arXiv:1704.03296v3 [] 10 Jan 2018

Interpretable Explanations of Black Boxes by Meaningful Perturbation
Ruth C. Fong
University of Oxford

Andrea Vedaldi
University of Oxford

ruthfong@robots.ox.ac.uk

vedaldi@robots.ox.ac.uk

flute: 0.9973

Abstract
As machine learning algorithms are increasingly applied
to high impact yet high risk tasks, such as medical diagnosis or autonomous driving, it is critical that researchers
can explain how such algorithms arrived at their predictions. In recent years, a number of image saliency methods
have been developed to summarize where highly complex
neural networks “look” in an image for evidence for their
predictions. However, these techniques are limited by their
heuristic nature and architectural constraints.
In this paper, we make two main contributions: First, we
propose a general framework for learning different kinds
of explanations for any black box algorithm. Second, we
specialise the framework to find the part of an image most
responsible for a classifier decision. Unlike previous works,
our method is model-agnostic and testable because it is
grounded in explicit and interpretable image perturbations.

flute: 0.0007

Learned Mask

Figure 1. An example of a mask learned (right) by blurring an
image (middle) to suppress the softmax probability of its target
class (left: original image; softmax scores above images).

be answered by providing interpretable rules that describe
the input-output relationship captured by f . For example,
one rule could be that f is rotation invariant, in the sense
that “f (x) = f (x0 ) whenever images x and x0 are related
by a rotation”.
In this paper, we make several contributions. First, we
propose the general framework of explanations as metapredictors (sec. 2), extending [18]’s work. Second, we identify several pitfalls in designing automatic explanation systems. We show in particular that neural network artifacts
are a major attractor for explanations. While artifacts are
informative since they explain part of the network behavior, characterizing other properties of the network requires
careful calibration of the generality and interpretability of
explanations. Third, we reinterpret network saliency in our
framework. We show that this provides a natural generalization of the gradient-based saliency technique of [15] by
integrating information over several rounds of backpropagation in order to learn an explanation. We also compare
this technique to other methods [15, 16, 20, 14, 19] in terms
of their meaning and obtained results.

1. Introduction
Given the powerful but often opaque nature of modern black box predictors such as deep neural networks [4,
5], there is a considerable interest in explaining and understanding predictors a-posteriori, after they have been
learned. This remains largely an open problem. One
reason is that we lack a formal understanding of what it
means to explain a classifier. Most of the existing approaches [19, 16, 8, 7, 9, 19], etc., often produce intuitive
visualizations; however, since such visualizations are primarily heuristic, their meaning remains unclear.
In this paper, we revisit the concept of “explanation” at
a formal level, with the goal of developing principles and
methods to explain any black box function f , e.g. a neural
network object classifier. Since such a function is learned
automatically from data, we would like to understand what
f has learned to do and how it does it. Answering the
“what” question means determining the properties of the
map. The “how” question investigates the internal mechanisms that allow the map to achieve these properties. We
focus mainly on the “what” question and argue that it can

2. Related work
Our work builds on [15]’s gradient-based method, which
backpropagates the gradient for a class label to the image layer. Other backpropagation methods include DeConvNet [19] and Guided Backprop [16, 8], which builds off
of DeConvNet [19] and [15]’s gradient method to produce
sharper visualizations.
Another set of techniques incorporate network activations into their visualizations: Class Activation Mapping
1

(CAM) [22] and its relaxed generalization Grad-CAM [14]
visualize the linear combination of a late layer’s activations
and class-specific weights (or gradients for [14]), while
Layer-Wise Relevance Propagation (LRP) [1] and Excitation Backprop [20] backpropagate an class-specific error
signal though a network while multiplying it with each convolutional layer’s activations.
With the exception of [15]’s gradient method, the above
techniques introduce different backpropagation heuristics,
which results in aesthetically pleasing but heuristic notions
of image saliency. They also are not model-agnostic, with
most being limited to neural networks (all except [15, 1])
and many requiring architectural modifications [19, 16, 8,
22] and/or access to intermediate layers [22, 14, 1, 20].
A few techniques examine the relationship between inputs and outputs by editing an input image and observing
its effect on the output. These include greedily graying out
segments of an image until it is misclassified [21] and visualizing the classification score drop when an image is occluded at fixed regions [19]. However, these techniques are
limited by their approximate nature; we introduce a differentiable method that allows for the effect of the joint inclusion/exclusion of different image regions to be considered.
Our research also builds on the work of [18, 12, 2]. The
idea of explanations as predictors is inspired by the work
of [18], which we generalize to new types of explanations,
from classification to invariance.
The Local Intepretable Model-Agnostic Explanation
(LIME) framework [12] is relevant to our local explanation
paradigm and saliency method (sections 3.2, 4) in that both
use an function’s output with respect to inputs from a neighborhood around an input x0 that are generated by perturbing the image. However, their method takes much longer to
converge (N = 5000 vs. our 300 iterations) and produces a
coarse heatmap defined by fixed super-pixels.
Similar to how our paradigm aims to learn an image perturbation mask that minimizes a class score, feedback networks [2] learn gating masks after every ReLU in a network to maximize a class score. However, our masks are
plainly interpretable as they directly edit the image while
[2]’s ReLU gates are not and can not be directly used as a
visual explanation; furthermore, their method requires architectural modification and may yield different results for
different networks, while ours is model-agnostic.

3. Explaining black boxes with meta-learning
A black box is a map f : X → Y from an input
space X to an output space Y, typically obtained from an
opaque learning process. To make the discussion more concrete, consider as input color images x : Λ → R3 where
Λ = {1, . . . , H} × {1, . . . , W } is a discrete domain. The
output y ∈ Y can be a boolean {−1, +1} telling whether
the image contains an object of a certain type (e.g. a robin),

the probability of such an event, or some other interpretation of the image content.

3.1. Explanations as meta-predictors
An explanation is a rule that predicts the response of a
black box f to certain inputs. For example, we can explain a behavior of a robin classifier by the rule Q1 (x; f ) =
{x ∈ Xc ⇔ f (x) = +1}, where Xc ⊂ X is the subset of all the robin images. Since f is imperfect, any such
rule applies only approximately. We can measure the faithfulness of the explanation as its expected prediction error:
L1 = E[1 − δQ1 (x;f ) ], where δQ is the indicator function
of event Q. Note that Q1 implicitly requires a distribution
p(x) over possible images X . Note also that L1 is simply
the expected prediction error of the classifier. Unless we did
not know that f was trained as a robin classifier, Q1 is not
very insightful, but it is interpretable since Xc is.
Explanations can also make relative statements about
black box outcomes. For example, a black box f , could
be rotation invariant: Q2 (x, x0 ; f ) = {x ∼rot x0 ⇒ f (x) =
f (x0 )}, where x ∼rot x0 means that x and x0 are related by
a rotation. Just like before, we can measure the faithfulness
of this explanation as L2 = E[1−δQ2 (x,x0 ;f ) |x ∼ x0 ].1 This
rule is interpretable because the relation ∼rot is.
Learning explanations. A significant advantage of formulating explanations as meta predictors is that their faithfulness can be measured as prediction accuracy. Furthermore, machine learning algorithms can be used to discover
explanations automatically, by finding explanatory rules Q
that apply to a certain classifier f out of a large pool of possible rules Q.
In particular, finding the most accurate explanation Q is
similar to a traditional learning problem and can be formulated computationally as a regularized empirical risk minimization such as:
n

min λR(Q) +

Q∈Q

1X
L(Q(xi ; f ), xi , f ), xi ∼ p(x). (1)
n i=1

Here, the regularizer R(Q) has two goals: to allow the explanation Q to generalize beyond the n samples x1 , . . . , xn
considered in the optimization and to pick an explanation Q
which is simple and thus, hopefully, more interpretable.
Maximally informative explanations. Simplicity and interpretability are often not sufficient to find good explanations and must be paired with informativeness. Consider
the following variant of rule Q2 : Q3 (x, x0 ; f, θ) = {x ∼θ
x0 ⇒ f (x) = f (x0 )}, where x ∼θ x0 means that x and x0
1 For rotation invariance we condition on x ∼ x0 because the probability of independently sampling rotated x and x0 is zero, so that, without
conditioning, Q2 would be true with probability 1.

chocolate sauce

mask

gradient

guided

contrast excitation

grad-CAM

occlusion

unicycle

impala

squirrel monkey sunglasses

CD player

pickup

Komodo dragon

street sign

cliff

Pekinese

orig img + gt bb

Figure 2. Comparison with other saliency methods. From left to right: original image with ground truth bounding box, learned mask subtracted from 1 (our method), gradient-based saliency [15], guided backprop [16, 8], contrastive excitation backprop [20], Grad-CAM [14],
and occlusion [19].

Stethoscope

Gradient

Soup Bowl

blur

Gradient

Figure 3. Gradient saliency maps of [15]. A red bounding box
highlight the object which is meant to be recognized in the image.
Note the strong response in apparently non-relevant image regions.

are related by a rotation of an angle ≤ θ. Explanations for
larger angles imply the ones for smaller ones, with θ = 0
being trivially satisfied. The regularizer R(Q3 (·; θ)) = −θ
can then be used to select a maximal angle and thus find an
explanation that is as informative as possible.2

3.2. Local explanations
A local explanation is a rule Q(x; f, x0 ) that predicts the
response of f in a neighborhood of a certain point x0 . If f
is smooth at x0 , it is natural to construct Q by using the
first-order Taylor expansion of f :
f (x) ≈ Q(x; f, x0 ) = f (x0 ) + h∇f (x0 ), x − x0 i.

(2)

This formulation provides an interpretation of [15]’s
saliency maps, which visualize the gradient S1 (x0 ) =
∇f (x0 ) as an indication of salient image regions. They
argue that large values of the gradient identify pixels that
strongly affect the network output. However, an issue is
that this interpretation breaks for a linear classifier: If
f (x) = hw, xi + b, S1 (x0 ) = ∇f (x0 ) = w is independent
of the image x0 and hence cannot be interpreted as saliency.
The reason for this failure is that eq. (2) studies the variation of f for arbitrary displacements ∆x = x − x0 from x0
and, for a linear classifier, the change is the same regardless
of the starting point x0 . For a non-linear black box f such
as a neural network, this problem is reduced but not eliminated, and can explain why the saliency map S1 is rather
diffuse, with strong responses even where no obvious information can be found in the image (fig. 3).
We argue that the meaning of explanations depends in
large part on the meaning of varying the input x to the
black box. For example, explanations in sec. 3.1 are based
on letting x vary in image category or in rotation. For
saliency, one is interested in finding image regions that
impact f ’s output. Thus, it is natural to consider perturbations x obtained by deleting subregions of x0 . If we
model deletion by multiplying x0 point-wise by a mask m,
2 Naively, strict invariance for any θ > 0 implies invariance to arbitrary
rotations as small rotations compose into larger ones. However, the formulation can still be used to describe rotation insensitivity (when f varies
slowly with rotation), or ∼θ ’s meaning can be changed to indicate rotation
w.r.t. a canonical “upright” direction for a certain object classes, etc.

constant

noise

Figure 4. Perturbation types. Bottom: perturbation mask; top: effect of blur, constant, and noise perturbations.

this amounts to studying the function f (x0 m)3 . The
Taylor expansion of f at m = (1, 1, . . . , 1) is S2 (x0 ) =
df (x0 m)/dm|m=(1,...,1) = ∇f (x0 ) x0 . For a linear
classifier f , this results in the saliency S2 (x0 ) = w x0 ,
which is large for pixels for which x0 and w are large simultaneously. We refine this idea for non-linear classifiers
in the next section.

4. Saliency revisited
4.1. Meaningful image perturbations
In order to define an explanatory rule for a black box
f (x), one must start by specifying which variations of the
input x will be used to study f . The aim of saliency is
to identify which regions of an image x0 are used by the
black box to produce the output value f (x0 ). We can do
so by observing how the value of f (x) changes as x is
obtained “deleting” different regions R of x0 . For example, if f (x0 ) = +1 denotes a robin image, we expect that
f (x) = +1 as well unless the choice of R deletes the robin
from the image. Given that x is a perturbation of x0 , this is
a local explanation (sec. 3.2) and we expect the explanation
to characterize the relationship between f and x0 .
While conceptually simple, there are several problems
with this idea. The first one is to specify what it means
“delete” information. As discussed in detail in sec. 4.3, we
are generally interested in simulating naturalistic or plausible imaging effect, leading to more meaningful perturbations and hence explanations. Since we do not have access
to the image generation process, we consider three obvious
proxies: replacing the region R with a constant value, injecting noise, and blurring the image (fig. 4).
Formally, let m : Λ → [0, 1] be a mask, associating each
pixel u ∈ Λ with a scalar value m(u). Then the perturbation
operator is defined as


constant,
m(u)x0 (u) + (1 − m(u))µ0 ,
[Φ(x0 ; m)](u) = m(u)x0 (u) + (1 − m(u))η(u), noise,

R
gσ0 m(u) (v − u)x0 (v) dv,
blur,
where µ0 is an average color, η(u) are i.i.d. Gaussian noise
samples for each pixel and σ0 is the maximum isotropic
3

is the Hadamard or element-wise product of vectors.

standard deviation of the Gaussian blur kernel gσ (we use
σ0 = 10, which yields a significantly blurred image).

espresso: 0.9964

espresso: 0.0000

Learned Mask

maypole: 0.9568

maypole: 0.0000

Learned Mask

4.2. Deletion and preservation
Given an image x0 , our goal is to summarize compactly
the effect of deleting image regions in order to explain the
behavior of the black box. One approach to this problem is
to find deletion regions that are maximally informative.
In order to simplify the discussion, in the rest of the paper we consider black boxes f (x) ∈ RC that generate a
vector of scores for different hypotheses about the content
of the image (e.g. as a softmax probability layer in a neural
network). Then, we consider a “deletion game” where the
goal is to find the smallest deletion mask m that causes the
score fc (Φ(x0 ; m))  fc (x0 ) to drop significantly, where
c is the target class. Finding m can be formulated as the
following learning problem:
m∗ = argmin λk1 − mk1 + fc (Φ(x0 ; m))

(3)

m∈[0,1]Λ

where λ encourages most of the mask to be turned off
(hence deleting a small subset of x0 ). In this manner, we
can find a highly informative region for the network.
One can also play an symmetric “preservation game”,
where the goal is to find the smallest subset of the image
that must be retained to preserve the score fc (Φ(x0 ; m)) ≥
fc (x0 ): m∗ = argminm λkmk1 − fc (Φ(x0 ; m)). The main
difference is that the deletion game removes enough evidence to prevent the network from recognizing the object in
the image, whereas the preservation game finds a minimal
subset of sufficient evidence.
Iterated gradients. Both optimization problems are
solved by using a local search by means of gradient descent
methods. In this manner, our method extracts information
from the black box f by computing its gradient, similar to
the approach of [15]. However, it differs in that it extracts
this information progressively, over several gradient evaluations, accumulating increasingly more information over
time.

4.3. Dealing with artifacts
By committing to finding a single representative perturbation, our approach incurs the risk of triggering artifacts
of the black box. Neural networks, in particular, are known
to be affected by surprising artifacts [5, 10, 7]; these works
demonstrate that it is possible to find particular inputs that
can drive the neural network to generate nonsensical or unexpected outputs. This is not entirely surprising since neural networks are trained discriminatively on natural image
statistics. While not all artifacts look “unnatural”, nevertheless they form a subset of images that is sampled with negligible probability when the network is operated normally.

Figure 5. From left to right: an image correctly classified with
large confidence by GoogLeNet [17]; a perturbed image that is
not recognized correctly anymore; the deletion mask learned with
artifacts. Top: A mask learned by minimizing the top five predicted classes by jointly applying the constant, random noise, and
blur perturbations. Note that the mask learns to add highly structured swirls along the rim of the cup (γ = 1, λ1 = 10−5 , λ2 =
10−3 , β = 3). Bottom: A minimizing-top5 mask learned by applying a constant perturbation. Notice that the mask learns to introduce sharp, unnatural artifacts in the sky instead of deleting the
pole (γ = 0.1, λ1 = 10−4 , λ2 = 10−2 , β = 3).

Although the existence and characterization of artifacts
is an interesting problem per se, we wish to characterize
the behavior of black boxes under normal operating conditions. Unfortunately, as illustrated in fig. 5, objectives
such as eq. (3) are strongly attracted by such artifacts, and
naively learn subtly-structured deletion masks that trigger
them. This is particularly true for the noise and constant
perturbations as they can more easily than blur create artifacts using sharp color contrasts (fig. 5, bottom row).
We suggests two approaches to avoid such artifacts in
generating explanations. The first one is that powerful
explanations should, just like any predictor, generalize as
much as possible. For the deletion game, this means not relying on the details of a singly-learned mask m. Hence, we
reformulate the problem to apply the mask m stochastically,
up to small random jitter.
Second, we argue that masks co-adapted with network
artifacts are not representative of natural perturbations. As
noted before, the meaning of an explanation depends on the
meaning of the changes applied to the input x; to obtain a
mask more representative of natural perturbations we can
encourage it to have a simple, regular structure which cannot be co-adapted to artifacts. We do so by regularizing m
in total-variation (TV) norm and upsampling it from a low
resolution version.
With these two modifications, eq. (3) becomes:
min λ1 k1 − mk1 + λ2

m∈[0,1]Λ

X

k∇m(u)kββ

u∈Λ

+ Eτ [fc (Φ(x0 (· − τ ), m))], (4)

0.610 => 0.351

0.610 => 0.015

chocolate sauce

Mask Overlay

0.717 => 0.850

0.717 => 0.018

pickup

Mask Overlay

Figure 6. Interrogating suppressive effects. Left to right: original
image with the learned mask overlaid; a boxed perturbation chosen
out of interest (the truck’s middle bounding box was chosen based
on the contrastive excitation backprop heatmap from fig. 2, row
6); another boxed perturbation based on the learned mask (target
softmax probabilities of for the original and perturbed images are
listed above).

P
where M (v) = u gσm (v/s − u)m(u). is the upsampled
mask and gσm is a 2D Gaussian kernel. Equation (4) can be
optimized using stochastic gradient descent.
Implementation details. Unless otherwise specified, the
visualizations shown were generated using Adam [3] to
minimize GoogLeNet’s [17] softmax probability of the target class by using the blur perturbation with the following
parameters: learning rate γ = 0.1, N = 300 iterations,
λ1 = 10−4 , λ2 = 10−2 , β = 3, upsampling a mask (28×28
for GoogLeNet) by a factor of δ = 8, blurring the upsampled mask with gσm =5 , and jittering the mask by drawing
an integer from the discrete uniform distribution on [0, τ )
where τ = 4. We initialize the mask as the smallest centered circular mask that suppresses the score of the original
image by 99% when compared to that of the fully perturbed
image, i.e. a fully blurred image.

5. Experiments
5.1. Interpretability
An advantage of the proposed framework is that the generated visualizations are clearly interpretable. For example,
the deletion game produces a minimal mask that prevents
the network from recognizing the object.
When compared to other techniques (fig. 2), this method
can pinpoint the reason why a certain object is recognized
without highlighting non-essential evidence. This can be
noted in fig. 2 for the CD player (row 7) where other visualizations also emphasize the neighboring speakers, and
similarly for the cliff (row 3), the street sign (row 4), and
the sunglasses (row 8). Sometimes this shows that only a
part of an object is essential: the face of the Pekenese dog
(row 2), the upper half of the truck (row 6), and the spoon
on the chocolate sauce plate (row 1) are all found to be minimally sufficient parts.
While contrastive excitation backprop generated

heatmaps that were most similar to our masks, our method
introduces a quantitative criterion (i.e., maximally suppressing a target class score), and its verifiable nature (i.e.,
direct edits to an image), allows us to compare differing
proposed saliency explanations and demonstrate that our
learned masks are better on this metric. In fig. 6, row 2,
we show that applying a bounded perturbation informed
by our learned mask significantly suppresses the truck
softmax score, whereas a boxed perturbation on the truck’s
back bumper, which is highlighted by contrastive excitation
backprop in fig. 2, row 6, actually increases the score from
0.717 to 0.850.
The principled interpretability of our method also allows
us to identify instances when an algorithm may have learned
the wrong association. In the case of the chocolate sauce
in fig. 6, row 1, it is surprising that the spoon is highlighted
by our learned mask, as one might expect the sauce-filled jar
to be more salient. However, manually perturbing the image reveals that indeed the spoon is more suppressive than
the jar. One explanation is that the ImageNet “chocolate
sauce” images contain more spoons than jars, which appears to be true upon examining some images. More generally, our method allows us to diagnose highly-predictive yet
non-intuitive and possibly misleading correlations by identified machine learning algorithms in the data.

5.2. Deletion region representativeness
To test that our learned masks are generalizable and robust against artifacts, we simplify our masks by further
blurring them and then slicing them into binary masks by
thresholding the smoothed masks by α ∈ [0 : 0.05 : 0.95]
(fig. 7, top; α ∈ [0.2, 0.6] tends to cover the salient part
identified by the learned mask). We then use these simplified masks to edit a set of 5,000 ImageNet images with constant, noise, and blur perturbations. Using GoogLeNet [17],
we compute normalized softmax probabilities4 (fig. 7, bottom). The fact that these simplified masks quickly suppress
scores as α increases for all three perturbations gives confidence that the learned masks are identifying the right regions to perturb and are generalizable to a set of extracted
masks and other perturbations that they were not trained on.

5.3. Minimality of deletions
In this experiments we assess the ability of our method
to correctly identify a minimal region that suppresses the
object. Given the output saliency map, we normalize its
intensities to lie in the range [0, 1], threshold it with h ∈
[0 : 0.1 : 1], and fit the tightest bounding box around the
resulting heatmap. We then blur the image in the box and
compute the normalized4 target softmax probability from
p − p0
, where p, p0 , pb are the masked, original, and fully
p0 − pb
blurred images’ scores
4 p0

=

Mean Norm. Softmax

Img

=0.2

Mask

=0.4

=0.6

=0.8

1.00

blur
constant
noise

0.75
0.50
0.25
0.00

0.0

0.2

0.4

0.6

0.8

Figure 7. (Top) Left to right: original image, learned mask, and
simplified masks for sec. 5.2 (not shown: further smoothed mask).
(Bottom) Swift softmax score suppression is observed when using
all three perturbations with simplified binary masks (top) derived
from our learned masks, thereby showing the generality of our
masks.

Figure 9. “tiger” (left two) and “bison” (right two) images with
eyes and feet annotations from [11]; our learned masks are overlaid. The mean average feet:eyes intensity ratio for “tigers” (N =
25) is 3.82, while that for bisons (N = 22) is 1.07.

keypoint. For all 76 classes, the mean average intensity of
eyes were lower and thus more salient than that of feet (see
supplementary materials for class-specific results).

5.5. Adversarial defense

GoogLeNet [17] of the partially blurred image.
From these bounding boxes and normalized scores, for
a given amount of score suppression, we find the smallest bounding box that achieves that amount of suppression.
Figure 8 shows that, on average, our method yields the
smallest minimal bounding boxes when considering suppressive effects of 80%, 90%, 95%, and 99%. These results
show that our method finds a small salient area that strongly
impacts the network.

Adversarial examples [5] are often generated using a
complementary optimization procedure to our method that
learns a imperceptible pattern of noise which causes an image to be misclassified when added to it. Using our reimplementation of the highly effective one-step iterative
method ( = 8) [5] to generate adversarial examples, our
method yielded visually distinct, abnormal masks compared
to those produced on natural images (fig. 10, left). We
train an Alexnet [4] classifier (learning rate λlr = 10−2 ,
weight decay λL1 = 10−4 , and momentum γ = 0.9) to
distinguish between clean and adversarial images by using
a given heatmap visualization with respect to the top predicted class on the clean and adversarial images (fig. 10,
right); our method greatly outperforms the other methods
and achieves a discriminating accuracy of 93.6%.
Lastly, when our learned masks are applied back to their
corresponding adversarial images, they not only minimize
the adversarial label but often allow the original, predicted
label from the clean image to rise back as the top predicted
class. Our method recovers the original label predicted on
the clean image 40.64% of time and the ground truth label
37.32% (N = 5000). Moreover, 100% of the time the original, predicted label was recovered as one of top-5 predicted
labels in the “mask+adversarial” setting. To our knowledge,
this is the first work that is able to recover originally predicted labels without any modification to the training set-up
and/or network architecture.

5.4. Testing hypotheses: animal part saliency

5.6. Localization and pointing

From qualitatively examining learned masks for different animal images, we noticed that faces appeared to be
more salient than appendages like feet. Because we produce dense heatmaps, we can test this hypothesis. From an
annotated subset of the ImageNet dataset that identifies the
keypoint locations of non-occluded eyes and feet of vertebrate animals [11], we select images from classes that have
at least 10 images which each contain at least one eye and
foot annotation, resulting in a set of 3558 images from 76
animal classes (fig. 9). For every keypoint, we calculate the
average heatmap intensity of a 5 × 5 window around the

Saliency methods are often assessed in terms of weaklysupervised localization and a pointing game [20], which
tests how discriminative a heatmap method is by calculating the precision with which a heatmap’s maximum point
lies on an instance of a given object class, for more harder
datasets like COCO [6]. Because the deletion game is meant
to discover minimal salient part and/or spurious correlation,
we do not expect it to be particularly competitive on localization and pointing but tested them for completeness.
For localization, similar to [20, 2], we predict a bounding box for the most dominant object in each of ∼50k

Mean Minimal Bounding Box Size (pixels^2)

25000
20000
15000

mask
occlusion
contrast exc
excitation
guided
grad-CAM
gradient

10000
5000
0

80%

90%
95%
Percent of Normalized Softmax Score Suppression

99%

Figure 8. On average, our method generates the smallest bounding
boxes that, when used to blur the original images, highly suppress
their normalized softmax probabilities (standard error included).

Val-α* Err (%)

Figure 10. (Left) Difference between learned masks for clean
(middle) and adversarial (bottom) images (28 × 28 masks shown
without bilinear upsampling). (Right) Classification accuracy
for discriminating between clean vs. adversarial images using
heatmap visualizations (Ntrn = 4000, Nval = 1000).

ImageNet [13] validation images and employ three simple thresholding methods for fitting bounding boxes. First,
for value thresholding, we normalize heatmaps to be in the
range of [0, 1] and then threshold them by their value with
α ∈ [0 : 0.05 : 0.95]. Second, for energy thresholding [2],
we threshold heatmaps by the percentage of energy their
most salient subset covered with α ∈ [0 : 0.05 : 0.95]. Finally, with mean thresholding [20], we threshold a heatmap
by τ = αµI , where µI is the mean intensity of the heatmap
and α ∈ [0 : 0.5 : 10]. For each thresholding method, we
search for the optimal α value on a heldout set. Localization
error was calculated as the IOU with a threshold of 0.5.
Table 1 confirms that our method performs reasonably and shows that the three thresholding techniques affect each method differently. Non-contrastive, excitation
backprop [20] performs best when using energy and mean
thresholding; however, our method performs best with
value thresholding and is competitive when using the other
methods: It beats gradient [15] and guided backprop [16]
when using energy thresholding; beats LRP [1], CAM [22],
and contrastive excitation backprop [20] when using mean
thresholding (recall from fig. 2 that the contrastive method
is visually most similar to mask); and out-performs GradCAM [14] and occlusion [19] for all thresholding methods.
For pointing, table 2 shows that our method outperforms
the center baseline, gradient, and guided backprop methods
and beats Grad-CAM on the set of difficult images (images
for which 1) the total area of the target category is less than
25% of the image and 2) there are at least two different object classes). We noticed qualitatively that our method did
not produce salient heatmaps when objects were very small.
This is due to L1 and TV regularization, which yield wellformed masks for easily visible objects. We test two variants of occlusion [19], blur and variable occlusion, to interrogate if 1) the blur perturbation with smoothed masks

Ene-α*

Err

Mea-α*

Err

Grad [15]
Guid [16, 8]
Exc [20]
C Exc [20]
Feed [2]
LRP [1]

0.25
0.05
0.15
—
—
—

46.0
50.2
46.1
—
—
—

0.10
0.30
0.60
—
0.95
—

43.9
47.0
38.7
—
38.8†
—

5.0
4.5
1.5
0.0
—
1.0

41.7§
42.0§
39.0§
57.0†
—
57.8†

CAM [22]
Grad-CAM [14]
Occlusion [19]
Mask‡

—
0.30
0.30
0.10

—
48.1
51.2
44.0

—
0.70
0.55
0.95

—
48.0
49.4
43.1

1.0
1.0
1.0
0.5

48.1†
47.5
48.6
43.2

Table 1. Optimal α thresholds and error rates from the weak
localization task on the ImageNet validation set using saliency
heatmaps to generate bounding boxes. † Feedback error rate are
taken from [2]; all others (contrastive excitation BP, LRP, and
CAM) are taken from [20]. § Using [20]’s code, we recalculated these errors, which are ≤ 0.4% of the originally reported
rates. ‡ Minimized top5 predicted classes’ softmax scores and used
λ1 = 10−3 and β = 2.0 (examples in supplementary materials).
Ctr Grad Guid Exc CExc G-CAM Occ Occ§ V-Occ† Mask‡
All 27.93 36.40 32.68 41.78 50.95 41.10 44.50 45.41 42.31 37.49
Diff 17.86 28.21 26.16 32.73 41.99 30.59 36.45 37.45 33.87 30.64

Table 2. Pointing Game [20] Precision on COCO Val Subset (N ≈
20k). § Occluded with circles (r = 35/2) softened by gσm =10 and
used to perturb with blur (σ = 10). † Occluded with variable-sized
blur circles; from the top 10% most suppressive occlusions, the
one with the smallest radius is chosen and its center is used as the
point. ‡ Used min. top-5 hyper-parameters (λ1 = 10−3 , β = 2.0).

is most effective, and 2) using the smallest, highly suppressive mask is sufficient (Occ§ and V-Occ in table 2 respectively). Blur occlusion outperforms all methods except
contrast excitation backprop while variable while variable
occlusion outperforms all except contrast excitation backprop and the other occlusion methods, suggesting that our
perturbation choice of blur and principle of identifying the
smallest, highly suppressive mask is sound even if our implementation struggles on this task (see supplementary materials for examples and implementation details).

6. Conclusions
We propose a comprehensive, formal framework for
learning explanations as meta-predictors. We also present
a novel image saliency paradigm that learns where an algorithm looks by discovering which parts of an image most affect its output score when perturbed. Unlike many saliency
techniques, our method explicitly edits to the image, making it interpretable and testable. We demonstrate numerous
applications of our method, and contribute new insights into
the fragility of neural networks and their susceptibility to artifacts.

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