# INTENT CLASSIFICATION USING PRE-TRAINED LANGUAGE AGNOSTIC EMBEDDINGS FOR LOW
RESOURCE LANGUAGES
Submitted to Interspeech 2022

###### Abstract

Building Spoken Language Understanding (SLU) systems that do not rely on
language specific Automatic Speech Recognition (ASR) is an important yet less
explored problem in language processing. In this paper, we present a
comparative study aimed at employing a pre-trained language agnostic acoustic
model to perform SLU in low resource scenarios. Specifically, we use three
different embedding settings extracted using Allosaurus, a pre-trained
universal phone decoder: (1) Phone-labels (2) Panphone, and (3) Allo
embeddings (proposed by us). These embeddings are then used in identifying the
spoken intent. We perform experiments across three different languages:
English, Sinhala, and Tamil each with different data sizes to simulate high,
medium, and low resource scenarios. Our system improves on the state-of-the-
art (SOTA) intent classification accuracy by absolute 2.11% for Sinhala and
7.00% for Tamil and achieves competitive results in English. Furthermore, we
also present a quantitative analysis to show how the performance scales with
the number of training examples.

Index Terms: Allosaurus, low resource, dilated CNNs, embeddings, Panphone.

##  1 Introduction

Spoken language understanding (SLU) systems are fundamental blocks when
building interactive technologies for new languages. A typical SLU system
consists of an Automatic Speech Recognition (ASR) module followed by a Natural
Language Understanding (NLU) module. ASR converts speech to textual
transcriptions and the NLU module performs downstream tasks like intent
recognition and slot filling from the transcripts obtained. However, building
high fidelity ASR systems requires a large amount labelled data which is
usually not available for most languages. Language specific ASR system thus
forms a bottleneck for creating SLU systems for low-resourced languages. To
circumvent this, phonetics based SLU systems have been proposed where the need
for language specific ASR is bypassed by typically using a universal phone
decoder. This allows creation of language and task specific, word-free, NLU
modules that perform intent recognition directly from phonetic transcriptions.

In this paper, we show that our proposed choice of method _i.e.,_ 1-D dilated
CNN coupled with Allo embeddings perform competitively with current state-of-
the-art (SOTA) SLU systems on English language, and we report new SOTA on
Sinhala and Tamil. We work with natural speech datasets in three languages -
English, Sinhala and Tamil each with different data sizes to simulate high,
medium, and low resource scenarios as shown in Table 1. Our contributions are
as follows: (i) We present a 1-D dilated CNN based method coupled with Allo
embeddings outperforms the previous approaches that employ phonetic
transcriptions (ii) We study the effect of $3$ different embeddings (explained
in Section 3) on the performance of the task _i.e.,_ \- (a) Phone, (b)
Panphone and, (iii) Allo embeddings and (3) We study how the performance
scales with the number of training examples.

##  2 Related Works

Intent recognition has been traditionally performed using textual transcripts
generated by ASR systems. Since building ASR technologies is not viable for
most languages, recent work has focused on creating such systems using
alternate methods. In [1], authors use spectral features of input speech such
as MFCCs for intent recognition. NLU modules have also been built for low
resourced languages using outputs of an English ASR system, for example, using
the softmax outputs of DeepSpeech [2]. DeepSpeech is a character level model
where the softmax outputs corresponding to the model vocabulary were used as
inputs to the intent classification model [3]. Similarly, softmax outputs of
an English phoneme recognition system [4] have also been used to build intent
recognition systems for Sinhala and Tamil [5].

On the other hand,[6][7][8] proposed to build NLU module using phones
extracted from Allosaurus [9]. Allosaurus is a universal phone recognizer and
therefore language independent. A prototypical naive-bayes intent classifier
was built using Allosaurus phonetic transcriptions as inputs in [6]. [7] was
the first extensive work on using phonetic transcriptions for intent
classification on multiple low resourced languages from two language families
- Romance and Indic languages. [8] was the first study on building intent
recognition systems for natural speech and achieved state-of-the-art results
on Tamil. Yet their work was unable to achieve competitive results for
languages like English or Sinhala with larger amounts of data. Building on the
work of [8], we propose to use Allosaurus to extract a sequence of dense
representation instead of the sequence of discrete phones given an audio file
as explained in Section 3. With this, we are able to achieve close to perfect
performance on English and significantly push the SOTA on Sinhala and Tamil.

##  3 Methodology

Figure 1: Our proposed choice of method. The reader must keep in mind that the
$3$ different embeddings are used independently _i.e.,_ one at a time, to
conduct the $3$ different experiments. Any block having a black shadow means
the parameters are trainable.

In this section we define our proposed method coupled with the $3$ different
input embeddings used for all the experiments. We propose to use an End-2-End
1D dilated convolution neural network (CNN) as shown in Figure 1. The network
consists of 4 CNN-BatchNorm-ReLU-Dropout layers. The CNN filters used are
dilated in an increasing linear order from first to fourth layer _i.e.,_ 1
dilation in the first layer to 4 dilation in the fourth layer. We apply
dilation to increase the overall context. Furthermore we also pad the input to
avoid any down-sampling in time dimension. This setup is followed by an
average pool and a dropout layer _i.e.,_ we map the variable input time steps
to a fixed number of time steps, which is $4$. Lastly we add a linear layer to
map the output probability distribution over the number of intents.

Let $X=\\{x_{1},x_{2},...,x_{n}\\}$ be the raw audio signal(input) and
$Y=\\{y_{1},y_{2},...,y_{n}\\}$ be the intent (output). In the first step we
map the input to a high-level representation using the Allosaurus tool
111https://github.com/xinjli/allosaurus [9]. The tool is used as a black-box,
fixed-weights, and we extract two information from the it (i) The output
sequence of phones [10] and (ii) The last layer outputs, before the logit
layer, corresponding to each sample $x_{i}$. Given these two information we
define three different embeddings for our proposed method.

  * •

Phone ($E_{1}$): Similar to the previous work[8], an embedding layer is learnt
during the training step such that it maps the individual phones to a
256-dimensional features.

  * •

Panphone ($E_{2}$): Instead of learning an embedding layer, we map the
individual phone units to a 26-dimensional features similar to the work by
[11]. Therefore the embedding is a 26-dimensional fixed features for each
phone.

  * •

Allo ($E_{3}$): This is our proposed choice of embedding. The embeddings are
language agnostic and our experiments show similar performance on the intent
classification task for different languages, Sinhala and Tamil, when the
dataset size is comparable as shown in Figure 2 and 4. To the best of our
knowledge this is a first work to use the pre-trained 640-dimensional last
layer of Allosaurus as an embeddings for the intent classification task. We
call it Allo embeddings.

From now on we will use the word embedding and input interchangeably _i.e.,_
input can be one of the $3$ embeddings explained earlier.

##  4 Dataset

Figure 2: The plot shows the accuracy vs context-size relation for each of the
$3$ different embeddings. Our proposed choice of embedding, Allo, performs the
best on all the $3$ languages compared to other two. Table 1: Dataset
statistics for $3$ different languages used in this work _i.e.,_ English,
Sinhala and,0 Tamil.

Language |  Number of Utterances |  Number of Speakers |  Number of Intents
---|---|---|---
English [4] |  30,043 |  97 |  31
Sinhala [1] |  7624 |  215 |  6
Tamil [12] |  400 |  40 |  6

Table 2: The table shows 5 different training configurations. Hyphens separate
the 4 CNN layers such that 3-5-7-9 means the architecture has a kernel size of
3,5,7,9 for 1,2,3,4 layer number respectively. We also compute the overall
context size for an experiment for an easier comparison between the different
experiments.

% |  kernel sizes |  Dilation rate |  Context size
---|---|---|---
C1 |  1-1-1-1 |  1-1-1-1 |  1
C2 |  3-3-3-3 |  1-1-1-1 |  9
C3 |  3-3-3-3 |  1-2-3-4 |  17
C4 |  3-5-7-9 |  1-1-1-1 |  21
C5 |  3-5-7-9 |  1-2-3-4 |  41

In this study, we experiment with $3$ different languages _i.e.,_ English,
Sinhala and, Tamil with varying training and test sizes and classify them as
high, medium and, low resource respectively. The complete statistics are shown
in Table 1. For English, we use the largest freely available Fluent Speech
Commands (FSC) dataset [4]. The dataset has 248 unique sentences spoken by 97
speakers and there is no overlap of speakers between train, valid and, test.
Similar to [8], we use the 31-class intent classification formulation of this
dataset.

Sinhala[1] and Tamil[12] datasets are of banking domain collected via crowd-
sourcing. Both the datasets have the 6-class intents. Similar to the previous
work [8], we also evaluate our models using 5-fold cross-validation technique
[12, 1], since there is no train, development and, test splits provided by the
authors.

Figure 3: We plot results on Sinhala language when the training size is
similar to Tamil. To study the language agnostic embedding claim made by us.
We see a similar trend for all the 3 embeddings, which again are derived from
Allosaurus tool which is a language independent phone decoder.

##  5 Experimental Setup

We train and evaluate our proposed model on three different languages of
varying dataset sizes as explained in Section 4. We fix the number of layers
to $4$ and experiment with 5 different configuration of kernel sizes with
varying dilation as shown in Table 2. Here context size refers to the total
context the output layer activation has just before the Global pooling layer.
Furthermore we map each experiment with its context size since it is more
intuitive to reason using context information. Therefore we will use context
size to differentiate the experiments instead of kernel size and dilation. We
experiment with three different embeddings independently as shown in Figure 1.
Note that out of these three only the Phone embedding ($E_{1}$) has learnable
parameters.

In all experiments, we use L2 weight decay, and dropout as regularization and
Adam as an optimizer with 0.0015 learning rate. We decay the learning rate
linearly up to 0.000001. The evaluation metrics we report is accuracy. All the
reported results are the average of 5 runs with different random seed.
Detailed training configurations and the code is available on GitHub222Will be
made available on GitHub upon acceptance..

##  6 Experimental Results

In this section we report results to show why CNN is a better architecture
choice than LSTM or Transformers in low resource settings _i.e.,_. We compare
our proposed method with previous work [8] using Phone embeddings which
employed LSTMs and Transformers. Secondly, we compare Phone, Panphone and,
Allo embeddings with increasing context size. Lastly we compare the
performance of these three embeddings as the number of training example are
increased.

Based on our experiment results, in almost all the settings we recommend to
use Allo embeddings for the intent classification task. We observe that
choosing a bigger context size is a necessity when using our proposed choice
of embedding, Allo, in low resource datasets.

Table 3: Accuracy on different architectural and training choices made in the
literature when using Phone embeddings. Gupta et al. [8] experiments using
LSTMs and Transformer architecture. Our proposed choice of 1-D dilated CNN
method shows accuracy gains as the amount of training data decreases. The
reader should keep in mind our method is most similar to [8]. Thus a fair
comparison would be to the second column.

Language |  End-2-End approaches |  Gupta et al. [8] |  Our Method
---|---|---|---
English |  99.71% [13] |  92.77% |  92.99%
Sinhala |  97.31% [5] |  96.33% |  97.05%
Tamil |  81.7% [5] |  91.50% |  97.25%

###  6.1 Comparison With Previous Work Using Phone Embedding

Similar to [8], we train our proposed choice of architecture _i.e.,_ 1-D
dilated CNN, on all the three languages using the phone embeddings (our
method). Our method performs better on all the three languages when compared
to the similar training settings using LSTM and Transformers by [8]. We also
compare our method with End-2End systems in the literature [13, 5]. The
accuracy gap increases as the dataset size decreases as shown in Table 3 and
we even beat End-2-End approach by a large margin on Tamil.

Based on these observations, we can now say that CNN is a better choice for
intent classification task in low resource nd performs on par in high resource
settings. For Tamil we report a new SOTA accuracy and for Sinhala we achieve
near SOTA accuracy when using the phone embeddings $E_{1}$ with our proposed
choice of architecture.

Figure 4: This plot shows the accuracy vs context size on using $3$ different
embeddings as we increase the number for training examples per intent.

###  6.2 Comparing the Phone, Panphone and, Allo Embeddings

Our proposed choice of embeddings, Allo, achieves the best accuracy on all the
three languages when compared to Phone and Panphone embeddings. 1-D dilated
CNN coupled with Allo embeddings achieve a new SOTA on Sinhala and Tamil as
shown in Table 4 and Figure 2. When compared to End-2-End system by [13] on
English language our method performs on par. The reader should keep in mind
that additional gains can be seen when fine-tuning the Allosaurus tool in an
End-2-End fashion with 1-D dilated CNN. It has to be noted that compared to
some of the earlier works [8, 5], our proposed method works exceptionally well
in case of medium and low resource languages _i.e.,_ Sinhala and Tamil because
of the proposed choice of architecture which is shown to be less prone to
overfitting.

Interestingly we observe that in case of Tamil, a low resource language in our
current setup, Allo embeddings does not provide significant gains in accuracy
compared to Phone and Panphone. We wanted to test if the cause for this
behaviour is the small training dataset or the language itself. Therefore we
sample training data from Sinhala of similar size to Tamil and repeat the same
experiments. As shown in Figure 3, we observe a similar pattern as before
across all the three embeddings _i.e.,_ as was seen for Tamil language as
shown in Figure 2. In extremely low resource setting Allo embeddings with
bigger context size performs on par to Phone and Panphone embeddings. And
therefore it should be the de-facto choice compared to Phone and Panphone
embeddings. Based on the similar performance on Tamil and Sinhala, we say that
the Allo embeddings are language agnostic as expected and the behavior is
dependent on the amount of training dataset used.

Finally, in high and medium resource settings, the Allo embeddings perform
similar no matter the context size. This shows that Allo embeddings has
contextual information too. In low resource we see the opposite behavior
behavior, our hypothesis is that the Allo embedding features are not
discriminative enough and a bigger context size somewhat compensates for
dataset size. We validate our hypothesis in the next Section.

###  6.3 Allo Embedding Performance VS Number Of Training Examples

Given our previous observations with Tamil, we were interested in the
correlation between the number of training examples and the our proposed
choice of model _i.e.,_ number of training example vs accuracy. Therefore we
scale the training dataset size such that $n$*split is the number of training
examples, where $n$ is the number of intents. For example in case of Sinhala
language we have $n$ equal to 6 and if the value of split is 32, this would
give us 192 training examples. We vary the value of split from 32 to 512 such
that the number of training examples ranges to 192 to 3072 as shown in Figure
3. We use the model with the highest context size.

We experiment on Sinhala to test the trend of accuracy using different
embeddings as we increase the number of training examples. We choose Sinhala
language and not English because the majority of Allosaurus training data was
English and therefore the results could be biased. As shown in Figure 4 we
observe that the performance of Allo embeddings is proportional to the number
of training examples and saturates after a certain point. Furthermore with
only 192 training examples (split=32) Allo perform on par with Phone Panphone
embeddings given that the model has a higher context size. After the split
value of 64 Allo embedding gains significant upper hand in accuracy compared
to Phone and Panphone embeddings. These experiments also validate our earlier
hypothesis that context size is compensating for the lack of dataset when
using Allo embeddings.

Table 4: Comparing the Allo embedding with the other two for each language.
Experiments are conducted using the configuration number $5$ as shown in Table
1 _i.e.,_ the biggest context size.

Language |  Phone |  Panphone |  Allo
---|---|---|---
English |  92.99% |  92.96% |  99.08%
Sinhala |  97.05% |  97.36% |  99.42%
Tamil |  97.25% |  97.75% |  98.50%

##  7 Conclusion and Future work

In this work we propose language agnostic embedding coupled with 1-D CNN based
architecture for the intent classification task which achieves new SOTA
accuracy in medium and low resource settings _i.e,_ for Sinhala and Tamil
language respectively and performs on par on high resource _i.e.,_ English. We
observe that for Allo embedding to perform on par with Phone and Panphone
embedding in low resource settings bigger context size is needed to compensate
for the dataset size. Similarly to [8], our proposed method can also be
extended to do slot identification task. For the future work, we would like to
explore how to make the Allo embeddings work better in extremely low resource
settings _i.e.,_ with a smaller context size.

## References

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# DiagViB-6: A Diagnostic Benchmark Suite for Vision Models in the Presence of
Shortcut and Generalization Opportunities

Elias Eulig Piyapat Saranrittichai Chaithanya Kumar Mummadi
Kilian Rambach William Beluch Xiahan Shi Volker Fischer11footnotemark: 1
Bosch Center for Artificial Intelligence (BCAI) Corresponding authors. Contact
via<EMAIL_ADDRESS>or<EMAIL_ADDRESS>
###### Abstract

Common deep neural networks (DNNs) for image classification have been shown to
rely on _shortcut opportunities_ (SO) in the form of predictive and easy-to-
represent visual factors. This is known as shortcut learning and leads to
impaired generalization. In this work, we show that common DNNs also suffer
from shortcut learning when predicting only basic visual object factors of
variation (FoV) such as shape, color, or texture. We argue that besides
shortcut opportunities, _generalization opportunities_ (GO) are also an
inherent part of real-world vision data and arise from partial independence
between predicted classes and FoVs. We also argue that it is _necessary_ for
DNNs to exploit GO to overcome shortcut learning. Our core contribution is to
introduce the Diagnostic Vision Benchmark suite _DiagViB-6_ , which includes
datasets and metrics to study a network’s shortcut vulnerability and
generalization capability for six independent FoV. In particular, _DiagViB-6_
allows controlling the type and degree of SO and GO in a dataset. We benchmark
a wide range of popular vision architectures and show that they can exploit GO
only to a limited extent.

##  1 Introduction

Despite their state-of-the-art performance on object classification tasks,
deep neural networks (DNN) are highly prone to shortcut learning [8, 33, 11].
Instead of learning holistic representations and decision rules that can
generalize beyond the training data, DNNs overly rely on so-called _shortcut
opportunities_ (SO), which occur when the target class is highly correlated to
one or very few easy-to-represent input factors [12]. This leads to poor
generalization on many out-of-distribution (OOD) settings, e.g. ImageNet
trained DNNs are biased towards texture and fail to generalize under texture-
shape cue conflict evaluation [8].

shape factor
---
(predicted)
243hue factorblueredgreen

$\star$

test (OOD) shortcut opp. gen. opp. train Figure 1:  Exemplar study in our
proposed benchmark. The network is trained to predict factor classes 2, 4, 3
for the shape factor with varying hue. All five depicted training combinations
are uniformly shown during training. The shape 2 co-occurs solely with the
blue class of the hue factor, which poses a shortcut opportunity. The shapes 4
and 3 occur uniformly with hue red and green; these combinations pose a
generalization opportunity, since they reduce the predictiveness of the hue
factor for the shape factor. Test accuracy is computed on examples from OOD
factor combinations to evaluate a model’s shortcut vulnerability in the
context of the given generalization opportunity.

While humans are also prone to shortcut learning in certain cases, such as
object classification under context-based cue conflict settings [29], the
biological model remains largely unaffected by shortcuts when it comes to
predicting _basic_ object factors of variation (FoV) such as shape, hue or
texture. This “shortcut-immunity” w.r.t. basic FoV is just as necessary for
intelligent systems; thus, efforts towards improving model generalization are
of utmost importance.

Existing literature studies shortcut behavior in DNNs mainly in the context of
object classification. In this work, we address a more fundamental variant of
shortcut learning, focusing specifically on the prediction of basic FoV
themselves, similar to [12]. In the context of FoV prediction, we refer to
different manifestations of a factor as _factor classes_. For example, “red”
and “green” are factor classes for the factor hue, whereas “circle” and
“elephant shape” are factor classes for the factor shape. Object classes such
as “elephant” or “car” are characterized by the co-occurrence of certain
factor classes, e.g. “gray”, “elephant shape”, and “elephant texture”
characterize an elephant. SO arise from this co-occurrence of different factor
classes.

As stated in [7], SO are a property of real-world vision data. In this work,
we propose to additionally consider _generalization opportunities_ (GO), which
are a relaxation of the strict correlation (or co-occurrence) between a target
class and an input FoV at training time. For instance, consider the object
“car” with factors shape and color; cars appearing in different colors during
training would induce a GO compared to cars appearing only in one color. We
refer to such cases as _compositional-based_ GO. Correlations can also be
violated in the form of outliers consisting of rare combinations of factor
classes, e.g. a “white elephant”. We refer to these cases as _frequency-based_
GO.

A straightforward approach to introduce GO in the training data is data
augmentation. For example, [21] applies random color transformations to the
training images, removing a potential correlation between the target class and
the factor color. However, data augmentation results in models that are
invariant with respect to the augmented factor. Such models lose important
information that may be needed to properly identify and reason about OOD
samples. In contrast to being invariant, we argue that a good vision model
needs to have an explicit representation of these FoV. Our work aims at
analyzing a model’s capability to exploit the GO already present in the data,
as opposed to adding more GO to a dataset, as done in data augmentation. While
several synthetic benchmark datasets for compositional generalization and cue-
conflict settings already exist [12, 2, 28], none of them enables sufficient
and systematic control over the SO and GO present in the dataset for a broad
set of different visual object FoV.

Inspired by prior work on shortcut learning and compositional generalization
[7, 12], we present a synthetic but diagnostic benchmark suite DiagViB-6
111https://github.com/boschresearch/diagvib-6that includes different studies
to evaluate a model’s shortcut vulnerability under varying degrees of GO.
Figure 1 illustrates an exemplar study in our benchmark. The benchmark suite
contains an image-generating function that allows direct and independent
control over the six basic, visual object FoV: position, hue, lightness,
scale, shape, and texture (Fig. 2). Additionally, our framework provides a
dataset-generating function that enables a user to control the nature of SO
and GO appearing in a dataset. This is achieved by introducing different
degrees of correlation between factors, and inducing co-occurrences of certain
factor class combinations. Furthermore, the benchmark suite provides metrics
to evaluate a model’s shortcut vulnerability under different GO for each
factor.

We evaluate a wide range of common deep learning vision models on our
benchmark and perform an exhaustive investigation of their shortcut
vulnerability w.r.t. the six stated FoV. We show that while they exploit
frequency GO, they exploit the more relevant compositional GO only to a
limited extent. This holds true also for approaches specifically designed to
counteract shortcut learning.

We admit that this benchmark suite does not _sufficiently_ and directly prove
a vision model’s ability to generalize on real-world data (e.g. object
classification on ImageNet). However, it serves as a critical diagnosis that
is _necessary_ in order to study a model’s shortcut vulnerability and
generalization ability under various controlled tasks and data setups.
Ultimately, the design of our benchmark suite allows a user to control
different degrees of SO and GO in a dataset (not commonly available in real-
world data), in order to assess a model’s behavior under different conditions.

Our contributions in this work can be summarized as follows: We propose a
benchmark suite to create datasets that enable the user to independently
combine six visual FoV, allowing explicit control over which SO and GO are
present in the resulting data. We establish suitable metrics to evaluate both
a model’s shortcut vulnerability, and its capability to exploit GO in the
data. Lastly, we provide empirical evidence that common vision architectures
exploit GO only to a limited extent, especially compositional-based GO.

Figure 2: Image space traversal across all six FoV and four corresponding
class labels used in this work. Along each column only the corresponding
factor is varied, while all others are fixed. Note that some factors have more
than the four classes shown here (refer Tab. 1).

##  2 Related work

#### Compositional generalization in visual attribute prediction

Our work is formulated under the general framework of visual attribute
prediction. In contrast to classical object recognition, the task is to learn
semantic attributes of a given object [6, 32]. By learning such class-agnostic
visual object attributes, models can make useful predictions about object
classes that have not been seen at training time. We are interested in
specific scenarios, called compositional generalization, in which only a
limited number of attribute combinations are provided during training. The
model needs to generalize during training in order to handle inputs with
unseen attribute combinations during testing [27, 28, 35, 24, 1, 2, 36]. While
most of these works use a multi-task setup requiring object and attribute
annotations for every image during training, we consider a single-task
prediction, in which only one factor is predicted. In addition, we consider
predicting more fundamental factors (e.g. lightness, scale, hue) that are
independent of each other and can be applied to all types of objects, as
opposed to high-level visual attributes (e.g. glossy, furry, smooth).

#### Shortcut learning

In recent years, multiple works have shown that DNNs are vulnerable to
shortcut learning on several real-world and synthetic datasets [3, 8, 11, 7,
12]. As a result, many studies investigate the reasons for shortcut learning,
and propose methods to alleviate such vulnerabilities. It is suggested in [11]
that differences between humans and DNNs may arise from differences in the
data that they see. [12] investigated how a model’s representations are shaped
by inductive biases in the presence of SO. A common attempt to overcome
shortcut learning is to augment the training data with handcrafted
transformations in order to reduce the importance of each individual factor
(e.g. shape or texture) [8, 23]. This approach has further been extended to
generative model-based augmentations [31, 34, 33]. Recently, [26] showed that
for self-supervised learning, certain shortcut features can be removed
automatically, under the assumption that such features are most vulnerable to
adversarial attacks. Most of the aforementioned works on shortcut learning are
benchmarked on black-box datasets (e.g. Stylized ImageNet [8]), leading to a
limited, implicit knowledge of the SO and GO introduced in both the training
and test data. In contrast, our DiagViB-6 enables explicit image generation,
allowing shortcuts learned from individual FoV to be evaluated.

#### Benchmarks

Numerous works exist on related areas, such as compositional generalization or
disentanglement of FoV, that introduce datasets which allow to control image
factors to some extent [12, 16, 2, 17, 20, 3, 13, 22, 15, 36]. However, these
all contain shortcomings that we try to address in our work.

In contrast to [12, 2, 17, 20, 3, 13], our dataset contains a richer intra-
factor class variation. E.g. for the shape factor, each digit class of MNIST
specifies a factor class. During image generation we use different instances
of each individual factor class. Similarly, for red, we use different tones of
red. Other datasets only provide for example a single cylinder shape or red
tone. For an overview of the variation provided for each of our six factors
see Sec. A.1.

Unlike the datasets in [12, 20, 3, 22], which use only 2-3 FoV, DiagViB-6
includes six fundamental FoV that all good vision models should be shortcut-
robust towards.

Some works, e.g. [12], investigate the internal feature representations of
DNNs at certain layers, whereas our benchmark depends only on a model’s
predictions and is therefore architecture agnostic.

In contrast to multi-attribute prediction [2, 15, 36], we evaluate a model’s
shortcut behavior w.r.t. a single factor under different correlations to other
FoV. This allows for a more structured and comprehensive analysis of shortcut
behavior, where the interaction between individual factors can be
investigated.

##  3 Diagnostic vision benchmark suite

This section describes our benchmark suite DiagViB-6, used to examine the
shortcut vulnerability and generalization capability of DNNs over six
different independent FoV. The suite consists of different image datasets
tailored for different diagnostic studies, as well as suitable metrics for
measuring shortcut vulnerability. We begin with an overview of the image
generation process in our benchmark studies (Sec. 3.1), describe the benchmark
setup in Sec. 3.2, introduce different studies in Sec. 3.3, and lastly
establish the metrics to evaluate DNNs on our studies in Sec. 3.4.

###  3.1 Prerequisite

$\mathcal{F}_{i}$ | $\mathcal{S}_{i}$ | $\mathcal{N}_{i}$ | $\mathcal{C}_{i,1}$ | $\mathcal{S}_{i,1}$
---|---|---|---|---
position | $[0,1]^{2}$ | 9 | top-left | $[0.1,0.3]\times[0.1,0.3]$
hue | $[0,2\pi)$ | 6 | red | $[$$,$$]$
lightness | $[0,1]^{2}$ | 4 | dark | $[0,0.1]\times[0.4,0.5]$
scale | $[0.69,1.45]$ | 5 | small | $[0.69,0.74]$
shape | MNIST | 10 | ‘0’ | digits ‘0’
texture | textures | 5 | tiles | tiles texture crops
Table 1: Overview of factors $\mathcal{F}_{i}$, respective factor spaces
$\mathcal{S}_{i}$, and number of classes $\mathcal{N}_{i}$.
$\mathcal{C}_{i,1}$ & $\mathcal{S}_{i,1}$ are exemplary factor classes and
factor space regions of class label 1.

#### Fundamentals

Datasets in our benchmark suite consist of images with a single object
described by a fixed, pre-defined set of six independent factors. Each factor
$\mathcal{F}_{i},\,i\in\\{1,\ldots,6\\}$ corresponds to a semantically
meaningful image attribute: shape, texture, hue, lightness, position and
scale. Every factor $\mathcal{F}_{i}$ is associated with a certain _factor
space_ $\mathcal{S}_{i}$ from which factor realizations
$f_{i}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widehat{=}f_{\mathcal{F}_{i}}}\in\mathcal{S}_{i}$
that describe the objects are sampled. For example, lightness of an object
$f_{\text{lightness}}$ can be realized as a scalar sampled from
$\mathcal{S}_{\text{lightness}}=[0,1]$. We assign $\mathcal{N}_{i}$ discrete
factor class labels for each factor, denoted as $\mathcal{C}_{i,j}$,
$j\in\\{1,\ldots,\mathcal{N}_{i}\\}$, which correspond to regions
$\mathcal{S}_{i,j}\subset\mathcal{S}_{i}~{}\forall j$ and
$\mathcal{S}_{i,j}\cap\mathcal{S}_{i,k}=\emptyset,j\neq k$. Similar to
factors, each factor class $\mathcal{C}_{i,j}$ corresponds to a semantically
meaningful attribute class (e.g. $\mathcal{C}_{\text{lightness},1}$ refers to
“dark” and $\mathcal{C}_{\text{hue},3}$ refers to “green”). Note that our
choice of classes for each factor is arbitrary and based on human intuition
(similar to [12]) and thus our work does not evaluate a factor’s general
ability to act as a shortcut over another.

Table 1 provides an overview of the list of factors, their corresponding
factor spaces, and the number of factor classes used throughout this work. As
examples, the factor spaces $\mathcal{S}_{i,1}$ and the factor classes
$\mathcal{C}_{i,1}$ are also provided. A comprehensive overview of factor
spaces for all factor classes is provided in Sec. A.1.

#### Image generation

An image is generated by defining a combination of factor classes
$\mathcal{C}\,=\,(\mathcal{C}_{1,j},\mathcal{C}_{2,k},\ldots,\mathcal{C}_{6,l})$,
one from each factor $\mathcal{F}_{i}$. This is done by first sampling factor
realizations $f_{i}$ from the corresponding regions
($\mathcal{S}_{1,j},\mathcal{S}_{2,k},...,\mathcal{S}_{6,l}$) and then
generating unique images from those sampled $f_{i}$ using an image-generating
function $I:(f_{1},f_{2}...,f_{6})\rightarrow\mathcal{I}$ with
$\mathcal{I}\in[0,1]^{H\times W\times 3}$, where $H=W=128$. The six FoV are
incorporated into the image as follows: $f_{\text{shape}}$ is always an
instance of a digit in the MNIST dataset, which is then thresholded to yield a
binary segmentation. From the two lightness values
$f_{\text{lightness}}=(f_{\text{lightness}}^{(1)},f_{\text{lightness}}^{(2)})$
two colors with equal hue $f_{\text{hue}}$ are generated. The
$f_{\text{texture}}$’s are crops of histogram-normalized grey-scale _texture_
images and their pixel values serve as the coefficients of a convex
combination between the two aforementioned colors, resulting in the final
color. The object is then either up or down-sampled depending on the chosen
_scale_ realization $f_{\text{scale}}$, and placed at a specific _position_
$f_{\text{position}}$ on a greyscale background.

Figure 2 illustrates the variation over the six different FoV. In particular,
the first column is generated with factor class labels
$\mathcal{C}=(\mathcal{C}_{pos.,j},\,\mathcal{C}_{hue,1},\,\mathcal{C}_{lightn.,3},\,\mathcal{C}_{scale,5},\\\
\mathcal{C}_{shape,1},\mathcal{C}_{text.,2})$ with $j\in\\{1,\,2,\,3,\,4\\}$.
Similarly, the other columns show images varying one factor while keeping the
classes of all other factors fixed. Note that the six different FoV are
independent, and thus different images can be generated for each of the
$\prod_{i}\mathcal{N}_{i}=54000$ different factor class combinations.

###  3.2 Benchmark Setup

In this section we lay out the fundamentals used in Sec. 3.3 to design and
conduct studies analyzing different aspects of a DNN’s shortcut behavior.

For all studies, we select a subset of three factor classes from the
$\mathcal{N}_{i}$ available for each factor $\mathcal{F}_{i}$, which are then
used during training, validation and testing. For example, as depicted in Fig.
1 and 3, we select the factor classes
$\\{\text{Blue},\text{Red},\text{Green}\\}$ and
$\\{\texttt{2},\,\texttt{4},\,\texttt{3}\\}$ for factors $\mathcal{F}_{hue}$
and $\mathcal{F}_{shape}$, respectively, and similarly for the other factors.
This results in $3^{6}=729$ different factor class combinations out of the
mentioned $54000$ being used for a single experiment. Choosing the same number
of factor classes across factors simplifies evaluation and removes the number
of factor classes as a source of variation. To account for randomness
introduced by this selection of factor classes, we draw five random _dataset
samples_ of selected factor classes, over which reported results are averaged.

As mentioned in Sec. 1, we want to evaluate a network’s shortcut behavior when
predicting an FoV in the presence of varying amounts of SO and GO in the
training data. In this work, we consider the interplay between _all possible_
pairings of two factors $(\mathcal{F}_{i},\mathcal{F}_{j})$. For each pairing,
the task is to predict the class of the first factor $\mathcal{F}_{i}$, where
SO and GO are induced by a specified co-occurrence pattern of factor class
combinations for $\mathcal{F}_{i}$ and $\mathcal{F}_{j}$. For example,
consider the setup in Fig. 1, with $\mathcal{F}_{i}$: shape and
$\mathcal{F}_{j}$: hue. Only five out of the nine possible combinations of
factor classes (dashed rectangles in Fig. 1) are shown during training, which
induces a shortcut opportunity for the blue “$2$”. During testing, we then
evaluate the networks performance on the OOD combinations (solid rectangles in
Fig. 1). The other $4$ factors are not correlated with $\mathcal{F}_{i}$ or
$\mathcal{F}_{j}$; their factor classes all appear with the same uniform
probability. We conduct an exhaustive analysis of all possible factor
pairings, resulting in $6\times(6-1)=30$ different settings, i.e. $6-1$
possible pairings for each target factor.

We generate validation data following the same distribution as the training
data. In contrast, test data is designed to analyze a model’s shortcut
behaviour by violating the pairwise correlations existing in the training
data, and thus contain OOD samples in the $3\times 3$ matrix (cf. $\times$ for
training and $\bigcirc$ for testing in Fig. 3). We always generate $43\,740$
training, $8748$ validation, and $10\,000$ test samples.

$\times$ train$\bigcirc$ test predicted
---
factor $\mathcal{F}_{i}$
(e.g. shape)
$\mathcal{F}_{j}$ (e.g.
hue)layout423BRG$\times$$\bigcirc$$\times$$\bigcirc$$\times$$\bigcirc$$\times$$\bigcirc$$\times$$\bigcirc$$\times$$\bigcirc$$\times$$\bigcirc$$\times$$\bigcirc$$\times$$\bigcirc$ZSO$\times$

$\times$

$\bigcirc$

$\times$

$\bigcirc$

$\times$

$\bigcirc$$\times$

$\times$

$\bigcirc$

$\times$

$\bigcirc$

$\times$

$\bigcirc$$\times$FGO-5$\times$

$\times$

$\bigcirc$

$\times$

$\bigcirc$

$\times$

$\bigcirc$$\times$

$\times$

$\bigcirc$

$\times$

$\bigcirc$

$\times$

$\bigcirc$$\times$FGO-10$\times$

$\times$

$\bigcirc$

$\times$

$\bigcirc$

$\times$

$\bigcirc$$\times$

$\times$

$\bigcirc$

$\times$

$\bigcirc$

$\times$

$\bigcirc$$\times$FGO-20$\times$$\bigcirc$$\bigcirc$$\bigcirc$$\times$$\bigcirc$$\bigcirc$$\bigcirc$$\times$ZGO$\times$$\times$$\times$$\times$$\times$$\bigcirc$$\bigcirc$CHGO$\times$$\times$$\times$$\times$$\bigcirc$$\bigcirc$$\bigcirc$$\bigcirc$$\bigcirc$CGO-1$\times$$\times$$\times$$\times$$\times$$\bigcirc$$\bigcirc$$\bigcirc$$\bigcirc$CGO-2$\times$$\times$$\times$$\times$$\times$$\times$$\bigcirc$$\bigcirc$$\bigcirc$CGO-3
Figure 3:  Schematic overview of conducted studies (cf. Fig. 1). For a
combination of two factors $\mathcal{F}_{i},\mathcal{F}_{j}$ ($x$\- and
$y$-axis), we select three factor classes (rows and columns) for each factor
and predict the first ($y$-axis) factor. Different selections from the
resulting nine combinations of factor classes for training and test datasets
($\times$ and $\bigcirc$) yield different GO and SO settings: The zero SO
(ZSO), the three degrees of frequency GO (FGO) (increasing size of crosses
$\times$ ,  $\times$ ,  $\times$ indicate increasing OOD frequency during
training), the zero GO (ZGO), the three compositional GO (CGO-1/2/3) settings
and the compositional hybrid GO (CHGO).

###  3.3 Benchmark Studies

Having introduced the general structure of our benchmark, we now provide an
overview of the five studies it enables (see Fig. 3 for illustration).

#### Zero Shortcut Opportunities (ZSO)

The goal of this first study is to measure a network’s factor classification
performance in the absence of any SO. In the ZSO training data, each class of
the target factor $\mathcal{F}_{i}$ co-occurs uniformly with all possible
classes of $\mathcal{F}_{j}$, as indicated by the positions of $\times$s in
the top-right of Fig. 3. Thus, $\mathcal{F}_{j}$ is not predictive for
$\mathcal{F}_{i},~{}\forall i\neq j$. After training, the factor-
classification performance of the model is tested on a dataset that shares the
same distribution as the training data (see $\bigcirc$ in Fig. 3).

#### Zero Generalization Opportunities (ZGO)

This study can be considered an “opposite” of the ZSO study. In the ZGO
training data, the target FoV $\mathcal{F}_{i}$ is perfectly correlated with
$\mathcal{F}_{j}$, i.e. each class of $\mathcal{F}_{i}$ can only co-occur
exclusively with one particular class of $\mathcal{F}_{j}$ (see ZGO in Fig.
3). Since both factors contain redundant information for the prediction task,
$\mathcal{F}_{j}$ can be exploited as a shortcut to predict $\mathcal{F}_{i}$.

#### Compositional Generalization Opportunities (CGO)

With the ZSO and ZGO studies capturing the two extremes, we continue with more
realistic setups where both SO and GO are present in the training data. As
mentioned in Sec. 1, one way to introduce GO is through compositions. The CGO
study does exactly this, addressing compositional GO by successively adding
more factor class combinations for $\mathcal{F}_{i},~{}\mathcal{F}_{j}$ into
the training set. In particular, we generate three sub-studies
CGO-$c,~{}c\in\\{1,2,3\\}$ with increasing degree of GO (Fig. 3); the added
combination is picked at random. For each class of the target factor, we hold
out at least one unseen combination for testing, as indicated by the
$\bigcirc$ symbols in each row in the CGO-$c$ diagrams of Fig. 3. For $c=0$,
no GO are provided and thus CGO-0 is identical to the ZGO study discussed
above. The goal of the CGO studies is to quantify a model’s generalization
capability, i.e. the capability to efficiently exploit the GO present in the
data. We would expect a good vision model to perform well on these
generalization tasks, especially as the number of SO decreases and the number
of GO increases.

#### Compositional Hybrid GO (CHGO)

This is a special case of CGO-2, for which both SO and GO are present, but
explicitly separated, as depicted in Fig. 3 (bottom-left). One can think of
this as a mixture of ZSO and ZGO, in which one of the classes of
$\mathcal{F}_{i}$ is exclusively coupled with a certain class of
$\mathcal{F}_{j}$ (ZGO), while the remaining two target factor classes co-
occur uniformly with the remaining two classes of $\mathcal{F}_{j}$ (ZSO). The
goal of CHGO is to examine whether a model can become immune to an extreme SO
by exploiting GO available for other classes of the same factors.

As pointed out in Sec. 1, this heterogeneous mixture of SO and GO comes
closest to real-world data. For instance, school buses might always be
“yellow” while other cars appear in arbitrary colors besides yellow.
Considering a model that predicts object shape, the yellow buses allow the
model to shortcut its bus-shape prediction by solely relying on object color.
A good vision model should now be able to exploit the provided GO, in the form
of differently colored cars, to also predict the correct shape for an OOD
“red” bus.

#### Frequency-based Generalization Opportunities (FGO)

Orthogonal to the CGO study, GO may also be introduced in a frequency-based
manner i.e., controlling the proportion of a correlation introduced in the
training data (cf. [12, 4]). These sub-studies can also be seen as a
transition from ZGO to ZSO by gradually increasing the frequency of
correlations in the training data. We generate three sub-studies
FGO-$f,~{}f=\\{5,\,10,\,20\\}$, where the strict correlation from the ZGO is
relaxed by means of low frequency violations (in $f\,$\mathrm{\char
37\relax}$$ of the samples) of this combination during training. Unlike CGO,
all combinations are seen during training; we test on those combinations that
are underrepresented.

(a)

(b)

Figure 4: Best viewed in color. (a) For the RN18 baseline, we show mean
accuracies $\text{P}_{i}$ for all factors $\mathcal{F}_{i}$ on the ZSO study
and mean accuracies $\text{P}_{i,j}$ for all factor pairings
$(\mathcal{F}_{i},\mathcal{F}_{j}),i\neq j$ on the CGO-c and ZGO studies. (b)
Aggregating model behavior over the studies, we compare all baselines using
our benchmark metrics FAAvg and FAMin. Note that in (b) error bars are omitted
to improve visibility of individual baselines; standard errors are reported in
Sec. A.3 and A.4.

###  3.4 Metrics

We benchmark a model on the studies described above, starting with evaluating
the mean per-class accuracy $\text{acc}_{i,j}$ for the factor pair
$(\mathcal{F}_{i},\mathcal{F}_{j})$, $i,j\in\\{1,\ldots,6\\}$, $i\neq j$,
where $\mathcal{F}_{i}$ gets predicted. We define the prediction accuracy
$\text{P}_{i,j}$ on the test dataset in a given study as the expectation over
the five corresponding dataset samples:
$\text{P}_{i,j}=\mathbb{E}\left[\text{acc}_{i,j}\right]$. In the case of ZSO,
the prediction accuracy of $\mathcal{F}_{i}$ is defined as
$\text{P}_{i}=\mathbb{E}\left[\text{acc}_{i}\right]$, with $\text{acc}_{i}$
being the accuracy of predicting $\mathcal{F}_{i}$ on a single dataset sample.
We also define two different metrics to summarize a factor’s shortcut
vulnerability with respect to the other factors:

i) The Factor-Aggregated Average accuracy (FAAvg)

$\displaystyle\text{FAAvg}_{i}=\mathbb{E}\left[\underset{j\in\\{1,\ldots,6\\},j\neq
i}{\text{mean}}\text{acc}_{i,j}\right]$ (1)

measures the average accuracy of predicting factor $\mathcal{F}_{i}$ over all
possible correlations with other factors
$\mathcal{F}_{j},j\in\\{1,\ldots,6\\},j\neq i$. It can also be seen as a
measure of the average shortcut vulnerability of the factor on a given study.

ii) The Factor-Aggregated Minimum accuracy (FAMin)

$\displaystyle\text{FAMin}_{i}$
$\displaystyle=\mathbb{E}\left[\underset{j\in\\{1,\ldots,6\\},j\neq
i}{\text{min}}\text{acc}_{i,j}\right]\vspace{-0.5em}$ (2)

measures the minimum accuracy among the correlations, and can therefore be
used as a measure of the maximum shortcut vulnerability of the factor on a
given study. In the case of ZSO,
$\text{FAAvg}_{i}=\text{FAMin}_{i}=\text{P}_{i}$.

##  4 Baseline setup

This section provides an overview of the baseline network architectures we
evaluate on our benchmark. We consider popular, vision-based DNN architectures
(not pretrained) from PyTorch’s _torchvision_ package [30]: AlexNet [21],
ResNet18 (RN18), ResNet50 (RN50) [10], Wide ResNet50-2 (WideRN) [37], and
DenseNet-161 [14]. We also include RN50 with pretrained weights on ImageNet
[5] (RN50-IN), with frozen convolutional layers and a randomly initialized
fully connected layer to adapt to our benchmark. We also use the prior work
Automatic Shortcut Removal (ASR) [26] as a baseline for our studies. In
addition, since generative models, in particular those with interpretable
factorised latent representations, are promising approaches to overcome
shortcut learning on classification tasks [7], we evaluate a standard VAE [19]
and a Factor-VAE [17] on our benchmark. More details on the training and setup
of baselines are provided in Sec. A.2.

##  5 Results

We evaluate the baselines described in Sec. 4 on our benchmark suite described
in Sec. 3.3. We start with the extreme cases of zero SO (ZSO) and zero GO
(ZGO), followed by the other studies that introduce compositional and
frequency-based GO to the training dataset.

#### ZSO and ZGO

Results for the ResNet18 baseline on both these studies are presented in Fig.
4(a). A comparison of all baselines using the metrics described in Sec. 3.4 is
provided in Fig. 4(b). The factor-wise results for each baseline are presented
in Sec. A.3 and A.4.

The ZSO study evaluates how well a model can predict each factor in the
absence of SO. This yields a single mean accuracy $\text{P}_{i}$ for each
factor $\mathcal{F}_{i}$, visualized as a vector on the very left in Fig.
4(a). Here, RN18 achieves high accuracies ($\geq$99\text{\,}\mathrm{\char
37\relax}$$) for predicting all factors except texture, for which the mean
accuracy is lower $62\pm 6.\text{\,}\mathrm{\char 37\relax}$. A similar
behavior can be observed for most other baselines (leftmost point in each plot
of Fig. 4(b)) with a few exceptions: VAE, Factor-VAE and ASR are worse at
predicting texture, with $34\text{\,}\mathrm{\char 37\relax}$,
$34\text{\,}\mathrm{\char 37\relax}$, and $45\text{\,}\mathrm{\char 37\relax}$
accuracy, respectively. This is likely due to a limited capability in
representing high-frequency structures (Sec. A.3). In summary, considering the
aforementioned exceptions, all baselines can predict all factors to a
reasonable extent.

The ZGO study evaluates a model’s ability to predict a factor
$\mathcal{F}_{i}$ when each of its three factor classes co-occurs exclusively
with a certain factor class of another factor $\mathcal{F}_{j}$ during
training. Each such pairing $(\mathcal{F}_{i},\mathcal{F}_{j})$ corresponds to
a single cell in the rightmost matrix plot in Fig. 4(a), where
$\mathcal{F}_{i}$ are the rows and $\mathcal{F}_{j}$ are the columns. The
color coded value in each cell indicates mean accuracy
$\text{P}_{i,j},~{}i\neq j$ on OOD test samples, for which this correlation is
violated.

For RN18, we observe that for some factor combinations (e.g.
$(\mathcal{F}_{i},\mathcal{F}_{j})$: (shape, texture)) the model does not
exploit the provided SO, and instead preferentially learns the actual task,
leading to similar mean accuracies on OOD test samples compared to the
corresponding ZSO accuracies for this factor. On the other hand, for other
factor combinations (e.g. (texture, shape)) the model does exploit the
provided SO, yielding OOD accuracies close to zero thus below chance level.
texture and position factors play a special role: While position is exploited
as a shortcut when predicting any other factor, texture is never exploited as
shortcut, but is easy to shortcut by any other factor. Interestingly,
$\text{P}_{i,j}\approx 1-\text{P}_{j,i}$ seems to hold for all factor
combinations, i.e., _if $\mathcal{F}_{i}$ is exploited as a shortcut for
predicting $\mathcal{F}_{j}$, then $\mathcal{F}_{j}$ is not exploited as
shortcut for predicting $\mathcal{F}_{i}$_.

We argue that during training either (a) the representation gets dominated by
a single factor ignoring the other or (b) is a superposition of both factors,
i.e., both factors share capacity of the representation. The extent a factor
is represented depends on its overall ”difficulty”, i.e., texture appears
relatively difficult to learn, and how strongly $\mathcal{F}_{i}$ and
$\mathcal{F}_{j}$ compete over the same capacities. During testing,
$P_{i,j}\approx 1-P_{j,i}$ holds now for (a) but not for (b). For lightness
and position, ZGO findings indicate (b), and a superposition of both factors
is learnt, utilising different capacities.

This yields a dataset-dependent ranking among factors, which we use to order
the rows of plots in Fig. 4(a), starting with the most shortcut-robust factor
position, and ending with the most shortcut-vulnerable factor texture. Since a
model’s ability to represent a factor in the presence of SO is related to its
inductive bias, the ZGO study could be used to diagnose the efficiency of
inductive biases of vision models in future work.

ZSO and ZGO findings transfer to most baselines, with a few exceptions: On
RN50-IN, position is less often exploited as shortcut; the model instead
exploits hue as shortcut when predicting position (Sec. A.4, matrix plots).

#### CGO

(a)

(b)

Figure 5: Best viewed in color. Similar to Fig. 4(a) and 4(b), we show mean
accuracies for the RN18 (a) and a comparison of all baselines (b) on the ZSO,
FGO and ZGO studies. Error estimates are provided in Sec. A.3 and A.5.

We now evaluate the capability of common vision models to exploit GO provided
in the form of additional factor-class combinations. Here, the three benchmark
datasets (CGO-1,2,3) successively add factor-class combinations to the ZGO
study. Results for RN18 are shown in Fig. 4(a), and a comparison of all
baselines using aggregated metrics is presented in Fig. 4(b). The factor-wise
results for each baseline are shown in Sec. A.4.

For RN18, we find that most factor combinations exhibit a monotonic
improvement with increasing number of GO, e.g.
$(\texttt{scale},\texttt{hue}):0.01,0.19,0.42,0.53$ for ZGO, CGO-1,2,3,
respectively. However, for some factor combinations the accuracy on OOD cases
does not improve with an increasing number of GO, and for some factor
combinations, e.g. $(\texttt{texture},\texttt{hue})$ or
$(\texttt{shape},\texttt{position})$ shortcut learning occurs also for CGO-3.

This can also be seen for RN18 in the two aggregated benchmark metrics FAAvg
and FAMin in Fig. 4(b). We find that FAAvg stays far below ZSO accuracy for
most baselines up until three GO are added. For texture, FAAvg is below chance
level for all baselines and studies, reflecting our earlier finding that all
other factors are exploited as shortcuts when predicting texture. Moreover,
FAMin stays below chance level for all baselines for texture, shape and scale
factors, up until three GO are added.

In contrast to texture, for position, none of the baselines exploit shortcuts,
with RN50-IN being the only exception (Fig. 4(b), leftmost column). This
phenomenon is likely due to ImageNet containing limited explicit localization
information [9], facilitating a partial invariance of the learned
representations in RN50-IN. From the CGO study, we conclude that all baselines
are unable to exploit compositional-based GO for most FoV used in this work.

#### CHGO

| pos. | hue | lightn. | scale | shape | text.
---|---|---|---|---|---|---
RN18 | $0\pm 0.$ | $2\pm 3.$ | $27\pm 21.$ | $-1\pm 14.$ | $2\pm 5.$ | $-1\pm 2.$
RN50 | $0\pm 0.$ | $3\pm 5.$ | $26\pm 21.$ | $-1\pm 16.$ | $2\pm 5.$ | $-2\pm 2.$
RN50-IN | $-1\pm 11.$ | $5\pm 6.$ | $15\pm 17.$ | $-7\pm 11.$ | $0\pm 8.$ | $-1\pm 1.$
AlexNet | $-4\pm 8.$ | $1\pm 4.$ | $24\pm 19.$ | $-1\pm 12.$ | $3\pm 6.$ | $-2\pm 2.$
DenseNet | $2\pm 3.$ | $-3\pm 5.$ | $24\pm 21.$ | $-3\pm 13.$ | $2\pm 5.$ | $-1\pm 1.$
WRN | $1\pm 1.$ | $-2\pm 4.$ | $23\pm 22.$ | $1\pm 14.$ | $3\pm 6.$ | $-2\pm 2.$
ASR | $1\pm 1.$ | $12\pm 15.$ | $32\pm 18.$ | $1\pm 15.$ | $1\pm 2.$ | $0\pm 2.$
F-VAE | $0\pm 0.$ | $2\pm 5.$ | $19\pm 17.$ | $-3\pm 11.$ | $1\pm 8.$ | $2\pm 1.$
VAE | $0\pm 0.$ | $2\pm 5.$ | $18\pm 15.$ | $-3\pm 12.$ | $0\pm 9.$ | $2\pm 1.$
Table 2: Mean accuracy improvement and its standard deviation of the CHGO
study over the ZGO study.

A special case of the CGO-2 study is the CHGO study, in which two classes of
the predicted factor are provided with GO, while one class is presented with
SO only (see Fig. 1 and 3). Comparing the accuracy on the OOD test samples for
the latter with the average accuracy on the ZGO study (Tab. 2), we find no
improvement for most factors on all baselines, and only minor improvements for
lightness, albeit with large variance across the five dataset subsets. We
conclude that all the baselines evaluated in this work do not transfer GO
across classes.

#### FGO

An orthogonal approach to providing GO through additional class-combinations
during training is to relax the strict correlation between factors by means of
low-frequency violations of this correlation at training time. We generated
three benchmark datasets (FGO-$f,~{}f\in\\{5,10,20\\}$), with $f$ indicating
the frequency of the correlation violation at training time. Results for RN18
and a comparison of all baselines are presented in Fig. 5(a) and 5(b). The
factor-wise results for each baseline are presented in Sec. A.5.

For the RN18, the classification for all factors improves significantly with
only $5\text{\,}\mathrm{\char 37\relax}$ of the training samples for each
class of the predicted factor violating the strict correlation from ZSO (e.g.
$(\texttt{shape},\texttt{hue}):0.,0.98$ for ZGO and FGO-5, respectively). When
$f\geq 10$, none of the factors except texture are vulnerable to shortcut
learning. A similar behavior can be observed for most factors and baselines
(Fig. 4(b)), where we find that FAAvg is above chance for all factors except
texture for $f=5$. However, we find that ASR fails to exploit the presented GO
for hue and lightness when compared to other baselines, a behavior which is
likely attributed to the pixel-wise reconstruction loss that is used as a
regularizer in the ASR objective. In summary, most baselines are able to
exploit frequency-based GO to a certain extent on our benchmark.

##  6 Conclusion

We addressed the lack of a suitable benchmark to evaluate the shortcut
behavior of vision models on a diverse set of basic visual FoV. To this end,
we introduced DiagViB-6, a benchmark suite designed to evaluate two crucial
model performance criteria: shortcut vulnerability and generalization
capability. Our framework allows the user to create benchmark datasets by
independently combining six visual FoV, thereby precisely controlling the SO
and GO present in the dataset. Upon evaluation of the most commonly used
vision architectures on DiagViB-6, we discovered that while they usually can
exploit frequency-based GO, their ability to exploit compositional-based GO is
limited. This finding also holds for some recent, promising approaches that
have been proposed to overcome shortcut learning.

The design of DiagViB-6 is versatile, and leads to certain natural extensions.
Three such promising future directions are: (a) including more complex
correlation structures, potentially between more than two FoV; (b)
transferring the benchmark design to other research areas like domain
generalization and multitask learning; (c) extending to additional factors
like background and natural corruptions. Lastly, we believe that our benchmark
suite will inspire and help to build shortcut-robust solutions for vision
models.

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DiagViB-6: A Diagnostic Benchmark Suite for Vision Models in the Presence of
Shortcut and Generalization Opportunities

Elias Eulig Piyapat Saranrittichai Chaithanya Kumar Mummadi
Kilian Rambach William Beluch Xiahan Shi Volker Fischer
Bosch Center for Artificial Intelligence (BCAI)

##  Appendix A Supplementary material

###  A.1 Image generation

We provide a comprehensive overview of all factor classes and respective
factor space regions in Tab. A1. The five textures from which the texture
crops $f_{\text{text.}}$ are sampled are shown in Fig. A1.

$\mathcal{F}_{i}$ | $\mathcal{S}_{i}$ | $\mathcal{N}_{i}$ | $\mathcal{C}_{i,j}$ | $\mathcal{S}_{i,j}$
---|---|---|---|---
position | $[0,1]^{2}$ | 9 | top-left | $[1/7,2/7]\times[1/7,2/7]$
top-center | $[1/7,2/7]\times[3/7,4/7]$
top-right | $[1/7,2/7]\times[5/7,6/7]$
center-left | $[3/7,4/7]\times[1/7,2/7]$
center-center | $[3/7,4/7]\times[3/7,4/7]$
center-right | $[3/7,4/7]\times[5/7,6/7]$
bottom-left | $[5/7,6/7]\times[1/7,2/7]$
bottom-center | $[5/7,6/7]\times[3/7,4/7]$
bottom-right | $[5/7,6/7]\times[5/7,6/7]$
hue | $[0,2\pi)$ | 6 | red | $[$$,$$]$
yellow | $[$$,$$]$
green | $[$$,$$]$
cyan | $[$$,$$]$
blue | $[$$,$$]$
magenta | $[$$,$$]$
lightness | $[0,1]^{2}$ | 4 | dark | $[0,1/11]\times[4/11,5/11]$
darker | $[2/11,3/11]\times[6/11,7/11]$
brighter | $[4/11,5/11]\times[8/11,9/11]$
bright | $[6/11,7/11]\times[10/11,1.]$
scale | $[1/1.45,1.45]$ | 5 | small | $[1/1.45,1/1.35]$
smaller | $[1/1.25,1/1.15]$
normal | $[1/1.05,1.05]$
larger | $[1.15,1.25]$
large | $[1.35,1.45]$
shape | MNIST | 10 | ‘0’ | digits ‘0’
‘1’ | digits ‘1’
$\vdots$ | $\vdots$
texture | textures | 5 | tiles | tiles texture crops
wood | wood texture crops
carpet | carpet texture crops
bricks | bricks texture crops
lava | lava texture crops
Table A1: Overview of factors $\mathcal{F}_{i}$, respective factor spaces
$\mathcal{S}_{i}$, number of classes $\mathcal{N}_{i}$, factor classes
$\mathcal{C}_{i,j}$ and respective factor space regions $\mathcal{S}_{i,j}$
used in this work.  Figure A1: Overview of the textures used in this work. All
textures are from CC0Textures.com, licensed under CC0 1.0 Universal.

###  A.2 Baselines

We train all baseline architectures using standard cross entropy loss and Adam
optimizer [18] using early stopping. All test accuracies reported in this work
correspond to the network with lowest validation loss. Batch size and learning
rate were optimized on the ZSO study and weight initialization is fixed across
all experiments. The output size of all the architectures are modified to
predict the factor classes in our studies.

Following ASR, we use a variant of their U-Net architecture for the lens and
ResNet18 as feature extractor. Instead of an auxiliary task, we train both the
lens and feature extractor using the classification task directly. The lens is
trained using least likely adversarial loss using the classification objective
and also the reconstruction loss, both with equal weighting. Output of lens is
used as inputs for training the feature extractor.

Both VAE methods use architectures similar to [25] with a latent size of $12$.
We train both VAE with batch size $64$ and Adam optimizer [18] with learning
rate $0.0001$. For the Factor-VAE, we use $\gamma=20$ and train the
discriminator with learning rate $0.00001$ throughout our experiments. Some
examples of latent traversals from VAE and Factor-VAE are shown in Fig. A2. We
experimented with $\beta$-VAEs but observed, that the reconstructions of are
poor when we use higher $\beta$ in our settings. One potential reason is the
discretely distributed positions in the trained datasets. The position factor
of DiagViB-6 has nine possible values with large gap among them violating the
Gaussian distribution assumption in the KL-divergence term during training. To
empirically demonstrate this claim, we train $\beta$-VAEs with different
$\beta$ on two settings: (1) with position factor freely assigned to three
different values (2) with position factor fixed to the center of the images.
According to the results shown in Fig. A3, the reconstructions from the
training with fixed position are significantly better, supporting the
aforementioned argument.

###  A.3 ZSO and ZGO

The mean accuracies $\text{P}_{i}$ for all factors $\mathcal{F}_{i}$ on the
ZSO together with the mean accuracies $\text{P}_{i,j}$ for all factor pairings
$(\mathcal{F}_{i},\mathcal{F}_{j}),i\neq j$ on the ZGO study for all baselines
are shown in Fig. A5 and A6. Additionally, the aggregated benchmark metrics
$\text{FAAvg}_{i}$ and $\text{FAMin}_{i}$ for all factors
$\mathcal{F}_{i},i\in\\{0,1,...,6\\}$ together with their respective standard
errors for all baselines on the ZSO and ZGO studies are presented in Tab. A3
and A3.

From the exemplar VAE decoder reconstructions (Fig. A3) and ASR lens outputs
(Fig. A4) we find that the high-frequency texture information is not
preserved, leading to the observed low classification accuracies on the
texture task.

###  A.4 Compositional-based generalization opportunities

The mean accuracies $\text{P}_{i}$ for all factors $\mathcal{F}_{i}$ on the
ZSO together with the mean accuracies $\text{P}_{i,j}$ for all factor pairings
$(\mathcal{F}_{i},\mathcal{F}_{j}),i\neq j$ on the CGO and ZGO studies for all
baselines are shown in Fig. A5 and A6. Additionally, the aggregated benchmark
metrics $\text{FAAvg}_{i}$ and $\text{FAMin}_{i}$ for all factors
$\mathcal{F}_{i},i\in\\{0,1,...,6\\}$ together with their respective standard
errors for all baselines on the CGO studies are presented in Tab. A6, A6 and
A6.

###  A.5 Frequency-based generalization opportunities

The mean accuracies $\text{P}_{i}$ for all factors $\mathcal{F}_{i}$ on the
ZSO together with the mean accuracies $\text{P}_{i,j}$ for all factor pairings
$(\mathcal{F}_{i},\mathcal{F}_{j}),i\neq j$ on the FGO and ZGO studies for all
baselines are shown in Fig. A7 and A8. Additionally, the aggregated benchmark
metrics $\text{FAAvg}_{i}$ and $\text{FAMin}_{i}$ for all factors
$\mathcal{F}_{i},i\in\\{0,1,...,6\\}$ together with their respective standard
errors for all baselines on the FGO studies are presented in Tab. A9, A9 and
A9.

As discussed in Sec. 5, we find that ASR fails to exploit the presented GO for
hue and lightness. This is likely attributed to the pixel-wise reconstruction
loss that is used as a regularizer in the ASR objective. We present
qualitative results of this behaviour in Fig. A9.

(a) Latent traversal plot from VAE

(b) Latent traversal plot from Factor-VAE

(c) Latent traversal plot from VAE

(d) Latent traversal plot from Factor-VAE

Figure A2: Comparison of latent traversals from VAE and Factor-VAE

(a) Input image (Sample 1)

(b) Reconstructed images
(Sample 1)

(c) Input image (Sample 2)

(d) Reconstructed images
(Sample 2)

(e) Input image (Sample 3)

(f) Reconstructed images
(Sample 3)

(g) Input image (Sample 4)

(h) Reconstructed images
(Sample 4)

Figure A3: Reconstruction images from the $\beta$-VAE networks trained with
datasets of free and fixed positions from different $\beta$ values Figure A4:
Examples of Automatic shortcut removal Lens output for ZSO study with factor
texture.

| position | hue | lightness | scale | shape | texture
---|---|---|---|---|---|---
RN18 | $100\pm 0.$ | $100\pm 0.$ | $99\pm 0.$ | $99\pm 0.$ | $100\pm 0.$ | $62\pm 6.$
RN50 | $100\pm 0.$ | $100\pm 0.$ | $89\pm 9.$ | $99\pm 0.$ | $100\pm 0.$ | $77\pm 3.$
RN50-IN | $99\pm 0.$ | $99\pm 0.$ | $94\pm 1.$ | $74\pm 2.$ | $83\pm 3.$ | $54\pm 1.$
AlexNet | $100\pm 0.$ | $100\pm 0.$ | $98\pm 1.$ | $97\pm 0.$ | $99\pm 0.$ | $68\pm 3.$
DenseNet | $100\pm 0.$ | $100\pm 0.$ | $99\pm 0.$ | $99\pm 0.$ | $100\pm 0.$ | $70\pm 3.$
WRN | $100\pm 0.$ | $100\pm 0.$ | $99\pm 0.$ | $99\pm 0.$ | $100\pm 0.$ | $86\pm 2.$
ASR (RN18) | $100\pm 0.$ | $77\pm 6.$ | $82\pm 2.$ | $92\pm 1.$ | $91\pm 1.$ | $45\pm 1.$
Factor-VAE | $100\pm 0.$ | $97\pm 1.$ | $93\pm 2.$ | $84\pm 1.$ | $91\pm 2.$ | $34\pm 0.$
VAE | $100\pm 0.$ | $98\pm 1.$ | $94\pm 1.$ | $86\pm 2.$ | $93\pm 2.$ | $34\pm 0.$

Table A2: FAAvg and respecive standard error for all baselines on the ZSO
study.

| position | hue | lightness | scale | shape | texture
---|---|---|---|---|---|---
| FAAvg | FAMin | FAAvg | FAMin | FAAvg | FAMin | FAAvg | FAMin | FAAvg | FAMin | FAAvg | FAMin
RN18 | $100\pm 0.$ | $99\pm 1.$ | $74\pm 2.$ | $0\pm 0.$ | $57\pm 4.$ | $31\pm 5.$ | $33\pm 2.$ | $0\pm 0.$ | $41\pm 2.$ | $0\pm 0.$ | $2\pm 1.$ | $0\pm 0.$
RN50 | $100\pm 0.$ | $100\pm 0.$ | $72\pm 2.$ | $0\pm 0.$ | $58\pm 4.$ | $27\pm 1.$ | $32\pm 2.$ | $0\pm 0.$ | $42\pm 2.$ | $0\pm 0.$ | $3\pm 1.$ | $0\pm 0.$
RN50-IN | $68\pm 3.$ | $18\pm 2.$ | $85\pm 2.$ | $64\pm 3.$ | $72\pm 4.$ | $50\pm 6.$ | $28\pm 2.$ | $4\pm 1.$ | $33\pm 5.$ | $8\pm 4.$ | $10\pm 1.$ | $1\pm 0.$
AlexNet | $100\pm 0.$ | $100\pm 0.$ | $72\pm 2.$ | $0\pm 0.$ | $62\pm 6.$ | $31\pm 5.$ | $32\pm 2.$ | $0\pm 0.$ | $39\pm 2.$ | $0\pm 0.$ | $3\pm 1.$ | $0\pm 0.$
DenseNet | $98\pm 1.$ | $88\pm 6.$ | $74\pm 3.$ | $7\pm 3.$ | $61\pm 6.$ | $34\pm 8.$ | $34\pm 1.$ | $0\pm 0.$ | $40\pm 2.$ | $0\pm 0.$ | $2\pm 0.$ | $0\pm 0.$
WRN | $99\pm 1.$ | $96\pm 3.$ | $73\pm 3.$ | $4\pm 3.$ | $61\pm 6.$ | $37\pm 8.$ | $32\pm 2.$ | $0\pm 0.$ | $41\pm 2.$ | $0\pm 0.$ | $2\pm 1.$ | $0\pm 0.$
ASR (RN18) | $99\pm 0.$ | $96\pm 1.$ | $27\pm 3.$ | $1\pm 0.$ | $43\pm 2.$ | $25\pm 0.$ | $51\pm 2.$ | $0\pm 0.$ | $60\pm 1.$ | $1\pm 0.$ | $8\pm 0.$ | $0\pm 0.$
Factor-VAE | $100\pm 0.$ | $100\pm 0.$ | $65\pm 4.$ | $0\pm 0.$ | $64\pm 4.$ | $27\pm 0.$ | $32\pm 1.$ | $0\pm 0.$ | $35\pm 2.$ | $0\pm 0.$ | $4\pm 0.$ | $0\pm 0.$
VAE | $100\pm 0.$ | $100\pm 0.$ | $65\pm 4.$ | $0\pm 0.$ | $64\pm 4.$ | $26\pm 0.$ | $33\pm 2.$ | $0\pm 0.$ | $37\pm 3.$ | $0\pm 0.$ | $4\pm 0.$ | $0\pm 0.$

Table A3: FAAvg and FAMin together with their respective standard errors for
all baselines on the ZGO study. Figure A5: Mean accuracies $\text{P}_{i}$ for
all factors $\mathcal{F}_{i}$ on the ZSO study and mean accuracies
$\text{P}_{i,j}$ for all factor pairings
$(\mathcal{F}_{i},\mathcal{F}_{j}),i\neq j$ for the RN18, RN50, RN50-IN,
AlexNet and DenseNet on the CGO and ZGO studies. Figure A6: Mean accuracies
$\text{P}_{i}$ for all factors $\mathcal{F}_{i}$ on the ZSO study and mean
accuracies $\text{P}_{i,j}$ for all factor pairings
$(\mathcal{F}_{i},\mathcal{F}_{j}),i\neq j$ for the WideRN, ASR (RN18),
Factor-VAE and VAE on the CGO and ZGO studies.

| position | hue | lightness | scale | shape | texture
---|---|---|---|---|---|---
| FAAvg | FAMin | FAAvg | FAMin | FAAvg | FAMin | FAAvg | FAMin | FAAvg | FAMin | FAAvg | FAMin
RN18 | $100\pm 0.$ | $100\pm 0.$ | $83\pm 1.$ | $29\pm 5.$ | $71\pm 8.$ | $49\pm 9.$ | $47\pm 3.$ | $6\pm 2.$ | $51\pm 3.$ | $2\pm 1.$ | $8\pm 2.$ | $0\pm 0.$
RN50 | $100\pm 0.$ | $100\pm 0.$ | $82\pm 1.$ | $24\pm 6.$ | $69\pm 8.$ | $46\pm 8.$ | $48\pm 4.$ | $9\pm 3.$ | $51\pm 3.$ | $0\pm 0.$ | $6\pm 2.$ | $0\pm 0.$
RN50-IN | $80\pm 2.$ | $42\pm 4.$ | $88\pm 2.$ | $73\pm 3.$ | $75\pm 5.$ | $59\pm 8.$ | $31\pm 3.$ | $8\pm 2.$ | $37\pm 6.$ | $12\pm 5.$ | $11\pm 1.$ | $1\pm 0.$
AlexNet | $97\pm 2.$ | $87\pm 12.$ | $81\pm 2.$ | $18\pm 7.$ | $72\pm 8.$ | $52\pm 10.$ | $41\pm 3.$ | $2\pm 1.$ | $45\pm 3.$ | $0\pm 0.$ | $3\pm 0.$ | $0\pm 0.$
DenseNet | $100\pm 0.$ | $99\pm 1.$ | $79\pm 1.$ | $19\pm 1.$ | $71\pm 8.$ | $50\pm 10.$ | $48\pm 4.$ | $9\pm 3.$ | $49\pm 2.$ | $2\pm 1.$ | $3\pm 0.$ | $0\pm 0.$
WRN | $100\pm 0.$ | $100\pm 0.$ | $81\pm 2.$ | $27\pm 4.$ | $72\pm 9.$ | $51\pm 10.$ | $48\pm 3.$ | $10\pm 3.$ | $51\pm 3.$ | $4\pm 2.$ | $4\pm 1.$ | $0\pm 0.$
ASR (RN18) | $99\pm 1.$ | $94\pm 3.$ | $34\pm 3.$ | $2\pm 1.$ | $54\pm 4.$ | $40\pm 5.$ | $61\pm 2.$ | $2\pm 1.$ | $64\pm 3.$ | $2\pm 1.$ | $10\pm 1.$ | $0\pm 0.$
Factor-VAE | $99\pm 0.$ | $93\pm 2.$ | $65\pm 3.$ | $0\pm 0.$ | $67\pm 6.$ | $41\pm 5.$ | $38\pm 3.$ | $0\pm 0.$ | $40\pm 3.$ | $0\pm 0.$ | $4\pm 0.$ | $0\pm 0.$
VAE | $98\pm 1.$ | $92\pm 3.$ | $64\pm 3.$ | $0\pm 0.$ | $67\pm 6.$ | $40\pm 6.$ | $39\pm 3.$ | $1\pm 0.$ | $43\pm 4.$ | $0\pm 0.$ | $3\pm 0.$ | $0\pm 0.$

Table A4: Aggregated benchmark metrics $\text{FAAvg}_{i}$ and
$\text{FAMin}_{i}$ for all factors $\mathcal{F}_{i},i\in\\{0,1,...,6\\}$
together with their respective standard errors for all baselines on the CGO-1
study.

| position | hue | lightness | scale | shape | texture
---|---|---|---|---|---|---
| FAAvg | FAMin | FAAvg | FAMin | FAAvg | FAMin | FAAvg | FAMin | FAAvg | FAMin | FAAvg | FAMin
RN18 | $100\pm 0.$ | $100\pm 0.$ | $89\pm 4.$ | $53\pm 16.$ | $76\pm 11.$ | $51\pm 14.$ | $59\pm 5.$ | $21\pm 6.$ | $64\pm 2.$ | $4\pm 2.$ | $17\pm 4.$ | $0\pm 0.$
RN50 | $100\pm 0.$ | $100\pm 0.$ | $91\pm 4.$ | $59\pm 17.$ | $76\pm 10.$ | $56\pm 14.$ | $59\pm 4.$ | $18\pm 7.$ | $66\pm 3.$ | $13\pm 5.$ | $16\pm 4.$ | $0\pm 0.$
RN50-IN | $88\pm 2.$ | $67\pm 6.$ | $91\pm 3.$ | $80\pm 4.$ | $77\pm 8.$ | $63\pm 12.$ | $35\pm 5.$ | $15\pm 4.$ | $46\pm 6.$ | $18\pm 5.$ | $15\pm 2.$ | $2\pm 1.$
AlexNet | $97\pm 2.$ | $87\pm 12.$ | $87\pm 2.$ | $45\pm 7.$ | $75\pm 9.$ | $55\pm 14.$ | $49\pm 4.$ | $11\pm 5.$ | $55\pm 2.$ | $0\pm 0.$ | $7\pm 2.$ | $0\pm 0.$
DenseNet | $100\pm 0.$ | $98\pm 2.$ | $89\pm 4.$ | $56\pm 14.$ | $77\pm 10.$ | $58\pm 15.$ | $58\pm 5.$ | $23\pm 4.$ | $62\pm 2.$ | $8\pm 3.$ | $12\pm 3.$ | $0\pm 0.$
WRN | $100\pm 0.$ | $100\pm 0.$ | $91\pm 4.$ | $62\pm 17.$ | $78\pm 11.$ | $58\pm 14.$ | $59\pm 4.$ | $19\pm 3.$ | $67\pm 3.$ | $11\pm 6.$ | $17\pm 4.$ | $0\pm 0.$
ASR (RN18) | $99\pm 1.$ | $96\pm 3.$ | $39\pm 6.$ | $6\pm 6.$ | $55\pm 7.$ | $39\pm 9.$ | $62\pm 2.$ | $4\pm 2.$ | $65\pm 3.$ | $2\pm 1.$ | $14\pm 2.$ | $0\pm 0.$
Factor-VAE | $97\pm 2.$ | $89\pm 6.$ | $67\pm 4.$ | $1\pm 0.$ | $68\pm 8.$ | $43\pm 10.$ | $40\pm 4.$ | $1\pm 0.$ | $48\pm 3.$ | $0\pm 0.$ | $5\pm 1.$ | $0\pm 0.$
VAE | $99\pm 1.$ | $97\pm 3.$ | $69\pm 4.$ | $1\pm 0.$ | $68\pm 7.$ | $39\pm 8.$ | $41\pm 4.$ | $1\pm 0.$ | $51\pm 4.$ | $0\pm 0.$ | $4\pm 0.$ | $0\pm 0.$

Table A5: Aggregated benchmark metrics $\text{FAAvg}_{i}$ and
$\text{FAMin}_{i}$ for all factors $\mathcal{F}_{i},i\in\\{0,1,...,6\\}$
together with their respective standard errors for all baselines on the CGO-2
study.

| position | hue | lightness | scale | shape | texture
---|---|---|---|---|---|---
| FAAvg | FAMin | FAAvg | FAMin | FAAvg | FAMin | FAAvg | FAMin | FAAvg | FAMin | FAAvg | FAMin
RN18 | $100\pm 0.$ | $100\pm 0.$ | $95\pm 3.$ | $81\pm 9.$ | $86\pm 6.$ | $65\pm 13.$ | $62\pm 7.$ | $22\pm 3.$ | $76\pm 3.$ | $28\pm 7.$ | $19\pm 4.$ | $1\pm 0.$
RN50 | $100\pm 0.$ | $100\pm 0.$ | $96\pm 2.$ | $85\pm 8.$ | $86\pm 7.$ | $66\pm 17.$ | $69\pm 7.$ | $29\pm 9.$ | $72\pm 2.$ | $8\pm 3.$ | $20\pm 4.$ | $1\pm 0.$
RN50-IN | $92\pm 2.$ | $74\pm 7.$ | $94\pm 1.$ | $81\pm 5.$ | $84\pm 5.$ | $70\pm 10.$ | $35\pm 5.$ | $12\pm 3.$ | $51\pm 8.$ | $23\pm 10.$ | $15\pm 2.$ | $3\pm 1.$
AlexNet | $97\pm 2.$ | $87\pm 12.$ | $86\pm 2.$ | $38\pm 7.$ | $79\pm 7.$ | $54\pm 11.$ | $56\pm 7.$ | $6\pm 2.$ | $62\pm 4.$ | $3\pm 2.$ | $10\pm 2.$ | $0\pm 0.$
DenseNet | $100\pm 0.$ | $100\pm 0.$ | $94\pm 2.$ | $80\pm 6.$ | $86\pm 7.$ | $71\pm 13.$ | $66\pm 7.$ | $23\pm 4.$ | $72\pm 4.$ | $29\pm 11.$ | $16\pm 3.$ | $1\pm 0.$
WRN | $100\pm 0.$ | $100\pm 0.$ | $97\pm 2.$ | $87\pm 6.$ | $87\pm 6.$ | $69\pm 13.$ | $74\pm 6.$ | $32\pm 7.$ | $82\pm 3.$ | $48\pm 9.$ | $24\pm 6.$ | $2\pm 0.$
ASR (RN18) | $100\pm 0.$ | $99\pm 0.$ | $45\pm 1.$ | $2\pm 1.$ | $62\pm 8.$ | $34\pm 8.$ | $67\pm 3.$ | $10\pm 5.$ | $69\pm 3.$ | $11\pm 7.$ | $14\pm 1.$ | $0\pm 0.$
Factor-VAE | $100\pm 0.$ | $100\pm 0.$ | $70\pm 3.$ | $1\pm 1.$ | $73\pm 6.$ | $36\pm 7.$ | $43\pm 4.$ | $1\pm 1.$ | $49\pm 3.$ | $0\pm 0.$ | $4\pm 1.$ | $0\pm 0.$
VAE | $100\pm 0.$ | $100\pm 0.$ | $72\pm 3.$ | $0\pm 0.$ | $72\pm 7.$ | $36\pm 8.$ | $44\pm 5.$ | $1\pm 0.$ | $53\pm 4.$ | $0\pm 0.$ | $3\pm 1.$ | $0\pm 0.$

Table A6: Aggregated benchmark metrics $\text{FAAvg}_{i}$ and
$\text{FAMin}_{i}$ for all factors $\mathcal{F}_{i},i\in\\{0,1,...,6\\}$
together with their respective standard errors for all baselines on the CGO-3
study. Figure A7: Mean accuracies $\text{P}_{i}$ for all factors
$\mathcal{F}_{i}$ on the ZSO study and mean accuracies $\text{P}_{i,j}$ for
all factor pairings $(\mathcal{F}_{i},\mathcal{F}_{j}),i\neq j$ for the RN18,
RN50, RN50-IN, AlexNet and DenseNet on the FGO and ZGO studies. Figure A8:
Mean accuracies $\text{P}_{i}$ for all factors $\mathcal{F}_{i}$ on the ZSO
study and mean accuracies $\text{P}_{i,j}$ for all factor pairings
$(\mathcal{F}_{i},\mathcal{F}_{j}),i\neq j$ for the WideRN, ASR (RN18),
Factor-VAE and VAE on the FGO and ZGO studies.

| position | hue | lightness | scale | shape | texture
---|---|---|---|---|---|---
| FAAvg | FAMin | FAAvg | FAMin | FAAvg | FAMin | FAAvg | FAMin | FAAvg | FAMin | FAAvg | FAMin
RN18 | $100\pm 0.$ | $100\pm 0.$ | $100\pm 0.$ | $100\pm 0.$ | $93\pm 2.$ | $85\pm 8.$ | $93\pm 1.$ | $89\pm 1.$ | $99\pm 0.$ | $98\pm 0.$ | $39\pm 2.$ | $35\pm 3.$
RN50 | $97\pm 2.$ | $87\pm 12.$ | $100\pm 0.$ | $100\pm 0.$ | $95\pm 1.$ | $93\pm 2.$ | $93\pm 1.$ | $90\pm 1.$ | $99\pm 0.$ | $98\pm 0.$ | $41\pm 2.$ | $36\pm 1.$
RN50-IN | $95\pm 1.$ | $91\pm 2.$ | $96\pm 1.$ | $94\pm 1.$ | $85\pm 3.$ | $78\pm 4.$ | $41\pm 4.$ | $28\pm 4.$ | $51\pm 8.$ | $37\pm 9.$ | $14\pm 1.$ | $4\pm 1.$
AlexNet | $100\pm 0.$ | $100\pm 0.$ | $97\pm 2.$ | $87\pm 12.$ | $82\pm 4.$ | $64\pm 8.$ | $82\pm 3.$ | $69\pm 8.$ | $94\pm 3.$ | $82\pm 11.$ | $14\pm 2.$ | $6\pm 3.$
DenseNet | $100\pm 0.$ | $100\pm 0.$ | $100\pm 0.$ | $100\pm 0.$ | $94\pm 1.$ | $92\pm 2.$ | $92\pm 1.$ | $88\pm 1.$ | $98\pm 0.$ | $97\pm 1.$ | $32\pm 2.$ | $26\pm 2.$
WRN | $100\pm 0.$ | $100\pm 0.$ | $100\pm 0.$ | $100\pm 0.$ | $95\pm 1.$ | $94\pm 1.$ | $95\pm 1.$ | $92\pm 1.$ | $98\pm 0.$ | $98\pm 0.$ | $47\pm 3.$ | $42\pm 2.$
ASR (RN18) | $99\pm 0.$ | $98\pm 0.$ | $34\pm 3.$ | $1\pm 1.$ | $53\pm 4.$ | $28\pm 1.$ | $69\pm 2.$ | $35\pm 6.$ | $70\pm 2.$ | $26\pm 7.$ | $10\pm 1.$ | $0\pm 0.$
Factor-VAE | $100\pm 0.$ | $100\pm 0.$ | $84\pm 4.$ | $61\pm 9.$ | $80\pm 5.$ | $69\pm 7.$ | $55\pm 4.$ | $40\pm 4.$ | $62\pm 5.$ | $44\pm 7.$ | $5\pm 0.$ | $0\pm 0.$
VAE | $100\pm 0.$ | $100\pm 0.$ | $86\pm 3.$ | $63\pm 8.$ | $82\pm 4.$ | $74\pm 5.$ | $57\pm 4.$ | $42\pm 4.$ | $68\pm 5.$ | $52\pm 6.$ | $4\pm 0.$ | $0\pm 0.$

Table A7: Aggregated benchmark metrics $\text{FAAvg}_{i}$ and
$\text{FAMin}_{i}$ for all factors $\mathcal{F}_{i},i\in\\{0,1,...,6\\}$
together with their respective standard errors for all baselines on the FGO-5
study.

| position | hue | lightness | scale | shape | texture
---|---|---|---|---|---|---
| FAAvg | FAMin | FAAvg | FAMin | FAAvg | FAMin | FAAvg | FAMin | FAAvg | FAMin | FAAvg | FAMin
RN18 | $100\pm 0.$ | $100\pm 0.$ | $100\pm 0.$ | $100\pm 0.$ | $96\pm 1.$ | $95\pm 1.$ | $96\pm 1.$ | $95\pm 1.$ | $99\pm 0.$ | $99\pm 0.$ | $47\pm 3.$ | $41\pm 4.$
RN50 | $100\pm 0.$ | $100\pm 0.$ | $100\pm 0.$ | $100\pm 0.$ | $96\pm 1.$ | $95\pm 1.$ | $97\pm 0.$ | $95\pm 1.$ | $99\pm 0.$ | $99\pm 0.$ | $49\pm 3.$ | $41\pm 2.$
RN50-IN | $97\pm 1.$ | $95\pm 1.$ | $97\pm 1.$ | $96\pm 1.$ | $87\pm 2.$ | $83\pm 3.$ | $45\pm 3.$ | $35\pm 4.$ | $59\pm 7.$ | $48\pm 9.$ | $17\pm 2.$ | $8\pm 1.$
AlexNet | $97\pm 2.$ | $87\pm 12.$ | $95\pm 3.$ | $73\pm 15.$ | $90\pm 3.$ | $75\pm 9.$ | $89\pm 1.$ | $85\pm 2.$ | $95\pm 3.$ | $84\pm 11.$ | $28\pm 2.$ | $22\pm 2.$
DenseNet | $100\pm 0.$ | $100\pm 0.$ | $100\pm 0.$ | $100\pm 0.$ | $96\pm 1.$ | $95\pm 1.$ | $95\pm 1.$ | $94\pm 1.$ | $99\pm 0.$ | $98\pm 1.$ | $40\pm 1.$ | $36\pm 1.$
WRN | $100\pm 0.$ | $100\pm 0.$ | $100\pm 0.$ | $100\pm 0.$ | $97\pm 1.$ | $96\pm 1.$ | $97\pm 0.$ | $95\pm 0.$ | $99\pm 0.$ | $99\pm 0.$ | $54\pm 5.$ | $52\pm 5.$
ASR (RN18) | $99\pm 0.$ | $98\pm 0.$ | $45\pm 4.$ | $11\pm 4.$ | $53\pm 3.$ | $31\pm 2.$ | $78\pm 1.$ | $50\pm 3.$ | $74\pm 4.$ | $32\pm 7.$ | $10\pm 0.$ | $0\pm 0.$
Factor-VAE | $100\pm 0.$ | $100\pm 0.$ | $89\pm 3.$ | $72\pm 9.$ | $83\pm 4.$ | $76\pm 6.$ | $62\pm 4.$ | $51\pm 5.$ | $71\pm 5.$ | $60\pm 5.$ | $5\pm 0.$ | $0\pm 0.$
VAE | $100\pm 0.$ | $100\pm 0.$ | $91\pm 2.$ | $76\pm 7.$ | $86\pm 3.$ | $79\pm 5.$ | $64\pm 4.$ | $53\pm 4.$ | $78\pm 5.$ | $70\pm 6.$ | $4\pm 0.$ | $0\pm 0.$

Table A8: Aggregated benchmark metrics $\text{FAAvg}_{i}$ and
$\text{FAMin}_{i}$ for all factors $\mathcal{F}_{i},i\in\\{0,1,...,6\\}$
together with their respective standard errors for all baselines on the FGO-10
study.

| position | hue | lightness | scale | shape | texture
---|---|---|---|---|---|---
| FAAvg | FAMin | FAAvg | FAMin | FAAvg | FAMin | FAAvg | FAMin | FAAvg | FAMin | FAAvg | FAMin
RN18 | $100\pm 0.$ | $100\pm 0.$ | $100\pm 0.$ | $100\pm 0.$ | $98\pm 1.$ | $97\pm 1.$ | $95\pm 2.$ | $84\pm 11.$ | $99\pm 0.$ | $99\pm 0.$ | $54\pm 5.$ | $51\pm 5.$
RN50 | $100\pm 0.$ | $100\pm 0.$ | $100\pm 0.$ | $100\pm 0.$ | $98\pm 1.$ | $97\pm 1.$ | $98\pm 0.$ | $98\pm 0.$ | $100\pm 0.$ | $99\pm 0.$ | $55\pm 5.$ | $52\pm 5.$
RN50-IN | $98\pm 0.$ | $98\pm 1.$ | $98\pm 0.$ | $97\pm 1.$ | $90\pm 2.$ | $88\pm 2.$ | $51\pm 3.$ | $44\pm 3.$ | $67\pm 6.$ | $60\pm 7.$ | $23\pm 2.$ | $15\pm 2.$
AlexNet | $97\pm 2.$ | $87\pm 12.$ | $100\pm 0.$ | $100\pm 0.$ | $90\pm 4.$ | $77\pm 10.$ | $91\pm 3.$ | $80\pm 11.$ | $93\pm 5.$ | $85\pm 11.$ | $38\pm 2.$ | $34\pm 2.$
DenseNet | $100\pm 0.$ | $100\pm 0.$ | $100\pm 0.$ | $100\pm 0.$ | $97\pm 1.$ | $96\pm 1.$ | $97\pm 0.$ | $97\pm 0.$ | $99\pm 0.$ | $99\pm 0.$ | $47\pm 3.$ | $43\pm 3.$
WRN | $100\pm 0.$ | $100\pm 0.$ | $100\pm 0.$ | $100\pm 0.$ | $98\pm 0.$ | $97\pm 1.$ | $98\pm 0.$ | $97\pm 0.$ | $99\pm 0.$ | $99\pm 0.$ | $61\pm 5.$ | $57\pm 6.$
ASR (RN18) | $100\pm 0.$ | $99\pm 0.$ | $59\pm 4.$ | $26\pm 8.$ | $60\pm 3.$ | $38\pm 2.$ | $84\pm 2.$ | $67\pm 5.$ | $82\pm 3.$ | $60\pm 7.$ | $16\pm 1.$ | $1\pm 1.$
Factor-VAE | $100\pm 0.$ | $100\pm 0.$ | $94\pm 2.$ | $84\pm 5.$ | $88\pm 3.$ | $82\pm 5.$ | $71\pm 3.$ | $63\pm 3.$ | $81\pm 4.$ | $75\pm 5.$ | $5\pm 0.$ | $0\pm 0.$
VAE | $100\pm 0.$ | $100\pm 0.$ | $96\pm 1.$ | $89\pm 4.$ | $89\pm 3.$ | $84\pm 4.$ | $72\pm 3.$ | $65\pm 3.$ | $85\pm 3.$ | $81\pm 4.$ | $5\pm 0.$ | $0\pm 0.$

Table A9: Aggregated benchmark metrics $\text{FAAvg}_{i}$ and
$\text{FAMin}_{i}$ for all factors $\mathcal{F}_{i},i\in\\{0,1,...,6\\}$
together with their respective standard errors for all baselines on the FGO-20
study. Figure A9: Examples of Automatic shortcut removal Lens output for
different studies in our benchmark on datasets with following factor
correlations: $(\texttt{shape},\texttt{hue})$ (first row),
$(\texttt{hue},\texttt{shape})$ (second row),
$(\texttt{scale},\texttt{lightness})$ (third row),
$(\texttt{lightness},\texttt{scale})$ (fourth row).