One Wave To Explain Them All: A Unifying Perspective On Feature Attribution
Authors: Gabriel Kasmi
Amandine Brunetto
Thomas Fel
Jayneel Parekh
Venue: ICML 2025 poster (2025)
Keywords: interpretability
feature attribution
wavelet
images
audio
3D shapes
Primary area: social_aspects->accountability_transparency_and_interpretability
TL;DR: We generalize gradient-based explainability to the wavelet domain; expanding it to new modalities beyond images and show that it yields novel insights on model understanding.

Abstract: Feature attribution methods aim to improve the transparency of deep neural networks by identifying the input features that influence a model's decision. Pixel-based heatmaps have become the standard for attributing features to high-dimensional inputs, such as images, audio representations, and volumes. While intuitive and convenient, these pixel-based attributions fail to capture the underlying structure of the data. Moreover, the choice of domain for computing attributions has often been overlooked. This work demonstrates that the wavelet domain allows for informative and meaningful attributions. It handles any input dimension and offers a unified approach to feature attribution. Our method, the **W**avelet **A**ttribution **M**ethod (WAM), leverages the spatial and scale-localized properties of wavelet coefficients to provide explanations that capture both the *where* and *what* of a model's decision-making process. We show that WAM quantitatively matches or outperforms existing gradient-based methods across multiple modalities, including audio, images, and volumes. Additionally, we discuss how WAM bridges attribution with broader aspects of model robustness and transparency. Project page: https://gabrielkasmi.github.io/wam/

One Wave To Explain Them All: A Unifying Perspective On Feature Attribution
Gabriel Kasmi 12
Amandine Brunetto
Thomas Fel Jayneel Parekh* 4
Abstract
This opacity motivated the rise of Explainable Artifi- cial Intelligence (XAI) techniques to provide human- understandable explanations of model decisions. While XAI has been predominantly applied in image classification, it is also extending into other fields, such as audio and vol- umes classification (Parekh, 2023; Paissan et al., 2024; Chen et al., 2021; Zheng al., 2019). Currently, the most popular XAI methods are feature attribution methods (Shrikumar et al., 2017; Sundararajan al., 2017; Smilkov et al., 2017; Fel et al., 2021; Novello et al., 2022; Muzellec et al., 2024) which consist in generating saliency maps, i.e., heatmaps highlighting the important pixels on the input image (Zeiler & Fergus, 2014b). The same principle has been expanded for audio and volumes, where most feature attributions are computed on 2D projections of the input modality. However, the pixel domain does not inherently capture the structural properties of signals. So far, existing works have overlooked the question of the domain in which feature attribution is computed. However, signal processing theory states that meaningful features often exist at multiple scales and are better represented in transformed domains that account for spatial and frequency information. This underlines the need for an alternative representations that preserves these prop- erties to provide more faithful explanations.
Feature attribution methods aim to improve the transparency of deep neural networks by identify- ing the input features that influence a model's deci- sion. Pixel-base heatmaps have become the stan- dard for attributing features to high-dimensional inputs, such as images, audio representations, and volumes. While intuitive and convenient, these pixel-based attributions fail to capture the under- lying structure of the data. Moreover, the choice of domain for computing attributions has often been overlooked. This work demonstrates that the wavelet domain allows for informative and mean- ingful attributions. It handles any input dimension and offers a unified approach to feature attribution. Our method, the Wavelet Attribution Method (WAM), leverages the spatial and scale-localized properties of wavelet coefficients to provide ex- planations that capture both the where and what of a model's decision-making process. We show that WAM quantitatively matches or outperforms existing gradient-based methods across multiple modalities, including audio, images, and volumes. Additionally, we discuss how WAM bridges at- tribution with broader aspects of model robust- ness and transparency. Project page: https: // //gabrie kkammi. . github. io/ wwam/.
Wavelets offer a hierarchical decomposition that retains both spatial and frequency information, unlike pixel-based methods that lose structural context. This makes wavelets a stronger foundation for interpreting model decisions across diverse modalities (or signals), as wavelets are inherently low-level features defined across various signal dimen- sions. This work introduces the Wavelet Attribution Method (WAM), which leverages wavelet coefficients as the basis features for computing attributions. Therefore, WAM pro- vides a unification of the feature attribution domain on com- ponents (wavelet coefficients) that are interpretable as low- level features of the input signal. We expand popular feature attribution methods, namely SmoothGrad (Smilkov et al., 2017) and Integrated Gradients (Sundararajan et al., 2017), within the wavelet domain, thus providing generaliza- tion of feature attribution to any square-interrable modality. WAM preserves the original structure of the data without converting it into 2D representation.
1. Introduction
Deep neural networks are increasingly being deployed in various applications. However, their opacity poses a signifi- cant challenge, especially in safety-critical domains, where understanding the decision- -aaiing process is crucial, for instance, in tumor detection (Pooch et al., 2020) or obstacle identification (Sun et al., 2022).
Miines Paris- PSL University, Paris, France ²RTE France, Paris La Défense, France "Kempner Institute. Harvard Uni- versity, Cambridge, MA, United-States "ISIR, Sorbonne Uni- versité, Paris, France. Correspondence to: Gabriel Kasmi seabriel.kasmijant jinmiiaaamiiessrs..........
Proceedings of the nnnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
Figure illustrates our approach, which computes the gradi- ent of classification model with respect to the wavelet co-

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One Wave To Explain Them All: A Unifying Perspective On Feature Attribution
Wavelet domain
Input domains
Model logits
Structural information Decomposition across
Z
X
f
y
Gradients Activations
scales
Common domain to compute attributions
Audio
Images
Volumes
Prequeccy (He))
3k
Time (6)
Interpretation audio spectrograms
Multiscale heatmap decomposition
Attribution on voxels
384
Precato
Intmmetin acsas
Csana 2C381
Emphasis on key audio parts
Identification of minimal images
Decomposition across scales
Figure 1. Explaining any input modality by decomposing the model's decision in the wavelet domain. WAM computes the gradient of the model's prediction with respect the wavelet coefficients the input modality (audio, image, volume). Unlike pixels, wavelet coefficients preserve structural information about the input signal, offering deeper insights into the model's behavior and going beyond where it focuses.
efficients of the input signal. These coefficients defined for any square-integrable function, regardless of dimension, and are localized in both space and scale, allowing them to capture information across different seales. For images and volumes, wavelet coefficients capture shapes, edges, and textures, while for audio signals, they represent transient patterns. Consequently, feature attribution the wavelet do- main enables us interpret audio, highlight important areas and patterns in images, or identify key regions in volumes, all in a post-hoc manner, i.e., using only a trained model and single backward pass.
filtering the important part of audio signal without requiring to train an explanation method.
In summary, the following are the key contributions of this work:
We introduce WAM, novel feature attribution method that operates in the wavelet domain, ensuring that the attributions align with the intrinsic properties of the data, whether it be audio, images, or volumes.
We demonstrate the effectiveness of WAM through extensive empirical evaluations across diverse models, datasets, and performance metrics.
We demonstrate the superiority of WAM through exten- sive empirical evaluations across diverse of metrics, model architectures, and datasets, showing its robustness and versatility. We also discuss the novel insights and con- nections that our method permits, such as connecting feature attribution and the characterization of model's 's robustness
We illustrate the relevance of transitioning the wavelet domain for feature attribution by showcas- ing how WAM offers novel insights into model trans- parency.
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One Wave To Explain Them All: A Unifying Perspective On Feature Attribution
2. Related works
nations into the two-dimensional pixel domain for 1D audio or 3D volumes further distorts these relationships, resulting in explanations that fail to capture the full complexity of the decision-making
2.1. Feature attribution methods
Images. Computer vision has supported the development of numerous post-hoe feature attribution methods (Baehrens et al., 2010). These techniques are applied to a trained model and estimate the importance of each pixel or region of an image based on its contribution to the model's pre- diction, i.e., estimate a saliency map. Post-hoc methods either leverage internal model information, such as gradi- ents (Baehrens et al., 2010; Simonyan et al., 2014; Zeiler & Fergus, 2014b; Springenberg et al., 2014; Sundararajan et al., 2017: Smilkov al., 2017; Muzellec al., 2024; Han et al., 2024; Binder et al., 2016) or apply perturbations to the input space (Lundberg & Lee, 2017; Ribeiro et al., 2016; Petsiuk et al., 2018; Fel et al., 2021; Novello et al., 2022; Kasmi et al., 2023a).
model's decision making process, ultimately reducing the relevance of the attributions.
2.2. Wavelet-based explainability
From an explainability perspective, we argue that the wavelet domain is more informative for feature attribution than the pixel domain, as it enables the extraction of inter- pretable low-level features from input signals. In particular, the dyadic wavelet decomposition of a signal isolates dis- tinct patterns within the input signal. In images, wavelet coefficients capture edges, textures, and patterns at different scales and orientations, aligning with human perception. For instance, wavelets coincide in some cases with Gabor filters (Gabor, 1946), which have been widely used for pattern analysis. In audio, they separate transient components from slowly varying structures, facilitating the identification of speech formats sudden bursts of sound.
Audio. The methods introduced in computer vision have been extended to audio classification in three directions. The first one explores the use of saliency methods to highlight key features for audio classifiers processing spectrograms (Becker et al., 2024; Won al., 2019) waveforms (Muck- enhirn et al., 2019). The second direction involves variants of LIME (Ribeiro et al., 2016) algorithm, proposed for differ- ent types of audio classification tasks (Mishra et al., 2017: 2020; Haunschmid et al., 2020; Chowdhury et al., 2021; Wullenweber et al., 2022). The third pursues the develop- ment of methods to generate listenable interpretations for audio classifiers by leveraging the hidden representations (Parekh al., 2022; Paissan et al., 2024).
Surprisingly, hardly a handful of works explored the use of wavelets for post-hoc explainability. CartoonX (Kolek et al., 2022) and ShearletX (Kolek et al., 2023) extended the Meaningful Perturbation framework (Fong & Vedaldi, 2017) into the wavelet and shearlet domains, respectively, allowing to highlight not just salient pixel regions but also relevant textures and edge structures. Recently, the Wavelet Scale Attribution Method (WCAM, Kasmi et al., 2023a), trans- posed Sobol-based feature attribution (Fel et al., 2021) to the wavelet domain, demonstrating the feasibility of multi- scale attributions for images. Wavelet coefficients have been used to construct scattering transforms (Bruna & Mallat, 2013; Andén & Mallat, 2014), i.e., fixed feature extractors that create translation-invariant representation of an input, stable to deformations. This aligns with the properties of trained feature extractors like convolutional neural networks (CNNs) without any training. This model and its descen- dants, such as the scattering spectra (Cheng et al., 2024), can be considered intrinsically interpretable models (Flora et al., 2022) and have been applied in fields where model understanding is crucial, such as finance (Leonarduzzi et al., 2019) and astrophysics (Cheng & Ménard, 2021).
Volumes. 3D data can be represented as point clouds or voxels. Point clouds offer an exact representation of the data but are unstructured. Voxels, conversely, are discretized but structured representation of the data, making them suit- able for processing with techniques such as 3D convolutions. Most explainability techniques for 3D data focused on ex- plaining point clouds. Chen et al. (2021); Schinagl et al. (2022); Gupta al. (2020); Zheng et al. (2019) introduced techniques to generate visual explanations for interpretabil- ity of 3D object detection and classification networks. They highlight critical features in point cloud by adapting 2D image-based saliency techniques (Gupta et al., 2020; Zheng et al., 2019), by using a perturbation- based approach (Schi- nagl et al., 2022) or by proposing a 3D variant of LIME (Tan& & Kotthaus, 2022). Explainability on volumes remains limited. A few works (Yang et al., 2018; Mamalakis al., 2023; Gotkowski et al., 2021) have proposed attention maps on 2D slices of 3D) medical scans using Gr.dC..C
Finally. Ha et al. (2021) proposed method which aims to distill information from a trained neural network into a wavelet transform, resulting in a highlight predictive, con- cise and computationally efficient model, whose properties (e.g., multi-scale structure) make it casy to interpret. This model was applied cosmological parameter inference and molecular-pare -aalttrrrrreerrrrr prediction.
Current feature attribution methods rely on a feature space that overlooks the inherent temporal, spatial, or geometric relationships within the data. Even worse, projecting expla-
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One Wave To Explain Them All: A Unifying Perspective On Feature Attribution
a) Original image
b) Wavelet heatmap
c) Image reconstruction
No details needed the background
High-resolution detail is essential in the center area
d) Heatmap and decomposition across scales
Fine scales
Intermediate scales
Coarse scales
Figure 2. WAM for images. Our method decomposes important components at multiple scales, revealing what the model focuses The same spatial location encodes different structural elements at different scales: the model predicts the elephant because of its trunk and ears but needs both the fine texture details (blue), the intemeedite-scall contours (red), and the coarse edges (yellow). Such decomposition not possible with traditional feature attribution in the pixel domain, and this hierarchical decomposition provides an interpretable view of model attributions.
3. Feature attribution in the wavelet domain
waves, which are localized in frequency but not in space, wavelets are localized in both space and frequency domains. analyze a function or a signal g E H. we define a family of functions D obtained by dilating and translating a mother wavelet vb:
Notations. Throughout, we let X' (2, be mea- sure space with set Ω o-algebra F, and measure μ. H (2(,,,) denotes the Hilbert space of inttrrli functions on X. Let f & H represent a predictor function (e.g.. a classifier), which maps an input æ E X to an output f(x) E V. We denote g H a square-integrable function.
D D= (MAAOE)) An/2 sb
(1)
DERR A30
where b is translation and À is scale factor. This family gives rise to the continuous wavelet transform (CWT) of defined as
3.1. Multiscale decompositions
Overview. Wavelets decompose signals into components that retain both spatial and frequency information, unlike Fourier transforms, which capture only frequency and dis- card spatial temporal context. The choice of the wavelet determines what kind of structural components will be high- lighted on the image. Its choice is therefore grounded in the application at hand. We refer the reader to Appendix A.2 for a discussion of the choice of the mother wavelet and its impact on the interpretation of the WAM.
W(g)(A,b) (g. v/,0))
b
g(x)
dx:
An/2
A
This operation can be interpreted as convolution with time-reversed and dilated version of the complex-conjugate z of ab (i.e., linear filtering operation), and thus defines a linear operator on H (Mallat, 2008).
Definition. A waveler is an integrable function vb E H. (RR) that is normalized, centered at the origin, and has zero mean (i.e., fga (bxx)) da 0). In contrast to sine
Discrete Wavelet Transform. When the scale parameter
is restricted to dyadic values A (2-5 j E Z) and

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the gradient of the classifier output with respect to the input One Wave To Explain Them All: A Unifying Perspective On Feature Attribution
translations lie on discrete grid, the continuous family becomes countable set. Under appropriate admissibility and regularity conditions on 1,, this discrete family forms an orthonormal basis or a tight frame for H. giving rise to the Discrete Wavelet Transform (DWT).
of the gradient) components in the input for determining the model's f decision. The higher the value, the greater the importance of the corresponding region.
However, moving from one pixel to the next is only a shift in the spatial domain and does not capture relationships between scales or frequencies. Therefore, we argue that the pixel domain not well suited for explaining what the model sceing on the image. On the other hand, the wavelet decomposition of an image and more broadly of any differ- entiable modality provides information on the structural components of the modality. Therefore, computing the gra- dient of f with respect to the wavelet transform of x will enable us to understand the model's reliance on features such as textures, edges, or shapes the case of images and volumes and transients, or harmonics in sounds.
The DWT is defined by projecting a signal g E (L" onto a dyadic family of wavelets obtained by discretizing the scale and translation parameters. For dyadic scales À = 2- and integer translations b = 2- define the family
[)))(())= = 2/2/2 k) KEZ",
(3)
The DWT coefficients are given by the inner products
Wbwr(g) k) = (g. x.,xa) = / g(z)) dx. (4)
Denoting x W(x) the discrete wavelet transform of æ, since W is invertible, we can define the saliency map in the wavelet domain as
These coefficients represent the content of at scale 2- and position 2- Under the appropriate admissibility and regularity conditions on *b, the DWT is an invertible linear transform: the signal can be exactly reconstructed from its wavelet coefficients Wowr(g)( k) via the inverse transform
(fc(a) Oz
@fo(a) dx
(x)
Tsa
(6)
Oz
(z) and where 0fc(z)
using the fact that æ W-1
denotes
Ox
image and (=) is the Jacobian matrix of the inverse wavelet transform. In practice, to retrieve Equation (6), we require the gradients on W(z) and directly evaluate a (f-WW
Vx R", g(x)= Eg. -))
(5)
Oz
jEZ
KE2"
Mallat (1989) showed that the dyadic wavelet transform of a signal g can be computed by applying a high-pass filter h, followed by downsampling by a factor of two, to obtain the detail coefficients, and similarly applying a low-pass filter g. followed by downsampling, obtain the approximation coefficients. Iteratively applying this to the approximation coefficients yields multilevel transform, where the jth level captures information at scales corresponding to frequency bands between 2 and 2/-1 octaves. For input signals g of dimension greater than one, the detail coefficients can be decomposed into directional components. In the 2D case (e.g., images), these typically correspond to vertical, horizontal, and diagonal orientations.
Oz
A remarkable property of this framework is that it accommo- dates any input dimension, and thus is modality-annostic. Therefore, we can apply it - and leverage its properties to numerical signals such as audio (ID signals), images (2D signals), or volumes (3D signals). this paper, we demon- strate the superiority of this method in the 1D, 2D and 3D settings compared to other domain-specific methods.
Smoothing. Saliency maps computed following Equa- tion (6) can fluctuate sharply at small scales as fc is not continuously differentiable. To yield smoother explanations, smoothing consists in averaging the explanation over a set of noisy samples (Smilkov et al., 2017). Analogously, we propose to calculate
3.2. Feature attribution in the wavelet domain
Problem formalization. Let f be a classifier and an in- put (e.g., an image, an audio, volume). The classifier f maps the input to class ye = arg max f (x) fc
1
(z)(2)=
V V&f v+f(W-
with slight abuse of notation. We recall that the original saliency map of the classifier f for class is then given by "/aa (x) = V fx)) where e denotes the class of interest. The saliency map is defined under the condition that the fe's 's are piecewise differentiable (Simonyan et al., 2014). It highlights the most influential (in terms of the absolute value
where ă = W(x + 5) and 8 ~ N(0, The number of samples, n., needed to compute the approximation of the smoothed gradient and the standard deviation o² are hyper- parameters. To transpose this method to the wavelet domain, we add noise to the input before computing its wavelet trans- form. We refer to this method as WAM SG throughout the
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One Wave To Explain Them All: A Unifying Perspective On Feature Attribution
rest of the paper. In Appendix A.I, we illustrate the en- hancement of the quality of the explanation after applying the smoothing to the gradients as described in Equation (7).
et al., 2016; Li et al., 2022). Insertion and deletion (Pet- siuk et al., 2018) have been widely used in XAI to evaluate the quality of feature attribution methods (Fong & Vedaldi, 2017). The deletion measures the evolution of the predic- tion probability when one incrementally removes features by replacing them with a baseline value according to their attribution score. Insertion consists gradually inserting features into baseline input.
Path integration. Another approach to derive smooth ex- planations from the model's gradients consists in averaging the gradient values along the path from bascline state to the current value. The baseline state is often set to zero, rep- resenting the complete absence of features. This technique, Integrated Gradients (Sundararajan et al., 2017), satisfies two axioms, sensitivity and implementation invariance. Sen- sitivity states that "for every input and baseline that differ in one feature but have different predictions, then the dif- fering feature should be given non-zero attribution" and implementation invariance that "the attributions are always identical for two functionally equivalent networks". We adapt the Integrated Gradient method from the image do- main to the wavelet domain. Denoting = = W(x), we evaluate
Given a model f and an explanation functional 7, the Faith- fulness F is given by
F(f. 7) = Ins(f.7) 7) - -Del(f 7)
(9)
We provide a detailed derivation of the Insertion and the Delction scores in Appendix C.2. They were initially de- fined in the context of images, but we propose to expand them audio and volumes.
Evaluation setting. To the best of our knowledge, no prior work has evaluated feature attribution method across multiple input modalities. To address this gap, we designed a comprehensive evaluation framework spanning diverse datasets: ESC-50 (Piczak, 2015) for audio, ImageNet (Rus- sakovsky et al., 2015) for images, and MedMNIST3D (Yang et al., 2023) for volumes. For each modality, we picked arbitrary model and applied the same feature attribu- tion methods to ensure consistency. We considered four feature attribution methods: SmoothGrad (Smilkov et al., 2017), Saliency (Simonyan et al., 2014), Integrated Gradi- ents (Sundararajan et al., 2017) and GradCAM (Selvaraju et al., 2017). We use the Python library Captum (Kokhlikyan et al., 2020) for consistently implementing existing methods on our datasets for audio and images. For volumes, we inte- grated WAM as additional method within the evaluation framework LATEC (Klein et al., 2024).
Mi = (z -0)- (o'
0fc (W-1 (zo + tazz (zo (W~-
zo))) da,
(8)
Oz
where zq denotes the baseline state of the wavelet decom- position of æ. We refer to this implementation of WAM as WAM1G-
Appendix A.1 illustrates the enhancement of the quality of the explanation after applying the smoothing to the gradi- ents as described in Equation (7) and after integrating the gradients, as described in Equation (8).
Theoretical properties. WAM extends feature attribution into the wavelet domain. The main focus of the paper is the proposal and the practical usage of WAM, however, it ap- pears that WG inherits from the theoretical properties of the vanilla integrated gradients method, and especially sensitivity. We refer the reader to Appendix B for a discus- sion on the theoretical aspects of WAM.
Results. Table presents the performance of WAM. Our method consistently outperforms traditional attribu- tion methods, especially for images and volumes, while achieving competitive results for audio. We can that WAM outperforms WAMsG. As further discussed in Appendix A. this improved performance arises because path integration captures inter-scale dependencies, revealing the relative importance of each scale in the model's 's pre- diction. Overall, these results highlight the relevance of using wavelets, rather than pixels, as the basis for feature attribution.
4. Evaluation and applications
4.1. Quantitative evaluation
Evaluation metrics. We evaluate WAM against gradient- based methods, as they are more reliable and efficient than perturbation-based methods (Crabbé & van der Schaar, 2023). We evaluate the Faithfulness, which we define the difference between the Insertion and Deletion scores, thereby following the definition proposed by Muzellec et al. (2024). We refer the reader to Chan et al. (2022) for a dis- cussion of the alternative definitions of Faithfulness. Faith- fulness is effective in evaluating attribution methods (Samek
In Appendix C.3, we evaluate WAM using additional datasets, metrics, and alternative model topologies, and com- pare its accuracy with more methods across all three input modalities. Our results further emphasize that WAM con- sistently matches or surpasses the performance of existing methods.
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One Wave To Explain Them All: A Unifying Perspective On Feature Attribution
Table Faithfulnees (Faith), Insertion (Ins) and Deletion (Del) scores across modalities. Evaluations are conducted on 400 samples from ESC-50 (fold 1). 1,000 images from ImageNet's validation set and the full AdrenaIMNIST3D test set (298 samples), WAM outperforms all other methods on image and volume modalities for both Insertion and Deletion metrics, meaning that is good identifying which features influence the most the model's decision. For audio, it achieves state-of-the-art performance in Insertion and performs as well as other methods for Deletion and Faithfulness. Best results are bolded and second best underlined.
Audio
Images
Volumes
Model
ResNet
EfficientNet
3D Former
Dataset
ESC-50
ImageNer
AdrenalMNIST3D
Ins (t)
Del (L)
Faith (f)
Ins(1)
Del (4)
Faith (1)
Ins (t)
Del (U)
Faith (1)
Integrated Gradients
0.267
0.047
0.264
0.113
0.113
0.000
0.666
0.743
-0.077
SmoothGrad
0.251
0.067
0.184
0.129
0.119
0.010
0.680
0.731
-0.051
GradCAM
0.274
0.201
0.072
0.364
0.303
0.061
0.689
0.744
-0.055
Saliency
0.220
0.154
0.066
0.148
0.140
0.008
0.751
0.742
0.009
IG (ours)
0.436
0,260
0.176
0.447
0.049
0.370
0.719
0.621
0.098
WAMsG (ours)
0.449
0,252
0.197
0.419
0.097
0.350
0.718
0.648
0.070
4.2. Exploring how WAM renews feature attribution
2018), ADV-Fast (Wong et al., 2020), ADV-Free (Shafahi et al., 2019).
Beyond its strong quantitative performance, WAM also offers new qualitative insights by disentangling features at different scales. This section highlights three representative use cases. We provide more examples in Appendix D.
Figure 3 averages the importance of each scale for 1,000 explanations computed from ImageNet. For each image, we normalize the importance to highlight the relative reliance different scales. As expected, the vanilla ResNet relies more fine scales (high frequencies) than the adversarially trained models. WAM consistently retrieves existing results from the robustness literature while avoiding the need for complex experimental settings, thus paving the way for casier assessment of the robustness of a model.
Images: bridging attribution and robustness. In com- puter vision, several works document correlation between the reliance on low frequency components of the image and the robustness of the model to adversarial or natural corruptions (Zhang et al., 2022; Chen et al., 2022; Wang et al., 2020). This property is grounded in the frequency or spectral bias of models (Rahaman et al., 2019) and has been highlighted by filtering out frequency bands from the Fourier transform of the input images. As the Fourier mag- nitude discards spatial information, frequencies are removed in all the image.
Importance of the Frequency Components
Vaailla ResNel
0.4
ADV
ADV-Free
0.3
ADV-Fast
Importancc
0.2
WAM quantifies the importance of each scale in the final prediction. Scales in the wavelet domain correspond to dyadic frequency ranges in the Fourier domain. Therefore, by summing the importance of wavelet coefficients at given scale, we can deduce the importance of the corre- sponding frequency range in the model's prediction. This estimation can be straightforwardly derived from the expla- nation and does not require multiple perturbations as done in previous works (Zhang al., 2022: Chen et al., 2022). As result, WAM can be used evaluate the robustness of model.
0.1
0.0
5
3
2
Scale (expressed 2 piseels in the frequency spectrum) (The smaller the scale, the higher the frequency)
Figure 3. Model robustness assessment with WAM Each bar indicates the importance of each scale in the model's 's prediction. Importances averaged over 1,000 explanations computed from ImageNet and normalized. The adversarially robust models rely more on coarse-scabe scale (low frequency) features than the vanilla ResNet, showing that WAM recovers results from the literature.
To show this property, we compare a set of ResNet models. The baseline is trained using the standard empirical risk minimization (Vanilla ResNet). The others are trained using three adversarial training methods: ADV (Madry et al.,
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One Wave To Explain Them All: A Unifying Perspective On Feature Attribution
Volumes:
revealing multi-scale structures. We retrieve
Target Audio
voxels the same decomposition for images audio. Figure 4 highlights the significance of the edges at larger scales, whereas, at smaller scales, the importance becomes increasingly concentrated at the center of the digit. This multi-scale decomposition is particularly valuable for volu- metric data, as it allows us to disentangle fine-grained details from broader structural patterns, which is crucial in many applications. For instance, in medical imaging, larger-scale features may correspond to organ boundaries or lesion con- tours, while finer scales capture subtle textural variations indicative of disease. Similarly, in 3D object recognition, coarse scales help identify overall shapes, whereas finer scales refine the details essential for distinguishing between similar objects. To the best of our knowledge, WAM is the first method demonstrate such decomposition perspective how models process
10k
Frequency
5k
Time
Input Audio
10k
Frequency
shapes, offering hierarchical spatial information in 3D data.
0
1
2
3
Heatmap and disentanglement across scales Heatmap
Fine scales
Interpretation Audio (Wavelet)
Frequency
2
Intermediate scales
Coarse scales
Time
Figure 5. Qualitative illustration of WAM for audio in a noise experiment. We add dB white noise the audio of the target class ("Crow")) as input to the classifier. Audio reconstructed using important wavelet coefficients effectively removes noise and highlights key parts of the target class (marked with green box). WAM identifies these relevant parts fully post-hoc manner.
cial for improving model interpretability and ensuring it focuses meaningful signal components. Traditionally, this task has been addressed using trained models such as Non-Negative Matrix Factorization (NMF, see for instance Parekh et al., 2022), which requires training to extract rele- vant features and explain decisions.
Figure Multi-scale decomposition feature importance on a volume using W.M. Coarse scales (yellow) highlight the edges of the number, capturing its global structure. Fine scales (blue) focus the digit's center, capturing high- geghffreyeecyy details and localized variations.
In contrast, WAM performs this identification in post- hoc manner, simplifying the task by directly highlighting the important audio components without the need for prior training. The interpretation audio in Figure 5, generated using top wavelet coefficients, provides further insights into the decision process and consistent with the fact that the model has not been affected by the addition of noise. Therefore, WAM effectively filters out the corrupted audio
Audio: identifying key sound components. Figure 5 qualitatively illustrates application of WAM for audio signals. We perform a noise experiment by adding 0 dB white noise to target audio to form the input audio. The model's 's prediction remains unchanged after adding noise, indicating it continues to focus on the relevant parts of the input audio. Identifying key parts of the audio input cru-
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One Wave To Explain Them All: A Unifying Perspective On Feature Attribution
while highlighting the key parts of the audio. Similarly, we discuss in Appendix D.3 how WAM also retrieves the key parts of an audio signal that has been corrupted by another source (overlap experiment).
Impact Statement
This work addresses the growing need for grounded. in- terpretable AI systems, especially in light of increasing regulatory demands for transparency in machine learning, By providing unified approach to explainability across diverse modalities (audio, image, and volumes) our method ensures that model explanations remain faithful to the inher- ent structure of the data. In parallel, as multimodal models become more prevalent, our work offers a solution that can applied to any input type, facilitating more consistent and robust explanations.
5. Discussion
Conclusion. We bring a novel perspective on feature at- tribution by computing explanations in the wavelet domain rather than the pixel domain, providing framework appli- cable to audio, images, and volumes. This method shifts away from traditional pixel-based decompositions used in saliency mapping, offering more precise insights into model decisions by leveraging the wavelet domain's ability to pre- serve inter-scale dependencies. This ensures that critical aspects like frequency and spatial structures are maintained, resulting in richer explanations compared to traditional fea- ture attribution methods.
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A. Implementational details and consistency checks
A.1. Effect of smoothing and integration on the explanations.
Figure 6 illustrates the improvement of the quality of the explanations by smoothing (third row) or integrating (fourth row) the gradients, compared to their raw values (second row). Smoothing follows Equation (7) and integration follows Equation (8). We visualize highlights the important locations within scales, while WAM emphasizes the relative importance of each scale. This suggests that the variants of WAM can be used for different applications. If one is interested in studying the inter-scale dependencies, the reliance on different frequency ranges, more broadly, the robustness of the model, then 5a should be preferred. On the other hand, if one wants to focus on the important details at each scale, for instance, to highlight the important patterns for the prediction a given class, then WAMsG appears as more relevant.
that both methods display complementary properties regarding the explanation. WAMsG enables
Image
Image
Image
Image
Image
Raw gradients
Raw gradients
Raw gradients
gradients
gradients
Averaging
Averaging
Averaging
Averaging
Averaging
Integrating
Integrating
Integrating
Integrating
Integrating
Figure 6. Effect of averaging or smoothing (3rd row) and integration (4th row) compared to the raw gradients (2nd) when computing the WAM of the images (Ist row). explanations are depicted the wavelet domain. "Smoothing" corresponds the application SroothGrad (Smilkov al., 2017) and "Integrating" to the application of Integrated Gradients (Sundararajan et al., 2017).
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A.2. Effect of the mother wavelet
Definition. The mother wavelet is the fundamental function & (2R())) used generate a family of wavelets through dilation and translation. This construction enables multiresolution analysis. From mother wavelet (b))),, the family of function is defined
1 An/2
a b
(
(10)
where À E I" \ {0} and b E I= are respectively the scale and translation parameters. The choice of the mother wavelet influences which features are captured, the quality of signal reconstruction, and the effectiveness of tasks like compression, denoising, and feature extraction.
Interpretability. WAM formally indicates that the model is sensitive to a given region the space-scale domain. Since the reconstructed image is the same, no matter the chosen basis for decomposition, the interpretation in terms of structural components (i.e., what those coefficients correspond to) depends the choice of the mother wavelet. The choice of the mother wavelet is therefore grounded in the application of interest, and the information that is invariant to the choice of the mother wavelet is the importance of of given scale at a given location.
Among the popular mother wavelet, the Haar wavelet (Haar, 1910) enables us to interpret the sensitivity of the model in terms of sharp changes, edges, or discontinuities on the image. can used for instance to detect grid-like structures, Daubechies wavelets (Daubechies, 1988) are smoother than Haar wavelets and highlight smooth patterns or fine textural details. Finally, the Biorthogonal (bior) wavelet (Cohen et al., 1992) underlines curves, contours, and symmetric structure, especially images.
Figure 7 illustrates the different structural features emphasized depending on the choice of the mother wavelet. In this example, we can see that the Haar wavelet decomposes the image in terms of sudden transitions, highlighting the edges of the input image. In contrast, using the Daubechies wavelet reveals more of the image textures. In Appendix D.4 we illustrate the usefulness of WAM a practical setting, where the chosen mother wavelet was the Haar wavelet, since the object of interest was to understand the model's behavior with respect to the detection of sharp objects on images.
Input Image
Haar Coefficients
Daubechies Coefficients
Figure 7. Impact of the mother wavelet. Different mother wavelets emphasize different features of the image.
Quantitative checks. Table 2 empirically checks that the explanations generated by WAM are invariant to the choice of the mother wavelet. For images and volumes, the insertion and deletion scores remain unchanged. Results are reported up to the 3rd decimal, but results remain the same up to the 8th decimal. For audio, the numerical results are close but not identical. This slight variation arises because the reconstruction of the input signal depends on the choice of the mother wavelet. Consequently, different wavelets can lead to minor differences in the results. However, when the reconstructions align, the resulting explanations are identical.
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Table Impact mother wavelet. Changing the mother wavelet does not affect IAM1G Insertion and Deletion scores for images and volumes, but it causes slight variations for audio.
Audio
Images
Volumes
Model Dataset
ResNet ESC-50
ResNet ImageNet
ResNet AdrenalMNIST
Insertion
Deletion
Insertion
Deletion
Insertion
Deletion
Bior
0.442
0.256
0.422
0.078
0.404
0.631
Daubechies
0,446
0.266
0,422
0.078
0,404
0.631
Haar (baseline)
0.438
0,263
0.422
0.078
0.404
0.631
A.3. Randomization checks
Motivation. The sanity checks introduced by Adebayo et al. (2018) aim at assessing whether an explanation depends on the model's parameters and the input labels. These tests assess the faithfulness of an explanation beyond visual evaluation. The randomization test evaluates whether an explanation depends on the model's parameters. Parameters have strong effect on a model's performance. Therefore, for a saliency method to be useful for debugging or analyzing model, should be sensitive its parameters. Adebayo et al. (2018) proposed different methods to randomize the model parameters. One particularly interesting implementation is the "cascading" randomization, in which the weights are randomized from the top to the bottom layers.
Method. We compute WAM for 1,000 ImageNet validation samples for set of increasingly randomized models. A randomized layer is a layer that reset at its initial value. We consider ResNet- 18 and randomize its layers from the shallowest convl to the deepest fc. We then compute the rank correlation (or Pearson correlation coefficient, Pearson, 1896) between the WAM of the original, fully-trained model (labeled orig) and the randomized models (labeled by the name of the layer until which they are randomized).
Randomiaation check ResNet-18 on ImageNet
1.04
WAM
WAM
Rack Cogrelation
1.00
0.98
0.94
0.92
arig
comel
bai
net1
maspead
layorl
layerd
layord
layord
elobal pool fr
Figure Cascading randomization WAM for explaining ResNet- on ImageNet. The y axis indicates the rank correlation between the original explanation and the explanation derived for randomization up that layer. The rank correlation is averaged over 1,000 ImageNet validation images.
Results. Figure 8 presents the results. The dotted line represents the average rank correlation across the 1,000 images, and the intervals represent the 95% confidence intervals. We can see that the correlation between WAM significantly decreases as the randomization increases, thereby showing that WAM is sensitive to the model's 's parameters. The lower decrease that we observe for WAM1G compared to WAMsG comes from the fact that WAM1G reflects more the inter-scales distribution of the importance than WAMsG does, pointed out by Rahaman et (2019) and Yin (2019), even random models
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One Wave To Explain Them All: A Unifying Perspective On Feature Attribution
exhibit spectral bias, i.e., natural tendency to favor lower frequencies over higher ones, which translates here into the fact of naturally putting more importance coarser scales rather than finer scales, matter the depth of the randomization. Figure 9 qualitatively illustrates how the explanations evolve the randomization depth increases.
Randomized up Randomized layer2
Randomized up
No randomization
conv1
Smoothed
Integrated
Figure Illustration of the randomization test cascade of randomizations of the layers of the classifier. From left right shows the explanation from WAM for an increasingly randomized ResNet-18.
B. Theoretical properties
Sundararajan al. (2017) highlighted axioms that feature attribution methods should satisfy, namely sensitivity and implementation invariance. We first recall the definitions of these axioms.
Axiom B.1 (Sensitivity (a)). If two inputs x and w' differ in one feature and have different predictions, then the differing feature must receive a non-zero attribution.
Axiom B.2 (Implementation Invariance). Two functionaalyy equivalent networks (i.e., computing the same function) must yield the same attributions.
Since WAM IG builds on integrated gradients, we may wonder whether inherits the properties of the original integrated gradients. In the following, let fo d + I be the class logit function in input space W denotes the wavelet transform operator and 1 its inverse. We define W(x) and let zu be fixed baseline in the wavelet space. We also define the composed function f = fo W-1, such that f(z) fa)
B.1. WAM1G satisfies completeness
We show that under some conditions on the choice of the mother wavelet 1b. WAM satisfies completeness, and thus sensitivity, Following Sundararajan et al. (2017), this implies that WAM satisfies sensitivity (a).
Definition B.1 (Completeness). An attribution method y satisfies completeness if
= F(ro))
(11)
for any given F; : Ra + R. differentiable almost everywhere. Theorem B.1. If W-1 is differentiable almost everywhere, then WAM1G satisfies completeness:
WAM1 f= fc(wa)
(12)
Proof. Proposition in Sundararajan et al. (2017) states that for some f differentiable almost everywhere,
Σ (G)(z) = f(x) - fxx)
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One Wave To Explain Them All: A Unifying Perspective On Feature Attribution
since by definition, I IG applied to f(z), we have
(2M fzz)).
Now taking / = fc W-1, Equation (x) can be re-written as
a WAMra((=)= [(z) ((20)
= =f.WW =z)) - (xo -f(()) - -f((z))
and fooW-1 remains differentiable (almost everywhere) provided W is differentiable. This can be enforeed by choosing mother wavelet 2a that is smooth (e.g., with Daubechies but not with Haar wavelets). Thus, provided zb is smooth, 1AM satisfies completeness.
B.2. Additional theoretical properties
It is also possible to show that WAM satisfies implementation invariance and linearity. As with completeness, it is enough to show that the corresponding composed function f "respects" implementation invariance or linearity of fc. And since IG satisfies both these properties, JA will satisfy as well as it IG applied to the corresponding f functions.
For implementation invariance, if f(') ;e
p!2) have different architectures but exactly same output for any input the
corresponding composed functions f(1): f(2) also have the same output for any input and thus WAMIG outputs are exactly the same for any given input æ for f!!) f !!
For linearity, one needs to show that rG output for a linear combination of any two functions = (1) bse (2) linear combination of WAMIG output for each of them. Let f(s) be the associated composed function of f! Then f(3) af(1) bf(2) and because of linearity of IG applied f(a) 's, WAM satisfies linearity well. Thus, it's easy to that WA also satisfies implementation invariance and linearity axioms. More generally, we believe that WAM is path method, and thus inherits the properties of the path methods proved by Friedman (2004). However, the focus of the paper is on the proposal and practical usage of wavelet attribution methods. We keep any formal study of theoretical properties WAMJG or other variants WAM as future work.
B.3. Feature attribution beyond the input space
More generally, our approach expands attribution methods on transformations of the input in an alternative domain. It seems that the properties satisfied by traditional attribution method generalize to the case where attribution is done on transformations of the input data. This work thus opens a promising direction in the systematic study of generalized attribution spaces, or transformed domains where gradients can be computed more expressively while preserving fidelity the original input.
C. Complements on the quantitative evaluation
C.1. Benchmark construction
Images.
Our models' parameterizations for benchmarking WAM on images are the following:
ResNet (He et al., 2016): unless specified otherwise, we consider the resnet 18 variant,
EfficientNet (Tan & Le, 2019): we consider the tf efficientnet b0. ns jft inlk variant,
ConvNext (Liu et al., 2022): we consider the convnext_smal www.ni2k_fttiiniimmmm
variant,
DeiT' (Touvron et al., 2021): we consider the deit. tiny patch16 224. in1k
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One Wave To Explain Them All: A Unifying Perspective On Feature Attribution
All models retrieved from the PyTorch Image Models (Wightman, 2019) repository. We load the model with the pre- trained weights and directly evaluate them the validation set of ImageNet. We implement the SmoothGrad, GradCAM, and GradCAM Plus Plus methods ourselves and use the Captum library (Kokhlikyan et al., 2020) for implementing the Intergrated Gradients and the Saliency methods.
Audio. For audio, we use the same technical infrastructure to evaluate our method. We use the CNN classification model of Kumar al. (2018) as our black-box model explain. We consider single model as alternative models (Huang Leanos, 2018; Wilkinghoff, 2021; Lopez-Meyer et al., 2021) only variations around the same topology. We add O dB white noise to the ESC-50 samples using the pseudocode displayed in Figure 10.
Input: audio (input audio signal, array of int16)
Output: neisy_audio (audio with added Gaussian noise, array of int16)
def gad_sian noise (audio)
# Convert the audio to float32 for safe computation
audio float convert_to 3 (audio)
I Calculate RMS (Root Mean Square) of the audio signal rms_signal - sqrt (mean (audio_flcat ** )
# Generate Gaussian noise noise sanddm_normaa_distte
12
(mean=0,
13
std-1, shope-audio_foat.chhepe
14
16
# Calculate RMS the generated noise
17
rms_noise- sqrt (mean (noise ** 2))
18
# Scale noise to have the same RMS the audio signal noise noise (rms signal
noise)
21
22
# Add noise to the audio signal auly__audi_ffoat = audio_float noise
23
25
clip the ncisy audio ensure it stays within the int16 range
26
clip moisy_audioo_loat,, -32768, 32767)
27
# Convert the clipped noisy audic back to intl6 audic convert_to intl6 moolay_addio_elppeed
30
31
return audio
Figure 10. Pseudo-code for adding Gaussian noise audio.
Volumes. For volumes, consider the LATEC benchmark (Klein al., 2024) and evaluate method two datasets of MedMNIST (Yang et al., 2023): AdrenalMNIST3ID and VesseIMNIST. We carry out our evaluation over the complete test set. We included WAM as an additional feature attribution method for fair comparisons with existing methods.
C.2. Definition of the evaluation metrics
Insertion and Deletion. Insertion and deletion are two evaluation metrics proposed by Petsiuk et al. (2018). These metrics are "area-under-curve" (AUC) metrics, which report the change in the predicted probability for the image class when inserting (resp. removing) meaningful information highlighted by the attribution method. Petsiuk et al. (2018) initially defined this metric for images, where the important features correspond to pixels. We expand this metric to wavelet coefficients, thus enabling computation of the Insertion and the Deletion for any modality.
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One Wave To Explain Them All: A Unifying Perspective On Feature Attribution
Both metrics consider an input in baseline state. Insertion consists in adding the most important features identified by the attribution method. Formally, step & with subset up of important features (which correspond in case in wavelet coefficients) at step k,
Insertion' (+)
je......... f =m=
(13)
where f() is the predicted probability and -u denotes the complementary set of We add features by decreasing order of importance and for 1 S k2, Uk1 Wky i.e., we gradually add more and more features until we eventually recover the full input
The deletion performs the opposite operation where we start from plain input with all variables and we gradually set features in the baseline state Zo, from the most important to the less important. We have
Deletion (k) = (a =
(14)
Finally, for the insertion and the deletion, we measure the AUC, which is comprised between and 1. Given K steps, the Insertion score of the feature attribution r for the model f is
K
K Σ k=1
Ins(f:
Insertion (4) A
f f n ox =m)) D..
(15)
where Ag is the width between two subintervals. The computation is analogous for Del(f, 7).
If the attribution method picks relevant features, then only including them (resp, only removing them) should result in large increase (resp. large decrease) in the predicted probability. Therefore, the AUC should be close to 1 for the insertion and close to for the deletion. We set the baseline to 2og = 0.
y--Fidelity. The y-Fidelity is correlation metric. It measures the correlation between the decrease of the predicted probabilities when features are in a baseline state and the importance of these features. We have
Eg(a;), f(x) - f(
J--Fidelity = Corr
(16)
"C Ky. Iulad
iiuu
where g is the explanation function (i.e., the explanation method), which quantifies the importance of the set of features
Faithfulness on Spectra. The Faithfulness on Spectra (FF, Parekh al., 2022) measures how important the generated interpretation for a classifier. The metric is calculated by measuring the drop in class-specific logit value f(r). when the masked out portion of the interpretation mask my is input to the classifier. This amounts to calculating.
FF. = f(x) -f (x (1 my)).
(17)
It should be noted that this strategy to simulate removal may introduce artifacts in the input that affect the classifier's s output unpredictably. Also, interpretations samples with poor fidelity can lead to negative FF These observations point to this metric's potential instability and outlying values. Thus, we report the final faithfulness of the system as the median of FF, over the test set, denoted by FF. A positive FF would signify that interpretations are faithful to the classifier.
Input Fidelity. The Input Fidelity (Fid-In, Paissan et al., 2023) measures if the classifier outputs the same class prediction on the masked-in portion of the input image. It is defined as,
n n Σ
Fid-In
aargmas max f(x))e = arg max fc (x, C m,,)
arg
(18)
where II denotes the indicator function and, again, larger values are better.
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One Wave To Explain Them All: A Unifying Perspective On Feature Attribution
C.3. Additional evaluation results
This section provides additional quantitative results. For audio, we evaluated WAM using additional metrics and on a noisy version of the dataset ESC-50 and SONY-UST, done by Parekh et al. (2022). We also evaluated WAM by computing explanations from the mel-spectrogram and the wavelet coefficients of the input audio. For images, we evaluated our method using different model topologies: ResNet, EfficientNet, Data Efficient Transformer, and ConvNext. We also included additional metrics and additional methods. For volumes, we evaluated the performance of our method on MNIST3D using a ResNet3D backbone and we included two additional datasets, AdrenalMNIST and VesseIMNIST, and three model topologies: ResNet3D, EfficientNet3D, and 3D Former. We also explored the effect of the number of decomposition levels on performance. Across all these variants, WAM demonstrated solid performance compared the baselines, showing its robustness and versatility across use cases and datasets.
AUDIO
Table 3 presents the full evaluation results for all audio metrics. We report the results for the clean and noisy dataset, as done by Parekh et al. (2022). For WAM, we consider two variants: when the explanations are computed on the mel-spectrogram of the input audio and when it is computed from the wavelet coefficients. We can see that when the explanations is computed from the wavelet coefficients, accuracy higher. We report the results on two datasets, ESC-50 (Piczak, 2015) and SONY-USC (Cartwright al., 2019)
Table 3. Evaluation scores of WAM and comparison with baselines on 400 audio samples from ESC-50 (fold 1) and on the validation set of the SONY-UST dataset. The column "_" indicates that the samples are unaltered. The column "+WN" indicates that the samples have 0 dB Gaussian white noisc. Bolded results are the best and underlined values are the second best.
Metric Faithfulness (t)
Insertion (1)
Deletion({) + +WN
FF (T)
Fid-In (t) +WN
+ WW
+ +WN
+WN
IntegratedGGradienss
0.264
0.310
0.267
0.312
0.047
0.045
0.207
0.207
0.220
0.225
GradCAM
0.072
0.073
0,274
0.274
0.201
0.201
0.137
0.135
0.517
0.542
ESC-50
Saliency
0.066
0.065
0,220
0.221
0.154
0.156
0.166
0.168
0.253
0.245
SmoothGrad
0.184
0.184
0,251
0.251
0.067
0.067
0.193
0.194
0.177
0.175
WAMsG (ours, wavelet)
0.197
0.205
0.449
0.452
0.252
0.246
0.132
0.130
0.718
0.690
IA (ours, wavelet)
0.176
0.182
0.436
0.442
0.260
0.261
0.118
0.124
0.652
0.657
WAMsG (ours, melspec)
0.009
0.007
0.169
0.166
0.159
0.161
0.152
0.149
0.117
0.122
WAM1c (ours, melspec)
0.000
0.004
0.168
0.171
0.168
0.167
0.149
0.149
0.105
0.128
IntegeatedGrraiiens
0.151
0.151
0.732
0.732
0.581
0.582
-0.122
-0.121
0.000
0.002
GradCAM
0.027
0.026
0.600
0.598
0.573
0.571
-0.004
-0.004
0.800
0.774
CONY-USS USS
Saliency
0.000
0.003
0.706
0.708
0.706
0.705
-0.224
-0.225
0.000
0.002
SmoothGrad
0.115
0.116
0.682
0.682
0.567
0.566
-0.098
-0.098
0.000
0.002
WAM sG (ours, wavelet)
-0.076
-0.076
0.598
0.600
0.674
0.675
0.029
0.026
0.882
0.878
IA (ours, wavelet)
-0.075
-0.080
0.591
0.591
0.666
0.671
0.037
0.036
0.866
0.840
WAMsG (ours, melspec)
0.003
0.010
0.677
0.679
0.674
0.669
-0.188
-0.183
0.000
0.002
WAM1G (ours, melspec)
0.028
0.023
0.728
0.723
0.700
0.700
-0.202
-0.200
0.000
0.000
IMAGES
For images, we report the evaluation results across different model topologies and with additional methods. On a ResNet, we also compare our method with more recent feature attribution methods. All explanations are computed on 1,000 samples from the validation set of ImageNet.
Faithfulness and p-Fidelity. Table reports the evaluation results using different metrics across a wide range of topologies. We also include additional methods such as GradCAM ++ (Selvaraju et al., 2017) and Guided Brackpropagation
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One Wave To Explain Them All: A Unifying Perspective On Feature Attribution
(Springenberg al., 2014). We can see that in terms of Faithfulness, our method outperforms alternative approaches over all model topologies. On the other hand, the performance measured by the j--iidelity aligns WAM with existing approaches. We can see that the good results reported in Table 4 are mostly driven by our method's s very good results in Insertion. We can see that WAM systematically outperforms the other methods in both Insertion and Deletion.
Table 4. Evaluation results of WA M for images. We report the Faithfulness (Muzellec et al., 2024), j-Fidelity (Bhatt et al., 2021) ( and Insertion and Deletion (Petsiuk et al., 2018) scores obtained on 1,000 images from the validation set ImageNet and different model architectures. Bolded results are the best and underlined values the second best.
Model
ResNer
ConvNext
EfficientNe
DeïT
Mean
Saliency
0.025
0.032
0.008
0.038
0.025
Integrated Gradients
0,000
0,001
0,000
0.003
0,001
GradCAM 0.134
0.072
0.061
0.162
0.107
GradCAM++ 0.184
0.055
0.050
0.044
0.083
Faithfulness (t)
SmoothGrad
0.023
0.000
0.010
0.004
0.009
Guided-Backin Baiided-Backkropagaation
0.001
0.001
0.001
0.000
0.000
WAMsG (ours)
0.438
0.334
0.350
0.423
0.386
WAM IG (ours)
0.344
0.359
0.370
0.420
0.373
Saliency
0.154
0.186
0.180
0.195
0.179
Integrated Gradients
0.228
0.223
0.219
0.241
0.228
GradCAM
0.141
0.216
0.149
0.151
0.164
GradCAM ++
0.135
0.212
0.141
0.222
0.178
u-Fidelity (t)
SmoothGrad
0.220
0.227
0.211
0.230
0.222
Guided Backpropagation
0.216
0.229
0.234
0.226
0.226
WAMsG (ours)
0.215
0.208
0.213
0.216
0.213
WAM (ours)
0.170
0.166
0.165
0.182
0.171
Saliency
0.134
0.381
0.148
0.194
0.214
Integrated Gradients
0.087
0.305
0.113
0.095
0.150
GradCAM 0,413
0.495
0,364
0.352
0.406
GradCAM ++
0.452
0.562
0.350
0.313
0.419
Insertion (1)
SmoothGrad
0.106
0.253
0.129
0.108
0.149
Guided Backpropagation
0.090
0.332
0.117
0.093
0.158
WAMsG (ours) 0.557
0.606
0.447
0.546
0.539
WAM (ours)
0,422
0.557
0.419
0.492
0.473
Saliency 0.109
0.349
0.140
0.156
0.189
Integrated Gradients 0.087
0.304
0.113
0.092
0.149
GradCAM 0.279
0.423
0.303
0.190
0.299
GradCAM ++
0.268
0.507
0.300
0.269
0.336
Deletion (4)
SmoothGrad 0.083
0.253
0.119
0.104
0.140
Guided Backpropagation
0.089
0.331
0.116
0.092
0.157
WAMsG (ours)
0.119
0.272
0.097
0.123
0.153
WAM (ours) 0.078
0.198
0.049
0.072
0.099
Additional methods. Table compares WAM with more recent feature attribution methods, considering a ResNet as model backbone. We compare WAM against LayerCAM (Jiang et al., 2021), Guided Backpropagation (Springenberg et al., 2014), LRP (Binder et al., 2016) and SRD (Han et al., 2024).
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Table Evaluution of WAM on ImageNet with recent feature attribution methods using ResNet backbone. Best results are bolded.
second best underlined.
Metric
Faithfuleess (f)
Insertion (1)
Deletion (4)
p-Fidelity (t)
Guided Backprop
0.001
0.090
0.089
0.216
LayerCAM
0.139
0,432
0,293
0.143
LRP
0.039
0.277
0.238
0.126
SRD
0.002
0.102
0.101
1.151
WAM Ta (ours)
0.438
0.557
0.119
0.215
WAMsG (ours) 0.344
0.344
0.078
0.170
VOLUMES
Complementary results on MNIST3D. In Table 6 we evaluate our method against Saliency (Zeiler Fergus, 2014a), GradCAM (Selvaraju et al., 2017), GradCAM+ (Chattopadhay et al., 2018), IntegratedGradients (Sundararajan et al., 2017), GuideedBackroopagation (Springenherg et al., 2014) and SmoothGrad (Smilkov et al., 2017) on the dataset MNIST3D, We report the Insertion, the Deletion, and the Faithfulness. Overall, WAM performs well as better than the other methods.
Table 6. Evaluation scores (3D) on full test MNIST3D, Best results bolded, second best underlined.
Metric
Faithfulness (t)
Insertion (f)
Deletion(()
Saliency
-0.166
0.108
0.274
GradCAM
-0.225
0.106
0.330
GradCAM+
-0.250
0.104
0.354
IntegratedGradients
-0.123
0.108
0.232
GuidedBackprop
-0.110
0.108
0.218
SmoothGrad
-0.190
0.107
0.297
WAMsG (ours, 2)
-0.153
0.108
0.261
WA (ours, 2)
-0.094
0.106
0.200
WAMsG (ours, J=1]
-0.152
0.108
0.260
WAM (ours, J=
-0.096
0.107
0.203
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One Wave To Explain Them All: A Unifying Perspective On Feature Attribution
Evaluation MedMNIST. MedMNIST (Yang et al., 2023) is collection of standardized biomedical 2D and 3D images. We evaluated WAM two datasets from MedMNIST, AdrenalMNIST and VesseIMNIST. Table 7 shows the evaluation results of WAM on AdrenalMNIST and VesseIMNIST using different model topologies.
Table Evaluation scores (3D) on the full test set of AdrenalMNIST and VesSeLMNIST. Best results are bolded, second best underlined.
Model Metric
ResNer3D
EfficientNet3D
3D Former
Faith (?)
Ins (f)
Del (t)
Faith (1)
Ins (?)
Deletion (U)
Faith (f)
Ins (f)
Del (U)
Saliency
-0.136
0.633
0,769
-0.085
0.679
0.764
-0.027
0.715
0.742
GradCAM
-0.195
0.529
0.724
-0.052
0.693
0.745
-0.065
0.679
0.744
GradCAM++
-0.137
0.531
0.668
-0.076
0.695
0.771
-0.091
0.656
0.748
IntegratedGraddiens
-0,204
0.531
0.735
-0.056
0.705
0.761
-0.077
0.666
0.743
AdrenaMMNIST
Guided packpropagation
10.
0.585
0.717
0.003
0.683
0.680
-0.004
0.685
0.689
SmoothGrud
0.115
0.663
0.548
-0.070
0.686
0.756
-0.051
0.680
0.731
WAMsa (ours, J = 2)
-0.279
0,467
0,746
0.049
0.666
0.617
0.070
0.718
0.648
WAM (ours, 2)
-0.224
0.407
0.631
0.086
0.684
0.598
0.106
0.719
0.613
WAM (ours, J = I)
-0.273
0.474
0.746
0.026
0.661
0.635
0.050
0.717
0.667
WAM1a (ours, J = 1)
-0.224
0.478
0.702
0.013
0.699
0.686
0.026
0.697
0.671
Saliency
-0,068
0.833
0.901
-0.747
0.131
0.878
-0,099
0.762
0.861
GradCAM
-0.095
0.789
0.884
-0.728
0.126
0.854
-0.065
0.804
0.869
GradCAM++
0.011
0.871
0.860
-0.725
0.124
0.849
-0.066
0.797
0.863
IntegratedGGradients
-0.116
0.771
0.887
-0.743
0.127
0.870
-0.060
0.808
0.868
VesselMNIST
GuidedBackprop
-0.229
0.654
0.883
-0.717
0.121
0.838
0,002
0.843
0.841
SmoothGrad
0.137
0,861
0.724
-0.674
0.130
0.804
0.045
0.851
0.806
WAM sc (ours, = 2)
-0.054
0.848
0.902
-0.674
0.126
0.800
0.020
0.798
0.778
1G (ours, = 2)
-0.064
0.756
0.820
-0.630
0.126
0.756
0,010
0.793
0.783
WAM sa (ours, J 1)
-0,070
0.832
0.902
-0.692
0.128
0.820
0.001
0.799
0.798
WA pg (ours, 1)
-0.069
0.787
0.856
-0.691
0.128
0.819
-0.070
0.764
0.834
D. Additional use cases
D.1. Revisiting meaningful perturbations
Meaningful perturbations in the wavelet domain. Meaningful perturbations (Fong & Vedaldi, 2017: Fong et al., 2019) identify the most relevant regions of an input for model's prediction by learning an optimal mask that hides as little of the image as possible while significantly altering the model's output. This is done by optimizing a mask through gradient- updates to minimize the classifier's confidence while maintaining a smoothness constraint. The result highlights the most important features in a way that is more structured and localized than saliency maps. We revisit this framework by proposing to recast the problem in the wavelet domain as more suitable space for optimization. The wavelet domain inherently captures spatial and spectral information, providing natural structure for producing meaningful and interpretable solutions. Specifically, we solve the following optimization problem:
m arg min fo (w-'(x C m)) + a m//m/ m=|o.1]l
(19)
where fo represents the classification score, W-1 is the inverse wavelet transform, z is the wavelet transform of the input signal, @ denotes element-wis. multiplication, and a controls the sparsity of the mask m. To solve this problem, we initialize the mask as mg = (i.e., a mask that retains all coefficients) and iteratively update it using gradient descent:
mi+1 m, v/ (fo (w- @ m.)) + al m m.
(20)
where y) represents the step size and the gradient is taken with respect to m,, The optimization process continues until convergence is achieved. We call the resulting image the minimal image. In practice, we employ the Nadam (Dozat, 2016) optimizer, which combines the benefits of Nesterov acceleration and Adam optimization. Our approach consistently
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One Wave To Explain Them All: A Unifying Perspective On Feature Attribution
produces masks with controllable sparsity levels up 90%, meaning that 90% of the wavelet coefficients are zeroed out, while maintaining classification score comparable better than the original prediction. This high sparsity level suggests that the model's decision may rely on minimal subset of wavelet coefficients.
A) Sparsity-optimiized minimal images
B) Sparsity-Logit pareto front
10
100
10a3
80
10
Logit
60
Alpha
8
40
20
1077
0.2
0.4
0.8 1.0
Sparsity (%6
Figure 11. A) Sparsity minimal images. We revisit meaningful perturbation by optimizing the sparsity of the wavelet transform using masking, instead of optimizing the mask in pixel space. The displayed examples show that the resulting minimal images reveal the model's reliance textures. B) Sparsity Pareto front. As a increases, the sparsity of the wavelet coefficients increases (x-axis), but beyond a certain point, too much information is lost and the logit score drops to zero. However, we observe that many components can be removed before adversely affecting the model. Results averaged across 1,000 images optimized for 500 steps and for ranging in [0, 100] for each image.
Traditional meaningful perturbation methods (Fong & Vedaldi, 2017) focus on spatial localization, identifying clusters of pixels that answer the question of where the important features are located. However, this spatial emphasis alone provides a limited understanding of the underlying data structure. In contrast, by operating in the wavelet domain, our method captures both the what the relevant scales and the where their spatial locations. This dual information enriches the explanation by revealing the location and the nature of the features influencing the model's ' decision. These results also show that we qualitatively recover the results from Kolek et al. (2023).
Figure 11 illustrates that minimal images derived using WAM recover the texture bias of the vanilla ResNet models trained on ImageNet, highlighted by Geirhos et (2019). The examples demonstrate how the model relies heavily on texture information, which is effectively isolated through our wavelet-domain optimization.
Effect of the regularizer on the sparsity of the images. Figure 12 illustrates the effect of the parameter a on the sparsity of the minimal images. We can see that the stronger a, the sparser the image, but at the expense of higher logit value. We can see that the first components of the image that disappear are the background, then the colors, and eventually the shape of the target class.
0.01
2 0.1
1.0
10.0
100.0
Figure 12. Effect of varying values of a on the sparsity of the minimal images.
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One Wave To Explain Them All: A Unifying Perspective On Feature Attribution
Minimal images and applications. Figure 13 presents additional examples of minimal images. We can see that the color information does appear as important for maximizing the model's prediction. On the other hand, the texture and edge information essential. It would be interesting to replicate this method on a shape-biased model such as those proposed by Chen et al. (2020) or Geirhos et al. (2019) to see whether the behavior remains the same or not.
Figure 13. Additional examples of minimal images.
D.2. Multi-class classification
Evaluation in multi-class setting. We evaluate WAM in a multi-class setting to assert whether our method still informative when image depicts different objects. To evaluate our method, we rely on the Pointing Game (Zhang et al., 2018) benchmark. The Pointing Game assesses the spatial precision of saliency attribution maps. For each image, the location of the maximum activation in the saliency map (the "point") is compared against the ground-truth object location. If the point lies within the ground-truth bounding box (or segmentation mask), it is considered a "hit"; otherwise, it is a "miss".
The final accuracy is computed as:
#its
Pointing Game Accuracy = Hits + #Misses
This metric focuses on whether the most confident localization prediction corresponds to the true object location, without requiring full object delincation. We evaluate WAM alongside competing methods (GradCAM, GradCAM++, and SmoothGrad) on the PASCAL VOC 2012 test set (Everingham et al., 2010), a widely used benchmark for visual recognition tasks. The dataset contains 20 object classes and 1456 test images. Each image is annotated with object bounding boxes and class labels, making it suitable for evaluating weakly weakly-supervised localization via the Pointing Game.
Table presents the results: we can that WAM outperforms the competing baselines the Pointing Game, thereby showing its ability to identify the relevant spatial parts of the input image for given class. This further backs the fact that the wavelet domain is informative in multi-category images and suggests that WAM can be adapted to object detectors.
Visualizations. Once shown that WAM can effectively distinguish objects in multi-category settings, we can analyze which regions are highlighted as important by WAM on such images. Figure 14, we observe that the WAM successfully highlights "cat-related coefficients." as the cat's nose and its corresponding regions in the wavelet domain are more active in the rightmost image. Interestingly, although less prominent, parts of the dog face also remain significant. This suggests
26

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One Wave To Explain Them All: A Unifying Perspective On Feature Attribution
Table Pointing game experiment results. Evaluation carried out on the test set the PASCAL VOC 2012 test split. Best results are bolded and second best underlined.
Model
ResNet
EfficientNes
ConvNext
VGG
GradCAM
0.476
0.454
0.528
0.437
GradCAM++
0.476
0.454
0.528
0.437
SmoothGrad
0.429
0.432
0.430
0.430
WAMsG (ours)
0.606
0.584
0.584
0.538
1A (ours)
0.604
0.601
0.632
0.586
that while the spatial separation between the cat and dog is clear in the original image, the distinction becomes less apparent the wavelet domain.
Target class: Dog (predicted class) WAM reprojected in pixel space
Target class: Cat
WAM reprojected in pixel space
WAM the wavelet domain
WAM in wavelet domain
Figure Illustration multi-category images. Example on an image depicting a cat dog to highlight WAM points towards the regions corresponding to the target label (Dogs on the right, Cats on the left).
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One Wave To Explain Them All: A Unifying Perspective On Feature Attribution
D.3. Overlap experiment in audio classification
Figure 15 illustrates that WAM able filter relevant parts corrupted mixed audio signals. In addition, highlights the key part of the target signal without requiring any training. Figure 15 qualitatively illustrates application of WAM for audio signals. Herein, perform overlap experiment to mix corrupting audio with target audio form the input audio. The model's prediction does not alter after introducing the corruption, and thus, the model is expected to still rely on parts of input audio coming from the target audio for its decision. The interpretation audio in Figure 15 generated using top wavelet coefficients provides insights into the decision process and supports this hypothesis. In particular, it almost entirely filters out the corruption audio, and without requiring any training, also clearly emphasizes key parts target audio.
Target Class Audio
Frequency (Hz)
5k-
Input Audio
Interpretation Audio
10k
10k
Frequency (H2)
Frequency (Hz)
0
Time (s)
5k
10k
Frequency (Hz)
2
Time
(s)
5k
2
5
(s)
Corruption Audio
Figure 15. Qualitative illustration of WAM for audio via Overlap experiment. The audio of the target class ("Crow") is mixed with a corrupting audio ("Chirping birds") to form the input to the classifier. Interpretation audio reconstructed with important wavelet coefficients virtually climinates signal from the corrupting audio, and also clearly emphasizes parts of the target class audio (indicated with green boxes).
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One Wave To Explain Them All: A Unifying Perspective On Feature Attribution
D.4. Scales meet semantics: the case of remote sensing applications
The indexation in scales of the wavelet coefficients finds a natural use case in remote sensing applications (Kasmi et al., 2023b), where scales correspond to actual physical objects, the pixels correspond to distances the ground. Therefore, given acrial image whose ground sampling distance is 20 em/pixel, the 1-2 pixel scale (the finest details), corresponds to details that have a size comprised between 20 to 40 cm.
On Figure 16, we consider a case where a classification model is trained on a source domain and deployed on a target domain, thus mimicking situations where a model is used for regular updates. We can see that the OOD image is slightly different from the source image, in the sense that it is less noisy. While one would expect the model's prediction not to change, it turns out that the photovoltaic (PV) panel is longer recognized on the rightmost image. Attribution in the pixel domain only not very informative to why this happens. Turning into the wavelet domain, model no longer detects the PV panel. The background noise, located mostly in the finest scales, is no longer present on the OOD image, and thus the PV panel is no longer recognized. Kasmi et al. (2025) discussed how savelet-based attribution methods help improve the reliability of classifiers deployed in such operational settings.
better grasp why the
Image the original domain
Image the OOD domain
WAM
WAM
Figure 16. Application case of WAM to remote sensing, While both images depict the same photovoltaic panel, model no longer detects the panel when evaluated on images coming from a different distribution ("OOD" or cout-of-distribene images from the source distribution). Attribution in the wavelet domain thanks to the WAM gives hints as why the PV system is no longer recognized: on the image of original domain, which is more noisy, the model relies on noise at finest scales that are no longer present on the OOD image. Thus suggesting that the representation learned by the model lacks robustness. Based on Kasmi et al. (2023b)
image, compared to
29


Reviews:
[{"review_id": "bCVhJW3IGu", "reviewer": "Reviewer_KCeN", "summary": "Presents a feature attribution method that performs attributions on wavelets derived from input domain. This helps to naturally extend explanations that are outside the image domain, such as an audio input domain. The method essentially constructs a wavelet transform of the input, then applies standard gradient and IG attribution methods to wavelet-domain inputs. Insertion/deletion tests for faithfulness are performed to compare WAM to other popular methods across audio, 3d, and image domains.\n\n## update after rebuttal\nAfter reading the rebuttal, I am confident in my original assessment. I encourage the authors to include analysis of choice of $\\Lambda$ and various wavelet choices, which I could not find in the original paper. We also encourage the author to address if wavelets have any absolute edge over all other methods and are, in some analytic sense, better than all other possible methods, or if wavelets just have a comparative edge over some current popular methods. Upon further reflection, we also recommend an edit for readability.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We first would like to thank reviewer KCeN for reviewing our manuscript and for the comments on our work. The points raised by the reviewers will help us improve the quality of our work. \n\n**Question** *Pg 4, line 209, left: [...] Please expound / I can see how wavelets is another way to attribute to the input domain, so that attributions now incorporate a sense of scale. [...] can you provide any argument as to why wavelets are appropriate terms in which to attribute? What about super-pixels? What about Fourier transforms and attributing in k-space? Is this method more appropriate than those, theoretically or experimentally?*\n\nOur formulation \"varying pixel values provide no information to wht is changing on the image\" is unclear and we modified it for \"moving from one pixel to the next is only a shift in the spatial domain and does not capture relationships between scales or frequencies\". This sentence explains why in our view, the pixel domain is insufficient to interpret the decision of a model. **The wavelet domain captures both the spatial component (as done by pixels, or superpixels) and the spectral component, as done by the Fourier transform**. Either one of these approaches is unsuitable for attribution (see [1,2] for Fourier), as **for attribution a spatial or temporal dimension is required and a spectral dimension desirable** to assess what the model sees at a given location.\n\n**Question** *What wavelet functions were chosen for each experiment? What was chosen? Why were they chosen? [...] perhaps they should be presented more prominently. / How does the faithfulness score and the quality of explanations vary with the choice of wavelet function and choice of $\\Lambda$ ? Is it sensitive to these choices?*\n\n**The quality and faithfulness of explanations does not change with the choice of the mother wavelet**. the quality and faithfulness of the explanation stem from the multi-scale decomposition, which is a property of the $\\Lambda$, irrelevant of the choice of the mother wavelet.  The choice of the mother wavelet determines how the input signal is decomposed, influencing whether finer details or broader structures are emphasized in the wavelet coefficients By default, we considered the Haar wavelet, due to its semantic properties but also because it is fast to compute. \n\n**Additional experiments verified that the choice of the wavelet does not change the quantitative results**. We evaluated $WAM_{IG}$ on a ResNet 50 model using Daubechies and Bior wavelets. We updated the supplementary materials in to state more clearly our choice of wavelet and discuss what wavelets can be chosen. \n\nOn the other hand, **the choice of $\\Lambda$, remained fixed and we did not explore beyond the dyadic transform because the multi-scale property of our decomposition fundamentally relies on its dyadic structure**. Using non-dyadic decompositions could disrupt the natural hierarchy of scales, leading to a loss of spatial localization and a less structured frequency representation.\n\n**Question** *Your paper claims WAM is a unifying perspective of feature attribution. [...]. However, the paper seems to provide a method that is adaptable to multiple input domains. Perhaps some more up-front clarity/renaming would clarify this fact and prevent confusion.*\n\nWe acknowlege that we remained elusive on what we aimed at unifying, a concern shared by R3. We will edit our manuscript to explicitely state what we aim to unify in this work. \n\n**Question** *A major component of IG is completeness. [...] Does WAM_IG satisfy completeness?*\n\nFor completeness, only **showing that inverse wavelet transform keeps things differentiable and that the initial and final points are same is enough (see [3], Prop 1)** to show that $WAM_{IG}$ satisfies completeness.\n\nFormally, IG satisfies completeness if $F$ is differentiable almost everywhere and\n$$\n\\sum_{i=1}^n IntegratedGrad_i(x) = F(x) - F(x_0)\n$$\n\nWhere $x_0$ is a baseline black image such that $F(x_0)\\approx 0$. To ensure that $WAM_{IG}$ satisfies the condition for completeness, we have to ensure that (1) $F(W^{-1}(z))$ is differentiable almost everywhere and that (2) we can set $z_0$ such that such that $W^{-1}(z_0)$ is a black image. **(1) depends on the choice of the mother wavelet $\\psi$** (we need the mother wavelet to be smooth) and **(2) is obtained by setting the wavelet coefficients to 0**, so **$WAM_{IG}$ satisfies completeness if $\\psi$ is smooth**, e.g. with Daubechies but not with Haar wavelets.\n\n- [1] Yin et al (2019). A fourier perspective on model robustness in computer vision. NeurIPS\n- [2] Chen et al (2022). Rethinking and improving robustness of convolutional neural networks: a shapley value-based approach in frequency domain. NeurIPS\n- [3] Sundararajan et al (2017) Axiomatic attribution for deep networks. ICML", "claims_and_evidence": "The method, WAM, does appear to have good performance compared to other popular gradient-based attribution methods over a variety of tests.\n\nThe paper demonstrates some advantage to using wavelets, i.e. for attributing at different feature scales.", "methods_and_evaluation_criteria": "The faithfulness metric is an appropriate method, and the evaluation sets seem appropriate.", "theoretical_claims": "I looked over the appropriateness of eq 4.", "experimental_designs_or_analyses": "Insertion/deletion is an appropriate metric, and at least the visual dataset is standard, to my knowledge.", "relation_to_broader_scientific_literature": "The paper piggybacks off of the gradient-based attribution literature (Sundararajan IG paper), adapting the method to better handle models for volume and audio input domains, by using wavelets. Thus it contributes to (the small amount of) literature on wavelet based attributions.", "essential_references_not_discussed": "None known.", "other_strengths_and_weaknesses": "See questions and suggestions", "other_comments_or_suggestions": "Pg 4, line 209, left: \"Varying pixel values provide no information to what is changing on the image.\" Please expound. Changing on the image? No information to what?\n\nYour paper claims WAM is a unifying perspective of feature attribution. When I read this, I expected the paper to present an analysis that unifies multiple attribution methods under one theory. However, the paper seems to provide a method that is adaptable to multiple input domains. Perhaps some more up-front clarity/renaming would clarify this fact and prevent confusion.\n\nI can see how wavelets is another way to attribute to the input domain, so that attributions now incorporate a sense of scale. While this is an advantage, can you provide any argument as to why wavelets are $\\textit{the}$ appropriate terms in which to attribute? What about super-pixels? What about Fourier transforms and attributing in k-space? Is this method more appropriate than those, theoretically or experimentally?", "questions_for_authors": "What wavelet functions were chosen for each experiment? What $\\Lambda$ was chosen? Why were they chosen? Sorry, I did not catch this on my read though; perhaps they should be presented more prominently.\n\nHow does the faithfulness score and the quality of explanations vary with the choice of wavelet function and choice of $\\Lambda$? Is it sensitive to these choices?\n\nA major component of IG is completeness, i.e., the total change in function value at the baseline vs input value is accounted for by the attributions. This gives meaning to the value of the IG output: i.e., the value of an IG attribution is equivalent to function change. Does WAM_IG satisfy completeness?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "pla9apb4Rt", "reviewer": "Reviewer_EDJ8", "summary": "This paper proposes an explanation method for DNN by using wavelet coefficients as features for attribution instead of image features. The proposed WAM can be adopted across diverse modalities, including audio, images, and volumes. It unifies and extends existing methods, SmoothGrad and Integrated Gradients, within the wavelet domain by transporting the gradient w.r.t. the input to the transformed wavelet domain.\n\n## update after rebuttal\nThank the authors for the detailed rebuttal. My concerns are basically addressed. For the results of localization evaluation (Point Game) and analysis of multi-class cases, the authors haven't updated the rebuttal yet. Still, I strongly recommend adding these results to the final version of the paper. I will keep my rating.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank reviewer EDJ8 for the review and the comments on our work. We also thank the reviewer for pointing the duplicate reference, which we have corrected. \n\n### Regarding the evaluation criteria\n\nThe Reviewer said to include localization evaluation, Pointing Game, and visual explanation accuracy alongside faithfulness evaluation for assessing explanation methods for image inputs. We thank the reviewer for providing additional benchmarks. **We kindly refer the reviewer to the supplementary material, as we have evaluated our method accross a wide range of common metrics from the existing literature**. To complement these results, **we are currently implementing the Pointing Game and Visual Explanation Accurary, as suggested by the reviewer, and will update the rebuttal if we get results**.\n\n### Regarding the multiclass classification (Cats and Dogs examples)\n\nThe reviewer outlined the fact that from appendix A3, the distinction between dog and cat as target classes is less apparent in the wavelet domain and suggested providing more examples and discussion on multi-category interpretation challenges. **To be more conclusive regarding the assertion ''Does that demonstrate that when there is a multi-category in the image, the model cannot be interpreted well in the wavelet domain'', we are gathering more examples**. Localization benchmarks should also give us information on the behavior in multi-category settings. **We try our best to get the results and will update the rebuttal**.\n\n### Questions \n*1. adopting wavelet in the feature layer;*\n\nThis is an interesting question and was actually planned for future work. **To the best of our knowledge, no work has adopted wavelets in the feature layers**. Like [A,B] and contrary to the statement of the reviewer, the wavelet decomposition does not prevent us from using the intermediate layers of the model to compute the explanations. **In principle by applying wavelet decomposition to intermediate feature maps, we could obtain a \"multiscale interpretation\" of feature importance**. \n\n*2. wavelet-based explanation method for interpreting Transformer;*\n\nIf the reviewer refers to the interpretation of the attention mechanism using wavelets, it is an interesting suggestion but beyond the scope of this work. **If it is meant applying WAM to Transformer-based architectures, then we kindly refer the reviewer to the supplementary materials B2 were we apply the WAM to a wide variety of topologies (ViT, ConvNext, 3D Former)**. Results remain the same no matter the choice of the classification model. \n\n*3. wavelet-based explanation method for other task models like object detectors and VLMs*\n\nThese two comments are a very good suggestion and we would like to thank ther reviewer for these remarks. We remained focused on the classification task but **in principle, our method could be expanded to object detectors. The way to follow would be to expand ODAM to the wavelet domain**. Regarding VLMs, since our method unifies the domain for feature attribution it makes sense to look for interpreting multimodal models. However **VLMs handle text data, and this modality is unsuitable for the wavelet transform, so attribution in a suitable latent space for text data shoud be explored**. We are currently applying WAM to object detectors and will update the rebuttal if we get results.", "claims_and_evidence": "- Using wavelets to decompose image information is a novel view for interpreting the model’s decision for obtaining what information the model has seen when making the prediction. The methodology seems reasonable and claimed clearly.\n- The key part of the proposed wavelet method is transporting the gradient w.r.t. original input to the gradient w.r.t. the transformed input (wavelet domain). This limits that the proposed method cannot adopt the intermediate features generated from the model, which are used more by other gradient-based explanation methods [A, B, C]\n\n[A] Selvaraju R R, Cogswell M, Das A, et al. Grad-cam: Visual explanations from deep networks via gradient-based localization[C]//Proceedings of the IEEE international conference on computer vision. 2017: 618-626.\n\n[B] Jiang P T, Zhang C B, Hou Q, et al. Layercam: Exploring hierarchical class activation maps for localization[J]. IEEE Transactions on Image Processing, 2021, 30: 5875-5888.\n\n[C] Zhao C, Chan A B. ODAM: Gradient-based Instance-Specific Visual Explanations for Object Detection[C]//ICLR. 2023.", "methods_and_evaluation_criteria": "- For evaluating explanation methods for image inputs, except faithfulness evaluation, it’s also better to provide localization evaluation, Pointing Game [D], and visual explanation accuracy [E]. \n[D] Zhang J, Bargal S A, Lin Z, et al. Top-down neural attention by excitation backprop[J]. International Journal of Computer Vision, 2018, 126(10): 1084-1102.\n[E] Oramas J, Wang K, Tuytelaars T. Visual explanation by interpretation: Improving visual feedback capabilities of deep neural networks[J]. arXiv preprint arXiv:1712.06302, 2017.", "theoretical_claims": "Seems no problem.", "experimental_designs_or_analyses": "- About the visualization results in A.3, the distinction between dog and cat as target class is less apparent in the wavelet domain. Does that demonstrate that when there is a multi-category in the image, the model cannot be interpreted well in the wavelet domain? Should give more examples and discussion on this kind of case.", "relation_to_broader_scientific_literature": "This paper proposes a way to interpret the model with inputs of audio and volumes. Almost all previous explanation methods are designed for models with image inputs.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "N/A", "other_comments_or_suggestions": "- duplicated reference for Grad-CAM (Page 11 Line 593-604)", "questions_for_authors": "The questions mentioned above and\n\nIs that possible \n1) adopting wavelet in the feature layer; \n2) wavelet-based explanation method for interpreting Transformer; \n3) wavelet-based explanation method for other task models like object detectors and VLMs?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "z3PzAxdYCl", "reviewer": "Reviewer_UvKx", "summary": "The paper explores feature attribution by determining the importance of individual wavelets in prediction tasks. Unlike traditional vision-based attribution methods that assess pixel importance, this approach evaluates how wavelets contribute to model predictions. The key idea is to compute gradients with respect to wavelets to measure their significance. Extensive experiments are conducted to validate the effectiveness of the proposed method.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We first would like to thank reviewer UvKx for the reviews on our work and underlining the potential of our idea, regarded as innovative. We would like to address the comments raised by the reviewer on our work. These comments will be taken into account in our work and will help us improve the quality and the clarity of our work.\n\n### Interpretability of wavelet coefficients\n\nThe reviewer challenged our assertion that wavelet coefficients are interpretable coefficients and stated that we did not provide enough justification. We kindly refer the reviewer to **section 2.2, page 3, where we explain why wavelets are interpretable features for feature attribution**. We add the wavelet transform of a signal provides a more interpretable representation of this signal than raw pixels or Fourier transforms by **capturing spatial and frequency information**. Fourier transforms only encode frequency content whereas wavelets preserve localization, making them particularly effective for analyzing structured data like audio and images. In audio, for instance, wavelets isolate transient components corresponding to phonemes, yielding a more intuitive representation of meaningful patterns. Wavelets **extract high-level features that better convey the overall structure of a signal**, making them a more interpretable alternative to the original data representation. **For images, wavelet coefficients can be intuitively understood by viewing them as capturing details at different scales**, like how our vision works when we look at an image from different distances. **Wavelet coefficients align with the definition of interpretability by Kim et al. (2016, [1]), as the decomposition into different scales enables users to predict how the method will emphasize on specific features (e.g. edges or gridded patterns on images)**. We would like to thank the reviewer for pointing this out and will further enrich section 2.2. with the provided expanation.\n\n### On the scope and positioning of the paper and our claim on unification\n\nThe reviewer stated that our paper could be situated amond approaches that improve model performance or alternative forms of analysis. While this perspective is interesting, as our work indeed bridges the gap between explainability and model robustness for instance, **we believe that our work aligns well with explainability and feature attribution as we essentially propose to attribute an importance score to input features, taken as the wavelet coefficients of the signal of interest rather than its pixels** as in traditional feature attribution methods. \n\nThe reviewer mentioned that where our identification lies was unclear, a concern shared by R1. We thank the reviewer for this remark and acknowledge that indeed, we remain elusive on what we aimed at unifying. **Unlike SHAP, we do not unify attribution methods but rather unify the domain in which attribution is carried out in a single input representation, the wavelet transform of this signal**. We edited our manuscript to reflect the changes.\n\n### Clarifications on the evaluation of $\\partial f_c(W^{-1}(z))/\\partial z$\n\nThe reviewer mentioned that it was unclear how the computation of $\\partial f_c(W^{-1}(z))/\\partial z$ was achieved. In Pytorch, we **require the gradients on the wavelet transform of the signal $z$** and reconstruct the original image from its wavelet coefficients, $x=W^{-1}(z)$. We **carry out a forward and compute the derivative of the prediction with respect to the wavelet coefficients**. We then re-express the signal into its wavelet transform to retrieve the gradients of the model with respect to the wavelet coefficients of the input signal. **The novelty lies in expanding existing attribution methods (SmoothGrad and IntegratedGrad) into the wavelet domain**. \n\n### Discussion of the faithfulness metric & additional references\n\nThe reviewer highlighted an additional definition of the faithfulness, as discussed in \"Towards Faithful Model Explanation in NLP: A Survey\". We thank the reviewer for this reference and will add it to our manuscript. **Regarding our definition of the faithfulness, we acknowledge that there are several definitions of this notion**. For this work, we chose the definition of Muzellec et al (2023), but as **we believe that one metric is not sufficient to reflect the behavior of a method, we extensively evaluated our method with alternative metrics in the supplementary materials, section B.2, p.16**.\n\nThe reviewer underlines that we do not discuss references from NLP. We would like to recall that **since our method cannot be theoretically applied to text data, we did not considered works in this field (see Introduction, second column, l47-48), although we acknowledge that there have been many works in the field of XAI for NLP**. We thank the reviewer for the reference and will add it to our manuscript.\n\n- [1] Kim et al, 2016. “Examples Are Not Enough, Learn to Criticize! Criticism for Interpretability.” NIPS’16", "claims_and_evidence": "The paper claims to introduce a new method that improves  interpretability in feature attribution by identifying the importance of individual wavelets. However, while the method assigns importance scores to wavelets, it is unclear how this directly improves interpretability. Wavelets are not inherently interpretable features, and the paper does not provide sufficient justification for why identifying important wavelets leads to a more interpretable representation.", "methods_and_evaluation_criteria": "The authors select one dataset per modality out of the three considered, which is a reasonable choice given the scope of their study. Additionally, they assess faithfulness using the Insertion and Deletion metrics. While these provide valuable insights, it is worth noting that faithfulness can be defined in multiple ways, and alternative definitions may offer complementary perspectives. For example, in the context of NLP, the survey \"Towards Faithful Model Explanation in NLP: A Survey\" discusses various definitions of faithfulness, many of which share common ideas with vision-based approaches.", "theoretical_claims": "There are no theoretical claims in the paper", "experimental_designs_or_analyses": "--", "relation_to_broader_scientific_literature": "The paper presents an approach positioned within explainability research, contributing to the broader literature by examining the importance of different wavelets. Prior work in explainability has typically emphasized methods that rely on interpretable features, ensuring that the extracted insights align with human understanding. In contrast, this paper leverages non-interpretable features, which, while valuable in their own right, may not directly advance explainability in its conventional sense. Given this, it may be beneficial to consider situating the paper within a broader methodological context—perhaps as an approach that improves model performance or provides alternative forms of analysis—rather than as a direct contribution to explainability. Such a reframing could more accurately reflect its impact within the field.", "essential_references_not_discussed": "The paper's presentation may give the impression that XAI has been primarily focused on vision-based applications, as stated in “While XAI has been predominantly applied in image classification, it is also extending into other fields, such as audio and volume classification.” However, XAI has also been widely explored in other domains, such as NLP, where numerous explainability methods have been developed and studied. Citing relevant works from NLP and other areas could provide a more comprehensive view of the broader landscape of XAI research (e.g., Towards Faithful Model Explanation in NLP: A Survey, A Comparative Study of Faithfulness Metrics for Model Interpretability Methods, Faithfulness Tests for Natural Language Explanations)", "other_strengths_and_weaknesses": "**Strength**: The paper presents a nice idea that is innovative and has potential value in the field. The authors have clearly invested significant time and energy into the experimental aspects of the work, which is evident from the detailed experiments.\n\n**Weakness**: A primary weakness of the paper is that wavelets, as used in the proposed method, are not inherently interpretable features. This raises concerns about the overall interpretability of the method, as the use of wavelets may undermine the transparency of the approach. Additionally, the presentation of the algorithm could be improved. Specifically, when the authors mention \"directly evaluate ∂fc(W−1(z))/∂z” it is unclear how this is achieved. Providing more clarity on the exact steps of this evaluation would significantly improve the paper's presentation and make the methodology more accessible to readers.", "other_comments_or_suggestions": "The title suggests a unifying perspective, but it is unclear what exactly is being unified. At first glance, it resembles the SHAP paper, which unifies different explanation methods for various attribution techniques. Clarifying what is being unified—whether it's explanation methods, domains, or something else—would provide better clarity for the reader.", "questions_for_authors": "--", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The paper was reviewed by 3 experts, with mixed initial reviews 233. The major issues raised were: 

1) What is the "unifying perspective" of feature attribution? [KCeN, UvKx]
2) Why are wavelets the most appropriate to attribute compared to super-pixels, Fourier transform, etc.? [KCeN]
3) Which wavelet functions were selected and why? How does the evaluation/quality change with choice of wavelet?
4) Does WAM_IG satisfy completeness? [KCeN]
5) Method cannot be adopted to the intermediate features generated from a model (e.g., as with Grad-CAM methods). [EDJ8]
6) could use other metrics for image explanations, such as pointing game or visual explanation accuracy  [EDJ8]
7) regarding images in A.3, can multi-category images be interpret well in the wavelet domain? [EDJ8]
8) could the same method be used for a feature layer, transformers, or more complex models for other tasks? [EDJ8]
9) wavelets are not inherently interpretable features, how does identifying wavelets lead to more interpretable representations? [UvKx]
10) experiments could use other faithfulness metrics [UvKx]
11) contribution should be reframed since it uses non-interpretable features, which may not directly advance explainability in the conventional sense [UvKx]
12) presentation could be improved [UvKx, KCeN]

The authors wrote a response. After the response, KCeN was satisfied, maintained rating, and comments that the paper could be further edited. In particular: 
- "Well grounded progress in attributions must provide, in part, some analysis showing the unique advantages of the method or its well-suitedness to a problem domain. This is very common in the field; see axiomatic approaches. This method does not provide much in that regard; the justifying paragraph seems a bit undeveloped and perfunctory. On the other hand, many methods have been introduced that lacked such analysis and yet were interesting and enlightening to read. The lack of such analysis meant the method lacked theoretical maturity when it was introduced, but the idea and experiments may have been of practical interest or inspirational for later work. That being said, doing some sort of wavelet explainability method seems like a thing that should be done, and the analysis here has some merit. However, the lack of theoretical development (theorems, in-depth discussion of advantages) significantly limits the impact of the paper. Of course, I would like the paper that first applies wavelets to attributions to have more of a thorough and enlightening review of what a wavelet is, and a better discussion of the advantages."

On the other hand, UVkX was still not convinced that wavelets are interpretable features. 

Regarding KCeN's concern, the AC agreed that it would have been nice to present some analysis about how wavelets could be better than others (superpixels, etc). Some analysis probably exists in the literature, e.g., wavelets are commonly used for image/sound compression, and are good at representing transient signals (common for natural images and audio). Perhaps there is also work showing that wavelets are more attuned to the human visual system (via Gabor filters) than super-pixels. 

Regarding UvKx's concern about interpretability, it can be noted that wavelets are compact basis functions of different spatial locations and frequency component, so they could be interpreted as spatially-localized frequency content. The analytical form of each basis can be derived mathematically. Gabor filters are an instance of Wavelets, and Gabor filters are considered as interpretable low-level features (orientation- and scale-selective features).  Thus, the AC thinks that wavelets are also interpretable low-level features. The authors could have shown some examples of the wavelets they are using to make this clearer, as well as noting that they are not high-level interpretable features (concepts).

All reviewers kept their ratings after the response. The AC agreed with the two positive reviewers. The method is interesting and novel, and experiment results are compelling.  The idea of using wavelets is likely to inspire other follow-up works.  Thus, the AC recommends accept. The authors should revise the paper according to the comments, taking particular care in introducing wavelets and properly motivating them as interpretable low-level features.
Decision: Accept (poster)