclassifications with CartoonX on Imagenet and MobileNetV3-Small.
+
+
+Figure 16: Explaining misclassifications with CartoonX on Imagenet and MobileNetV3-Small.
+
+
+Figure 17: Comparing CartoonX on random ImageNet samples and VGG16.
+
+
+Figure 18: Comparing CartoonX on random ImageNet samples and VGG16.
+
+
+Figure 19: Comparing CartoonX on random ImageNet samples and VGG16.
+
+
+Figure 20: Comparing CartoonX on random ImageNet samples and VGG16.
+
+
+Figure 21: Failures of CartoonX.
\ No newline at end of file
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+[
+ {
+ "type": "text",
+ "text": "CARTOON EXPLANATIONS OF IMAGE CLASSIFIERS ",
+ "text_level": 1,
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+ "text": "Anonymous authors Paper under double-blind review ",
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+ "type": "text",
+ "text": "ABSTRACT ",
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+ 226
+ ],
+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "We present CartoonX (Cartoon Explanation), a novel model-agnostic explanation method tailored towards image classifiers and based on the rate-distortion explanation (RDE) framework. Natural images are roughly piece-wise smooth signals—also called cartoon images—and tend to be sparse in the wavelet domain. CartoonX is the first explanation method to exploit this by requiring its explanations to be sparse in the wavelet domain, thus extracting the relevant piece-wise smooth part of an image instead of relevant pixel-sparse regions. We demonstrate experimentally that CartoonX is not only highly interpretable due to its piece-wise smooth nature but also particularly apt at explaining misclassifications. ",
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+ {
+ "type": "text",
+ "text": "1 INTRODUCTION ",
+ "text_level": 1,
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+ 409
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+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "Powerful machine learning models such as deep neural networks are inherently opaque, which has motivated numerous explanation methods over the last decade (see for example the survey by Das & Rad (2020)). A significant fraction of the research literature has focused on explaining image classifications due to both the practical relevance of computer vision tasks and the ease at which heatmaps can communicate explanatory information. Despite the great variety in methods and explanation philosophies, all current methods share the following characteristic: they operate in pixel space. Roughly speaking, existing explanation methods for image classifiers either allocate additive attribution scores to each pixel or optimize a deletion mask on the pixel coefficients to mark a relevant set of pixels. The result is typically a pixel-sparse and jittery explanation. We challenge the conventional approach to explain in pixel space by successfully applying the rate-distortion explanation (RDE) framework (Macdonald et al., 2019; Heiß et al., 2020) in the wavelet domain of images. Our novel explanation method, CartoonX, extracts the relevant piece-wise smooth part of an image (see Figure 1). Instead of demanding sparsity in pixel space, as in (Macdonald et al., 2019; Chang et al., 2019), CartoonX demands sparsity in the wavelet domain, which produces piece-wise smooth explanations (cartoon-like images). Our work makes the following contributions: ",
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+ "Dog classified as Egyptian cat "
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+ "CartoonX of misclassification "
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+ "text": "Reformulation and reinterpretation of the RDE framework: We reformulate the RDE framework in a more general manner with enhanced flexibility in the input representation to accommodate complex interpretation queries such as “What is the piece-wise smooth part of the input signal that leads to its model decision?”. Thereby, we reinterpret RDE as a simplification of the input signal, which is interpretable to humans and adheres to a meaningful interpretation query. The simplification is achieved by demanding sparsity in a suitable representation system, which sparsely represents the class of explanations that are desirable for the interpretation query. ",
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+ "Slam dunk classified as basketball "
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+ "text": "CartoonX, a novel explanation method tailored to image classifiers: CartoonX is the first explanation method to extract the relevant piece-wise smooth part of an image instead of relevant pixel sparse regions. This is achieved by demanding sparsity in the wavelet domain of images, where ",
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+ "image_caption": [
+ "Figure 1: Examples of CartoonX explanations. "
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+ "text": "sparsity translates into piece-wise smooth images. We demonstrate that \nour piece-wise smooth explanations are more interpretable than jittery \npixel-sparse explanations and that they can reveal relevant piece-wise smooth patterns that are not easily visible with existing pixel-based methods. Surprisingly, we find that our method is particularly well-equipped to explain misclassifications, often showing “what the neural network actually saw” (see Figure 1). ",
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+ "type": "text",
+ "text": "2 RELATED WORK ",
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+ "text": "The Rate-Distortion Explanation (RDE) framework was first introduced in (Macdonald et al., 2019), and extended in (Heiß et al., 2020), as a mathematically well-founded and intuitive explanation framework. RDEs are model-agnostic explanations and inspired by rate-distortion theory, which studies lossy-data compression. An explanation in RDE consists of a relatively sparse mask over the input features, highlighting the relevant set of features. The mask is optimized to produce low distortion in the model output after applying perturbations to the unselected features in the input while remaining relatively sparse. Heiß et al. (2020) also applied RDE to non-canonical input representations to explain model decisions in challenging domains such as audio classification (Engel et al., 2017) and radio-map estimation (Levie et al., 2021; 2020). ",
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+ "text": "The explanation principle of optimizing a mask $s \\in [ 0 , 1 ] ^ { n }$ was first proposed by Fong & Vedaldi (2017) who explained image classification decisions by considering one of the two “deletion games”: (1) optimizing for the smallest deletion mask that causes the class score to drop significantly or (2) optimizing for the largest deletion mask that has no significant effect on the class score. The original RDE approach (Macdonald et al., 2019) is based on the second deletion game. ",
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+ "text": "Other explanation methods developed by the research community are typically either (1) gradientbased such as Smoothgrad (Smilkov et al., 2017), Integrated Gradients (Sundararajan et al., 2017), Image-Specific Class Saliency (Simonyan et al., 2014), and Guided Backpropagation (Springenberg et al., 2015), (2) surrogate models such as LIME (Ribeiro et al., 2016), (3) based on propagation of activations in neurons such as LRP (Bach et al., 2015; Shrikumar et al., 2017), and DeepLIFT (Shrikumar et al., 2017), (4) based on Shapely values from game-theory (Lundberg & Lee, 2017), (6) concept-based such as Concept Activation Vectors (Kim et al., 2018), or (7) based on generative causal explanations (O' Shaughnessy et al., 2020). Also related are methods that were developed to explain individual neurons such as in (Nguyen et al., 2016; Dhamdhere et al., 2019). To our knowledge, all existing explainability methods operate in pixel space and all methods looking for sparse explanations demand sparsity in pixel space (Macdonald et al., 2019; Fong & Vedaldi, 2017; Chang et al., 2019). ",
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+ "text": "3 BACKGROUND: RATE-DISTORTION EXPLANATION FRAMEWORK ",
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+ "text": "In this section, we review the rate-distortion explanation (RDE) framework, which was introduced by Macdonald et al. (2019) and later extended by Heiß et al. (2020) by applying RDE to noncanonical input representations. Suppose $\\Phi : \\mathbb { R } ^ { n } \\mathbb { R } ^ { m }$ is a pre-trained model, e.g., a classifier (with $m$ class labels) or a regression model (with $m$ -dimensional output), where $n$ denotes the dimension of the model input. RDE produces an explanation for a model decision $\\Phi ( x )$ with $x \\in \\mathbb { R } ^ { n }$ as a relatively sparse mask $s \\in \\{ 0 , 1 \\}$ marking the relevant input features in $x$ . More precisely, RDE aims to solve the following optimization problem over a mask $s \\in \\{ 0 , 1 \\} ^ { n }$ : ",
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+ "text": "$$\n\\operatorname* { m i n } _ { s \\in \\{ 0 , 1 \\} ^ { n } } \\quad \\operatorname { \\mathbb { E } } _ { v \\sim \\mathcal { V } } \\left[ d \\Bigl ( \\Phi ( x ) , \\Phi ( x \\odot s + ( 1 - s ) \\odot v ) \\Bigr ) \\right] \\quad \\mathrm { s . t . } \\quad \\| s \\| _ { 0 } \\leq \\ell ,\n$$",
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+ "text": "where $\\odot$ denotes the Hadamard product (element-wise multiplication), $d ( \\Phi ( x ) , \\cdot )$ is a measure of distortion (e.g. $d ( \\Phi ( x ) , \\cdot ) = \\lVert \\Phi ( { \\bar { x } } ) - \\cdot \\rVert _ { 2 } )$ , $\\nu$ is a distribution over input perturbations $v \\in \\mathbb { R } ^ { n }$ , and $\\ell \\in \\{ 1 , . . . , n \\}$ is a given sparsity level for the explanation mask $s$ . A solution $s ^ { * }$ to the optimization problem in (1) marks relatively few components in the model input $x$ that suffice to approximately retain the model output $\\Phi ( x )$ . This approach is in the spirit of rate-distortion theory, which deals with lossy compression of data. Therefore, Macdonald et al. (2019) coined such explanations ratedistortion explanations (RDEs). ",
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+ "text": "In practice, the optimization problem in (1) is relaxed to continuous masks $s \\in [ 0 , 1 ]$ solving ",
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+ "text": "$$\n\\operatorname* { m i n } _ { s \\in \\{ 0 , 1 \\} ^ { n } } \\quad \\operatorname { \\mathbb { E } } _ { v \\sim \\mathcal { V } } \\left[ d \\Bigl ( \\Phi ( x ) , \\Phi ( x \\odot s + ( 1 - s ) \\odot v ) \\Bigr ) \\right] + \\lambda \\left\\| s \\right\\| _ { 1 } ,\n$$",
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+ "text": "where $\\lambda > 0$ determines the sparsity level of the mask. The relaxed optimization problem can be solved with stochastic gradient descent in $s \\in [ 0 , 1 ]$ if $\\Phi$ is differentiable—as is the case for deep neural networks. Macdonald et al. (2019) applied the RDE method as described above to image classifiers in the pixel domain of images, where each mask entry $s _ { i } \\in [ 0 , 1 ]$ corresponds to the $i$ -th pixel values. We refer to this method as Pixel RDE throughout this work. ",
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+ "type": "text",
+ "text": "4 RDE REFORMULATED AND REINTERPRETED ",
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+ "text": "Instead of applying RDE to the standard input representation $\\boldsymbol { x } = [ x _ { 1 } \\dots x _ { n } ] ^ { T }$ , we can apply RDE to a different representation of $x$ to answer a particular interpretation query. For example, consider a 1D-signal $x \\in \\mathbb { R } ^ { n }$ : if we ask “What is the smooth part in the signal $x$ that leads to the model decision $\\Phi ( x ) ? ^ { , }$ , then we can apply RDE in the Fourier basis of $x$ . Since frequency-sparse signals are smooth, applying RDE in the Fourier basis of $x$ extracts the relevant smooth part of the signal. To accommodate such interpretation queries, we reformulate RDE in Section 4.1. Finally, based on the reformulation, we reinterpret RDE in Section 4.2. Later in Section 5, we use our reformulation and reinterpretation of RDE to derive and motivate CartoonX as a special case and novel explanation method tailored towards image classifiers. ",
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+ "text": "4.1 GENERAL FORMULATION ",
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+ "text": "An input signal $\\boldsymbol { x } = [ x _ { 1 } , \\dots , x _ { n } ] ^ { T }$ is represented in a basis $\\{ b _ { 1 } , \\ldots , b _ { n } \\}$ as a linear combination $\\textstyle \\sum _ { i = 1 } ^ { n } h _ { i } b _ { i }$ with coefficients $[ h _ { i } ] _ { i = 1 } ^ { n }$ . As we argued above and demonstrate later on, some choices for a basis may be more suitable than others to explain a model decision $\\Phi ( x )$ . Therefore, we define the RDE mask not only on the canonical input representation $[ x _ { i } ] _ { i = 1 } ^ { n }$ but also on a different representation $[ h _ { i } ] _ { i = 1 } ^ { n }$ with respect to a choice of basis $\\{ b _ { 1 } , \\ldots , b _ { n } \\}$ . Examples of non-canonical choices for a basis include the Fourier basis and the wavelet basis. This work is centered around CartoonX, which applies RDE in the wavelet basis, i.e., a linear data representation since $x$ is represented as a linear combination of basis vectors. Nevertheless, there also exist other domains and interpretation queries where applying RDE to a non-linear data representation can make sense (see the interpretation query “Is phase or magnitude more important for an audio classifier?” in (Heiß et al., 2020)). Therefore, we formulate RDE in terms of a data representation function $\\textstyle f : \\prod _ { i = 1 } ^ { k } \\mathbb { R } ^ { c } \\to \\mathbb { R } ^ { n }$ , $f ( h _ { 1 } , \\ldots , h _ { k } ) = x .$ , which does notlinear case and o be linear, we have $c$ ls in the, where e imare ortantfixed $c = 1$ $\\begin{array} { r } { f ( h _ { 1 } , \\ldots , h _ { k } ) = \\sum _ { i = 1 } ^ { k } h _ { i } b _ { i } } \\end{array}$ $\\{ b _ { i } , \\ldots , b _ { k } \\} \\subset \\mathbb { R } ^ { n }$ $k$ $c > 1$ \nchannels at once, e.g., all color channels of an image, to reduce the number of entries in the mask that will operate on $[ h _ { i } ] _ { i = 1 } ^ { k }$ . In the following, we introduce the important definitions of obfuscations, expected distortion, the RDE mask, and $R D E ' s \\ell _ { 1 }$ -relaxation, which generalize the RDE framework of (Macdonald et al., 2019) to abstract input representations. ",
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+ "text": "4.1.1 DEFINITIONS ",
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+ "text": "The first two key concepts in RDE are obfuscations and expected distortion, which are defined below. ",
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+ "text": "Definition 1 (Obfuscations and expected distortion) Let $\\Phi : \\mathbb { R } ^ { n } \\mathbb { R } ^ { m }$ be a model and $x \\in \\mathbb { R } ^ { n }$ a data point with a data representation $x = f ( h _ { 1 } , . . . , h _ { k } )$ as discussed above. For every mask $s \\in [ 0 , 1 ] ^ { k }$ , let $\\gamma _ { s }$ be a probability distribution over $\\textstyle \\prod _ { i = 1 } ^ { k } \\mathbb { R } ^ { c }$ . Then the obfuscation of $x$ with respect to s and $\\gamma _ { s }$ is defined as the random vector $y : = { f ( s \\odot h + ( 1 - s ) \\odot v ) }$ , where $v \\sim \\mathcal { V } _ { s }$ , $( s \\odot h ) _ { i } = s _ { i } h _ { i } \\in \\mathbb { R } ^ { c }$ and $( ( 1 - s ) \\odot v ) _ { i } = ( 1 - s _ { i } ) v _ { i } \\in \\mathbb R ^ { c }$ , for $i \\in \\{ 1 , \\ldots , k \\}$ . A choice for the distribution $\\gamma _ { s }$ is called obfuscation strategy. Furthermore, the expected distortion of $x$ with respect to the mask s and the perturbation distribution $\\gamma _ { s }$ is defined as ",
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+ "text": "$$\nD ( x , s , \\mathcal { V } _ { s } , \\Phi ) : = \\underset { v \\sim \\mathcal { V } _ { s } } { \\mathbb { E } } \\left[ d \\Big ( \\Phi ( x ) , \\Phi ( y ) \\Big ) \\right] ,\n$$",
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+ "text": "where $d : \\mathbb { R } ^ { m } \\times \\mathbb { R } ^ { m } \\to \\mathbb { R } _ { + }$ is a measure of distortion between two model outputs. ",
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+ "text": "In the RDE framework, the explanation is given by a mask that minimizes distortion while remaining relatively sparse. The rate-distortion explanation mask is defined as follows. ",
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+ "text": "Definition 2 (The RDE mask) In the setting of Definition $I$ we define the RDE mask as a solution $s ^ { * } ( \\ell )$ to the minimization problem ",
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+ "text": "$$\n\\operatorname* { m i n } _ { s \\in \\{ 0 , 1 \\} ^ { k } } \\quad D ( x , s , \\mathcal { V } _ { s } , \\Phi ) \\quad s . t . \\quad \\| s \\| _ { 0 } \\leq \\ell ,\n$$",
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+ "text": "where $\\ell \\in \\{ 1 , \\ldots , k \\}$ is the desired level of sparsity. ",
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+ "text": "Geometrically, the RDE mask $s$ is associated with a particular subspace. The complement mask $( 1 - s )$ can be seen as selecting a large stable subspace of $\\Phi$ , with each point representing a possible perturbation in unselected coefficients in $h$ . The RDE mask minimizes the expected distortion along its associated subspace, which requires non-local information of $\\Phi$ . We illustrate this geometric view of RDE in Figure 2 with a toy example for a hypothetical classifier $\\Phi : \\mathbb { R } ^ { 2 } \\mathbb { R } ^ { \\bar { m } }$ and two distinct input representations: (1) Euclidean coordinates, i.e., $f$ is the identity in $x = f ( h )$ , and (2) polar coordinates, i.e. $f ( h ) = ( h _ { 2 } \\cos h _ { 1 } , h _ { 2 } \\sin h _ { 1 } ) = x$ . In the example, we assume $\\gamma _ { s }$ to be a uniform distribution on $[ - 1 , 1 ] ^ { 2 }$ in the Euclidean representation and a uniform distribution on $[ - \\pi , \\pi ] \\times [ 0 , 1 ]$ in the polar representation. The expected distortion associated with the masks $s = ( 1 , 0 )$ and $s = ( 0 , 1 )$ is given by the red and green shaded area, respectively. The RDE mask aims for low expected distortion, and hence, in polar coordinates, the RDE mask would be the green subspace, i.e., $s = ( 0 , 1 )$ . On the other hand, in Euclidean coordinates, neither $s = ( 1 , 0 )$ nor $s = ( 0 , 1 )$ produces a particularly low expected distortion, making the Euclidean explanation less meaningful than the polar explanation. The example illustrates why certain input representations can yield more meaningful explanatory insight for a given classifier than others—an insight that underpins our novel CartoonX method. Moreover, the plot in polar coordinates illustrates why the RDE mask cannot be simply chosen with local distortion information, e.g., with the lowest eigenvalue of the Hessian of $\\bar { h } \\mathbin { \\stackrel { \\cdot } { \\mapsto } } d ( \\Phi ( x ) , \\Phi ( f ( h ) ) )$ : the lowest eigenvalue in polar coordinates belongs to the red subspace and does not see the large distortion on the tails. ",
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+ "Figure 2: The RDE mask can find low expected distortion in polar coordinates but not in Euclidean coordinates. Therefore, in this example, polar coordinates are more appropriate to explain $\\Phi ( x )$ , and RDE would determine that the angle $\\varphi$ , not the magnitude $r$ , is relevant for $\\Phi ( x )$ . "
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+ "text": "As was shown by Macdonald et al. (2019), the RDE mask from Definition 2 cannot be computed efficiently for non-trivial input sizes. Nevertheless, one can find an approximate solution by considering continuous masks $s \\in [ 0 , 1 ] ^ { k }$ and encouraging sparsity through the $\\ell _ { 1 }$ -norm. ",
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+ "text": "Definition 3 (RDE’s $\\ell _ { 1 }$ -relaxation with Lagrange multipliers) In the setting of Definition $I$ , we define RDE’s $\\ell _ { 1 }$ -relaxation with Lagrange multipliers as a solution $s ^ { * } ( \\lambda )$ to the minimization problem ",
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+ "text": "$$\n\\begin{array} { r l } { \\underset { s \\in [ 0 , 1 ] ^ { k } } { \\operatorname* { m i n } } } & { { } D ( \\boldsymbol { x } , s , \\mathcal { V } _ { s } , \\boldsymbol { \\Phi } ) + \\lambda \\| s \\| _ { 1 } , } \\end{array}\n$$",
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+ "text": "where $\\lambda > 0$ is a hyperparameter for the sparsity level. ",
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+ "text": "The $\\ell _ { 1 }$ -relaxation above can be solved with stochastic gradient descent (SGD) over the mask $s$ while approximating $D ( x , s , \\mathcal { V } _ { s } , \\Phi )$ with i.i.d. samples from $v \\sim \\mathcal { V } _ { s }$ . ",
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+ "type": "text",
+ "text": "4.1.2 OBFUSCATION STRATEGIES",
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+ "text": "An obfuscation strategy is defined by the choice of the perturbation distribution $\\mathcal { V } _ { s }$ . Common choices are Gaussian noise (Macdonald et al., 2019; Fong & Vedaldi, 2017), blurring (Fong & Vedaldi, 2017), constants (Fong $\\&$ Vedaldi, 2017), and inpainting GANs (Heiß et al., 2020; Chang et al., 2019). Inpainting GANs train a generator $G ( s , z , h )$ ( $z$ denotes random latent factors) such that for samples $v \\sim G ( s , z , h )$ the obfuscation $f ( s \\odot h + ( 1 - s ) \\odot v )$ remains in the data manifold. In our work, we refrain from using an inpainting GAN due to the following reason: it is hard to tell whether a GAN-based mask did not select coefficients because they are unimportant or because the GAN can easily inpaint them from a biased context. Instead, we choose a simple and wellunderstood obfuscation strategy, which we call Gaussian adaptive noise, making the explanation as transparent as possible. ",
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+ "text": "Gaussian adaptive noise works as follows: Let $A _ { 1 } , . . . , A _ { j }$ be a pre-defined choice of a disjoint partition of $\\{ 1 , \\ldots , k \\}$ (recall $s \\in [ 0 , 1 ] ^ { k } )$ . For $i = 1 , . . . , j$ , we compute the empirical mean and empirical standard deviation for each partition across all partition instances: ",
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+ "text": "$$\n\\mu _ { i } : = \\frac { 1 } { \\sum _ { a \\in A _ { i } } d _ { a } } \\sum _ { a \\in A _ { i } , t = 1 , \\ldots , d _ { a } } h _ { a t } , \\sigma _ { i } : = \\sqrt { \\frac { 1 } { \\sum _ { a \\in A _ { i } } d _ { a } } \\sum _ { a \\in A _ { i } , t = 1 , \\ldots , d _ { a } } ( \\mu _ { i } - h _ { a t } ) ^ { 2 } }\n$$",
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+ "text": "The adaptive Gaussian noise strategy then samples $v _ { a t \\_ } \\sim \\mathcal { N } ( \\mu _ { i } , \\sigma _ { i } ^ { 2 } )$ for all partition members $a \\in$ $A _ { i }$ and channels $t = 1 , . . . , d _ { a }$ . We write $v \\sim \\mathcal { N } ( \\mu , \\sigma ^ { 2 } )$ for the resulting Gaussian random vector $\\boldsymbol { v } \\in \\prod _ { i = 1 } ^ { k } \\mathbb { R } ^ { c }$ . Note that the distribution $\\gamma _ { s }$ chosen as Gaussian adaptive noise does depend on $s$ (unlike with an inpainting GAN). For Pixel RDE, we only use one set $A _ { 1 } = \\left\\{ 1 , . . . , k \\right\\}$ for all $k$ pixels. In CartoonX, which represents input signals in the discrete wavelet domain, we will partition $\\{ 1 , . . . , k \\}$ along the scales of the discrete wavelet transform. ",
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+ "text": "4.1.3 MEASURES OF DISTORTION ",
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+ "text": "There are various choices for the measure of distortion $d ( \\Phi ( x ) , \\Phi ( y ) )$ . For example, one can take the squared distance in the post-softmax probability of the predicted label for $x$ , i.e., ",
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+ "text": "$$\nd \\big ( \\Phi ( x ) , \\Phi ( y ) \\big ) : = \\big ( \\Phi _ { j ^ { * } } ( x ) - \\Phi _ { j ^ { * } } ( y ) \\big ) ^ { 2 } ,\n$$",
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+ "text": "where $j ^ { * } : = \\arg \\operatorname* { m a x } _ { i = 1 , \\ldots , m } \\Phi _ { i } ( x )$ and $\\Phi ( x )$ is assumed to be the post-softmax probabilities of a neural net. Alternatively, one could also choose $d ( \\Phi ( x ) , \\Phi ( y ) )$ as the $\\ell _ { 2 }$ -distance or the KLDivergence in the post-softmax layer of $\\Phi$ . In our experiments for CartoonX, we found that these choices had no significant effect on the explanation (see Appendix A.3.3). ",
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+ "text": "4.2 INTERPRETATION ",
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+ "text": "The philosophy of the generalized RDE framework is that an explanation for a decision $\\Phi ( x )$ on a generic input signal $x = f ( h )$ should be some simplified version of the signal, which is interpretable to humans. The simplification is achieved by demanding sparsity in a suitable representation system $h$ , which sparsely represents the class of explanations that are desirable for the interpretation query. This philosophy is the fundamental premise of CartoonX, which aims to answer the interpretation query “What is the relevant piece-wise smooth part of the image for a given image classifier?”. CartoonX first employs RDE on a representation system $x = f ( h )$ that sparsely represents piecewise smooth images and finally visualizes the relevant piece-wise smooth part as an image back in pixel space. In the following section, we explain why wavelets provide an appropriate representation system in CartoonX, present the CartoonX implementation, and finally provide experiments on ImageNet to demonstrate the capability of CartoonX. ",
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+ "text": "5 CARTOONX ",
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+ "text": "The focus of this paper is CartoonX, a novel explanation method—tailored to image classifications— that we obtain as a special case of our generalized RDE framework formulated in Section 4. CartoonX first performs RDE in the discrete wavelet position-scale domain of an image $x$ , and finally, visualizes the wavelet mask $s$ as a piece-wise smooth image in pixel space. Wavelets provide optimal representations for piece-wise smooth 1D functions (DeVore, 1998), and represent 2D piecewise smooth images, also called cartoon-like images (Kutyniok & Lim, 2011), efficiently as well (Romberg et al., 2006). In particular, sparse vectors in the wavelet coefficient space encode cartoonlike images reasonably well (Stephane, 2009a)—certainly better than sparse pixel representations. ´ Moreover, wavelets constitute an established tool in signal processing (Stephane, 2009c). ´ ",
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+ "Figure 3: CartoonX has many interesting parallels to wavelet-based image compression. Distortion is denoted as $d$ , $\\Phi$ is an image classifier, $h$ denotes the discrete wavelet coefficients, $\\tau$ is the discrete wavelet transform, and $\\ell$ is the coefficient budget. "
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+ "text": "The optimization process underlying CartoonX produces sparse vectors in the discrete wavelet coefficient space, which results in cartoon-like images as explanations. This is the fundamental difference to Pixel RDE, which produces rough, jittery, and pixel-sparse explanations. Cartoon-like images are more interpretable and provide a natural model of simplified images. Since the goal of the RDE framework is to generate an easy to interpret simplified version of the input signal, we argue that CartoonX explanations are more appropriate for image classification than Pixel RDEs. ",
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+ "text": "CartoonX exhibits interesting parallels to wavelet-based image compression. In image compression, distortion is minimized in the data domain, which is equivalent to selecting the $\\ell$ largest entries in the discrete wavelet transform (DWT) coefficients. In comparison, CartoonX minimizes distortion in the model output of $\\Phi$ , which translates to selecting the $\\ell$ most relevant entries in the DWT coefficients. The objective in image compression is efficient data representation, i.e., producing minimal data distortion with a budget of $\\ell$ entries in the DWT coefficients. Conversely, in CartoonX, the objective is extracting the relevant piece-wise smooth part, i.e., producing minimal model distortion with a budget of $\\ell$ entries in the DWT coefficients. We illustrate this connection in Figure 3—highlighting once more the rate-distortion spirit of the RDE framework. ",
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+ "text": "5.1 IMPLEMENTATION ",
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+ "text": "An image $x \\in [ 0 , 1 ] ^ { c \\times w \\times t }$ with $c \\in \\{ 1 , 3 \\}$ channels, width $w \\in \\mathbb { N }$ , height $t \\in \\mathbb N$ , and a total of $p = w t$ pixels can be represented in a wavelet basis by computing its discrete wavelet transform (DWT). The DWT of an image is defined by the number of scales $J \\in \\{ 1 , \\ldots , \\lfloor \\log _ { 2 } p \\rfloor \\}$ , the padding mode, and a choice of the wavelet family (such as the Haar or Daubechies family). For images, the DWT computes four types of coefficients: details in (1) horizontal, (2) vertical, and (3) diagonal orientation at scale $j \\in \\{ 1 , \\dots , J \\}$ , and (4) coefficients of the image at the very coarsest resolution. We briefly illustrate the DWT for an example image in Figure 4. ",
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+ "Figure 4: Left side: an image of a memorial arch dedicated to peace. Right side: visualization of the DWT coefficients for five scales. Three L-shaped sub-images describe coefficients for details in vertical, horizontal, and diagonal orientation at a particular scale. The largest sub-images (the outer L-shape) belong to the lowest scale, i.e., the highest resolution. The smaller L-shaped sub-images gradually build up to higher scales, i.e., lower resolution features. "
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+ "text": "CartoonX, as described in Algorithm 1 in Appendix A.1, computes the RDE mask in the wavelet domain of images. More precisely, for the data representation ${ \\bar { \\boldsymbol { x } } } = f ( h )$ , we choose $h$ as the concatenation of all the DWT coefficients along the channels, i.e., $\\boldsymbol { h } _ { i } \\in \\mathbb { R } ^ { c }$ . The representation function $f$ is then the discrete inverse wavelet transform, i.e., the summation of the DWT coefficients times the DWT basis vectors. We optimize the mask $s \\in [ 0 , 1 ] ^ { k }$ on the DWT coefficients $[ h _ { 1 } , \\ldots , h _ { k } ] ^ { T }$ to minimize RDE’s $\\ell _ { 1 }$ -relaxation from Definition 3. For the obfuscation strategy $\\gamma _ { s }$ , we use adaptive Gaussian noise with a partition by the DWT scale (see Section 4.1.2), i.e., we compute the empirical mean and standard deviation per scale. We measure distortion as the squared difference in the postsoftmax score of the predicted label for $x$ (see Section 4.1.3). To visualize the final DWT mask $s$ as a piece-wise smooth image in pixel space, we multiply the mask with the DWT coefficients of the greyscale image $\\hat { x } : = ( 1 \\breve { / c } \\sum _ { l = 1 } ^ { \\hat { c } } x _ { l a i } \\dot { ) } _ { a i }$ before inverting the product back to pixel space with the discrete inverse wavelet transform. The inversion is finally clipped into $[ 0 , 1 ] ^ { w \\times \\dot { t } }$ as are obfuscations during the RDE optimization to avoid overflow (we assume here the pixel values in $x$ are normalized into $[ 0 , 1 ] )$ . The clipped inversion in pixel space is the final explanation, which we call CartoonX. ",
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+ "text": "5.2 EXPERIMENTS AND ANALYSIS ",
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+ "text": "We compare CartoonX to the closely related Pixel RDE (Macdonald et al., 2019) and several other state-of-the-art explanation methods , that is, Integrated Gradients (Sundararajan et al., 2017), Smoothgrad (Smilkov et al., 2017), Guided Backprop (Springenberg et al., 2015), and LRP (Bach et al., 2015). Our experiments show that CartoonX carries the following strengths: Cartoon X is (1) highly interpretable due to its cartoon-like nature and (2) remarkably apt at explaining misclassifications, and highlighting meaningful patterns that are otherwise hard to see. Due to the fast implementation of the DWT, Cartoon RDE is not significantly slower than Pixel RDE. For the ImageNet classifier MobileNetV3-Small, an image of 256 times 256 pixels, and 2001 optimization steps, we reported a runtime of 81.56 seconds for CartoonX and 70.53 seconds for Pixel RDE on the NVIDIA Titan RTX GPU. However, like other perturbation-based methods, CartoonX is significantly slower than gradient or propagation-based methods, which only compute a single or few forward and backward passes and are very fast (Integrated Gradients computes an explanation in 0.48 seconds for the same image, model, and hardware). ",
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+ "text": "Our experiments use the pre-trained ImageNet classifiers MobileNetV3-Small (Howard et al., 2019) (top-1 accuracy of $6 7 . 6 6 8 \\%$ and VGG16 (Simonyan & Zisserman, 2015) (top-1 accuracy of $7 1 . 5 9 2 \\%$ ). We note that the open-source implementation of LRP did not implement propagation rules for certain layers in MobileNetV3-Small, therefore we compare CartoonX to LRP only for VGG16. Images were preprocessed to have 256 times 256 pixel values in [0, 1]. We provide further details about the choice of hyperparameters in the experiments in Appendix A.2. The three main hyperparameters for CartoonX are: (1) the sparsity level $\\lambda > 0$ , (2) the measure of distortion $d$ , and (3) the obfuscation strategy (perturbation distribution) $\\gamma _ { s }$ . We discuss the sensitivity of CartoonX to these hyperparameters in Appendix A.3. In Appendix A.5, we also shed light on the evolution of ImageNet classifiers from an explanation angle by comparing CartoonX explanations for classifiers of varying generalization power, i.e., AlexNet (Krizhevsky et al., 2012), VGG16 (Simonyan & Zisserman, 2015), InceptionV3 (Szegedy et al., 2016), ResNeXt50 (Xie et al., 2017). Moreover, in Appendix A.4, we argue experimentally why CartoonX is less susceptible than Pixel RDE to so-called explanation artifacts—an unwanted phenomenon that we observed empirically. ",
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+ "img_path": "images/0a1f842c7c2f27339003af5f5ca74ec6399ee2beaa7f81924ce3ed1fbaf9fc58.jpg",
+ "image_caption": [
+ "Figure 5: Each row compares CartoonX explanations of misclassifications by MobileNetV3-Smal to Pixel RDE (Macdonald et al., 2019), Integrated Gradients (Sundararajan et al., 2017), and Smoothgrad (Smilkov et al., 2017). The predicted label is depicted above each misclassified image. "
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+ "text": "In practice, explaining misclassifications is particularly relevant since good explanations can pinpoint model biases and causes for model failures. We observe that CartoonX is particularly good at explaining certain misclassifications, which we illustrate for three examples in Figure 5 and many more in Appendix A.6. In the first row in Figure 5, the input image shows a man holding a dog that was classified as a “diaper”. CartoonX shows the man not holding a dog but a baby, revealing that the neural net associated diapers with babies and babies with the pose with which the man is holding the dog. In the second row, the input image shows a dog sitting on an armchair with leopard patterns. The dog was classified as an “Egyptian cat”, which can exhibit leopard-like patterns. CartoonX exposes the Egyptian cat by connecting the dog’s head to parts of the armchair forming the cat’s torso and legs. In the last row, the input image displays the backside of a man wearing a striped sweater that was classified as a “screw”. CartoonX reveals how the stripe patterns look like a screw to the neural net. ",
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+ "img_path": "images/93b3a56308de3770000542d827bd0e4671770354a0f003b0ff69f08314095854.jpg",
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+ "Figure 6: CartoonX explanations for VGG16 compared to state-of-the-art methods, that is, Pixel RDE (Macdonald et al., 2019), Integrated Gradients (Sundararajan et al., 2017), Smoothgrad (Smilkov et al., 2017), Guided Backprop (Springenberg et al., 2015), and LRP (Bach et al., 2015). "
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+ "text": "6 CONCLUSION ",
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+ "text": "CartoonX is the first explainability method to extract the relevant piece-wise smooth part of an image and is based on our novel formulation of the RDE framework. We corroborated experimentally that CartoonX explanations are highly interpretable due to their cartoon-like nature and surprisingly well-suited to explain misclassifications. Nonetheless, Cartoon RDE is still computationally quite expensive, like other perturbation-based explanation methods. In the future, we hope to devise new techniques to speed up the runtime for CartoonX. Moreover, we are pursuing applications of CartoonX beyond explanation tasks, such as detecting adversarial examples. We believe CartoonX is a valuable new explanation method for practitioners and potentially a great source of inspiration for future explanation methods aiming to tailor their explanations to other data domains. Our reformulation and reinterpretation of the RDE framework provide a blueprint for such future work: First, formulate an interpretation query related to the underlying model task, then find a representation system that sparsely represents the class of desirable explanations for the interpretation query. ",
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+ {
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+ "text": "Saining Xie, Ross B. Girshick, Piotr Dollar, Zhuowen Tu, and Kaiming He. Aggregated residual ´ transformations for deep neural networks. 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 5987–5995, 2017. ",
+ "bbox": [
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+ "page_idx": 10
+ },
+ {
+ "type": "text",
+ "text": "A APPENDIX ",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 10
+ },
+ {
+ "type": "text",
+ "text": "A.1 CARTOONX ALGORITHM ",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 10
+ },
+ {
+ "type": "text",
+ "text": "The final CartoonX algorithm is depicted in Algorithm 1. ",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "Algorithm 1: CartoonX ",
+ "text_level": 1,
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+ },
+ {
+ "type": "table",
+ "img_path": "images/5dee94cf1bc128a847ce9d9adeca502f44bb0180b9f6f21a677be2ba06a66b60.jpg",
+ "table_caption": [],
+ "table_footnote": [],
+ "table_body": "| AigormmmrT.CartoonA Data: Image x ∈ [0,1]exwxt with c channels and wt pixels, classifier Φ. Result: CartoonX explanation ε ∈ [0,1]w×t for decision Φ(x). Hyperparameters: Sparsity level X > O, number of steps N, number of noise samples L. Initialize mask s := [1,..,1]T ∈ [0,1]k on DWT coefficients h = [h1,.., hk] with x = f(h), where f is the discrete inverse wavelet transform; Compute predicted label j* := arg maxi Φ(x); fori←1toNdo |
| Sample L adaptive Gaussian noise samples u(1),., u(L) ~ N(μ,o²); Compute obfuscations y(1), ),.,y(L) with y() := f(h ① s+ (1- s) ①u(i)); Clip obfuscations into [0,1]cx w ×t; |
| Approximate expected distortion D(𝑥x,s,Φ) :=∑𝑖=1(Φj+(x)- Φj(y())²/L; |
| Compute loss for the mask l(s) := D(x,s,Φ) + λ|lsll1 and gradient Vsl(s); Update mask s with gradient descent step and clip s back to [0,1]k ; |
| end Compute wavelet coefficients h for greyscale image x of x; |
| Invert wavelet mask s back to pixel space as & := f(h s) ; Clip the explanation ε into [0,1]w×t to obtain ε. Visualize ε; |
",
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+ {
+ "type": "text",
+ "text": "A.2 EXPERIMENT DETAILS ",
+ "text_level": 1,
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+ },
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+ "type": "text",
+ "text": "Throughout our experiments with CartoonX and Pixel RDE, we used a learning rate of $\\epsilon = 0 . 0 0 1$ , a sample size of $L = 6 4$ for the adaptive Gaussian noise, and $N = 2 0 0 0$ steps. Several different sparsity levels were used. We recommend specifying the sparsity level in terms of the number of mask entries $k$ , i.e., choosing the product $\\lambda k$ . Pixel RDE typically requires a smaller sparsity level than CartoonX. We chose $\\bar { \\lambda k } \\in [ \\bar { 2 0 } , 8 0 ]$ for CartoonX and $\\bar { \\lambda } k \\in [ 3 , 2 \\bar { 0 } ]$ for Pixel RDE. The obfuscation strategy for Pixel RDE was chosen as Gaussian adaptive noise with mean and standard deviation computed for all pixel values (see Section 4.1.2). In Appendix 8, we show that Gaussian adaptive noise produces much more interpretable explanations than using a zero baseline perturbation. We implemented the DWT for CartoonX with the Pytorch Wavelets package, which is compatible with PyTorch gradient computations, and chose the Daubechies wavelet system with $J = 5$ scales and zero-padding. For the Integrated Gradients method, we used 100 steps, and for the Smoothgrad method, we used 10 samples and a standard deviation of 0.1. ",
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+ },
+ {
+ "type": "text",
+ "text": "A.3 SENSITIVITY TO HYPERPARAMETERS ",
+ "text_level": 1,
+ "bbox": [
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+ {
+ "type": "text",
+ "text": "We compare CartoonX’s sensitivity to its main hyperparameters, i.e., the sparsity level $\\lambda$ , the perturbation distribution $\\gamma _ { s }$ , and the distortion measure $\\bar { d ( \\Phi ( x ) , \\Phi ( y ) ) }$ . For each experiment, we fix all but one of the three parameters. ",
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+ "type": "text",
+ "text": "A.3.1 SENSITVITY TO THE SPARSITY LEVEL $\\lambda$ ",
+ "text_level": 1,
+ "bbox": [
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+ {
+ "type": "text",
+ "text": "Figure 7 plots CartoonX explanations and Pixel RDEs for increasing $\\lambda$ —the hyperparameter determining the explanation’s sparsity in the respective representation system. We find that CartoonX is less sensitive than Pixel RDE to $\\lambda$ . In practice, this means one can find a suitable $\\lambda$ faster for CartoonX than for Pixel RDE. ",
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+ },
+ {
+ "type": "text",
+ "text": "A.3.2 SENSITVITY TO THE DISTRIBUTION $\\gamma _ { s }$ ",
+ "text_level": 1,
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "Figure 8 plots CartoonX explanations for two choices of $\\gamma _ { s }$ : (1) Gaussian adaptive noise (see Section 4.1.2) and (2) constant zero perturbations (i.e. $v = 0$ with probability one under $\\gamma _ { s }$ ). We observe that the Gaussian adaptive noise gives much more meaningful explanations than the simple zero baseline perturbations. ",
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+ },
+ {
+ "type": "image",
+ "img_path": "images/669376803cf15b48772f47190fc15e7b3958a45c1013353768d96a6ea5ac042a.jpg",
+ "image_caption": [
+ "Figure 7: We compare the sensitivity of CartoonX and Pixel RDE to the sparsity level $\\lambda$ . The top row depicts CartoonX, and the bottom row depicts Pixel RDE, for increasing values of $\\lambda$ . Note that for $\\lambda = 0$ , Pixel RDE is entirely yellow because the mask is initialized as $s ^ { \\check { = } } [ 1 \\ldots 1 ] ^ { T }$ and $\\lambda = 0$ provides no incentive to make s sparser. For the same reason, CatoonX is simply the greyscale image for $\\lambda = 0$ . "
+ ],
+ "image_footnote": [],
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+ },
+ {
+ "type": "image",
+ "img_path": "images/014f60850b885dee2e6d5318a68d96e55506ad31ed7a0b81c56dda8b69fa0b6c.jpg",
+ "image_caption": [
+ "Figure 8: We compare the sensitivity of CartoonX to the perturbation distribution $\\gamma _ { s }$ . The top image was classified as a fountain and the bottom image as a viaduct. The second column depicts CartoonX with $\\gamma _ { s }$ as Gaussian adaptive noise, and the third column depicts CartoonX with $\\gamma _ { s }$ as constant zero perturbations (zero baseline). We observe that Gaussian adaptive noise is much more interpretable than the zero baseline. "
+ ],
+ "image_footnote": [],
+ "bbox": [
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+ "page_idx": 12
+ },
+ {
+ "type": "text",
+ "text": "A.3.3 SENSITVITY TO THE DISTORTION MEASURE $d ( \\Phi ( x ) , \\Phi ( y ) )$ ",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 12
+ },
+ {
+ "type": "text",
+ "text": "Figure 9 plots CartoonX explanations for the following four choices of $d ( \\Phi ( x ) , \\Phi ( y ) )$ , where $x$ is the original input, $y$ is the RDE obfuscation, and $\\Phi$ outputs post-softmax probabilities: ",
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+ {
+ "type": "text",
+ "text": "1. $d ( \\Phi ( x ) , \\Phi ( y ) ) = ( \\Phi _ { j ^ { * } } ( x ) - \\Phi _ { j ^ { * } } ( y ) ) ^ { 2 }$ , where $j ^ { * } : = \\arg \\operatorname* { m a x } _ { j } \\Phi _ { j } ( x )$ (squared $\\ell _ { 2 }$ in label ) ",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "2. $d ( \\Phi ( x ) , \\Phi ( y ) ) = ( \\Phi _ { j ^ { * } } ( x ) - 1 ) ^ { 2 }$ , where $j ^ { * } : = \\arg \\operatorname* { m a x } _ { j } \\Phi _ { j } ( x )$ (maximize label) \n3. $d ( \\Phi ( x ) , \\Phi ( y ) ) = \\lVert \\Phi ( x ) - \\Phi ( y ) \\rVert _ { 2 }$ ( $\\ell _ { 2 }$ probabilities) \n4. $d ( \\Phi ( x ) , \\Phi ( y ) ) = K L ( \\Phi ( y ) , \\Phi ( x ) )$ (KL-Divergence) ",
+ "bbox": [
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+ "page_idx": 13
+ },
+ {
+ "type": "text",
+ "text": "The explanations for $d ( \\Phi ( x ) , \\Phi ( y ) )$ as “squared $\\ell _ { 2 }$ in label”, “maximize label”, and ${ } ^ { 6 6 } \\ell _ { 2 }$ probabilities” look indistinguishable. For $\\bar { d } ( \\Phi ( x ) , \\mathbf { \\bar { \\Phi } } ( y ) )$ as KL-Divergence, we see a slightly less smooth explanation, which may be due to the fact that the KL-Divergence is unbounded unlike the other measures of distortion. ",
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+ "page_idx": 13
+ },
+ {
+ "type": "image",
+ "img_path": "images/9187ae880a8b62675d03bda7eac035f50f4ff1e5be910d54d869700c0b8877f3.jpg",
+ "image_caption": [
+ "Figure 9: We compare the sensitivity of CartoonX to four measures of distortion $d ( \\Phi ( x ) , \\Phi ( y ) )$ . Each of the measures of distortion is marked at the top of each column. We observe almost no difference in the CartoonX explanations for the four distortion measures. "
+ ],
+ "image_footnote": [],
+ "bbox": [
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+ "page_idx": 13
+ },
+ {
+ "type": "text",
+ "text": "A.4 RELIABILITY AND EXPLANATION ARTIFACTS ",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 13
+ },
+ {
+ "type": "text",
+ "text": "We argue experimentally why CartoonX is more reliable than Pixel RDE for image data. More precisely, we show that CartoonX is less susceptible to so-called explanation artifacts than Pixel RDE. An explanation artifact is an unwanted phenomenon that we observed for Pixel RDE: instead of marking the relevant entries in $x$ , the mask $s$ creates artificial edges that end up making up an artificial class prototype. Explanation artifacts are problematic because they highlight not actual substructures but artificial structures that trigger the classification. Examples for explanation artifacts in Pixel RDE are given in Figure 11. ",
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+ },
+ {
+ "type": "image",
+ "img_path": "images/202f6780abcf94eb3a89b7f1a8d763a4bce9c8a43edc6cc45602844f237f5f61.jpg",
+ "image_caption": [
+ "Figure 10: CartoonX and Pixel RDE are both performed on the image of the blue sky. However, both methods are adjusted here to find evidence for the output probabilities of the image of the airplane instead of the blue sky. Pixel RDE, unlike CartoonX, can create an artificial airplane as evidence for an airplane in the smooth blue sky. "
+ ],
+ "image_footnote": [],
+ "bbox": [
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+ "page_idx": 13
+ },
+ {
+ "type": "text",
+ "text": "Pixel RDE can produce artificial edges in smooth regions for the following reason: When $s$ has a curve-like structure in some region, unselected points near $s$ are replaced with perturbations that tend to differ from the values of the curve-like structure in $s$ . Thus, the curve-like structure also appears in the obfuscation and can produce low distortion if the structure makes up a prototypical class feature (see, for example, the airplane in Figure 10). ",
+ "bbox": [
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+ "page_idx": 13
+ },
+ {
+ "type": "text",
+ "text": "We suspect CartoonX is inherently less susceptible to explanation artifacts for the following reason: Natural images tend to be piece-wise smooth, and piece-wise smooth images have sparse high-frequency DWT coefficients that cluster about the edges Stephane (2009b) (see for example ´ Figure 4). For a DWT mask to create artificial edges, it has to select a curve-like structure in the high-frequency coefficients (low-frequency coefficients cannot create edges) and replace surrounding unselected values with different values. However, in CartoonX, perturbations of high-frequency coefficients are Gaussian with low variance centered close to zero (see adaptive Gaussian noise in Section 4.1.2), which are not very different from the values along the selected curve due to the sparsity of the coefficients. ",
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+ {
+ "type": "text",
+ "text": "",
+ "bbox": [
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+ "page_idx": 14
+ },
+ {
+ "type": "text",
+ "text": "We illustrate our previous reasoning about explanation artifacts in a controlled example (see Figure 10). We take an image $x ^ { ( \\mathrm { s k y } ) }$ of a blue sky that is very smooth and an image $x ^ { \\mathrm { ( p l a n e ) } }$ of a airplane. The goal is to show that Pixel RDE, unlike CartoonX, can create artificial evidence for the class airplane on the image of the smooth blue sky. We perform CartoonX and Pixel RDE on the blue sky image with the distortion function ",
+ "bbox": [
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+ "page_idx": 14
+ },
+ {
+ "type": "equation",
+ "img_path": "images/dcde7f43f02e3222daa273a29250cf103f0a708496a49f569bc22f0173240521.jpg",
+ "text": "$$\n\\forall y \\in \\mathbb { R } ^ { n } : \\ d ( \\Phi ( x ^ { ( \\mathrm { s k y } ) } ) , \\Phi ( y ) ) = 1 0 ^ { 6 } \\| \\Phi ( x ^ { ( \\mathrm { p l a n e } ) } ) - \\Phi ( y ) \\| _ { 2 } ,\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 14
+ },
+ {
+ "type": "text",
+ "text": "and a sparsity level of $\\lambda = 8 0 0 0 0$ . As expected, we observe that Pixel RDE, unlike CartoonX, can create an artificial plane in the smooth blue sky(see Figure 10). ",
+ "bbox": [
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+ "page_idx": 14
+ },
+ {
+ "type": "text",
+ "text": "A.5 EXPLAINING THROUGH IMAGENET HISTORY: FROM ALEXNET TO RESNETXT50",
+ "bbox": [
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+ "page_idx": 14
+ },
+ {
+ "type": "text",
+ "text": "In the deep learning community, it is well-known that AlexNet (Krizhevsky et al., 2012) provided a major breakthrough in deep learning, improving the top-5 error on ImageNet from $2 5 \\%$ to $16 \\%$ . Since then, deep learning based ImageNet classifiers have continued to drastically improve on ImageNet—achieving less than $6 \\%$ top-5 error in 2016. In Figure 12, we compare CartoonX for four ImageNet classifiers with increasing performance, starting with AlexNet (top-1 accuracy $5 6 . 5 5 \\%$ , AlexNet), VGG16 (top-1 accuracy $7 1 . 5 9 \\%$ , Simonyan & Zisserman (2015)), InceptionV3 (top-1 accuracy $7 7 . 2 9 \\%$ , Szegedy et al. (2016)), and ResNeXt50 (top-1 accuracy $7 7 . 6 2 \\%$ , Xie et al. (2017).) Throughout the experiment, the CartoonX hyperparameters for a given image are not changed for any of the four classifiers. ",
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+ },
+ {
+ "type": "text",
+ "text": "A.6 EXPLAINING MISCLASSIFICATIONS WITH CARTOONX ",
+ "text_level": 1,
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+ "page_idx": 14
+ },
+ {
+ "type": "text",
+ "text": "In Figure 13, 14, 15, and 16, we provide further examples where CartoonX provides insightful explanations for misclassified images. ",
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+ },
+ {
+ "type": "text",
+ "text": "A.7 CARTOONX COMPARED ON RANDOM IMAGENET SAMPLES ",
+ "text_level": 1,
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+ },
+ {
+ "type": "text",
+ "text": "Figure 17, 18, 19, and 20 compares CartoonX to Pixel RDE (Macdonald et al., 2019), Integrated Gradients (Sundararajan et al., 2017), Smoothgrad (Smilkov et al., 2017), Guided Backprop (Springenberg et al., 2015), and (Bach et al., 2015) on random Imagenet samples classified by VGG16. ",
+ "bbox": [
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+ "page_idx": 14
+ },
+ {
+ "type": "text",
+ "text": "A.8 CARTOONX FAILURES ",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 14
+ },
+ {
+ "type": "text",
+ "text": "We also show failures of CartoonX in Figure 21. These are examples of explanations that are not interpretable and seem to fail at explaining the model prediction. Notably, most failure examples are also not particularly well explained by other state-of-the-art methods. It is challenging to state the underlying reason for the CatoonX failures with certainty (there is always the possibility that the neural net bases its decision on non-interpretable grounds). We intentionally also showed uninterpretable CartoonX explanations that were not too sparse (all or almost black explanations) since one can typically fix these explanations by decreasing $\\lambda$ . ",
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+ "page_idx": 14
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+ {
+ "type": "image",
+ "img_path": "images/4348604dba432b3c96ce1e643eb3ab00bb1b1360ac0508fa151d7742220d33a6.jpg",
+ "image_caption": [
+ "Figure 11: Explanation artifacts in Pixel RDE. We observe that Pixel RDE tends to create edges that are not a subset of the edges in the original input image. These edges can make prototypical artifact patterns such as wrinkles in the cloak (first row), coral tentacles (second row), or chain mail (third row). "
+ ],
+ "image_footnote": [],
+ "bbox": [
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+ "page_idx": 15
+ },
+ {
+ "type": "image",
+ "img_path": "images/6787c0907f2780a91eeba9a8c758e1e7e257500e6962b20c21edad4a037d379c.jpg",
+ "image_caption": [
+ "Figure 12: We compare CatoonX explanations for classifications by AlexNet (Krizhevsky et al., 2012), VGG16 (Simonyan & Zisserman, 2015), InceptionV3 (Szegedy et al., 2016), and ResNeXt50 (Xie et al., 2017). Green labels mark correct classifications and red labels mark wrong classifactions. "
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+ "content": ". The clipped inversion in pixel space is the final explanation, which we call CartoonX.",
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| Sample L adaptive Gaussian noise samples u(1),., u(L) ~ N(μ,o²); Compute obfuscations y(1), ),.,y(L) with y() := f(h ① s+ (1- s) ①u(i)); Clip obfuscations into [0,1]cx w ×t; |
| Approximate expected distortion D(𝑥x,s,Φ) :=∑𝑖=1(Φj+(x)- Φj(y())²/L; |
| Compute loss for the mask l(s) := D(x,s,Φ) + λ|lsll1 and gradient Vsl(s); Update mask s with gradient descent step and clip s back to [0,1]k ; |
| end Compute wavelet coefficients h for greyscale image x of x; |
| Invert wavelet mask s back to pixel space as & := f(h s) ; Clip the explanation ε into [0,1]w×t to obtain ε. Visualize ε; |
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| Sample L adaptive Gaussian noise samples u(1),., u(L) ~ N(μ,o²); Compute obfuscations y(1), ),.,y(L) with y() := f(h ① s+ (1- s) ①u(i)); Clip obfuscations into [0,1]cx w ×t; |
| Approximate expected distortion D(𝑥x,s,Φ) :=∑𝑖=1(Φj+(x)- Φj(y())²/L; |
| Compute loss for the mask l(s) := D(x,s,Φ) + λ|lsll1 and gradient Vsl(s); Update mask s with gradient descent step and clip s back to [0,1]k ; |
| end Compute wavelet coefficients h for greyscale image x of x; |
| Invert wavelet mask s back to pixel space as & := f(h s) ; Clip the explanation ε into [0,1]w×t to obtain ε. Visualize ε; |
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| Sample L adaptive Gaussian noise samples u(1),., u(L) ~ N(μ,o²); Compute obfuscations y(1), ),.,y(L) with y() := f(h ① s+ (1- s) ①u(i)); Clip obfuscations into [0,1]cx w ×t; |
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| end Compute wavelet coefficients h for greyscale image x of x; |
| Invert wavelet mask s back to pixel space as & := f(h s) ; Clip the explanation ε into [0,1]w×t to obtain ε. Visualize ε; |
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diff --git a/parse/dev/_XNtisL32jv/_XNtisL32jv.md b/parse/dev/_XNtisL32jv/_XNtisL32jv.md
new file mode 100644
index 0000000000000000000000000000000000000000..253f71d7ead0f54a1e488d4c6c9c5bb545215d8a
--- /dev/null
+++ b/parse/dev/_XNtisL32jv/_XNtisL32jv.md
@@ -0,0 +1,342 @@
+# TEMPORAL EFFICIENT TRAINING OF SPIKINGNEURAL NETWORK VIA GRADIENT RE-WEIGHTING
+
+Shikuang Deng1,2, Yuhang $\mathbf { L i ^ { 3 } }$ , Shanghang Zhang4 & Shi $\mathbf { G u } ^ { 1 , 2 , 5 \boxtimes }$
+
+1University of Electronic Science and Technology of China,
+2Shenzhen Institute for Advanced Study, UESTC
+3Yale University, 4Peking University ,5Peng Cheng Laboratory
+dengsk119@std.uestc.edu.cn, yuhang.li@yale.edu, gus@uestc.edu.cn
+
+# ABSTRACT
+
+Recently, brain-inspired spiking neuron networks (SNNs) have attracted widespread research interest because of their event-driven and energy-efficient characteristics. Still, it is difficult to efficiently train deep SNNs due to the nondifferentiability of its activation function, which disables the typically used gradient descent approaches for traditional artificial neural networks (ANNs). Although the adoption of surrogate gradient (SG) formally allows for the back-propagation of losses, the discrete spiking mechanism actually differentiates the loss landscape of SNNs from that of ANNs, failing the surrogate gradient methods to achieve comparable accuracy as for ANNs. In this paper, we first analyze why the current direct training approach with surrogate gradient results in SNNs with poor generalizability. Then we introduce the temporal efficient training (TET) approach to compensate for the loss of momentum in the gradient descent with SG so that the training process can converge into flatter minima with better generalizability. Meanwhile, we demonstrate that TET improves the temporal scalability of SNN and induces a temporal inheritable training for acceleration. Our method consistently outperforms the SOTA on all reported mainstream datasets, including CIFAR-10/100 and ImageNet. Remarkably on DVS-CIFAR10, we obtained $8 3 \%$ top-1 accuracy, over $\bar { 1 0 \% }$ improvement compared to existing state of the art. Codes are available at https://github.com/Gus-Lab/temporal_ efficient_training.
+
+# 1 INTRODUCTION
+
+The advantages of Spiking neuron networks (SNNs) lie in their energy-saving and fast-inference computation when embedded on neuromorphic hardware such as TrueNorth (DeBole et al., 2019) and Loihi (Davies et al., 2018). Such advantages originate from the biology-inspired binary spike transmitted mechanism, by which the networks avoid multiplication during inference. On the other hand, this mechanism also leads to difficulty in training very deep SNNs from scratch because the non-differentiable spike transmission hinders the powerful back-propagation approaches like gradient descents. Recently, many studies on converting artificial neuron networks (ANNs) to SNNs have demonstrated SNNs’ comparable power in feature representation as ANNs (Han & Roy, 2020; Deng & Gu, 2020; Li et al., 2021a). Nevertheless, it is commonly agreed that the direct training method for high-performance SNN is still crucial since it distinguishes SNNs from converted ANNs, especially on neuromorphic datasets.
+
+The output layer’s spike frequency or the average membrane potential increment is commonly used as inference indicators in SNNs (Shrestha & Orchard, 2018; Kim et al., 2019). The current standard direct training (SDT) methods regard the SNN as RNN and optimize inference indicators’ distribution (Wu et al., 2018). They adopt surrogate gradients (SG) to relieve the non-differentiability (Lee et al., 2016; Wu et al., 2018; Zheng et al., 2021). However, the gradient descent with SG does not match with the loss landscape in SNN and is easy to get trapped in a local minimum with low generalizability. Although using suitable optimizers and weight decay help ease this problem, the performance of deep SNNs trained from scratch still suffers a big deficit compared to that of ANNs Deng et al. (2020). Another training issue is the memory and time consumption, which increases linearly with the simulation time. Rathi & Roy (2020) initializes the target network by a converted SNN to shorten the training epochs, indicating the possibility of high-performance SNN with limited activation time. The training problem due to the non-differentiable activation function has become the main obstruction of spiking neural network development.
+
+
+Figure 1: Workflow of temporal efficient training (TET). To obtain a more generalized SNN, we modify the optimization target to adjust each moment’s output distribution.
+
+In this work, we examine the limitation of the traditional direct training approach with SG and propose the temporal efficient training (TET) algorithm. Instead of directly optimizing the integrated potential, TET optimizes every moment’s pre-synaptic inputs. As a result, it avoids the trap into local minima with low prediction error but a high second-order moment. Furthermore, since the TET applies optimization on each time point, the network naturally has more robust time scalability. Based on this characteristic, we propose the time inheritance training (TIT), which reduces the training time by initializing the SNN with a smaller simulation length. With the help of TET, the performance of SNNs has improved on both static datasets and neuromorphic datasets. Figure 1 depicts the workflow of our approach.
+
+The following summarizes our main contributions:
+
+• We analyze the problem of training SNN with SG and propose the TET method, a new loss and gradient descent regime that succeeds in obtaining more generalizable SNNs. • We analyze the feasibility of TET and picture the loss landscape under both the SDT and TET setups to demonstrate TET’s advantage in better generalization. • Our sufficient experiments on both static datasets and neuromorphic datasets prove the effectiveness of the TET method. Especially on DVS-CIFAR10, we report $8 3 . 1 \bar { 7 } \%$ top-1 accuracy for the first time, which is over $1 0 \%$ better than the current state-of-the-art result.
+
+# 2 RELATED WORK
+
+In recent years, SNNs have developed rapidly and received more and more attention from the research community. However, lots of challenging problems remain to be unsolved. In general, most works on SNN training have been carried out in two strategies: ANN-to-SNN conversion and direct training from scratch.
+
+ANN-to-SNN Conversion. Conversion approaches avoid the training problem by trading high accuracy through high latency. They convert a high-performing ANN to SNN and adjust the SNN parameters w.r.t the ANN activation value layer-by-layer (Diehl et al., 2015; 2016). Some special techniques have been proposed to reduce the inference latency, such as the subtraction mechanism (Rueckauer et al., 2016; Han et al., 2020), robust normalization Rueckauer et al. (2016), spike-norm (Sengupta et al., 2018), and channel-wise normalization (Kim et al., 2019). Recently, Deng & Gu (2020) decompose the conversion error to each layer and reduce it by bias shift. Li et al. (2021a) suggest using adaptive threshold and layer-wise calibration to obtain high-performance SNNs that require a simulation length of less than 50. However, converted methods significantly extend the inference latency, and they are not suitable for neuromorphic data (Deng et al., 2020).
+
+Direct training. In this area, SNNs are regarded as special RNNs and training with BPTT (Neftci et al., 2019). On the backpropagation process, The non-differentiable activation term is replaced with a surrogate gradient (Lee et al., 2016). Compared with ANN-to-SNN conversion, direct training achieves high accuracy with few time steps but suffers more training costs (Deng et al., 2020). Several studies suggest that surrogate gradient (SG) is helpful to obtain high-performance SNNs on both static datasets and neuromorphic datasets (Wu et al., 2019; Shrestha & Orchard, 2018; Li et al., 2021b). On the backpropagation process, SG replaces the Dirac function with various shapes of curves. Exceptionally, Wu et al. (2018) first propose the STBP method and train SNNs on the ANN programming platform, which significantly promotes direct training development. Zheng et al. (2021) further proposes the tdBN algorithm to smooth the loss function and first realize training a large-scale SNN on ImageNet. Zhang & Li (2020) proposes TSSL-BP to break down error backpropagation across two types of inter-neuron and intra-neuron dependencies and achieve low-latency and high accuracy SNNs. Recently, Yang et al. (2021) designed a neighborhood aggregation (NA) method to use the multiple perturbed membrane potential waveforms in the neighborhood to compute the finite difference gradients and guide the weight updates. They significantly decrease the required training iterations and improve the SNN performance.
+
+# 3 PRELIMINARY
+
+# 3.1 ITERATIVE LIF MODEL
+
+We adopt the Leaky Integrate-and-Fire (LIF) model and translate it to an iterative expression with the Euler method (Wu et al., 2019). Mathematically, the membrane potential is updated as
+
+$$
+\begin{array} { r } { \pmb { u } ( t + 1 ) = \tau \pmb { u } ( t ) + \pmb { I } ( t ) , } \end{array}
+$$
+
+where $\tau$ is the constant leaky factor, ${ \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } \mathbf { } { \mathbf { } } \mathbf { } { \mathbf { } } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf \Psi \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf \Psi \Psi \mathbf { } \mathbf { } \mathbf \Psi \mathbf { } \mathbf { } \mathbf \Psi \mathbf { } \mathbf { } \mathbf \Psi \mathbf { } \mathbf \Psi \mathbf { } \mathbf \Psi \mathbf { } \mathbf \mathbf { } \mathbf \Psi \mathbf { } \mathbf \mathbf { } \mathbf \mathbf \Psi \Psi \mathbf { } \mathbf \mathbf \Psi \mathbf { } \mathbf \mathbf \Psi \mathbf { } \mathbf \mathbf \mathbf \Psi \Psi \mathbf { \mathbf } \mathbf \mathbf \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \mathbf \Psi \mathbf \mathbf \Psi \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf $ is the membrane potential at time $t$ , and $\mathbf { } I ( t )$ denotes the pre-synaptic inputs, which is the product of synaptic weight $\mathbf { W }$ and spiking input ${ \mathbf { } } x ( t )$ . Given a specific threshold $V _ { t h }$ , the neuron fires a spike and ${ \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } \mathbf { } { \mathbf { } } \mathbf { } { \mathbf { } } \mathbf { } { \mathbf { } } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf \Psi \mathbf { } \Psi \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf \Psi { } \mathbf \Psi \mathbf { } \mathbf { } \mathbf \Psi \Psi \mathbf { } \mathbf { } \mathbf \Psi \mathbf { } \mathbf \Psi \mathbf { } \mathbf \Psi \Psi \mathbf { } \mathbf \mathbf { } \mathbf \Psi \mathbf { } \mathbf \mathbf { } \mathbf \mathbf \Psi \Psi \mathbf { } \mathbf \mathbf \Psi \Psi \mathbf { } \mathbf \mathbf \Psi \mathbf { } \mathbf \mathbf \Psi \mathbf \Psi \Psi \mathbf \Psi \mathbf { \mathbf } \mathbf \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \mathbf \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \mathbf \Psi \mathbf \Psi \mathbf \mathbf \Psi \mathbf \Psi \mathbf \mathbf \Psi \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf $ reset to 0 when the ${ \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } \mathbf { } { \mathbf { } } \mathbf { } { \mathbf { } } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf \Psi \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf \Psi \Psi \mathbf { } \mathbf { } \mathbf \Psi \mathbf { } \mathbf { } \mathbf \Psi \mathbf { } \mathbf { } \mathbf \Psi \mathbf { } \mathbf \Psi \mathbf { } \mathbf \Psi \mathbf { } \mathbf \mathbf { } \mathbf \Psi \mathbf { } \mathbf \mathbf { } \mathbf \mathbf \Psi \Psi \mathbf { } \mathbf \mathbf \Psi \Psi \mathbf { } \mathbf \mathbf \mathbf { } \mathbf \mathbf \Psi \mathbf \Psi \mathbf { \Psi \mathbf } \mathbf \mathbf \Psi \mathbf \Psi \mathbf \mathbf \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \mathbf \Psi \mathbf \Psi \mathbf \mathbf \Psi \mathbf \Psi \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \Psi \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf $ exceeds the threshold. So the firing function and hard reset mechanism can be described as
+
+$$
+\pmb { a } ( t + 1 ) = \pmb { \Theta } ( \pmb { u } ( t + 1 ) - V _ { t h } )
+$$
+
+$$
+\pmb { u } ( t + 1 ) = \pmb { u } ( t + 1 ) \cdot ( 1 - \pmb { a } ( t + 1 ) ) ,
+$$
+
+where $\Theta$ denotes the Heaviside step function. The output spike $\mathbf { \delta } \mathbf { \ } \mathbf { \em a } ( t + 1 )$ will become the post synaptic spike and propagate to the next layer. In this study, we set the starting membrane $\pmb { u } ( 0 )$ to 0, the threshold $V _ { t h }$ to 1, and the leaky factor $\tau$ to 0.5 for all experiments.
+
+The last layer’s spike frequency is typically used as the final classification index. However, adopting the LIF model on the last layer will lose information on the membrane potential and damage the performance, especially on complex tasks (Kim et al., 2019). Instead, we integrate the pre-synaptic inputs $\mathbf { } I ( t )$ with no decay or firing (Rathi & Roy, 2020; Fang et al., 2021). Finally, we set the average membrane potential as the classification index and calculate the cross-entropy loss for training.
+
+# 3.2 SURROGATE GRADIENT
+
+Following the concept of direct training, we regard the SNN as RNN and calculate the gradients through spatial-temporal backpropagation (STBP) (Wu et al., 2018):
+
+$$
+\frac { \partial L } { \partial \mathbf { W } } = \sum _ { t } \frac { \partial L } { \partial \pmb { a } ( t ) } \frac { \partial \pmb { a } ( t ) } { \partial \pmb { a } ( t ) } \frac { \partial \pmb { u } ( t ) } { \partial \pmb { I } ( t ) } \frac { \partial \pmb { I } ( t ) } { \partial \mathbf { W } } ,
+$$
+
+where the term $\frac { \partial \pmb { a } ( t ) } { \partial \pmb { u } ( t ) }$ is the gradient of the non-differentiability step function involving the derivative of Dirac’s $\delta$ -function that is typically replaced by surrogate gradients with a derivable curve. So far,
+
+there are various shapes of surrogate gradients, such as rectangular (Wu et al., 2018; 2019), triangle (Esser et al., 2016; Rathi & Roy, 2020), and exponential (Shrestha & Orchard, 2018) curve. In this work, we choose the surrogate gradients shaped like triangles. Mathematically, it can describe as
+
+$$
+\frac { \partial \pmb { a } ( t ) } { \partial \pmb { u } ( t ) } = \frac { 1 } { \gamma ^ { 2 } } \mathrm { m a x } ( 0 , \gamma - | \pmb { u } ( t ) - V _ { t h } | ) ,
+$$
+
+where the $\gamma$ denotes the constraint factor that determines the sample range to activate the gradient.
+
+# 3.3 BATCH NORMALIZATION FOR SNN
+
+Batch Normalization (BN) (Ioffe & Szegedy, 2015) is beneficial to accelerate training and increase performance since it can smooth the loss landscape during training (Santurkar et al., 2018). Zheng et al. (2021) modified the forward time loop form and proposed threshold-dependent Batch Normalization (tdBN) to normalize the pre-synaptic inputs $\pmb { I }$ in both spatial and temporal paradigms so that the BN can support spatial-temporal input. We adopt this setup with the extension of the time dimension to batch dimension 1. In the inference process, the BN layer will be merged into the pre-convolutional layer, thus the inference rule of SNN remain the same but with modified weight:
+
+$$
+\hat { \mathbf { W } } \gets \mathbf { W } \frac { \gamma } { \alpha } , \hat { \pmb { b } } \gets \beta + ( \pmb { b } - \mu ) \frac { \gamma } { \alpha } ,
+$$
+
+where $\mu , \alpha$ are the running mean and standard deviation on both spatial and temporal paradigm, $\gamma , \beta$ are the affine transformation parameters, and $\mathbf { W } , b$ are the parameters of the pre-convolutional layer.
+
+# 4 METHODOLOGY
+
+# 4.1 FORMULA OF TRAINING SNN WITH SURROGATE GRADIENTS
+
+Standard Direct Training. We use $O ( t )$ to represent pre-synaptic input $\mathbf { } I ( t )$ of the output layer and calculate the cross-entropy loss. The loss function of standard direct training ${ \mathcal { L } } _ { \mathrm { S D T } }$ is:
+
+$$
+\mathcal { L } _ { \mathrm { S D T } } = \mathcal { L } _ { \mathrm { C E } } \big ( \frac { 1 } { T } \sum _ { t = 1 } ^ { T } O ( t ) , { \pmb y } \big ) ,
+$$
+
+where $T$ is the total simulation time, $\mathcal { L } _ { \mathrm { C E } }$ denotes the cross-entropy loss, and $\textbf { { y } }$ represents the target label. Following the chain rule, we obtain the gradient of $\mathbf { W }$ with softmax $S ( \cdot )$ inference function :
+
+$$
+\frac { \partial \mathcal { L } _ { \mathrm { S D T } } } { \partial { \bf W } } = \frac { 1 } { T } \sum _ { t = 1 } ^ { T } [ S ( O _ { \mathrm { m e a n } } ) - \hat { \pmb { y } } ] \frac { \partial O ( t ) } { \partial { \bf W } } ,
+$$
+
+where $O _ { \mathrm { m e a n } }$ denotes the average of the output $O ( t )$ over time, and $\hat { y }$ is the one-hot coding of $\textbf { { y } }$ .
+
+Temporal Efficient Training. In this section, we come up with a new kind of loss function ${ \mathcal { L } } _ { \mathrm { T E T } }$ to realize temporal efficient training (TET). It constrains the output (pre-synaptic inputs) at each moment to be close to the target distribution. It is described as:
+
+$$
+\mathcal { L } _ { \mathrm { T E T } } = \frac { 1 } { T } \cdot \sum _ { t = 1 } ^ { T } \mathcal { L } _ { \mathrm { C E } } [ O ( t ) , { \pmb y } ] .
+$$
+
+Recalculate the gradient of weights under the loss function ${ \mathcal { L } } _ { \mathrm { T E T } }$ , and we have:
+
+$$
+\frac { \partial \mathcal { L } _ { \mathrm { T E T } } } { \partial { \bf W } } = \frac { 1 } { T } \sum _ { t = 1 } ^ { T } [ S ( \pmb { O } ( t ) ) - \pmb { \hat { y } } ] \cdot \frac { \partial \pmb { O } ( t ) } { \partial { \bf W } } .
+$$
+
+# 4.2 CONVERGENCE OF GRADIENT DESCENT FOR SDT V.S. TET
+
+In the case of SDT, the gradient consists of two parts, the error term $( S ( O _ { \mathrm { m e a n } } ) - \hat { \pmb y } )$ and the partial derivative of output $\partial { \cal O } \bar { ( } t ) / \partial { \bf W }$ . When the training process reaches near a local minimum, the term $( S ( O _ { \mathrm { m e a n } } ) - \hat { \pmb y } )$ approximates 0 for all $t = 1 , . . . , T$ , ignorant of the term $\partial O ( t ) / \partial \mathbf { W }$ . For traditional ANNs, the accumulated momentum may help get out of the local minima (e.g. saddle point) that typically implies bad generalizability (Kingma & Ba, 2014; Kidambi et al., 2018). However, when the SNN is trained with surrogate gradients, the accumulated momentum could be extremely small, considering the mismatch of gradients and losses. The fact that the activation function is a step one while the SG is bounded with integral constraints. This mismatch dissipates the momentum around a local minimum and stops the SDT from searching for a flatter minimum that may suggest better generalizability.
+
+In the case of TET, this issue of mismatch is relieved by reweighting the contribution of $\partial { \cal O } ( t ) / \partial { \bf W }$ . Indeed, considering the fact that the first term $( S ( O ( t ) ) - \hat { { \mathbf { y } } } )$ is impossible to be 0 at every moment of SNN since the early output accuracy on the training set is not $1 0 0 \%$ . So TET needs the second term $\partial { \cal O } ( t ) / \partial { \bf W }$ close to 0 to make the ${ \mathcal { L } } _ { \mathrm { T E T } }$ convergence. This mechanism increases the norm of gradients around sharp local minima and drives the TET to search for a flat local minimum where the disturbance of weight does not cause a huge change in $O ( t )$ .
+
+Further, to ensure that the convergence with TET implies the convergence of SDT, we prove the following lemma:
+
+Lemma 4.1. $\mathcal { L } _ { S D T }$ is upper bounded by $\mathcal { L } _ { T E T }$
+
+Proof. Suppose $O _ { i } ( t )$ and $\hat { y } _ { i }$ denote the i-th component of $O ( t )$ and $\hat { y }$ , respectively. Expand Eqn.9, we have:
+
+$$
+\begin{array} { r l } & { \mathcal { L } _ { \mathrm { T E T } } = - \displaystyle \frac { 1 } { T } \sum _ { t = 1 } ^ { T } \sum _ { i = 1 } ^ { n } \hat { y } _ { i } \log S ( \boldsymbol { O } _ { i } ( t ) ) = - \frac { 1 } { T } \sum _ { i = 1 } ^ { n } \hat { y } _ { i } \log ( \prod _ { t = 1 } ^ { T } S ( \boldsymbol { O } _ { i } ( t ) ) ) } \\ & { \quad \quad \quad = - \displaystyle \sum _ { i = 1 } ^ { n } \hat { y } _ { i } \log ( \prod _ { t = 1 } ^ { T } S ( \boldsymbol { O } _ { i } ( t ) ) ) ^ { \frac { 1 } { T } } \geq - \sum _ { i = 1 } ^ { n } \hat { y } _ { i } \log ( \frac { 1 } { T } \sum _ { t = 1 } ^ { T } S ( \boldsymbol { O } _ { i } ( t ) ) ) } \\ & { \quad \quad \quad \geq - \displaystyle \sum _ { i = 1 } ^ { n } \hat { y } _ { i } \log ( S ( \frac { 1 } { T } \sum _ { t = 1 } ^ { T } O _ { i } ( t ) ) ) = \mathcal { L } _ { \mathrm { S D T } } , } \end{array}
+$$
+
+where the first inequality is given by the Arithmetic Mean-Geometric Mean Inequality, and the second one is given by Jensen Inequality since the softmax function is convex. As a corollary, once the ${ \mathcal { L } } _ { \mathrm { T E T } }$ gets closed to zero, the original loss function ${ \mathcal { L } } _ { \mathrm { S D T } }$ also approaches zero. □
+
+Furthermore, the network output $O ( t )$ at a particular time point may be a particular outlier that dramatically affects the total output since the output of the SNN has the same weight at every moment under the rule of integration. Thus it is necessary to add a regularization term like $\mathcal { L } _ { \mathrm { M S E } }$ loss to confine each moment’s output to reduce the risk of outliers:
+
+$$
+\mathcal { L } _ { \mathrm { M S E } } = \frac { 1 } { T } \sum _ { t = 1 } ^ { T } \mathrm { M S E } ( \mathbf { O } ( t ) , \phi ) ,
+$$
+
+where $\phi$ is a constant used to regularize the membrane potential distribution. And we set $\phi = V _ { t h }$ in our experiments. In practice, we use a hyperparameter $\lambda$ to adjust the proportion of the regular term, we have:
+
+$$
+\mathcal { L } _ { \mathrm { T O T A L } } = ( 1 - \lambda ) \mathcal { L } _ { \mathrm { T E T } } + \lambda \mathcal { L } _ { \mathrm { M S E } } .
+$$
+
+It is worth noting that we only changed the loss function in the training process and did not change SNN’s inference rules in the testing phase for a fair comparison. This algorithm is detailed in Algo.1.
+
+| Algorithm1:Temporalefficienttrainingforoneepoch Input: SNN model; Simulation length: T; Threshold: Vth; Training dataset; Validation dataset; |
| total training iteration in one epoch: Itrain; total validation iteration in one epoch: Ival |
| for all i= 1,2,...Itrain iteration do Get mini-batch training data,and class label: Yi; |
| Compute the SNN output Oi(t) of eatch time step; |
| Calculate loss function: LTOTAL = (1-λ)LTET + 入LMSE = |
| (1-λ):¹∑t=1LcE(O²(t),Yi)+>·¹∑t=1 MSE(Oi(t),𝜙); |
| Backpropagation and update model parameters; |
| end for all i= 1,2,..Ival iteration do |
| Get mini-batch validation data,and class label: Yi; |
| T |
| Compute the SNN average output Omean = ∑T=1 O(t) over al time step; |
| Compare the clasification factor Omean and Yi for classification; end |
+
+# 4.3 TIME INHERITANCE TRAINING
+
+SNN demands simulation length long enough to obtain a satisfying performance, but the training time consumption will increase linearly as the simulation length grows. So how to shorten the training time is also an essential problem in the direct training field. Traditional loss function ${ \mathcal { L } } _ { \mathrm { S D T } }$ only optimizes the whole network output under a specific $T$ , so its temporal scalability is poor. Unlike the standard training, TET algorithm optimizes each moment’s output, enabling us to extend the simulation time naturally. We introduce Time Inheritance Training (TIT) to alleviate the training time problem. We first use long epochs to train an SNN with a short simulation time T, e.g., 2. Then, we increase the simulation time to the target value and retrain with short epochs. We discover that TIT performs better than training from scratch on accuracy and significantly saves the training time. Assuming that training an SNN with simulation length $T = 1$ cost $t s$ time per epoch, the SNN needs 300 epochs to train from scratch, and the TIT needs 50 epochs for finetuning. So we need $1 8 0 0 t s$ time to train an SNN with $T = 6$ from scratch, but following the TIT pipeline with the initial $T = 2$ only requires $9 0 0 t s$ . As a result, the TIT can reduce the training time cost by half.
+
+# 5 EXPERIMENTS
+
+We validate our proposed TET algorithm and compare it with existing works on both static and neuromorphic datasets. The network architectures in this paper include ResNet-19 (Zheng et al., 2021), Spiking-ResNet34 (Zheng et al., 2021), SEW-ResNet34 (Fang et al., 2021), SNN-5, and VGGSNN. SNN-5 (16C3-64C5-AP2-128C5-AP2-256C5-AP2-512C3-AP2-FC) is a simple convolutional SNN suitable for multiple runs to discover statistical rules (Figure A. 7). The architecture of VGGSNN (64C3-128C3-AP2-256C3-256C3-AP2-512C3-512C3-AP2-512C3-512C3-AP2- FC) is based on VGG11 with two fully connected layers removed as we found that additional fully connected layers were unnecessary for neuromorphic datasets.
+
+# 5.1 MODEL VALIDATION AND ABLATION STUDY
+
+Effectiveness of TET over SDT with SG. We first examine whether the mismatch between SG and loss causes the convergence problem. For this purpose, we set the simulation length to 4 and change the spike function $\Theta$ in Eqn.2 to Sigmoid $\sigma ( \bar { k } \cdot \mathrm { { i n p u t } } )$ . We find that the TET and SDT achieved similar accuracy (Table 2) when $k = 1 , 1 0 , 2 0$ . This indicates that both TET and SDT work when the gradient and loss function match each other. Next, we compare the results training with ${ \mathcal { L } } _ { \mathrm { S D T } }$ and ${ \mathcal { L } } _ { \mathrm { T E T } }$ on SNNs (ResNet-19 on CIFAR100) training with surrogate gradient for three runs. As shown in Table 1, our proposed new TET training strategy dramatically increases the accuracy by $3 . 2 5 \%$ when the simulation time is 4 and $3 . 5 3 \%$ when the simulation time is 6. These results quantitatively support the effectiveness of TET in solving the mismatch between gradient and loss in training SNNs with SG.
+
+
+Figure 2: Loss landscape of VGGSNN. The 2D landscape of ${ \mathcal { L } } _ { \mathrm { S D T } }$ and ${ \mathcal { L } } _ { \mathrm { T E T } }$ from two different training methods.
+
+Table 1: Comparison between SDT and TET. We adopt the SNN architecture ResNet-19 with SG on CIFAR100 and record the results with three different simulation lengths 2, 4, and 6.
+
+| Algorithm1:Temporalefficienttrainingforoneepoch Input: SNN model; Simulation length: T; Threshold: Vth; Training dataset; Validation dataset; |
| total training iteration in one epoch: Itrain; total validation iteration in one epoch: Ival |
| for all i= 1,2,...Itrain iteration do Get mini-batch training data,and class label: Yi; |
| Compute the SNN output Oi(t) of eatch time step; |
| Calculate loss function: LTOTAL = (1-λ)LTET + 入LMSE = |
| (1-λ):¹∑t=1LcE(O²(t),Yi)+>·¹∑t=1 MSE(Oi(t),𝜙); |
| Backpropagation and update model parameters; |
| end for all i= 1,2,..Ival iteration do |
| Get mini-batch validation data,and class label: Yi; |
| T |
| Compute the SNN average output Omean = ∑T=1 O(t) over al time step; |
| Compare the clasification factor Omean and Yi for classification; end |
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| Brightness&Contrast Impact | 0.966± 0.030 | 0.913 ±0.047 | 1.0±0.009 | 0.951 ±0.016 | 0.663±0.239 |
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| Brightness&Contrast Psimilarity | 0.0307±0.0039 | 0.011± 0.0003 | 0.2190± 0.0046 | 0.2147±0.0212 | 0.1045± 0.0031 |
| Blurring Psimilarity | 0.1672±0.0192 | 0.0707±0.0074 | 0.0351±0.0072 | 0.1442±0.0107 | 0.2014±0.0645 |
| Rotation Psimilarity | 0.0520±0.0070 | 0.0275±0.0016 | 0.1020±0.0115 | 0.0422± 0.0033 | 0.1020±0.0115 |
| Shifting Psimilarity | 0.0492±0.0046 | 0.0650±0.0092 | 0.2455±0.0432 | 0.0945±0.0032 | 0.1167±0.0121 |
| Compression Artifacts Psimilarity | 0.0240±0.0037 | 0.1325±0.0301 | 0.2506±0.0559 | 0.2250±0.0202 | 0.1592±0.0369 |
| Perspective Transform Psimilarity | 0.0398±0.0067 | 0.012±0.0007 | 0.0140±0.0018 | 0.0422±0.0016 | 0.0440±0.0050 |
| Carlini&WagnerRaw Scores | 15.0±2.549 | 285.0±25.495 | -20.8±0.189 | 1168.0± 140.696 | 4090.0±347.979 |
| Brightness&Contrast Raw Scores | 17.0±1.651 | 45.0±6.846 | -21.0±0.000 | 744.0±76.957 | 3180.0±711.027 |
| Blurring Raw Scores | 18.0±3.405 | 190.0±33.015 | -21.0±0.000 | 820.0±72.013 | 3880.0±329.484 |
| Rotation Raw Scores | 2.0±1.264 | 190.0± 27.203 | -20.6±0.209 | 873.0±201.866 | 3150.0±482.959 |
| Shifting Raw Scores | 13.0±1.449 | 70.0±20.248 | -21.0±0.000 | 988.0± 89.057 | 3560.0± 437.538 |
| Compression ArtifactsRaw Scores Perspective Transform Raw Scores | 17.0±3.478 | 60.0±18.439 | -19.4±0.428 | 2589.0±389.679 | 3980.0±593.936 |
| 1.0±0.948 | 75.0±12.649 | -20.9±0.126 | 486.0±29.127 | 3550.0±435.028 |
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| Rotation Degree | 1.4 | 1.6 | 3 | 1.8 | 5 |
| Shifting[ti,tj] | [1,1] | [0,1] | [2,1] | [1,2] | [2.2] |
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| Carlini&Wagner Impact | 0.982±0.009 | 0.451±0.231 | 0.995±0.014 | 0.928±0.030 | 0.567 ±0.159 |
| Brightness&Contrast Impact | 0.966± 0.030 | 0.913 ±0.047 | 1.0±0.009 | 0.951 ±0.016 | 0.663±0.239 |
| Blurring Impact | 0.979±0.009 | 0.635±0.200 | 1.0±0.000 | 0.946±0.015 | 0.589±0.150 |
| Rotation Impact | 0.997±0.004 | 0.635±0.189 | 0.99±0.015 | 0.942±0.042 | 0.581±0.158 |
| Shifting Impact | 0.985 ±0.005 | 0.865±0.140 | 1.0±0.00 | 0.935 ±0.023 | 0.623±0.199 |
| Compression Artifacts Impact | 0.980 ±0.013 | 0.884 ±0.128 | 0.962±0.032 | 0.803 ±0.051 | 0.578 ±0.271 |
| Perspective Transform Impact | 0.998±0.003 | 0.865±0.087 | 0.996±0.009 | 0.968±0.006 | 0.624±0.198 |
| Carlini&Wagner Psimilarity | 0.0657±0.0073 | 0.2622±0.0312 | 0.6134±0.0271 | 0.2714±0.0285 | 0.1336± 0.0231 |
| Brightness&Contrast Psimilarity | 0.0307±0.0039 | 0.011± 0.0003 | 0.2190± 0.0046 | 0.2147±0.0212 | 0.1045± 0.0031 |
| Blurring Psimilarity | 0.1672±0.0192 | 0.0707±0.0074 | 0.0351±0.0072 | 0.1442±0.0107 | 0.2014±0.0645 |
| Rotation Psimilarity | 0.0520±0.0070 | 0.0275±0.0016 | 0.1020±0.0115 | 0.0422± 0.0033 | 0.1020±0.0115 |
| Shifting Psimilarity | 0.0492±0.0046 | 0.0650±0.0092 | 0.2455±0.0432 | 0.0945±0.0032 | 0.1167±0.0121 |
| Compression Artifacts Psimilarity | 0.0240±0.0037 | 0.1325±0.0301 | 0.2506±0.0559 | 0.2250±0.0202 | 0.1592±0.0369 |
| Perspective Transform Psimilarity | 0.0398±0.0067 | 0.012±0.0007 | 0.0140±0.0018 | 0.0422±0.0016 | 0.0440±0.0050 |
| Carlini&WagnerRaw Scores | 15.0±2.549 | 285.0±25.495 | -20.8±0.189 | 1168.0± 140.696 | 4090.0±347.979 |
| Brightness&Contrast Raw Scores | 17.0±1.651 | 45.0±6.846 | -21.0±0.000 | 744.0±76.957 | 3180.0±711.027 |
| Blurring Raw Scores | 18.0±3.405 | 190.0±33.015 | -21.0±0.000 | 820.0±72.013 | 3880.0±329.484 |
| Rotation Raw Scores | 2.0±1.264 | 190.0± 27.203 | -20.6±0.209 | 873.0±201.866 | 3150.0±482.959 |
| Shifting Raw Scores | 13.0±1.449 | 70.0±20.248 | -21.0±0.000 | 988.0± 89.057 | 3560.0± 437.538 |
| Compression ArtifactsRaw Scores Perspective Transform Raw Scores | 17.0±3.478 | 60.0±18.439 | -19.4±0.428 | 2589.0±389.679 | 3980.0±593.936 |
| 1.0±0.948 | 75.0±12.649 | -20.9±0.126 | 486.0±29.127 | 3550.0±435.028 |
| Brightness&Contrast [α,β] | [1.2,40] | [0.9,20] | [1.7,40] | [2.4,-275] | [2.4,-260] |
| Blurring Kernel Size | 5 | 3 | 3 | 5 | 5 |
| Rotation Degree | 1.4 | 1.6 | 3 | 1.8 | 5 |
| Shifting[ti,tj] | [1,1] | [0,1] | [2,1] | [1,2] | [2.2] |
| Perspective Transform Norm | 1 | 1 | 3 | 2 | 3 |
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