# A Novel Loss Function Utilizing Wasserstein Distance to Reduce Subject-
Dependent Noise for Generalizable Models in Affective Computing

Nibraas A. Khan
Department of Computer Science
Vanderbilt University
Nashville, TN 37235
<EMAIL_ADDRESS>
&Mahrukh Tauseef
Department of Electrical Engineering
Vanderbilt University
Nashville, TN 37235
<EMAIL_ADDRESS>
&Ritam Ghosh
Department of Electrical Engineering
Vanderbilt University
Nashville, TN 37235
<EMAIL_ADDRESS>
&Nilanjan Sarkar
Department of Mechanical Engineering
Vanderbilt University
Nashville, TN 37235
<EMAIL_ADDRESS>

###### Abstract

Emotions are an essential part of human behavior that can impact thinking,
decision-making, and communication skills. Thus, the ability to accurately
monitor and identify emotions can be useful in many human-centered
applications such as behavioral training, tracking emotional well-being, and
development of human-computer interfaces. The correlation between patterns in
physiological data and affective states has allowed for the utilization of
deep learning techniques which can accurately detect the affective states of a
person. However, the generalisability of existing models is often limited by
the subject-dependent noise in the physiological data due to variations in a
subject’s reactions to stimuli. Hence, we propose a novel cost function that
employs Optimal Transport Theory, specifically Wasserstein Distance, to scale
the importance of subject-dependent data such that higher importance is
assigned to patterns in data that are common across all participants while
decreasing the importance of patterns that result from subject-dependent
noise. The performance of the proposed cost function is demonstrated through
an autoencoder with a multi-class classifier attached to the latent space and
trained simultaneously to detect different affective states. An autoencoder
with a state-of-the-art loss function i.e., Mean Squared Error, is used as a
baseline for comparison with our model across four different commonly used
datasets. Centroid and minimum distance between different classes are used as
a metrics to indicate the separation between different classes in the latent
space. An average increase of $14.75\%$ and $17.75\%$ (from benchmark to
proposed loss function) was found for minimum and centroid euclidean distance
respectively over all datasets.

_K_ eywords Machine Learning, Affective Computing, Optimal Transport Theory,
HCI, Wasserstein Distance

##  1 Introduction

Affective state can be defined as the underlying experience of feeling,
emotion, or mood [1]. Its importance stems from the influence affective states
have on a person’s thinking skills, decision making, and communication skills
as well as mental and physical well-being [2]. Accurate monitoring and
identification of affective states can lead to important applications in
behavioral training, computer-based emotional analysis (e.g., stress
detection), and human-computer interfaces that can cater to the emotional
needs of an individual [3, 4].

Affective state can be measured and monitored in two ways: intrusively and
non-intrusively. Intrusive methods measure the concentration of various
hormones in the blood stream that can be used to detect an affective state.
For instance, cortisol levels produced by the hypothalamic-pituitary-
adrenocortical (HPA) axis can be collected through samples of blood, urine,
hair or saliva and used to detect stress [5]. One of the challenges with
intrusive measurement is that it is invasive and cannot be used to monitor the
affective state in real-time.

Non-intrusive methods, on the other hand, include the analysis of behavioral
or physiological data that can lead to non-invasive real-time detection of
affective states. Behavioural actions such as blink rate, facial expression,
gesture, speech, and body pose have been commonly used for affective state
detection [6]. However, all these actions can be masked and voluntarily
controlled by a subject which reduces its reliability [4, 6]. Alternatively,
strong evidence suggests physiological signals to be more reliable for
affective state detection due to its involuntary nature and a strong
correlation of patterns in the data with different affective states [5].

Physiological signals are a response to the Autonomic Nervous System (ANS)
utilizing both motor and sensory neurons to communicate and operate between
the Central Nervous System (CNS) and the various organs or muscles. The ANS is
composed of the Sympathetic (SNS) and Parasympathetic (PNS) nervous systems.
The sympathetic nervous system prepares the body for emergency action, which
results in the “fight or flight” response [7]. The response leads to an
increase in measurable physiological signals, such as heart rate, blood flow,
and increased muscle activation which can be mapped to different affective
states. Whereas, the parasympathetic nervous system helps sustain homeostasis
during rest by decreasing physiological signals and maintaining them in
moderate ranges [8]. Since ANS is involuntarily stimulated, the response
cannot be manipulated or masked by an individual. This has led to the popular
field of affecting computing to focus on dynamically identifying different
affective states using non-invasive wearable sensors that can monitor changes
in physiological signals in response to a stimulus.

The ability to map patterns in physiological data with affective states has
allowed for utilization of machine learning techniques for the detection of
affective states. Research in psychophysiology has led to the compilation of a
comprehensive list of physiological data that can be used to monitor affective
state. This list includes heart activity (ECG), brain activity (EEG), skin
response (EDA), blood pressure variation (PPG), respiratory response, and
muscle activity (EMG) [5]. Several datasets have been compiled by collecting
physiological data while inducing an affective state. For instance, one of the
state-of-the-art datasets, Wearable Stress and Affect Detection (WESAD) [9],
induced amusement by making the subjects watch funny video clips and they
induced stress through public speaking and mental arithmetic tasks. Meanwhile,
their physiological data i.e., blood volume pulse, ECG, EDA, EMG, respiration,
body temperature, and three-axis acceleration was collected and classified
simultaneously. Other common datasets like Database for Emotional Analysis
using Physiological Signals (DEAP) [10], Affect, Personality and Mood Research
on Individuals and Groups (AMIGOS) [11], and Cognitive Load, Affect, and
Stress Recognition (CLAS) [12] follow suit by using video clips or stress-
inducing tasks to collect and classify physiological data for the induced
affective states.

Numerous machine learning techniques have been utilised to detect affective
states from physiological data. These approaches include support vector
machines (SVM), random forest (RF), k-nearest neighbors (KNN) and Linear
Discriminant Analysis (LDA) that need handcrafted features from the pre-
processed signal in order to remove noisy data [6]. However, there is no
consensus on the list of features extracted from physiological data that can
be accurately mapped to affective states which results in reduced performance
[13]. Alternatively, with the advancement of deep learning, there has been an
increasing interest in using deep learning techniques like long short-term
memory (LSTM), autoencoders, and convolutional neural networks (CNN) [3, 14].
These models allow for automatic feature extraction based on the model’s
ability to automatically comprehend patterns in data with respect to the
labels.

Even though deep learning techniques show promise, the lack of
generalisability still exists and contributes to poor performance. This is
because the subjects might not exhibit the same physiological response for a
stimuli [13]. Thus, subject dependent noise can lead to poor generalisability
of the model. This issue can be resolved by using a loss function that filters
out the features of data that are person specific and do not contribute much
to affective state detection. In other words, if each subject’s data is
treated as a distribution, the loss function assigns higher importance to
features that are closer in distance to the group distribution (distributions
of all subjects) while assigning lower importance to features that are much
further apart across from the group distribution. The distance between
distributions can be calculated by using the Wasserstein distance [15].

In this paper, we introduce a novel loss function that accounts for subject
dependent noise in the data in order to develop more generalizable models.
This is accomplished by training an autoencoder model with a loss function
that utilizes Wasserstein Distance to scale the importance of subject
dependent patterns to obtain a latent space with reduced dimensions and less
noisy subject independent features. The performance of the model is tested on
four different datasets (WESAD, AMIGOS, CLAS, and DEAP) using the centroid and
minimum distances between classes as metrics.

This paper is structured as follows. Section 2 presents an overview of
existing literature that informed the development of the proposed model. This
is followed by the 3 section that discusses the mathematical and algorithmic
details of the model. The results are shown and discussed in Section 4
followed by concluding remarks in Section 5.

##  2 Background

As mentioned before, machine learning techniques for affective state detection
using physiological data has been a topic of immense interest in the past few
years. Kolodyazhniy et al. [16] conducted a notable study in 2011 on the topic
by compiling a dataset and training several machine learning models like LDA,
Quaratic Discriminant Analyis (QDA), Multilayer Perceptron (MLP), Radial Basis
Function (RBF), and KNN. They used a variety of features extracted from the
physiological data to discover the features that are strongly correlated to
affective state. They reported maximum accuracy of 81.9% using KNN with 7
features on subject-dependent classification, but they reported the accuracy
to go down to 78.9% for subject independent classification.

Ever since, several novel techniques have been implemented to improve the
performance of subject independent classification. Bota et al. [17] conducted
a review of existing machine learning and deep learning techniques for
affective state detection from 2001 to 2019. They reported lower performance
for subject independent classification as compared to subject dependent
classification for most of the models.

Li et. al [18] proposed a technique that assigned variable learnable weights
to different physiological signals fed to an attention-based bidirectional
LSTM model. They reported an accuracy of 81.1% for subject-independent
classification using the AMIGOS dataset. However, the model’s accuracy across
other datasets is not known.

Reviews conducted by both Bota et al. [17] and Xin et al. [19] highlighted the
lack of generalisation to be a consistent issue with the existing techniques
for affective state detection. This has led to several works proposing novel
techniques to increase generalisation of affective state detection models.

Li et al. [20] used different automatic feature selection techniques such as
Chi-Squared-Based Feature Selection, Mutual Information-Based Feature
Selection, ANOVA F-Value-Based Feature Selection, Recursive Feature
Elimination (RFE), and L1-Norm Penalty-Based Feature Selection. These
techniques were used such that the most important features needed for
affective state detection can be extracted and used to train an SVM. They
reported an accuracy of 83.3% for subject independent classification using the
SEED dataset and 59.06% using the DEAP dataset.

In addition, several domain adaptation techniques are being used in order to
select features such that the difference between the distributions of each
subject’s physiological data is minimized. Chai et al. [21] proposed a
Subspace Alignment Autoencoder to minimize the distribution mismatch of the
physiological data of each subject by minimizing the maximum mean discrepancy
of reproduced kernel Hilbert space (RKHS). This allowed the alignment of
different distributions by mitigating the subject-specific noise from the
data. They reported an accuracy 77.8% using the SEED dataset.

Based on the literature survey, we identified the potential of decreasing the
distance between each subject’s distribution of physiological data and use the
extracted features to train a more generalised model. However, this entails
that a robust method needs to be chosen to calculate the distance between
distribution. One such method, Wasserstein Distance, can be retrieved from
Optimal Transport Theory (OTT) [15]. Wasserstein Distance allows for the
calculation of the distance between distributions while also considering the
geometry of the distribution (symmetrical and triangular) [22]. Kolouri et al.
[22] demonstrated the potential of Wasserstein Distance to minimize the
distance between an input and target distribution for generative modeling
using autoencoders.

Hence, in this paper, we propose a novel loss function using Wasserstein
Distance for automatic feature extraction by assigning more weight to features
that are closer in distance to each other across all subjects. An autoencoder
model is used to generate a latent space that excludes all the features that
are influenced by subject-dependent nuances. The goal is to increase the
separation between different affective states in the latent space such that a
more generalised classification model can be trained.

##  3 Methods

The datasets are publicly available through their respective organizations,
and the code used for the presented work is publicly provided
(https://anonymous.4open.science/status/NeurIPS-F252).

###  3.1 Optimal Transport Theory

Optimal Transport Theory was first formalized by the French mathematician,
Gaspard Monge, in 1781, right before the French revolution [15]. The
motivation to study this was to come up with a transport plan to move a mass
from one point to another with minimum cost. The outcome of this study
formulated a method for measuring the distance between two probability
distributions. In the formulation, consider two probability measures $\mu$ and
$v$ defined on spaces X and Y respectively. $\mu$ and $v$ have density
functions $f$ and $g$ where $d\mu=f(x)dx$ and $dv=g(y)dy$, where $(x,y)\in
X$x$Y$. A cost function is defined, $c(x,y)$ which finds the distance between
a point x and y. The initial choice by Monge was Euclidean distance. A
transport plan $T:X\longrightarrow Y$ allows for moving a mass, M, from X to Y
such that the measure $\mu$ of the mass in space X is the same as the measure
$v$ of the mass in space Y after the mass is transported from X to Y:

$\mu[M]=v[T(M)]$ (1)

Once a plan T has been found, the cost associated with it is:

$I(T,f,g,c)=\int_{X}c(x,T(x))f(x)dx$ (2)

While there can be many solution to T, the aim is to minimize the overall cost

$I(f,g,c)=\inf_{T\in M}\int_{X}c(x,T(x))f(x)dx,$ (3)

where $M$ is the set of all transport plans T that transfers $f$ to $g$. In
the formulation, the optimal cost of transporting $x$ to $y$ is also known as
the Wasserstein Distance.

$W_{p}(\mu,v)=(\inf_{T\in
M}\int_{\mathbb{R}^{n}}|x-T(x)|^{p}d\mu(x))^{\frac{1}{p}},\mu,v\in\mathcal{P}_{p}(X),$
(4)

where $\mathcal{P}_{p}(X)$ is the set of probability measures with finite
moments of order $p$ [23]. In this paper, we will focus on using the
Wasserstein Distance to measure the distance between a subjects’ individual
distribution and the group’s distribution.

Figure 1: Architecture of the Autoencoder model with a classifier attached to
the latent space.

###  3.2 Models

In this work, we focus on using an autoencoder architecture to reduce noise
along with our novel cost function through dimensionality reduction. In
addition to a reconstruction loss, a classifier is attached to the latent
space to ensure it is discriminant. Figure 1 shows the architecture of our
model, and both components (encoder and classifier) are trained
simultaneously. The proposed cost function is not restricted to the
autoencoder model and can be used with any suitable machine learning
algorithm. The following section will address how the loss function can be
integrated in the autoencoder architecture. We use a standard autoencoder
model with Mean Squared Error (MSE) for a baseline to compare it with our
novel cost function.

###  3.3 Autoencoder Error Decomposition

The group reconstruction error, $r_{g}$, of the autoencoder for the subject-
independent model is given as

$r_{g}=\frac{1}{N}\sum\limits_{n=1}^{N}\mathcal{L}_{a}(a;g_{n,x},g^{\prime}_{n,x})$
(5)

where $N$ is the total number of samples across all subjects,
$\mathcal{L}_{a}$ is any loss function for reconstruction (Mean-Squared
Error), and $a$ is the encoder and decoder architecture of the network. The
reconstruction error will be used in conjunction with the classifier loss. The
classifier head, attached to the latent space, is used to calculate the group
classifier error and subject specific classifier error. The group classifier
uses all labeled samples available across all subjects.

$c_{g}=\frac{1}{N}\sum\limits_{n=1}^{N}\mathcal{L}_{c}(c;g_{n,x},g_{n,y})*\lambda_{g}$
(6)

where $\mathcal{L}_{c}$ is any classification loss function (Categorical
Cross-Entropy) and $\lambda_{g}$ is a regularizer term used to scale the
importance of the group classifier error during backpropagation. The
definition of $\lambda_{g}$ is given in equation 11. Subject $i$’s specific
loss is computed using all labeled samples available $S_{i}$ and any loss
function $\mathcal{L}_{c}$.

$c_{s,i}=\frac{1}{N_{i}}\sum\limits_{n=1}^{N_{i}}\mathcal{L}_{c}(f;s_{n,x},s_{n,y})*\lambda_{s,i}$
(7)

$\lambda_{s,i}$ is used to scale the importance of a specific subject in the
calculation of the overall loss for backpropogation. The overall
classification loss is given as:

$c_{s}=\sum\limits_{i=i}^{S}c_{s,i}$ (8)

where $S$ is the number of subjects. Optimal Transport Theory, specifically,
Wasserstein Distance, is used to calculate $\lambda_{s,i}$ by measuring the
distance between the group distribution and subject distribution. Equations 9
and 10 denote the formulation for $\lambda_{s,i}$.

$\alpha_{s,i}=EMD(G,S_{i})=\inf_{\gamma\in\phi(G,S_{i})}\mathbb{E}_{(G,S_{i})\sim\gamma}[||x-y||]$
(9)

$\alpha_{s,i}$ is normalized and subtracted from 1 to ensure that the further
away a subject $i$ is from the group, the smaller the scaling factor,
$\lambda_{s,i}$, is.

$\lambda_{s,i}=1-\frac{\alpha_{s,i}}{|\sum\limits_{i=1}^{S}\alpha_{s,i}|}$
(10)

Finally, $\lambda_{g}$ is determined by ensuring that the regularizer terms
sum to 1.

$\lambda_{g}+\sum\limits_{i=1}^{S}\lambda_{s,i}=1$ (11)

###  3.4 Datasets

Experiments in this work were conducted on popular, public physiological
datasets: WESAD, AMIGOS, CLAS, and DEAP.

WESAD is a multimodal dataset for wearable stress and affect detection with
data recorded using a wrist-worn device (sensors: PPG, accelerometer,
electrodermal activity, and body temperature) and a chest-worn device
(sensors: ECG, accelerometer, EMG, respiration, and body temperature). The
dataset consists of 15 subjects (aged 24–35 years) with 100 min of data each.
The dataset was recorded with the goal of detecting and distinguishing between
three affective states: neutral, stress, amusement which were denoted using
self-reports for subjective experience during the emotional stimulus along
with the study protocol of inducing emotion [9].

The AMIGOS dataset consists of data from sensors (EEG, ECG and GSR), full-body
videos, and depth videos from 40 volunteers. Data was collected while watching
16 short videos and 37 of the 40 volunteers watching 4 long-videos. Both self-
assessment (valence-arousal, control, familiarity, like-dislike, and selection
of basic emotions) and external assessment of participants’ levels of valence
were used to annotate the dataset. The participants were also asked to fill
forms with Personality Traits and Positive and Negative Affect Schedule
(PANAS) questionnaires [11].

DEAP contains EEG, GSR, RESP, SKT, EMG, EOG, BVP data from 32 volunteers while
they were watching 40 one minute-long music videos. Additionally, frontal face
video was recorded for 22 of the participants. The dataset was self-annotated
after each video with the following labels: arousal, valence, like-dislike,
familiarity, and dominance by the volunteers [10].

PPG, EEG and GSR data from 60 patients engaged in cognitively difficult tasks
from was collected in the CLAS dataset. The cognitively difficult tasks used
in this dataset is: Stroop test, math test, logic problem test, and
emotionally evoking stimuli [24]. The dataset was labeled based on the study
protocol (i.e., high cognitive load during logic and math problems and low
cognitive load during neutral stimuli).

##  4 Results

The results presented in this work use a standard 10-fold Leave-One-Subject-
Out (LOSO) where the models were trained on all subjects excluding one and
testing on the excluded subject.

The dimensionality of the latent space is too high to be visualized, so
Principal Component Analysis (PCA) is used to condense the space into three
components. Figure 2 compares the latent space using MSE with our cost
function. Figure 2(a) - 2(d) shows a large overlap of classes which would lead
to imprecise decision boundaries. Specifically consider Figure 2(b) where all
classes are highly interrelated. In contrast, the latent space in 2(f) shows a
clear separation of classes by utilizing Wasserstein Distance. While the
advantages of the novel cost function is distinct in three dimensions, the
latent space has higher dimensionality, and the improvements are expressed
using the euclidean distance between centroids of classes and the euclidean
distance between closest points between different classes.

Figure 3 highlights the percent increase between the centroids of classes in
all datasets with the novel cost function for training and testing data. For
all datasets, there is a significant increase in the distance over MSE.
Additionally, Figure 4 shows the increase in the minimum distance between
samples of classes.

The accuracy of our models is presented in Table 1 through a comparison of MSE
with our novel cost function using LOSO. The table shows that the proposed
cost function is able to achieve better performance on most datasets and
performed the same as the MSE in the worst case scenario. The experiments
presented in this work were conducted on a computer with a AMD Ryzen 9 5900HX,
NVIDIA GeForce RTX 3070, and 32 GB of RAM with an approximate run time of 45
minutes per dataset.

(a) WESAD MSE

(b) DEAP MSE

(c) CLAS MSE

(d) AMIGOS MSE

(e) WESAD Custom

(f) DEAP Custom

(g) CLAS Custom

(h) AMIGOS Custom

Figure 2: Autoencoder latent space visualization using PCA

(a) WESAD

(b) DEAP

(c) CLAS

(d) AMIGOS

Figure 3: Distance increase when using the novel cost function over MSE
expressed as a percentage for training and testing data (LOSO) for class
centroids.

(a) WESAD

(b) DEAP

(c) CLAS

(d) AMIGOS

Figure 4: Distance increase when using the novel cost function over MSE expressed as a percentage for training and testing data (LOSO) for the minimum distance between classes. Table 1: A comparison of the accuracy between using the standard MSE for reconstruction loss and the novel cost function. | MSE Accuracy | Novel Cost Accuracy
---|---|---
WESAD | 80.31% $\pm$ 1.13% | 84.11% $\pm$ 1.05%
CLAS | 77.23% $\pm$ 0.76% | 79.63% $\pm$ 0.86%
AMIGOS | 79.56% $\pm$ 1.44% | 79.97% $\pm$ 1.75%
DEAP | 66.22% $\pm$ 1.87% | 71.64% $\pm$ 1.81%

##  5 Conclusion and Future Works

The lack of generalizability of affective state detection models has been a
consistent issue. It is often due to a large variation in the distribution of
physiological data for a set of individuals in reaction to the same stimuli.
This leads to subject dependent noise that affects the performance of a model.
Our approach employs autoencoders that can inherently reduce noise and extract
relevant features through dimensionality reduction. In order to further reduce
the subject-dependent noise, we introduce a novel cost function that uses
Optimal Transport Theory to lower the importance of uncommon patterns across
individuals. This results in a latent space of affective state classes with
increased separability.

The model was trained and tested on four different datasets and the
performance of the proposed cost function was compared to the state-of-the-art
MSE. From our study, we show the proposed cost function significantly
increases the distance between classes in the latent space across all
datasets. An average increase of $14.75\%$ and $17.75\%$ (from benchmark to
proposed loss function) was found for minimum distance between classes and
centroid euclidean distance respectively.

Additionally, our proposed cost function increases the robustness of affective
computing models allowing for generalizability as the population increases.
The larger the population, the more diverse the physiological responses to
stimuli. This results in overlap between class clusters leaving no space to
draw accurate decision boundaries. Since state-of-the-art models do not have a
mechanism to accommodate this increased noise, their performance might
decreases as the population size increase. Alternatively, our proposed loss
function mitigates the affect of increased subject-dependent noise while
increasing the performance by allowing the model to train on more data.

However, commonly used public physiological datasets do not contain samples
from a large population as the compilation of a large dataset is expensive. In
our experiments, we used datasets with a smaller population size and were
still able to show an increase in the distance between classes. Even though
there was a significant increase in the separation between the classes, the
accuracy of a multilayer perceptron trained on the latent space was nearly
equivalent to that of the state-of-the-art models (see Table 1). We were not
able to highlight the full extent of the cost function due to the lack of a
large scale physiological dataset. Future works include implementation of more
complex encoder and decoder models (for e.g., LSTMs and CNNs), exploration of
different classifiers on the latent space in order to further increase the
accuracy, and investigation of the cost function on a large scale dataset.

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