AdaptGrad: Adaptive Sampling to Reduce Noise
Authors: Linjiang Zhou
Chao Ma
Zepeng Wang
Libing Wu
XIAOCHUAN SHI
Venue: NeurIPS 2025 poster ()
Keywords: Explainable Artificial Intelligence
Neural Networks
Gradient Smoothing
Primary area: social_and_economic_aspects_of_machine_learning
TL;DR: AdaptGrad proposed in this paper aims to reduces noise in gradient-based model explanations by controlling out-of-range sampling behavior, outperforming existing techniques like SmoothGrad in noise reduction and visualization quality.

Abstract: Gradient smoothing is an efficient approach to reducing noise in gradient-based model explanation methods. SmoothGrad adds Gaussian noise to mitigate much of this noise. However, the crucial hyperparameter in this method, the variance $\sigma$ of the Gaussian noise, is often set manually or determined using a heuristic approach. This results in the smoothed gradients containing extra noise introduced by the smoothing process. In this paper, we aim to analyze the noise and its connection to the out-of-range sampling in the smoothing process of SmoothGrad. Based on this insight, we propose AdaptGrad, an adaptive gradient smoothing method that controls out-of-range sampling to minimize noise. Comprehensive experiments, both qualitative and quantitative, demonstrate that AdaptGrad could effectively reduce almost all the noise in vanilla gradients compared to baseline methods. AdaptGrad is simple and universal, making it a practical solution to enhance gradient-based interpretability methods to achieve clearer visualization.

AdaptGrad: Adaptive Sampling to Reduce Noise
Linjiang Zhou'
Chao Ma² Zepeng Wang² Libing Wu² Xiaochuan Shi
Key Laboratory of Aerospace Information Security and Trusted Computing. Ministry of Education School of Cyber Science and Engineering, Wuhan University, Wuhan 430072, China School of Cyber Science and Engineering, Wuhan University, Wuhan 430072, China {linjiang, chaoma, wangzepeng, wu, shixiaochuan edu. cn
Abstract
Gradient smoothing is efficient approach to reducing noise in gradient-based model explanation methods. SmoothGrad adds Gaussian noise to mitigate much of this noise. However, the crucial hyperparameter in this method, the variance a of the Gaussian noise, is often set manually or determined using a heuristic approach. This results in the smoothed gradients containing extra noise introduced by the smoothing process. In this paper, we aim to analyze the noise and its connection to the out-of-range sampling in the smoothing process of SmoothGrad. Based on this insight, we propose AdaptGrad, an adaptive gradient smoothing method that controls out-of-range sampling minimize noise. Comprehensive experiments, both qualitative and quantitative, demonstrate that AdaptGrad could effectively reduce almost all the noise in vanilla gradients compared to baseline methods. AdaptGrad is simple and universal, making it practical solution to enhance gradient- based interpretability methods achieve clearer visualization. All code would be found in https: //gt//gttttttttt com/ AiShare- WHU/Adap AdaptGrap
Introduction
Explanation of the deep learning model critical part of applications of artificial intelligence (AI) with human interaction. For example, explanation methods are crucial in these data-sensitive and decision-sensitive fields such as medical image analysis (4, financial data analysis (55] and autonomous driving [1. Additionally, given the prevalence of personal data protection laws in most countries and regions, fully black-boo box Al models may face intense legal scrutiny [1].
In recent years, some explanation methods have attempted to explain neural network decisions by visualizing the decision rationale and feature importance 31 These local explanation techniques aim to provide explanations for individual samples. Moreover, the explanation process of these methods often leverages the gradients of the neural network. For example, Grad-CAM 35), Grad-CAM++
[1I. and CAM [66] use gradients to generate the weights of class activation maps.
Gradients of input samples are critical information for analyzing deep neural networks [41]. However, these gradients often contain significant amount of noise, primarily due to the complex structure
and numerous parameters in neural networks [30]. As highlighted in [2] (25 [5I) this noise can significantly affect the ability of explanation methods to extract latent learning features. Furthermore, caused by both local noise and gradient saturation, sample gradients may fail accurately explain the influence of input values on model decisions 42). Therefore, reducing the noise in gradients is an important step toward improving the interpretability of deep learning models.
LLijjang Zhou and Chao Ma contribute equally to this work. Xiaochuan Shi is the corresponding author of this paper.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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SroothGrad 40 is currently the most widely applied and empirically proven simplest yet effective method for gradient smoothing. Although other methods, such as NoiseGrad [9], have also been proposed for gradient smoothing, SmoothGrrd remains the most widely used due to its universality and practicality. However, as mentioned have not been thoroughly explored. Additionally, its key parameter a, the variance of Gaussian noise, typically set empirically. We found that this setup causes SmoothGrad to introduce additional noise during its sampling process, which consequently leads to the smoothed gradient still retaining significant amount noise.
31 the underlying principles of SmoothGrad
(a) Original image
(b) Vanilla gradient
(c) SmoothGrad
(d) AdaptGrad (ours)
Figure An example to compare the visual performance between different gradient smoothing methods.
In this paper, we rethink the noise sources SmoothGrad utilizing the convolution formula. We discover the relationship between the out-of-range sampling behavior caused by its key hyperpa- rameter settings and the noise introduced during the sampling process. This discovery enables us to theoretically analyze the shortcomings of SmoothGrad and subsequently design adaptive gradient smoothing method, AdaptGrad. Figure T iillustrates an example of AdaptGrad. retically prove that AdaptGrad outperforms SmoothGrad, but also demonstrate that our AdaptGrad capable of eliminating almost all the noise while the smoothed gradients reveal richer detailed features. Similarly to SmoothGrad, AdaptGrad is also simple and universal, making it applicable for improving gradient-based interpretability methods. In addition, we comprehensively demonstrate the superiority of AdaptGrad through experiments with other gradient-based interpretability methods.
not only theo-
2 Related Work
In general, a neural network can be considered as a function F(x: 8) R. - with the trainable parameters and its output could be probability, logit, etc. Here, D is the input dimension of the neural network, and C is the output dimension. the example of a classification function, the neural network will output score for each class c, where c {1. C). To simplify the analysis, focus on the output of the neural network for single class c, and the neural network can be viewed
can
function fe (x; 0) R R. which maps the input gD to the 1-dimensional IR space. For
simplicity, we use Fxx) to represent the neural network and only consider the output of the neural network on single class
The gradient of F(x) could be presented as Equation
G(x) OF(x)
(I)
0x
possible explanation [30] for the large amount of noise the original gradient is that, considering the local surroundings of sample, neural networks tend not to present an ideal linearity but rather have a very rough and nonlinear decision boundary. So the complexity of neural networks and the input features usually leads to the unreliability of the vanilla sample gradients.
Similarly, gradient maps such as those in Figure I are commonly referred heatmaps. For simplicity, this paper uniformly refers to them as saliency maps and also refers to the results generated by other explanation methods saliency maps. The highlighted areas in these maps indicate the relevant features learned by the neural network or the basis for the decisions.
saliency maps
The methods for reducing gradient noise can be roughly categorized into the following two categories:

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Adding noise to reduce noise. SmoothGrad proposed by 40 introduces randomness to smooth the noisy gradients. SmoothGrad averages the gradients of random samples in the neighborhood of the input Xo. This could be formulated as shown in Equation 2
N Σ G(x+ wheree No Esg); Eng ID o² (xg(x)= = N
1
(2)
In Equation 2 N is the sample times, and is distributed over D-dimension A (O, Esg). Similarly to SmoothGrad, NoiseGrad, and FusionGrad presented in [9], additionally add perturbations to the model parameters. And FusionGrad a mixup NoiseGrad and SmoothGrad. These simple methods are experimentally verified to be efficient and robust [13
Improving backpropagation to reduce noise. Deconvolution [33 and Guided Backpropagation
directly modify the gradient computation algorithm of the ReLU function. Integrate Gradient (IG) 52] was proposed to replace the original gradient interpretation and was shown to have axiomatic completeness. Some other methods such Feature Inversion [13], Layerwise Relevance Propagation 77], DeepLift [36]. Occlusion [5], DeepTaylor [30] employ some additional features to approximate or improve the gradient for precise visualization.
42 23
Original
(b) N
(c) N 20
(d) N 50
(c) N 70 (f) N 100 (g) AdaptGrad with 100
image
Figure The visual saliency map G.g of SmoothGrad with different sampling number N and
= 0.2. The classification model is VGG16 [37], and this image is from IISVRR2012 (26
3 Convolution for Smoothing
SmoothGrad is a simple yet effective gradient smoothing method. However, some its underly- ing principles have not been fully discussed. As mentioned in (9. SmoothGrad is essentially Monte Carlo method. Thus, we could further derive the definition of SmoothGrad to gain a deeper understanding of noise in gradient smoothing.
3.1
Monte Carlo Approximation for Convolution
In Section 2
SmoothGrad is formulated in Equation 2 SmoothGrad essentially a type of convolution. In the form of Monte Carlo integration, SmoothGrad could be redefined as Equution 3]
As summarized in previous work [47].
N Σ G(x+)) N
N N Σ p(e)
G(x + =)p(() where (f-) = p(-)
Gxy())
In Equation 3 the p(.) is the Probability Density Function (PDF) of D-dimensional distribution N (O, ,.y). And according the Monte Carlo integration principle, the limit of Gag can be estimated as N approaches infinity (sampling an infinite number of times). So we could obtain the upper limit of Gag in Equation 4.
N Σ
G(x G(x+))p(()) p(E)
= / (x)+t) (G (x)
lim (x) lim N
(4)
N
N
In Equation 4 * is the convolution operator. From Equation could observe that the function p(-) (equal to sp(:)) acts as both a convolution kernel sp(:) and sampling distribution p(-). The reason why sp(-) and p(-) must be equal arises from computational considerations, as the PDF values of

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high-ding sigh-dmeensooall distributions are typically very small, which can lead to floating-pon underflow. For ease of distinction, in the following text, we will use p(.) to denote both the sampling distribution and the convolution kernel. So we extend the definition of SmoothGrad as shown in Equation 5
(x) = (G (x),, p PDF(N (0. Egg)
(5)
Original
Bias=0.2
Dias=0.4
Bias=0.6
Bias=0,8
Dias=1.0
Original
SenoohGGad
2.71%
3.39%
4.00%/2547.06
5.27%/3347.1
7.27%/788106
10.04%/7230663
AdaepGrred
0.62 0.62% 0.61%
0.56% 62
0.60% 0.61%/167.06
0.69%
Figure 3: An example of the relationship between out-of-bound sampling behavior and extra noise. By adding a bias to the input image (illustrated as Bias in the figure), the out-of-bound sampling behavior of SmoothGrad increases progressively (the proportion of out-of-bound pixels / the value of out-of-bound and labeled at the bottom of the saliency map).
3.2
Noise in the Smoothed Gradients
As implied by Equation 5] when the sampling number N is sufficiently large, Gag has an upper limit. In Figure 2 a visual example of SmoothGrad, as N increases, around 50 70, Gag gradually converges. However, even as Gag approaches the limit, the residual noise in Gag could still affect the details in the saliency map.
Based on our findings in Section 3.1 SmoothGrad can be understood as a convolution of gradient functions. The convolution of the Gaussian kernel camnot completely remove all noise. Therefore, there will inevitably be some inherent noise in the smoothed gradient. However, we find that in the SmoothGrad method, there is also new noise caused by the sampling range, this part of the noise is generated by the SmoothGrad method itself, so we call it extra noise. Next, we will analyze and prove the existence of extra noise.
The a, a key parameter in SmoothGrad, the variance of sample distribution (O, Eng); Eng = ID * 0². SmoothGrad employs a simple strategy select an appropriate value for σ. Assume that the minimum value of the input data is denoted as T'min and the maximum value as Zmarz SmoothGrad introduces a new variable a (set to 0.2 as recommended by 40 to compute a. The relationship between them is expressed in Equation 6
(Xmax Tminn)
66)
However, we believe that this setup has led significant amount of out-of-range sampling behavior during the sampling process, thereby generating extra noise. In fact, SmoothGrad overlooks the fact that the integral in Equation 4 which is not performed over RD but rather over a bounded domain Ω [Tmin 2 ] which is determined by the statistical features of the dataset. In Equation 4| the domains of G(-) and p(-) are inconsistent: the former is defined over 9 while the latter is defindd over "R Input samples that fall outside the bounds of Ω are considered meaningless because they do not align with the statistical properties the dataset. Consequently, during the smoothing process, SmoothGrad samples values that lie outside Ω2 and this out-of-bounde sampling behavior introduces a significant amount of extra noise into G G.g. Figure provides an example illustrating the presence of this extra noise.

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Table 1: The Spearman correlation test results between OBA and OBV with Sparseness under different hyperparameter settings
Correlation coefficient (p-value)
Hyperparametertoo
Variable
0.2
0.3
0,5
OBA
-001990011177
10.055(0.0813))
-0.06 0533
000562000777)
-0.0S RR0
OBV
-0.051500 (0051500.133))
.0.0592(000655)
-0.0625(0.0482)
-0057990.0673))
0.008660.0657
Further, we quantitatively define the extra noise the probability of sampling points that fall outside the domain of the dataset in gradient smoothing methods. This allows us to mathematically express the extra noise introduced in SmoothGrad.
Given an input sample x = I2, of x. Clearly, for the one-dimensional case, the smoothed gradient G of the gradient with respect to the variable 2; can be represented in Equation
I2, ... ,DD] Ω for simplicity, we only focus on one variable I;
Geg
G( + (xxx))xx))x))) ~N(,00²) p- PDDFNN,
(7)
Notice that H, + is also random variable and follows the distribution ((x o²). Therefore, quantify extra noise as the probability that 22 + a falls outside the sampling interval [Tmin: Fmar]. The extra noise the i-th dimension A' can be expressed Equation 8
Bman 21
p())dl
(8)
By substituting the expression of SmoothGrad (Equation 5) into Equation 8 mathematical expression for the extra noise A SmoothGrad, as presented Equation 9 where erf(-) denotes the Gaussian Error Function.
we can derive the
A erf(
Tmar Zj
1 erf(
Tmin Ij
22 (I (mma Imin
2
V2a( [maaz 2 Zmión
To investigate the correlation between extra noise and out-of-bounds behavior, we employed noise metric, Sparseness [10), and conducted hypothesis testing on this relationship. For detailed explanation of the Sparseness metric, see Section 5.1
We utilize two metrics to quantify out-of-bounds behavior: the proportion of out-of-bounds pixels in SmoothGrad (denoted as OBA, representing the statistical value of A.g) and the sum of values for out-of-bounds pixels (denoted as OBV). Additionally, we employ Sparseness and VGG16 to evaluate the amount of noise, where higher Sparseness value indicates less noise. We conduct experiments on 1000 samples from ILSVRC2012 and vary the a parameter in SmoothGrad. (Talle I presents the Spearman correlation test results between OBA and OBV with Sparseness under all hyperparameter settings. Across all settings, the variables OBA and OBV show a negative correlation with Sparseness. Although the absolute value of the Spearman correlation coefficient is very low, which is mainly due to the fact that Sparseness is not perfectly positively correlated with the amount of noise, we can accept our hypothesis with 90% confidence the setting of 0.2. This indicates relationship between out-of- out-of-bounds behavior and the presence of noise in the smoothed gradients, thereby validating our hypothesis regarding extra noise.
Adapted Sampling to Reduce Noise
The inconsistency between the noise sampling distribution and the domain of the input data leads to the fact that the smoothed gradient still retains a certain amount of extra noise. Therefore, we propose a gradient smoothing method called AdaptGrad, which adaptively adjusts the noise sampling distribution according to the input data to alleviate this problem and significantly improve the performance of the gradient smoothing.
According to the analysis in [Section 3.2, our goal to control the extra noise. Therefore, one the most direct methods is to set a minimum upper limit on the amount of extra noise. In fact, this goal is
5

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conceptually similar to parameter estimation parameters under a given confidence level. Following this idea, design new gradient smoothing method, AdaptGrad. to generate the smoothed gradient
Gag with specified extra noise level c. The Gag is computed using Eqqation 10 Equation 12 where erfinv(-) represents the Inverse Gaussian Error Function (the inverse function erf(-)) and diag denotes the diagonal matrix.
Gag Σ
G(x cr c~ Eag)
(10)
Eag = diag (
n
(11)
min( (yy Smen l Rmaxl)
if + Tmin and H; o Tmax
0i
Veriiow
(12)
i; Jmin or I; Jmax
To explain the design idea of AdaptGrad, we continue to focus on one variable Iy. Our goal is to calculate o; such that the random variable æ; + 8, falls within the sampling interval [Imins Tmar. at extra noise level of This implies that the maximum allowable extra noise is directly limited to c. Therefore, we need to solve the variable a; Equation 13
Tmar Fj
erf( Pmin Ii )1)= =c
erf(
(13)
V20,
V20,
A
However, the o; in Equation 13 does not have simple analytical solution, making it impractical to implement the corresponding algorithm directly. To address this issue, we leverage the sym- metry of the normal distribution and define the sampling interval [- min([ xı &Emin Ial) min Fmaz =,,, [mmin- - xz ]- This sampling interval ensures that the half-length determined the boundaries. Using this approach, Eqqutiinn 12] In the extended case D-dimensions, we construct the covariance matrix Eag. thereby reducing the noise caused by out-of-bounds behavior via sampling from the distribution. 2 / Eag).
by the shortest distance from
derive o, from
(a) Original
(b) 0.95
(c) 0.99
(d) 0.995
(c) 0.999
Figure The visual saliency map Gag of AdaptGrad with different extra noise level c. Other settings are the same as Figure
In AdaptGrad, the extra noise level is defined following the concept of low-probability events in probability theory, with typical values such 0.95, 0.99, 0.9 0.999. Figure 4 illlustrates the visual results of AdaptGrad at different extra noise levels, highlighting its effectiveness. In Appendix A perform a limited hyperparameter search. The results show that AdaptGrad is not only robust to hyperparameter variations but also outperforms SmoothGrad across nearly all hyperparameter configurations. Based these findings, we recommend using c == 0.95 For consistency, we adopt = 0.95 in all subsequent evaluations. Furthermore, to verify the effectiveness of the AdaptGrad design, which is based on probabilistic inference, compared its performance with that of a smoothing method that directly clip the sampling according to the sampling interval [mmin Tmax] in Appendix
0.99.
5 Experiments
In this section, we evaluate AdaptGrad and baseline methods from both qualitative and quantitative perspectives, well as their performance when combined with other explanation methods. To further

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evaluate AdaptGrad, we designed indirect experiments, detailed in Appendix and Appendix F which provide additional insights into its effectiveness and efficiency.
5.1 Experimental Settings
All experimental codes and detailed results can be found in the Supplementary Material, and will be released the public code platform under the anonymous policy. And more experimental details can be found in Appendix C
5.1.1 Metrics
Gradient smoothing methods are often applied as explanation techniques in the field of computer vision. As such, the quality of visualization is a critical metric for evaluating the effectiveness of these methods. However, there is currently standardized framework metric to systematically measure the quality of visualizations. To address this gap, we aim to provide a set visualization examples in Secciinn 5.2 and Appendix G objectively demonstrate the effectiveness of AdaptGrad.
Additionally, recent studies 19 50
22 proposed wide range of evaluation metrics that incorporate human evaluation. Among these, we adopt four specific metrics to assess the explanation performance, as they align with the two fundamental objectives of model explanation: understanding
model decisions 39 J6
34 and enhancing human understanding (8 48) 21 6 29]. Con-
sistency 4 evaluates whether the explanation method aligns with the model's 's learning capability. Invariance [25] ensures that the explanation method maintains output invariance in the presence of
constant data offsets within datasets sharing the same model architecture. Sparseness [I0] measures the distinguishability and identifiability of the saliency map. Faithfulness [19] quantifies the fidelity of salience map to reflect the model's ' decision Docision-making process. We report the variance of these metrics through 5 independent experiments.
Original ("hammer")
30
50
SmmotbCCod
Figure 5: The visual saliency map from VGG16 of SmoothGrad and AdaptGrad with different sample
times N.
5.1.2 Datasets and Models
To apply these metrics for comprehensive evaluation, following the experimental setup in 25 2 we
choose MNIST (27] for experiments on Consistency and Invariance, ILSVRC2012 (ImageNet) 26 for experiments on Sparseness and Faithfulness.
Correspondingly, construct MLP model for Consistency and Invariance check, VGG16 37 ResNet50 [18] and Inception V3 43 for visualization, Sparseness and Faithfulness experiments. VGGI6, ResNet50, and Inception V3 are constructed by pre-trained models released in Torchvision The MLP architecture consists of two linear layers with 200 and 10 units, respectively. The MLP was trained on MNIST by SGD optimizer with 20 epochs, and the learning rate was set to 0.01.
5.1.3 Explanation Methods
AdaptGrad, similar to SmoothGrad, is model-agnostic and can be applied to any gradient- based interpretability methods. Therefore, in addition to using smoothed gradients, we will also use

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AdaptGrad alongside other specific methods to generate saliency maps. However, due to the large number of gradient-base based explanation methods available, applying AdaptGrad to all of them computationally challenging. Thus, following 9 and 40 we select three different explanation
methods for our experiments.
Gradient x Input (GI) (38), is a simple explanation method that generates saliency maps by directly multiplying the image gradients with the input image. GI the representative of the methods that directly use gradients to generate saliency maps. Integrated Gradients (IG) 42 generates saliency maps using global integrated gradients, which can avoid the gradient saturation problem. IG the representative of the methods that partially use gradients in their computation. IG has different options for the baseline background. We have chosen black and white as the baseline backgrounds, which are labeled as IG( and IG(W) respectively. NoiseGrad (NG) [9] is another gradient smoothing method. Unlike SmoothGrad and AdaptGrad, NG reduces noise by perturbing model parameters. However, this approach incurs significant computational costs and does not substantially improve visualization quality. NG represents other gradient smoothing methods. To denote the combination of SmoothGrad and AdaptGrad with other methods, use the prefixes S- and A-, respectively. Grad, SG, and AG denote the original gradient, SmoothGrad, and AdaptGrad.
Referring to the setup in 2 [] Consistency and Invariance are employed to validate SmoothGrad, AdaptGrad, and their combinations with NG. While Sparseness applied to evaluate all the explanation methods. Since Faithfulness is divided into two types of scores: insertion scores and deletion scores, we use Faithfulness-I and Faithfulness-D to label these two types of scores respectively.
5.2 Qualitative Evaluation
As shown in Figure 5) we compare the visualization effects of SmoothGrad and AdaptGrad using different sampling numbers N. The saliency maps generated by AdaptGrad demonstrate better visualization quality than those produced by SmoothGrad, particularly terms of the clarity and detail of object representations. Even at a low sampling number (N = 10), AdaptGrad can exhibit a clear noise reduction capability.
we present an example of saliency maps generated by applying SmoothGrad and AraptGrad to the methods GI, IG(B), IG(W), and NG. The results demonstrate that AdaptGrad has a clear advantage over SmoothGrad visualizing latent features. Specifically, AdaptGrad provides a more nuanced and detailed representation. Furthermore, the enhancement effect of AdaptGrad on the IG method is pronounced, with a notable reduction in noise in the saliency map and the presentation of intricate detail features, such as the facial features of the object.
In Figure 6
Original ("red fox")
(G(B)
16(W)
NG
Codint
SanehhCod
Adipsrani
Figure 6: The visual saliency map from VGG16 of Gradient, SmoothGrad, and AdaptGrrd combined with GI, IG(B), IG(W) and NG.

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Table 2: Results of Consistency and Invariance checks for SmoothGrad (SG) and AdaptGrad (AG)
Methods
Grad
SG
AG
Consistency
0.02076(0.0028))
0.01911(0.00014) 0.361 (0.0300)0000))
0.020239(0.0000)))
Invariance
(0.000))))
(0.482)
Table 3: Results of Sparseness (SS), Faithfulness-I (FI) and Faithfulness-D (FD) evaluation for VGGI6. The f indicates the higher is better.
Vaizz
LGTW
S-12W)
3:10:W3
1t:81
X-1:20
RC
S-N11
X:NG
0.6417
0030)
F377
1N9c
(0,0007)
103330
(0.0033)
118729
13740
(0.000))
10
638972
00.00000
(000055)
00 000015
03772
1000000
03510
(0.0000
02311
(010000)
00.0000
02339
(000011
03557
10623
(000073)
(0.001)
63872
(0.0005)
11504
(0.0037)
1L0728
0.0147
1L302K
8.487
106055
0AKT1
83663
118774
Table 4: Results of Sparseness (SS), Faithfulness-I (FI) and Faithfulness-D (FD) evaluation for
Inception V3. The f indicates the higher is better.
Menic
Valae
585
540W
3.10/W:
51018)
A.10
J.NG
(0,0030)
10,0000)
(0.0000)
0000)
00.0000)
10,0000)
(0.0000)
(00017)
(0,0165)
(0.0016)
1979
102a
N10
(8,000))
106366
(0.020)
013155
00,0001
10635
10.004
80338
(R0003)
U33317
00 000024
03098
100001
033807
(0,0000)
035920
(0,005)
10033
(00000)
03337
(00004)
037705
(80005)
106257
(00087)
8.50
(8,002)
01501
(8 18,0400
1.5075
FDLD
10:40
033277
1030XX
0389
188138
03337
C33000
Table 5: Results of Sparseness (SS), Faithfulness-I (FI) and Faithfulness-D (FD) evaluation for
ResNet50. The indicates the higher better.
310131
0,5535
6.4785
18kr3
(0.0003)
(0.00
00.0031)
(0.0001)
10,00000
(0.0031)
(00001)
(0.000))
(0.0000)
(0.0001)
(0.0000)
(6,000)
(0.001)
(66067)
(9.0107)
(0.0055)
197)
Moan
(80000))
142767
10 00.00133
0.2626
0.0010
0.2002
10 3007))
8.7587
(0:0030 90
8. 1938
10000
0.1866
(0:0003
0.390
(0.9005)
0.2717
(00022)
0.2940
00:0000
0.3668
(0000))
0.3701
0.2954
(00007))
(00013)
0.3618
(8:001:
0.2733
(10014)
0.3775
0.3626
02756
63518
5.3 Quantitative Evaluation
The settings outlined in Section 5.1 are employed to initially assess the Consistency and Invariance of AdaptGrad. The metric values are evaluated to determine whether they exhibit any unusual deviations compared to the original gradients. The results, as shown in Table 2 indicate that both AdaptGrad and SmoothGrad fall within the normal range for Consistency and Invariance. This suggests that AdaptGrad successfully meets the criteria for these two metries.
Table Table 4 and Table reeeeass that AdaptGrad demonstrates significant improvement in the evaluation of Sparseness and Faithfulness. In terms of Sparseness, AdaptGrad shows clear advan- tage over the SmoothGrad method across all 3 models and 5 types of interpretability methods. For the Faithfulness, AdaptGrrad also mostly outperforms SmoothGrad. This indicates that AdaptGradd capable to achieve better balance between the visual quality of the significance maps and the fidelity of the model.
However, noted by 149 the Faithfulness metric is highly dependent the model's inherent performance and suffers from pronounced long-tail effect. In addition, this metric involves numerous hyperparameter choices, which further make its evaluation process less fair and consistent. Therefore, we argue that Faithfulness may not serve as an accurate indicator of an explanation method's 's true quality. The details and limitations of this metric are further discussed in Amppendix B.
These metrics rely on certain assumptions about the quantitative analysis of visualization effects, which means they can only indirectly evaluate the performance of interpretability methods. To provide more direct evidence, Appenddx G inccudess numerous illustrative visualizations that highlight AdaptGrad's 'ss superior denoising capability.
Conclusion
In this paper, we reconsider the principles of gradient smoothing methods by applying the convolution formula, which helps identify the presence of extra noise in gradient smoothing. Based on this analysis, we propose an adaptive gradient smoothing method, AdaptGrad, designed to mitigate extra noise. Theoretical analyses of extra noise, supported by qualitative and quantitative experiments,

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demonstrate that AdaptGrad is an effective alternative to SmoothGrad. Specifically, AdaptGrad outperforms SmoothGrad in terms of noise reduction and robustness. terms of implementation, AdaptGrad, like SmoothGrad, is computationally efficient and model-agnostic. It can also be integrated with other gradient-based explanation methods enhance their performance.
Limitations and Future works
Selection of hyperparameters. In fact, the extra noise level in AdaptGrad is still an empirical choice, despite its widespread use in the field of probability theory. For different datasets or artificial intelligence tasks, there should be different optimal parameter choices.
we used the Sparseness metric to evaluate the amount of noise present in the saliency map. Since there are no established methods for directly assessing noise, we relied on a single indirect evaluation approach. Moreover, we argue that the Faithfulness metric based on insertion and deletion scores is not reliable measure of saliency map's explanatory capability. This limitation may result in insufficient experimental evidence to empirically demonstrate the relationship between out-of-bounds sampling behavior and noise.
Measurement of noise. In [Secction 3.2]
Evaluation of explanation methods. comprehensive and fair evaluation of explanation methods is common challenge in current research on explanation methods. As a result, we also face this issue in our work. To advance research in this field, we hope to develop a unified and widely accepted evaluation framework to provide clear guidelines for assessment.
Broader impact of AdaptGrad. Through comprehensive experimental design, we demonstrate that AdaptGrrd provides a more efficient yet simple approach to gradient denoising. Our comparative experiments show that integrating AdaptGrad with existing interpretation methods such Integrated Gradients and NoiseGrad can significantly improve their performance. This suggests that AdaptGrad can enhance the explanatory power of nearly all gradient-based interpretation methods. Beyond the three gradient-dependent methods validated in our experiments, other approaches such as Smooth- CAM++ [3), Smooth Moore-CAM(L. 45). [DG1[49], and TAIG can also directly replace their gradient smoothing modules with AdaptGrad to achieve better denoising performance. Therefore, we hope that by actively contributing the open-source community, we can provide new solutions to advance the development of the XAI field.
Acknowledgments and Disclosure of Funding
This work is supported by National Natural Science Foundation of China (No.6227255) 63441237, U24A20336). The authors thank for the helpful discussions of anonymous reviewers and area chairs.
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A Hyperparameters Selection
AdaptGrrd only contains two hyperparameters, sample times N and extra noise level c. And SmoothGrad also only contains two hyperparameters, sample times N and All examples and experiments in this article use the settings of N = 50, c 0.95, a 0.2. The settings for SmoothGrad refer to [40], while the settings for AdaptGrad simply follow conventions in probability theory. did not employ any hyperparameter optimization or search methods in the experiments.
As presented in Equation 10 Equation 12
Clearly, the setting of hyperparameterss can affect the performance of AdaptGrad and SmoothGrad. However, the excellent performance of AdaptGrad is robust to the selection of hyperparameters. To prove this, we use Sparseness, as mentioned in Section 5.2 (which is related to visualization performance). as indicator to measure the performance for different hyperparameter configurations, Table 6/shos the experimental results, and all experimental configurations are consistent with those in /Section 5.2 From the results in (aall 6 under the same number of sampling times, the performance of AdaptGrra outperiorms that of SmoothGrrad with any setting of c.
So, intuitively applied c 0.95 the setting for all examples and experiments in this paper. Although we do not believe this is the best hyperparameter selection, believe that regardless of the chosen value of long as falls within reasonable range, like 0.9 0.999, AdaptGrad is likely to achieve convincing performance.
Another set of hyperparameters Min and Xmar. Xmin and X mar are generally independent of the dataset. In the field of image processing, it is almost conventional to set the value corresponding to black pixels as Xmin and the value corresponding to white pixels as X Therefore, whether it's '''s the SmoothGrad method (seen in Equation 6 or the Integrated Gradients method (seen in (Section 5.1.3), this is used as default assumption in these methods. Therefore, in AdaptGrad, we assume by default that X and Xmar are universal information, requiring neither special computational procedures nor manual configuration.
Table 6: The performance (Sparseness 1) of AdaptGrad (AG) and SmoothGrad (SG) with different hyperparameter combinations. We marked the maximum value SG performance, the minimum value of AG performance, and the maximum value of AG performance using red, blue, and green, respectively, under the same sampling number (n).
SG
Sparseness Score (?) tested on VGG16
10
20
30
50
70
100
0.1
0.5304
0.5330
0.5363
0.5427
0.5475
0.5533
0.2
0.5370
0.5345
0.534
0.5334
0.5338
0,5343
0.3
0.5361
0.5313
0.5279
0.5235
0.5209
0,5174
0.4
0.5334
0.5266
0.5221
0.5154
0.5109
0.5057
0.5
0.5309
0.5237
0.5185
0.5113
0.5061
0.5001
AG
Sparseness Score (*) tested on VGG16
10
20
30
50
70
100
c
0.9
0.5496
0.5493
0.5511
0.5535
0.5561
0.5588
0.95
0.5511
0.5532
0.5558
0.5608
0.5644
0.5687
0.99
0.5526
0.5576
0.5621
0.5691
0.5748
0.5809
0.995
0.5527
0.5582
0.563
0.5711
0.5772
0.5838
0.999
0.5526
0.5597
0.5655
0.5746
0.5807
0.5881
B
Metrics Details
Currently, there is no unified and widely accepted system for the quantitative evaluation of explanation methods. In fact, nearly every related study employs different evaluation metrics, making it difficult to follow consistent standard for selecting assessment metrics. We conducted a relatively comprchensive evaluation from two perspectives: the axiomatic properties of explanation methods (understanding model decisions) and their visualization effects (enhancing human understanding). Below, we provide detailed introduction to the origins and implementations of the four quantitative metrics used in this paper. And all the implementations of the metrics are included in our source code.
Consistency from the Sanity Check experiment in 2 Two types of check experiment, model parameter randomization test and data randomization test, were designed to evaluation Gradient, SmoothGrad, Gradient x Input, Guided Back propaaarooaa,.nn GradCAM. Guided GradCAM, Integrated Gradients, and Integrated Gradients- SG. The model parameter randomization test primarily check whether the explanation method can remain
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Table 7: Results of SIC evaluation for SmoothGrad(((SG) and AdaptGrad(AG) combined
combinie with (IG).
The f indicates the higher better.
Methods
VGG16
Inception
ResNet50
IG(W)
0.5636
0.6677
0.5491
S-IG(W)
0.5741
0.7179
0.5718
A-IG(W)
0.5846
0.7221
0.5849
IG(B)
0,5823
0.6751
0,5355
S-IG(B)
0.6035
0.7166
0,5731
A-IG(B)
0.6121
0.7193
0.5673
consistent with the model's randomization process through visualization effects. While the data randomization test evaluates the consistency, quantified using rank correlation, of the interpretation method hefore and after randomization by permuting the training labels and training model the randomized training data. Therefore, we selected the data randomization test as evaluation metric this paper and, following 19] named Consistency.
Invariance is from the the axiom input invariance in [25 They designed a experiment aimed at validate the input shift invariance of explanation methods. Since [25] did not provide a clear name for it, and both [9] and [19] refer to this metric as Robustness. To distinguish it from the adversarial example generation experiments in
Appendix named it Invariance.
Sparseness from the Assumption LOSS-CVX and sparseness of an attribution vector experiment in [O] For an explanation method, Gini Index is used to quantify the sparseness of absolute values of saliency maps. Given an input of saliency map [f1, Jn its Gini Index C can be calculated using Equation 14
Σ (2i 71 >ド=1 22.
1)x7
(14)
Faithfulness is evaluated using the Area Under the Curve (AUC) of the insertion and deletion metrics, which are widely adopted for assessing the reliability of model explanations. In our experiments, we set the insertion and deletion ratio to 5% of the total pixels, and used 0 the baseline filling value. Although prior work [49] has pointed out that such straightforward ixertion-ddlltion procedures may not fully capture the faithfulness of an explanation method, we employ them here for fair comparison with existing studies. Interestingly, when pixels are removed in batches according to their saliency ranking, noisier saliency maps may achieve deceptively higher deletion scores. This is because, after removing pixels with high importance. the residual noise tends to "smooth out" the deletion process by also eliminating pixels surrounding truly important regions. This phenomenon aligns with our empirical observations reported in Table
Table
and Table
Therefore, for Integrated Gradients-based explanation methods, we additionally adopt an improved variant of the cnsertion-deletton
insertion- deletion metric, namely the Softmax Information Curve (SIC) proposed by 23). Unlike conventional insertion and deletion metrics that directly replace pixels with zeros, SIC progressively removes or restores pixels identified as important based on the amount of information they contribute to the model's prediction. We report the experimental results in Table 7 which show that AdaptGrad consistently outperforms other explanation methods under the SIC metric.
C Experimental Details
In the quantitative evaluation, due to the time-consuming computation of the experiment, we randomly sampled 1,000 samples instead of all validation or test set samples for the comparison experiments. This also led to difficulties reporting the statistical significance of our experimental results. Our experiments on a server with 4 NVIDIA RTX 4090 and 2 Intel Xeon Gold 6128. Although AdaptGrad consumes little computational resources, combining with other rendering methods results in exponential consumption (similar to SmoothGrad), so we recommend using GPU devices with at least 12G memory to run the reproducible code.
conducted
The MLP architecture in Consistency and Invariance check consists of two linear layers with 200 and 10 units, respectively. The MLP was trained on MNIST by the SGD optimizer with 20 epochs, and the learning was 0.01. The CNN architecture contains few layers follows: [Conv(6), Maxpool(2). Conv(16), Maxpool(2), Linear(120), Linear(84), Linear(10)).
We set the hyperparameters of IG and NG explanation methods as described 4p and 9 number of Riemann integration samples in the IG was set to 50, and the number of parameter perturbations in the NG was set to 50. The variance of the corresponding perturbation noise was set to 0.2, and numerical overflow perturbations were excluded.
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To ensure the stability of the results of metrics Faithfulness, each sample was tested 5 times. The values reported in Table 3 Table 4 and Table 5) are the average of 1000 samples tested times. And its hyperparameler, the saliency threshold, was set to [0.05, 0.1, 0.15, 0.2, 0.25, 0.3] according to [2]]
D Discussion of Hard Threshold
As discussed in Section 3.2] we identified the presence of extra noise in SmoothGrad. Correspondingly. one might consider directly applying hard constraint the input samples a potential way to suppress this extra noise. However, we argue that such straight thrd-threshold operation can hinder the convergence of the smoothing convolution, thereby introducing additional artifacts and noise into the saliency maps.
In contrast, AdaptGrad maintains a convergent sampling process and adaptively adjusts the sampling range, allowing it to better control the extra noise and emphasize the visualization performance. To validate this hypothesis, implemented simple variant of SmoothGrad that clips the sampled values within the range [zmin, Zmaz], which we refer as ClipGrad. We then compared ClipGrad and AdaptGrad across evaluation metrics.
As shown Table and Table 9] the results consistently demonstrate that AdaptGrad achieves superior performance in both noise suppression and target feature enhancement, confirming its effectiveness over the simple clipping-based alternative.
Table 8: Experimental result of Consistency and Invariance checks of AdaptGrad (AG) and ClipGrad (CG).
Methods
AG
CG
Consistency
0.20239(0.00022) 0.01773(0.0001))
Invariance
0.3484(0.0002))
0.3625(0.0008)
Table 9: Results of Sparseness (SS), Faithfulness-I (FI) and Faithfulness-D (FD) evaluation for AdaptGrad (AG) and ClipGrad (CG). The indicates the higher is better.
Models
VGGT6
I3ccption
ResNc150
Metrics
AG
CG
AG
CG
AG
CG
SS(1)
0.5740(0.0000)
(5294(0.0001)
0.5584(0.0001)
(..44100.0001)
0.5721(0.0001)
0.5607(0.000) 0001)
FI(1)
0.6748(0.0002)
0.6740 0000))
0.6145(0.0007)
0.5999 0000))
0.2692(0.0011)
0.2605(0.0020
FD(L)
(.6747)0.0002)
0.6739(0.0006)
0.6146(0.0008)
0.009))000006)
0.2693(0.001))
0.2605(0.0019)
E Evaluation on Visual Task
In this section. we compare the differences between AdaptGrad and SmoothGrad in two common visual tasks that utilize saliency maps: object localization and adversarial sample generation. These indirect visual tasks could assist in evaluating the performance of the AdaptGrad method more objectively.
E.1 Evaluation on Object Localization Task
The performance improvement in weakly supervised object localization tasks is often used to evaluate class activation map methods as type of explanation method. [54] 35 1T
46 22 generate saliency maps based on
explanation methods and use "heuristic" approaches create series of object localization candidate hounding boxes. Then the localization accuracy of these candidate bounding boxes is used to evaluate explanation methods. Due to the numerous uncertainties associated with these "heuristic" methods, we employ a publicly available and widely used candidate bounding box generation algorithm called selective search 44). The selective search algorithm has very high recall rate but lower precision rate. Therefore, used its precision rate to measure the performance of AdaptGrad and SmoothGrad
The hyperparameter settings of this experiment are consistent with other experiments. Specifically, the parameters of the selective scarch algorithm are set as follows: clusters number in felzenszwalb segmentation is 500 (scalc=500), width of Gaussian kernel is 0.9 (sigma=0.9). and minimum component size is 10 (min_ miin__sze==1).. Figure Jillustrates example of candidate bounding boxes using the selective search algorithm. It is clearly observed that the candidate bounding hoxes generated based AdaptGrad more concentrated on the ground truth box.
settingg
'The saliency map generated by the vanilla gradients contains huge amount of noise, which causes the selective search algorithm unable to generate usable bounding candidate boxes. Most of these bounding boxes only contains few pixels. So exclude the testing for Vanilla Gradient.
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Original
Saliency Map
Bounding Boxes
SeadthCred
Figure 7: The visual bounding boxes from VGG16 of SmoothGrad and AdaptGrad with selective search algorithm. The generated candidate bounding boxes are marked in red, and the ground truth box is marked in blue.
Table 10:
The object localization bounding boxes precision, which is generated by selective search algorithm using saliency maps from SmoothGrad and AdaptGrad.
Localization Precision (f)
Models
VGG16
Inception V3
ResNet50
SmoothGrad
0.01937
0.1048
0.05528
AdaptGirad
0.05951
0.1237
0.08108
Table 10 summarizes the performance of SmoothGrad and AdaptGrad in our designed object localization capability evaluation. can be observed that AdaptGrad outperforms SmoothGrad across all three test models. This indicates that AdaptGrad could more accurately represent the neural network's learning capability for object features, and also suggests that AdaptGrad may have better potential applications in weakly supervised object localization algorithms.
E.2 Evaluation on Adversarial Sample Generation Task
In this section, we indirectly evaluate the performance of different explanation methods by comparing their performance in the adversarial sample generation task. We use the pixel-level Fast Gradient Sign Method (FGSM) to generate adversarial samples. Specifically, we select the pixel corresponding the maximum value the saliency map (i.e., the gradient map in FGSM) a sample one by one and change the value of this pixel to make the neural network incorrectly classify the sample.
We adopt pixel-level FGSM because we generate adversarial samples directly based on the entire gradient map, it would lead to noisier gradients performing better (i.c., causing a greater drop in model accuracy), as this noise causes more pixels to change in the adversarial sample. However, this is elearly not the sole objective of the adversarial sample generation task. The quality evaluation of adversarial samples should also consider the similarity between the adversarial sample and original image, as well as the subjective image quality. Therefore, to avoid these complex adversarial sample quality evaluation issues, we generate adversarial samples based on pixel-level FGSM. By setting uniform attack target, we then measure the performance of different explanation methods counting the number of pixels that need to be changed based on the saliency map generated by each explanation method. In our experiment, we set the attack target make the model's output for the target class (Softmax output) less than 0.5.
Figure 8 shows an example of an adversarial sample. Due to the high randomness of adversarial attacks, we chose a relatively simple VGG16 model as the attack target. We comprehensively evaluate the performance of different explanation methods in the adversarial sample generation task by conducting 10 independent experiments. The rest of the experimental settings remain consistent with other experiments in this paper. [Tigure 9 illustrates the experimental results based based on the original gradient exhibited significant volatility. However, it can be observed that the FGSM method based on AdaptGrad requires significantly fewer pixel changes compared to the FGSM method based on SmoothGrad. This indicates that AdaptGrad
evaluation of the adversarial sample generation task. The FGSM method
accurately reveal the internal decision mechanisms of the
23

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model compared to SmoothGrad. and suggests that AraptGrad may have potential applications in adversarial sample generation.
Odginal
NOrasee Sanple
Gradier.
SmooohhGm
AdapSSmm
Figure Comparison of adversarial samples generated by different explanation methods. The attacked model is VGG16.
Number Changed pixels
400
36885836
300
250
200
Figure 9: Statistics the number of pixels that need to be changed generate adversarial samples based on different interpretation methods. AG, SG and VG represent AdaptGrad, SmoothGrad and Vanilla Gradient respectively.
Evaluation on Computational Cost
Compared to SmoothGrad, AdaptGrad introduces only a minimal amount of additional computation. This extra computation almost entirely concentrated in Egquaiinn 12 However, it's worth noting that in |Equation 12] the denominator be practically treated constant, while the computational complexity of the numerator only depends the dimension of the input, with complexity of just (NV) Therefore, AdaptGrad only adds an O(N) level complexity, which virtually negligible compared the computation of the gradients neural network itself.
To demonstrate this, we tested the computational time of SmoothGrad and AdaptGrad on benchmark models (VGG16, InceptionV3, ResNet50) using 1,000 images. The hardware and other parameter settings were entirely consistent with the other experiments. The Table II below presents experimental results.
Table 11: Execution time (s) of AdaptGrad and baselines.
Method
VGG16
Inception V3
ResNet50
Grad
0.0075 + 0.0044
0.0387 = 0.0092
0.01828 = 0.0069
SmoothGrad
0.5914 + 0.0162
2.6255 + 0.0724
1.9955 0.0877
AdaptGrad
0.6054 + 0.0201
2.6673 = 0.0829
1.9426 = 0.0873
results show that AdaptGrad incurs only very small extra computational overhead. This demonstrates the strong versatility of AdaptGrad and SmoothGrad, We will also include the corresponding experimental results and analysis in the paper, and provide detailed discussion on the potential computational costs when dealing with large-scale data and complex models.

---

G More Visualization Examples
We provide more visualization examples to compare AdaptGrad with the Baseline method, including experiments the VGGI6, ResNet50, and Inception V3 models. And we provide visualization examples images with low contrast, high contrast, small targets, large targets, and multiple targets
Figure 10 Figure 11 and Figure 12 show the explanation performance of AdaptGrad and baselines on VGGI6, ResNet50, and Inception models. It can be noticed that AdaptGrad could improve the visualization perfor- mance.
Onginal
Grad
Codbett
SmoothGrdd
AdapiSred
Figure 10: The visual saliency map from VGG16 of Gradient, SmoothGrad, and AdaptGrad combined with Grad, GI, IG(B), IG(W) and NG.
Original
Goad
GI
IGIBI
RG/W)
NG
Onder
AduxGled
The visual saliency map from ResNer50 of Gradient, SmoothGrad, and AdaptGrad combined with Grad, GI, IG(B), IG(W) and NG.
Figure 11:
We also conducted visual demonstrations on various types of images, including high-contrast images Figure 13 low-contrast images Figure 14 large-object images Figure 15 small-object images Figure 16 and multi-ohject images|Figure SaaagesFiggus 17 jsiig VGG16. 6. All these images were sourced from the ImageNet et dataset. Specifically, the high-contrast images were selected from among the highest coctest- images in the ImageNet validation set, and the same applies to the low-contrast images, large-object images, and small-object images. Almost all the examples demonstrate that GraptGrad can achieve better visualization results than the baseline on different types of images.
25

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Original
Goad
G1
18:B1
KGTW
NG
Godex
Adappian
Figure 12: The visual saliency map from Inception V3 of Gradient, SmoothGrad, and AdaptGrad combined with Grad, GI, IG(B), IG(W) and NG.
RMS-114 2003
RM5-13.7746
RMS=11966757 Original Image
Gradient
SmoothGrad
A daptGraa
Figure 13: The visual salieney map of high-contrast images comparison between Gradient, Smooth- Grad, and AdaptGrad
26

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RMS-3
RMSE7 5073
333- Original Image
Gradient
SmoothGrad
A daptG
Figure 14: The visual saliency map of low-contrast images comparison between Gradient, Smooth- Grad, and AdaptGrad
Oblect
Objaci Size 1898
Object Size 449440
Original Image
Gradient:
SmoothGrad
daptGGrad
Figure 15: The visual saliency map of large-object images comparison between Gradient, Smooth- Grad, and AdaptGrad
27

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Objject Sizee30
Object Size =45
Object Size -48
Original Image
Gradient
SmoothGrad
AapttG rad
Figure 16: The visual saliency map of small-object images comparison between Gradient, Smooth- Grad, and AdaptGrad
Original Image
Gradient
SmoothGrad rad
A daptGrad
Figure 17: The visual saliency map multi- images comparison between Gradient, Smooth- Grad, and AdaptGrad
28


Reviews:
[{"review_id": "0E0Bvig7B0", "reviewer": "Reviewer_AKq9", "summary": "The authors investigate the problem of excessive noise when using the SmoothGrad feature attribution method. In this work, they propose a metric for quantifying the excess noise as well as a novel method AdaptGrad which limits the amount of excess noise based on a given threshold. AdaptGrad extends SmoothGrad by adaptively selecting the smoothing parameter $\\sigma$.  Quantitative and Qualitative experiments show that AdaptGrad reduces noise compared to SmoothGrad and other gradient-based attribution methods.", "questions": "* How does the confidence interval defined in line 172 relate to AdaptGrad? This seems to be a range of values (which could be [0,0] if x_i = x_max) rather than relating to probability.\n* Did you explore simply hard thresholding the samples to be within [x_min, x_max], rather than adaptively reducing $\\sigma$? \n* What was the statistical test performed in Line 150? Do you have results for all experiment settings?", "limitations": "Limitations are discussed in section 6.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 1, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Strengths:**\n* The method empirically seems to improve the sparseness and information level of explanations as compared with taking gradients or smoothgrad.\n* The proposed method is fast and simple to implement. It can be used in conjunction with other gradient-based methods.\n\n**Weaknesses:**\n* Manuscript could be made more rigorous and has issues with clarity (see \"other comments\" below).\n* While the AdaptGrad saliency maps seem to be more \"distinctive\" compared with SmoothGrad, it's not clear to me that the explanation is actually better. Since the smoothing parameter $\\sigma$ decreases for features that are closer to the boundary of [x_min, x_max] does this not simply highlight the dark / light pixels or pixels close to the boundary values?\n* No error bars or measures of variability in Table 2 and 3.\n* Limited theoretical contribution.\n\n**Other Comments**\n* The connection between SmoothGrad and convolutions / expectations (Section 3.1) is well known and used in previous works (e.g. proposition 1 in [1]). It's not clear if section 3 is intended to be background or part of this paper's contribution -- it would be helpful if this was more clearly delineated.\n* The paragraph under equation 6 (lines 116-121) is very confusing. In particular, \"the integral in Equation 4 which is not performed over $R^D$ but rather over a bounded domain\". However, Equation 4 is an indeed an integral over $R^D$ (or at least, it's not otherwise specified). The next line states that the domain of $G$ is defined over $\\Omega$ rather than $R^D$, however, this was not previously established in the paper (and the domain of $F$ is $R^D$).\n* In Equation 7, the notation $G(x_i + \\epsilon_i; x$ \\ $x_i)$ is not defined anywhere and conflicts with the previous definition of $G$.\n* Equation 10-12 is undefined when $x_i = x_{min}$ or  $x_{max}$\n* The symbol * indicates multiplication in Eq. 2 and convolution in Eq. 4, which is confusing.\n* It would be helpful if the metrics (e.g. consistency, invariance, sparseness) were defined in the main text, given that they are mentioned frequently. In addition, the definitions in the supplement could be made more formal. It's difficult to understand what each metric is measuring without referencing the original paper.\n* In Eq. 14, I believe the $x_i$ values need to be non-negative and sorted in non-decreasing order?\n\n[1] https://arxiv.org/pdf/2006.06643", "was_revised": true, "final_justification": "My original score was 3. During the discussion period, the authors included additional results on clipgrad and statistical error, which addressed many of my original concerns. I still have some concerns regarding notation, given that the reviewers cannot see the revised manuscript. Overall, I think the improvements justify increasing the score to a 4.", "rebuttal": "Dear reviewer:\n\nWe greatly appreciate your valuable comments. We are delighted that you have recognized our efforts and provided such insightful feedback. We have carefully considered your remarks and will respond according to the following points.\n\n## Explanation of effects and contributions\n\nSince neural networks are extremely complex, nonlinear functions, and the input features are highly correlated with each other, SmoothGrad tends to out-of-bound sampling significantly, leading to the extra noise we mentioned in Section 3.2. Therefore, rather than simply highlighting brighter or darker pixels, AdaptGrad carefully **controls the out-of-bound sampling behavior across all pixels**, including those in intermediate positions. As can be clearly observed from the images provided in our paper (we highly recommend zooming in on the saliency maps for visual comparison), AdaptGrad smooths out a significant amount of noise outside the image's main objects and offers a noticeable enhancement over other gradient-based methods.\n\nTherefore, we want to once again highlight **the contribution of extra noise**. We have, for the first time, identified the presence of extra noise in gradient smoothing methods. Methods like SmoothGrad and NoiseGrad, on the other hand, simply introduce perturbations to the inputs and parameters. This not only allows us to design a simple and efficient approach like AdaptGrad, but also assists other research focused on noise reduction in saliency maps.\n\n## The statistical values in Table 2 and Table 3\n\nWe apologize for the oversight in providing complete experimental results. We have now included partial error metrics in Table 2 and Table 3. However, we do apologize that, due to the time constraints of the experiment, we were unable to collect all the statistical metrics in such a short period. The NoiseGrad and Integrate Gradient methods require a significant amount of computational resources to test, so we can only provide partial statistical indicators for now.\n\nWe re-conducted 5 separate experiments and calculated the average and standard error. We will continue to gather more experimental data in the coming days and will include the full set of statistical metrics in the final version.\n\nTable 2. Partial results of Consistency and Invariance checks for SmoothGrad(SG)， AdaptGrad(AG) and ClipGrad(CG).\n\n| Method      | Grad             | SG               | AG                | CG               |\n| ----------- | ---------------- | ---------------- | ----------------- | ---------------- |\n| Consistency | 0.02076(0.00028) | 0.01911(0.00014) | 0.020239(0.00026) | 0.01773(0.00013) |\n| Invariance  | 0.3483(0.0002)   | 0.3613(0.0009)   | 0.3484(0.0002)    | 0.3625(0.0008)   |\n\nTable 3. Partial results of Sparseness (SS) evaluation of SmoothGrad(SG), AdaptGrad(AG) and ClipGrad(CG).\n\n| Method | VGG16          | InceptionV3    | ResNet50       |\n| ------ | -------------- | -------------- | -------------- |\n| Grad   | 0.5421(0.0000) | 0.5546(0.0000) | 0.5626(0.0000) |\n| SG     | 0.5334(0.0001) | 0.5421(0.0000) | 0.5580(0.0001) |\n| AG     | 0.5605(0.0002) | 0.5621(0.0001) | 0.5731(0.0000) |\n| CG     | 0.5538(0.0000) | 0.5424(0.0001) | 0.5579(0.0003) |\n\nHere, we need to clarify that the values we reported in the Rebuttal differ from those presented in the paper. This is because we conducted the experiments again. Specifically, the ResNet50 model parameters varied due to differences in the code libraries referenced. Pytorch recently updated the ResNet50 model parameters in the Torchvision library, which has led to different results in our experiments compared to before. We will update these results in the final version.\n\n## The definition of confidence interval\n\nWhat we want to explain is that the confidence interval defined in line 172 corresponds to the interval in Equation 12. Theoretically, the confidence interval we should use is $[x_i - x_{min}, x_{max} - x_i]$, but since this interval is **not symmetric with respect to $x_i$**, it creates difficulty in solving for the corresponding sample $\\sigma_i$ given the confidence level $c$. This is because our sampling distribution is a normal distribution, which is symmetric. We are unable to obtain an **analytical solution** for the following Equation:\n\n$$\n\\int_{x_i - x_{min}}^{x_{max} - x_i} \\frac{1}{\\sqrt{2\\pi}\\sigma_i} e^{-\\frac{(x_i - x)^2}{2\\sigma_i^2}} dx = c\n$$\n\nSo, we need to further narrow this interval to $[- \\min(|x_i - x_{\\text{max}}|, |x_i - x_{\\text{min}}|), \\min(|x_i - x_{\\text{max}}|, |x_i - x_{\\text{min}}|)]$, in other words, we choose the \"shortest\" side as the half-length of our confidence interval. This results in a symmetric confidence interval, allowing us to solve for the desired value. This is essentially the calculation method for $\\sigma_i$ in formula 12 in the current AdaptGrad.\n\n## Discussion of the hard thresholding method\n\nOur goal is to eliminate extra noise, rather than simply limiting the sampling to a fixed range. A simple hard thresholding approach would make smooth convolution kernel is not strictly smooth (The term \"smooth\" here refers to a function that is smooth in the mathematical sense.), resulting in a significant amount of noise in the saliency maps. In contrast, AdaptGrad, by adaptively adjusting the sampling range, is better at highlighting the features of visual targets.\n\nTo verify this, we compared AdaptGrad with the simple hard thresholding method, which we named ClipGrad (marked as CG in experimental results). ClipGrad specifically implements direct clipping of the SmoothGrad sampled values within the range $[x_{min}, x_{max}]$. We supplemented the experimental results of this method on the three baseline models, as shown in Table 2 and Table 3. The results demonstrate that AdaptGrad performs better in eliminating noise and highlighting target features.\n\n## The statistical test performed in Line 150\n\nThe experiment we conducted here is to examine whether the extra noise we proposed correlates with the noise in the saliency map. We used a t-test to assess the correlation between the extra noise and the noise in the saliency map. The experiment was carried out on 1,000 samples, and due to space limitations, we are only presenting the results of the final statistical test along with the corresponding p-value. We have provided all the statistical values in the main text, and the detailed experimental results will be included in the supplementary materials.\n\n## Clarification of the rigor and clarity of the manuscript\n\n### The connection between SmoothGrad and convolutions / expectations\n\nWe acknowledge that the connection is well known and has been used in previous works. This serves merely as preliminary theory to help readers understand our findings regarding extra noise. We will refine this section and cite relevant literature to clearly define this point.\n\n### The confusion in the paragraph under Equation 6\n\nWe apologize if the explanation here caused any misunderstanding for readers. The integral in Formula 4 refers to the integral associated with the SmoothGrad method, whose domain is $ R $. However, what we intend to highlight is that the \"correct\" domain of the integral in the SmoothGrad method should actually be a bounded domain. The original domain $ R $ is an \"incorrect\" setting, which leads to a significant amount of extra noise. We will revise the relevant descriptions to improve clarity and readability.\n\n### The notation $G(x_i+\\sigma_i;x \\backslash x_i)$ in Equation 7\n\nWe would like to clarify that the notation $G(x_i+\\sigma_i;x \\backslash x_i)$ in Equation 7 originates from Equation 4. The previous definition was made for all dimensions of $x$, while here we redefine $G$ for a single dimension $x_i$ to simplify subsequent formula derivations. There is no conflict with the previous definition; they simply differ in the dimensions defined.\n\n### Equation 10-12 being undefined when $x_i=x_{min}$ or $x_{max}$\n\nWe would like to clarify that when $ x*i = x*{min} $ or $ x\\_{max} $, in equation 12, $ \\sigma_i = 0 $, which means we will not sample the feature $ x_i $ in this dimension. This is consistent with our theory of extra noise.\n\n### The symbol usage of \\* in the equations\n\nWe have adopted the notation for convolution symbols from the field of mathematics. We regret that this may be unfamiliar to some readers, and we will provide additional explanations and context in the manuscript to clarify its usage.\n\n### Metrics definition and clarity\n\nWe acknowledge that the definitions of metrics in the main text may not be clear enough. Due to space limitations in the main text, we were only able to provide the relevant definitions in the appendix. Thank you for pointing this out. We will include definitions for certain metrics such as consistency, invariance, and sparsity in the main text, and make these definitions more formal in the appendix. We will ensure that readers can understand what each metric specifically measures without having to refer to the original paper.\n\n### Values $x_i$ in Equation 14\n\nThe values of $ x_i $ need to be non-negative (which is typically achieved through normalization) and arranged in non-decreasing order. This is determined by the calculation of the Gini coefficient. We will include a clarification of this point in the paper."}, {"review_id": "12UntwY6f5", "reviewer": "Reviewer_G7TC", "summary": "This paper studies a problem about generating visual saliency maps with better quality. A novel method named AdaptGrad is proposed to reduce the noise issue in gradient computation and improve the performance for better explanation.", "questions": "What’s the difference of statistical behaviour when increasing N? How to decide an optimal N with good explanation quality but minimum computation cost?\n\nThe proposed method is pixel level-based computation. While the visual information is usuall redundant, it would be better to run the algorithm with group style.", "limitations": "See weakness and question. Authors also provide a discussion section about the limitation at the end of paper.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strength:\n\n1) The paper is well-organized and easy to follow. \n\n2) The related works are listed in detail with clear background, as well as the pros/cons. \n\n3) The analysis on SmoothGrad is interesting and motivated. The design of per-pixel variance in gradient computation is simple but novel. \nExperiment results show the effectiveness of the proposed method. The visual explanation map seems better than baseline methods. \n\nWeakness:\n\n1) Experiments are limited. Only 1000 images from the ImageNet dataset is collected for evaluation. It is unsure about the stability and robustness of the proposed method  against different set of domain of images. No statistic significance is provided. \n\n2) Metrics are limited. Previous work usually use the AUC/deletion&insertion score to evaluate the quality of saliency map, while this work uses the sharpness and information level score, which are rare to use. \n\n3) The confidence is still required to manually tuned for better performance.", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer,\n\nWe greatly appreciate your constructive comments and the effort you put into reviewing our work. We are pleased that you have acknowledged our contributions and provided such insightful feedback. We have carefully considered your remarks and will respond according to the following points.\n\n## Limitations of the experiment\n\nWe acknowledge that the current experiments are limited to 1,000 images from the ImageNet dataset. At this time, we have not thoroughly tested the stability and robustness of AdaptGrad across different collections of images. We plan to expand our work to other datasets in the future and include the results in the revised version. In our response to reviewer AKq9, we have provided some experimental results that include statistical metrics. Due to the limitations in the efficiency of certain interpretation methods and the calculation of metrics, we are currently only able to provide statistical results for some of the indicators. We will include the complete experimental results in the revised version.\n\nIn some related work, such as SmoothGrad, visualization experiments were conducted on only a very small number of samples, and NoiseGrad was tested using just 250 images from ImageNet. In the revised version, we will provide additional experimental results to verify the performance of AdaptGrad on a broader range of data.\n\n## Limitations of the metrics\n\nWe would like to clarify that the information level score we used is a more quantitative metric for AUC/deletion&insertion score, which was first proposed in [1] and is widely used to test the saliency maps of IG methods. This metric quantifies the deletion score as an information entropy-based measure. Based on the entropy value, it calculates the area of the accuracy information curves, thereby obtaining the information level score. This can avoid issues in experimental results caused by \"human-selective\" deletion and addition. **In short, the information level is an AUC/deletion&insertion score based on information entropy**.\n\nAdditionally, the Adversarial Sample Generation Task we designed in the appendix is a supplement to the AUC/deletion&insertion score. We will supplement relevant discussions in the revised version.\n\n## Manual tuning of confidence $c$\n\nWe acknowledge that the confidence level $c$ in AdaptGrad requires manual tuning. However, we have found that the performance of AdaptGrad is relatively stable across a range of confidence levels, as proven by our experiments in Appendix A. In our experiments, we used a confidence level of 0.95, which yielded satisfactory results. We will clarify this point in the revised version and provide more details on how to select the confidence level.\n\n## Discussion of choice of N\n\nAs we mentioned in Section 3.1, increasing N tends to make the Saliency Map converge. Generally speaking, referring to methods like SmoothGrad, NoiseGrad, and Integrated Gradients, we empirically choose N to be 50.\n\nHowever, we also note that, as shown in Appendix A Table 4, AdaptGrad can achieve good results with a smaller N because it largely eliminates extra noise. We will further discuss the choice of N in the revised version.\n\n## Other comments\n\nWe appreciate your suggestions regarding the group style running algorithm. We will conduct relevant experiments and discuss the potential for this improvement.\n\n[1] A. Kapishnikov, T. Bolukbasi, F. Viegas, and M. Terry, “XRAI: Better Attributions Through Regions,” in 2019 IEEE/CVF International Conference on Computer Vision (ICCV), Oct. 2019, pp. 4947–4956. doi: 10.1109/ICCV.2019.00505."}, {"review_id": "bXpMN0VOpo", "reviewer": "Reviewer_Dez6", "summary": "The paper introduces AdaptGrad, a novel adaptive gradient smoothing method designed to reduce noise in gradient-based interpretability techniques such as SmoothGrad. By reinterpreting SmoothGrad as a convolution and analyzing its behavior, the authors identify a key source of “extra noise” caused by out-of-bound sampling. AdaptGrad addresses this by adaptively restricting the sampling range using confidence intervals based on data bounds. The method is theoretically motivated, computationally efficient, and empirically shown to improve saliency map quality across multiple models and interpretability methods.", "questions": "1.  How sensitive is AdaptGrad’s performance to the choice of the confidence level, especially in domains with different data distributions or input scales?\n\n2.  Can AdaptGrad be effectively extended to non-vision domains (e.g., NLP or tabular data), where input bounds and semantics differ significantly?", "limitations": "NA", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths\n-The authors provide a clear theoretical reinterpretation of SmoothGrad using convolution, which leads to identifying a new source of noise (extra noise) and motivates the development of AdaptGrad.\n-AdaptGrad is conceptually straightforward, easy to implement, model-agnostic, and computationally efficient—making it broadly usable across existing gradient-based interpretability pipelines.\n- The method is evaluated using a wide range of metrics, datasets , models, and explanation methods , including indirect tasks such as object localization and adversarial robustness.\n\nWeaknesses\n- The evaluation of noise relies primarily on the Sparseness metric, which is not a direct measure of explanation quality or ground truth relevance, and may not capture all noise-related phenomena.\n-The evaluation is conducted on standard image classification tasks and does not demonstrate performance or applicability in more complex or high-stakes real-world domains", "was_revised": true, "final_justification": "I am convinced of the merits of this work and suggest accepting.", "rebuttal": "Dear Reviewer,\n\nWe sincerely appreciate your thoughtful feedback and the acknowledgment of our contributions. Your insights are invaluable to us, and we are committed to addressing your comments thoroughly.\n\n## More evaluation and merits of AdaptGrad\n\nWe acknowledge that selecting the right metrics is a challenging and complex task. We opted to rely on the Sparseness metric because it intuitively reflects the clarity of the visualized Saliency Map. Additionally, we referred to other relevant metrics used in related work, such as SmoothGrad and NoiseGrad methods. Furthermore, we introduced two additional evaluations based on visual tasks to further assess the performance of AdaptGrad.\n\nWe are grateful for the suggestions you've provided, especially regarding evaluations on real-world data. However, due to the significant workload involved, we are currently limited to assessments based on the ImageNet dataset. In our future work, we will consider conducting evaluations in more complex real-world domains and incorporate those results into the revised version.\n\n## The sensitivity of AdaptGrad to confidence level\n\nIn Appendix A, we discuss the condition of confidence level $c$ in AdaptGrad. Our experiments demonstrate that AdaptGrad's performance remains relatively stable across a range of confidence levels.\n\nHowever, due to the limited selection of experimental datasets, our current experiments are confined to the ImageNet dataset. This does not fully represent AdaptGrad's performance across other domains with different data distributions and input scales. We plan to expand our work to other datasets in the future and include those results in the revised version.\n\n## Discussion of broader applications of AdaptGrad\n\nWe appreciate your suggestion to extend AdaptGrad to non-visual tasks. Currently, the majority of gradient-based explanation methods focuses almost exclusively on visual tasks. As a result, we haven't yet applied AdaptGrad to non-visual tasks. However, we believe that the concepts and methods behind AdaptGrad can be extended to other domains, especially when dealing with tasks that involve complex structures and high-dimensional inputs. We will explore these possibilities in our future work."}, {"review_id": "gYOUy7km55", "reviewer": "Reviewer_CbLr", "summary": "The paper identifies a key limitation in SmoothGrad. SmoothGrad reduces noise in gradient-based saliency maps (e.g., Grad-CAM) by averaging gradients from inputs perturbed with Gaussian noise. However, its fixed noise variance hyperparameter causes out-of-range sampling, introducing extra noise into smoothed gradients. To address this, the authors propose AdaptGrad, an \"adaptive\" method that dynamically adjusts σ per input dimension. This ensures sampling stays within the valid data domain with a user-defined confidence level.", "questions": "Please address the 3 weakness points;", "limitations": "yes", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 1, "clarity": 3, "significance": 1, "originality": 2, "strengths_and_weaknesses": "Pros:\n1. the paper is easy to follow;\n2. it identifies some interesting limitations of SmoothGrade\n3. it appears to be a practical and versatile XAI tool.\n\nCons:\n1. The evaluation is not sufficent. First, very limited XAI methods are compared. Second, the metrics are insufficient and non-standard. some very common and established faithfulness metrics like Deletion/Insertion score and Pointing Game (for localization accuracy) are missing. Third, it neglects common XAI benchmarks like OpenXAI or Quantus.\n2. The proposed method is very incremental to SmoothGrad without broader XAI implications, thus the novelty and significance of the method is very limited. It reads better suited as a workshop paper rather than NeurIPS, IMHO.\n3. The table 1 shows very limited correlation.", "was_revised": true, "final_justification": "I thank the authors for their reply. My concerns are partially addressed, thus I raised my score from 2 to 3. \n\nI still believe the effectiveness of the work is not clear without comparing to other more general XAI methods in publich benchmarks. The author confine the paper only to \"gradient-smoothing\" XAI which is a very niche topic that is not significant enough for top conferences like NeurIPS (more like a workshop paper IMHO).", "rebuttal": "Dear Reviewer,\n\nWe greatly appreciate your comments and the effort you put into reviewing our work. We have carefully considered your remarks and will respond according to the following points.\n\n## The limitations of comparison with other XAI methods\n\nOur comparative analysis of XAI methods focuses exclusively on established gradient-smoothing baselines. We contend that comparing AdaptGrad with non-gradient-smoothing XAI approaches would be inappropriate, as their underlying principles and implementation mechanisms are fundamentally distinct from AdaptGrad’s core framework.\n\nWhile acknowledging the inherent limitations of our current evaluation scope, we emphasize that the near-universal reliance on gradient information across XAI methodologies renders comprehensive coverage of all XAI techniques practically infeasible within a single study.\n\nOur directly relevant comparisons center on baseline gradient-smoothing works:\n\n- SmoothGrad: The original study limited quantitative comparisons to SmoothGrad and SmoothGrad+Integrated Gradients (IG).\n- NoiseGrad: Evaluated SmoothGrad, NoiseGrad, and their hybrid variant, alongside combinations with Integrated Gradients and Gradient Shap.\n\nThus, our evaluation framework aligns with established methodological conventions in gradient-smoothing research. We maintain that this approach provides a rigorously controlled basis for assessing AdaptGrad’s contributions relative to its conceptual predecessors. In the revised manuscript, we will elaborate on this rationale and explicitly contextualize our comparative strategy within the broader XAI literature.\n\n## The limitations of the metrics\n\nWe would like to clarify that the Information Level score we employ represents a more quantified metric for AUC/deletion & insertion scores. This methodology was first introduced in [1] and has been widely adopted for evaluating saliency maps in Integrated Gradients-based XAI methods. Specifically, it quantifies deletion scores through an information entropy-based approach, calculating the area under the accuracy-information curve derived from entropy values. This formulation effectively circumvents experimental artifacts arising from \"human-selective\" deletion and insertion processes. In summary, the Information Level score constitutes an entropy-based refinement of AUC/deletion & insertion scores. Additionally, the adversarial example generation task presented in our appendix serves as a complementary evaluation to these standard metrics.\n\nRegarding common XAI benchmarks, we acknowledge the current limitations in our evaluation scope. However, we wish to clarify that our evaluation framework is primarily derived from Quantus. The Quantus framework encompasses 27 distinct evaluation metrics, rendering its full implementation computationally prohibitive for our research context. This complexity is further exacerbated when integrating AdaptGrad with other gradient-based XAI methods (e.g., NoiseGrad and Integrated Gradients), which significantly amplifies both evaluation intricacy and computational demands. Consequently, we selectively adopted key metrics from Quantus:\n\n- Input Invariance (termed Invariance in our paper),\n- Completeness (termed Consistency in our paper), and\n- Sparseness from the Complexity category (retained as Sparseness in our paper).\n- For localization-related metrics (e.g., Pointing Game and AUC), we utilized the Information Level Score—a more robust and stable alternative—as our primary evaluation tool.\n\nWe further note that Quantus lacks efficient mechanisms for controlling computational overhead in such assessments. To address this, we independently designed a dedicated Object Detection task to rigorously evaluate AdaptGrad’s performance.\n\nWe recognize that our current evaluation does not incorporate robustness-focused metrics (e.g., those in Quantus’ Robustness category) or benchmarks like OpenXAI. While we acknowledge this as a potential limitation, we maintain that our streamlined evaluation framework—centered on theoretically grounded metrics and task-specific validation—sufficiently comprehensively characterizes AdaptGrad’s capabilities across critical XAI dimensions.\n\n## The limitations of the novelty and significance\n\nThe primary contribution of AdaptGrad lies in proposing a novel gradient smoothing method that effectively eliminates extra noise while demonstrating superior performance across multiple benchmark models. We acknowledge that AdaptGrad builds upon SmoothGrad, but we contend that this advancement constitutes a meaningful innovation. While SmoothGrad achieves notable success in gradient smoothing, it still suffers from residual extra noise. AdaptGrad resolves this issue through the introduction of an adaptive sampling range, thereby significantly enhancing performance across diverse benchmark models.\n\nCritically, we theoretically establish the concept of extra noise for the first time and empirically validate its existence. We assert that this contribution holds substantial significance for the XAI field, as it provides new theoretical foundations and methodological directions for gradient smoothing techniques. Both theoretically and experimentally, our work offers fresh perspectives for XAI research—particularly through the discovery that AdaptGrad can be seamlessly integrated with other gradient-based XAI methods, yielding substantial performance improvements. Consequently, we find it difficult to concur with the characterization of our work as merely \"incremental.\"\n\n## The correlation between extra noise and noise in saliency maps\n\nWhat we need to clarify is that the correlations in Table 1 were assessed using a t-test. The Sparseness metric itself does not directly measure the amount of noise. Therefore, the correlation values in Table 1 do not directly indicate the relationship between the additional noise and the noise in the significance map. However, they can indirectly reflect this relationship based on whether they pass the t-test. If the p-value is less than 0.10, it can be inferred that there is a significant correlation between the additional noise and the noise in the significance map.\n\n[1] A. Kapishnikov, T. Bolukbasi, F. Viegas, and M. Terry, “XRAI: Better Attributions Through Regions,” in 2019 IEEE/CVF International Conference on Computer Vision (ICCV), Oct. 2019, pp. 4947–4956. doi: 10.1109/ICCV.2019.00505."}]

Author rebuttal:

Meta-review: The paper proposes AdaptGrad, an improved version of SmoothGrad, where the noise level is adapted for each input dimension.  Experimental results show that AdaptGrad significantly improves the explainability of SmoothGrad.

Reviewers acknowledged significant contributions in improving the gradient-based explanation methods with a simple idea, but also raised several concerns, including clarity issues, limited experiment, missing criteria, and missing baselines.  The authors' rebuttal addressed those concerns with promises to include additional experimental results.  

This is a borderline paper, and the reviewer-AC discussion concluded that, although the scope of the paper is niche, the proposed method is useful for many people who use gradient-based explanation methods.  The main experiment comparing the proposed method with the other gradient-based methods is fine, but the performance comparison with other types of sota explanation methods is also important.  The proposed method may not be sota in terms of the explanation quality.  Then, the authors should discuss why the proposed method is still important and useful.
Decision: Accept (poster)