SmoothGrad: removing noise by adding noise

Daniel SmilkovNikhil ThoratBeen KimFernanda ViégasMartin Wattenberg

article2017ICML2,724 citations

Proposes SmoothGrad, a simple method that cleans up noisy gradient-based sensitivity maps by averaging the gradients of randomly perturbed inputs, producing clearer visual explanations of deep learning predictions.

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The article addresses the challenge of interpreting decisions made by deep neural networks for image classification. Gradient-based sensitivity maps, which highlight influential pixels, often appear visually noisy and fail to clearly align with human-perceived meaningful regions. This limits their usefulness for debugging models or gaining insight in high-stakes domains such as health care.

The article set out to evaluate a simple technique called SmoothGrad that reduces this noise by averaging sensitivity maps computed on multiple slightly perturbed versions of the input image. It also examined the combined effects of this approach with other gradient-based methods and with adding noise during model training.

The authors conducted qualitative experiments using an Inception v3 model trained on ImageNet and a convolutional network on MNIST. They compared SmoothGrad against vanilla gradients, Integrated Gradients, and Guided Backpropagation across hundreds of images, varying noise levels and sample sizes, and assessed visual coherence and the ability to discriminate between multiple objects in a scene.

SmoothGrad produced noticeably more coherent and less noisy maps than vanilla gradients and Integrated Gradients, with the strongest improvements on images having uniform backgrounds. It also enhanced discriminativity between competing classes. Adding noise during training further sharpened maps, and the two smoothing techniques together yielded the best results. Applying SmoothGrad on top of Guided Backpropagation or Integrated Gradients improved those methods as well.

These findings indicate that much of the apparent noise in gradient maps stems from local fluctuations in the class-score function rather than from the model's actual decision process. Smoother maps can therefore give stakeholders a more reliable view of what drives classifications, supporting better model validation and refinement without requiring new architectures.

Practitioners should consider applying SmoothGrad as a post-processing step for any gradient-based explanation method and explore training with noise when map legibility is a priority. Further work is needed to develop quantitative metrics for map quality, test generalization beyond image classification, and investigate whether explicit penalties on derivative variation during training can produce even smoother explanations.

Google svm visualization SmoothGrad Sensitivity Map Gradient

arXiv: 1706.03825 PAIR-code/saliency

Paper: Axiomatic Attribution for Deep Networks, Mukund Sundararajan et al. (2017). Introduces Integrated Gradients, one of the primary gradient-based baseline methods directly benchmarked and enhanced by SmoothGrad.
Paper: Deep Inside Convolutional Networks: Visualising Image Classification Models and Saliency Maps, Karen Simonyan et al. (2013). Establishes foundational vanilla gradient saliency maps for convolutional networks, which form the direct baseline and subject of noise-reduction in SmoothGrad.
Paper: Grad-CAM: Visual Explanations from Deep Networks via Gradient-Based Localization, Ramprasaath R. Selvaraju et al. (2016). Introduces Grad-CAM and highlights the role of Guided Backpropagation in gradient-based visual explanations, providing essential context for the methods SmoothGrad modifies.
Paper: Visualizing and Understanding Convolutional Networks, Matthew D. Zeiler et al. (2014). Provides fundamental background on interpreting deep convolutional representations in pixel space using backpropagation-based visualization.
Paper: Learning Deep Features for Discriminative Localization, Bolei Zhou et al. (2016). Pioneers Class Activation Mapping for discriminative visual localization, establishing the paradigm of attributing image classification decisions to visual regions.
Paper: Learning Important Features Through Propagating Activation Differences, Avanti Shrikumar et al. (2017). Develops DeepLIFT as a backpropagation-based attribution method addressing gradient saturation, serving as standard comparative background for neural interpretability.
Paper: The Mythos of Model Interpretability, Zachary C. Lipton (2016). Clarifies the theoretical terminology and underlying goals of post-hoc interpretability methods such as saliency maps.

Paper: Sanity Checks for Saliency Maps, Julius Adebayo et al. (2018). Conducts crucial sanity checks on popular saliency methods—including gradients and Integrated Gradients—critically evaluating whether smoothed visual maps faithfully reflect model parameters.
Paper: Grad-CAM++: Improved Visual Explanations for Deep Convolutional Networks, Aditya Chattopadhyay et al. (2017). Extends gradient-based visual explanations using higher-order derivatives to produce sharper, class-discriminative saliency maps for multi-object scenes.
Paper: Methods for interpreting and understanding deep neural networks, Grégoire Montavon et al. (2018). Provides a comprehensive taxonomy and formal quality metrics—such as continuity and selectivity—to systematically compare gradient sensitivity analysis against decomposition techniques.
Paper: Interpretability Beyond Feature Attribution: Quantitative Testing with Concept Activation Vectors (TCAV), Been Kim et al. (2018). Moves beyond local pixel-level gradient attribution methods to quantify model sensitivity to human-understandable high-level concepts.
Paper: A Unified Approach to Interpreting Model Predictions, Scott M. Lundberg et al. (2017). Presents SHAP, establishing a unified game-theoretic framework that connects additive attribution techniques with gradient and difference-from-reference methods.
Paper: Explaining Explanations: An Overview of Interpretability of Machine Learning, Leilani H. Gilpin et al. (2018). Surveys the broader interpretability literature and contextualizes saliency map smoothing within a formal framework of explainability versus interpretability.
Paper: GNNExplainer: Generating Explanations for Graph Neural Networks, Rex Ying et al. (2019). Generalizes attribution-based explanation techniques from image grid domains to relational graph neural networks.
Paper: The Dead Salmons of AI Interpretability, Maxime Méloux et al. (2025). Examines non-identifiability and statistical fragility across post-hoc feature attribution methods, evaluating the theoretical limits of visual saliency explanations.

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Abstract

Explaining the output of a deep network remains a challenge. In the case of an image classifier, one type of explanation is to identify pixels that strongly influence the final decision. A starting point for this strategy is the gradient of the class score function with respect to the input image. This gradient can be interpreted as a sensitivity map, and there are several techniques that elaborate on this basic idea. This paper makes two contributions: it introduces SmoothGrad, a simple method that can help visually sharpen gradient-based sensitivity maps, and it discusses lessons in the visualization of these maps. We publish the code for our experiments and a website with our results.

Topics

svm visualization

Keywords

SmoothGrad Sensitivity Map Gradient Gaussian Noise Image Classification Integrated Gradient Guided Backpropagation Inception MNIST Training With Noise

1 Introduction
2 Gradients as sensitivity maps
2.1 Previous work on enhancing sensitivity maps
2.2 Smoothing noisy gradients
3 Experiments
3.1 Visualization methods and techniques
3.2 Effect of noise level and sample size
3.3 Qualitative comparison to baseline methods
3.4 Combining SmoothGrad with other methods
3.5 Adding noise during training
4 Conclusion and future work
References

Knowls

Knowl 1 — SmoothGrad Algorithm
algorithm
SmoothGrad computes an enhanced sensitivity map by stochastically approximating the Gaussian smoothing of a gradient-based sensitivity function in the neighborhood of an input image. Given an input image $x \in \mathbb{R}^d$ and a class score function $S_c(x)$ for class $c$ , a base sensitivity map $M_c(x)$ is typically defined as the gradient $\partial S_c(x) / \partial x$ (or another gradient-based attribution method). SmoothGrad estimates the locally smoothed sensitivity map $\hat{M}_c(x)$ via Monte Carlo sampling:

$\hat{M}_c(x) = \frac{1}{n} \sum_{i=1}^n M_c\left(x + \mathcal{N}(0, \sigma^2 I)\right)$

where $n$ is the number of samples and $\sigma$ is the noise standard deviation, typically chosen relative to the image dynamic range as a fraction $\sigma / (x_{\max} - x_{\min})$ .
Input: Input image $x$ , base sensitivity function $M_c$ , noise level $\sigma$ , sample count $n$
Output: Smoothed sensitivity map $\hat{M}_c(x)$
Initialize accumulator map $A \leftarrow 0$
for $i \leftarrow 1$ to $n$ do
    Sample noise vector $\epsilon \sim \mathcal{N}(0, \sigma^2 I)$ of the same dimension as $x$
    Compute perturbed image $\tilde{x} \leftarrow x + \epsilon$
    Compute base sensitivity map $M \leftarrow M_c(\tilde{x})$
     $A \leftarrow A + M$
end for
return $\hat{M}_c(x) \leftarrow \frac{1}{n} A$
Knowl 2 — Local Gradient Fluctuations as the Cause of Saliency Noise
empirical result
In deep neural networks with piecewise linear activation functions (such as ReLU), the class score function $S_c(x)$ is not continuously differentiable, and its gradient $\partial S_c(x) / \partial x$ fluctuates sharply at small spatial scales. When moving along a short trajectory $x + t\epsilon$ parameterized by $t \in [0, 1]$ , where $\epsilon \sim \mathcal{N}(0, 0.01^2 I)$ is a imperceptible Gaussian noise vector such that classification remains unchanged and $x + \epsilon$ is indistinguishable from $x$ to a human, the partial derivatives $\partial S_c(x + t\epsilon) / \partial x_i$ for individual pixel channels fluctuate by significant fractions of the maximum gradient entry $\max_i \partial S_c / \partial x_i$ . Consequently, raw gradients are noisy local point samples, and a local average over a Gaussian neighborhood provides a more meaningful attribution map.
Knowl 3 — Hyperparameter Calibration for Noise Scale and Sample Count in SmoothGrad
empirical result
SmoothGrad depends on two hyperparameters: the noise ratio $\sigma / (x_{\max} - x_{\min})$ (the standard deviation of Gaussian perturbations divided by the dynamic range of the input) and the sample count $n$ .
- Noise level $\sigma$ : Setting $\sigma / (x_{\max} - x_{\min})$ between $10\%$ and $20\%$ balances the removal of background noise against preserving the structural features of the underlying object on deep vision models such as Inception v3 on ImageNet. Setting noise too low retains high-frequency noise, while setting it too high degrades object contours.
- Sample count $n$ : Increasing $n$ yields progressively smoother sensitivity maps, but visual improvements exhibit diminishing returns beyond $n = 50$ samples.
Knowl 4 — Combining SmoothGrad with Other Attribution Methods
model/method
SmoothGrad operates as a meta-method that can wrap any gradient-based attribution technique. When applied to Integrated Gradients or Guided Backpropagation, SmoothGrad substitutes the base attribution calculation $M_c(x)$ with an ensemble average of that attribution method applied across $n$ noise-perturbed inputs $x + \mathcal{N}(0, \sigma^2 I)$ . In both cases, this smoothing substantially reduces visual noise and sharpens object boundaries compared to using Integrated Gradients or Guided Backpropagation alone.
Knowl 5 — Additive Denoising Effect of Training with Noise and SmoothGrad Inference
empirical result
Regularizing deep neural networks during training by adding Gaussian noise to training samples induces smoother class score functions, which inherently reduces noise in post-hoc sensitivity maps. Furthermore, training with input noise and applying inference-time smoothing via SmoothGrad have an additive effect: models trained with noise and evaluated using SmoothGrad produce the sharpest and most visually coherent sensitivity maps compared to applying either technique alone or applying neither.
Knowl 6 — Class Discriminativity in Saliency Maps
empirical result
For images containing multiple objects belonging to distinct classes $c_1$ and $c_2$ , the class discriminativity of a sensitivity map method can be evaluated by normalizing the two class maps to $[0, 1]$ and computing their difference:

$\Delta M(x) = \text{scale}(M_{c_1}(x)) - \text{scale}(M_{c_2}(x))$

visualized on a diverging color scale $[-1, 1] \mapsto [\text{blue}, \text{gray}, \text{red}]$ . SmoothGrad produces high visual discriminativity, assigning distinctly positive values to the regions corresponding to class $c_1$ and negative values to regions corresponding to class $c_2$ . In contrast, methods such as Guided Backpropagation exhibit weak discriminativity despite producing sharp visual features.
Knowl 7 — Gradient Outlier Capping in Sensitivity Heatmap Visualization
model/method
Raw gradient maps typically contain a small number of outlier pixels with partial derivatives significantly higher in magnitude than the mean gradient. If heatmap color scales are linearly mapped from the minimum to maximum gradient value, these extreme outliers cause the visualization to render almost entirely black. Capping gradient magnitudes at a high percentile threshold (e.g., the 99th percentile) before normalising to $[0, 1]$ prevents scale compression and yields visually interpretable heatmaps.
Knowl 8 — Dataset-Dependent Selection of Signed vs Absolute Gradient Visualizations
model/method
The choice between visualizing signed gradients $\partial S_c(x) / \partial x$ or absolute gradients $|\partial S_c(x) / \partial x|$ depends on the contrast properties of the dataset:
- In datasets with fixed contrast polarity across classes (e.g., MNIST, where digits are uniformly white on a black background), positive gradients specifically denote supportive evidence for the target class, making signed visualizations appropriate.
- In natural image datasets (e.g., ImageNet), visual classification is invariant to illumination and background contrast changes (e.g., a dark object on a bright background yields negative gradients, whereas a bright object on a dark background yields positive gradients). Taking the absolute value of the gradient eliminates polarity artifacts and produces clearer object attribution.
Knowl 9 — Trade-offs of Gradient Multiplied by Input Visualization
model/method
Multiplying gradient sensitivity maps point-wise by the input image values ( $x \odot M_c(x)$ ) produces visually sharper maps by incorporating input structure, analogous to computing feature contributions $x_i w_i$ in linear models $y = Wx$ . However, this multiplication has structural limitations: pixels with a value of zero are strictly mapped to zero attribution regardless of their gradient magnitude (preventing the identification of dark objects on light backgrounds), and high-contrast edges in the input image can introduce sharp visual contours into the attribution map even when the underlying sensitivity function contains no edge information.

Coverage note — None was omitted; all key contributions, algorithms, visualization lessons, empirical findings, and hypotheses from the paper are represented.

References

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Citation

MLA

Smilkov, D., et al. “SmoothGrad: Removing Noise by Adding Noise”. arXiv, 2017, https://doi.org/10.48550/arxiv.1706.03825.

APA

Smilkov, D., Thorat, N., Kim, B., Viégas, F., & Wattenberg, M. (2017). SmoothGrad: removing noise by adding noise. arXiv. https://doi.org/10.48550/arxiv.1706.03825

Chicago

Smilkov, D., N. Thorat, B. Kim, F. Viégas, and M. Wattenberg. 2017. “SmoothGrad: Removing Noise by Adding Noise”. Preprint, ArXiv. https://doi.org/10.48550/arxiv.1706.03825.

Harvard

Smilkov, D. et al. (2017) “SmoothGrad: removing noise by adding noise”. arXiv. Available at: https://doi.org/10.48550/arxiv.1706.03825.

Vancouver

1. Smilkov D, Thorat N, Kim B, Viégas F, Wattenberg M (2017) SmoothGrad: removing noise by adding noise. https://doi.org/10.48550/arxiv.1706.03825

BibTeX

@misc{https://doi.org/10.48550/arxiv.1706.03825,
  doi = {10.48550/ARXIV.1706.03825},
  url = {https://arxiv.org/abs/1706.03825},
  author = {Smilkov, Daniel and Thorat, Nikhil and Kim, Been and Viégas, Fernanda and Wattenberg, Martin},
  keywords = {Machine Learning (cs.LG), Computer Vision and Pattern Recognition (cs.CV), Machine Learning (stat.ML), FOS: Computer and information sciences, FOS: Computer and information sciences},
  title = {SmoothGrad: removing noise by adding noise},
  publisher = {arXiv},
  year = {2017},
  copyright = {arXiv.org perpetual, non-exclusive license}
}

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SmoothGrad: removing noise by adding noise

Abstract

Topics

Keywords

Table of Contents

Knowls

Knowl 1 — SmoothGrad Algorithm

Knowl 2 — Local Gradient Fluctuations as the Cause of Saliency Noise

Knowl 3 — Hyperparameter Calibration for Noise Scale and Sample Count in SmoothGrad

Knowl 4 — Combining SmoothGrad with Other Attribution Methods

Knowl 5 — Additive Denoising Effect of Training with Noise and SmoothGrad Inference

Knowl 6 — Class Discriminativity in Saliency Maps

Knowl 7 — Gradient Outlier Capping in Sensitivity Heatmap Visualization

Knowl 8 — Dataset-Dependent Selection of Signed vs Absolute Gradient Visualizations

Knowl 9 — Trade-offs of Gradient Multiplied by Input Visualization

References

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