# Uncertainty-Aware PPG-2-ECG for Enhanced Cardiovascular Diagnosis using
Diffusion Models

David S. Hippocampus
Department of Computer Science
Cranberry-Lemon University
Pittsburgh, PA 15213
<EMAIL_ADDRESS>
Use footnote for providing further information about author (webpage,
alternative address)— _not_ for acknowledging funding agencies.

###### Abstract

Analyzing the cardiovascular system condition via Electrocardiography (ECG) is
a common and highly effective approach, and it has been practiced and
perfected over many decades. ECG sensing is non-invasive and relatively easy
to acquire, and yet it is still cumbersome for holter monitoring tests that
may span over hours and even days. A possible alternative in this context is
Photoplethysmography (PPG): An optically-based signal that measures blood
volume fluctuations, as typically sensed by conventional “wearable devices”.
While PPG presents clear advantages in acquisition, convenience, and cost-
effectiveness, ECG provides more comprehensive information, allowing for a
more precise detection of heart conditions. This implies that a conversion
from PPG to ECG, as recently discussed in the literature, inherently involves
an unavoidable level of uncertainty. In this paper we introduce a novel
methodology for addressing the PPG-2-ECG conversion, and offer an enhanced
classification of cardiovascular conditions using the given PPG, all while
taking into account the uncertainties arising from the conversion process. We
provide a mathematical justification for our proposed computational approach,
and present empirical studies demonstrating its superior performance compared
to state-of-the-art baseline methods.

##  1 Introduction

Cardiovascular diseases are a significant public health problem, affecting
millions of people worldwide and remaining a leading cause of mortality
tsao2022heart. The analysis of cardiovascular conditions using
Electrocardiography (ECG) signals has emerged as one of the most widely
utilized diagnostic tool kligfield2007recommendations, and its practice and
efficiency have been continuously refined over many decades.

While ECG sensing is non-invasive and relatively easy to acquire, it is time-
consuming and requires the expertise of trained professionals with specialized
skills in order to ensure accurate diagnosis. Consequently, there has been a
surge in the development of numerous contemporary wearable ECG systems in
recent decades for digital diagnosis. However, the materials used to deliver a
high-quality signal via electrodes can often lead to skin irritation and
discomfort during extended usage, thereby limiting the long-term viability of
these devices zhu2019ecg.

A possible alternative to ECG sensing and diagnosis is the use of
Photoplethysmography (PPG), which is an optically-based non-invasive signal
associated with rhythmic changes in blood volume within tissues
reisner2008utility. Unlike ECG, PPG signals are easier to obtain, convenient,
and cost-effective, as they are widely available in clinics and hospitals and
can be sensed through finger/toe clips. Moreover, the popularity of PPG is
increasing with the emergence of wearable devices like smart watches, enabling
continuous long-term monitoring without causing skin irritations.

Naturally, the PPG and ECG signals are inter-related, as the timing,
amplitude, and shape characteristics of the PPG waveform contain information
about the interaction between the heart and the blood vessels. These features
of the PPG have been leveraged to measure heart rate, heart rate variability,
respiration rate karlen2013multiparameter, blood oxygen saturation
aoyagi2002pulse, blood pressure payne2006pulse, and to assess vascular
function marston2002ppg; allen1993development. Despite these capabilities,
using PPG-based monitoring during daily activities and light physical
exercises poses unavoidable uncertainty challenges due to signal inaccuracies
and inherent noise, stemming from their indirect nature, compared to ECG
signals sourced directly from the heart.

As the use of wearable devices capturing PPG signals is becoming widespread,
there is an emerging interest in accurately converting PPG sources into ECG
signals chiu2020reconstructing; li2020inferring; zhu2021learning;
sarkar2021cardiogan; vo2021p2e; tang2022robust; ho2022quickly; tian2022cross;
vo2023ppg; lan2023performer; abdelgaber2023subject; hinatsu2023heart;
shome2023region. However, such a mapping is not injective111An injective
mapping implies a one-to-one correspondence between the two domains. As an ECG
is richer than PPG in the information it carries about the Cardiovascular
system, any PPG signal may correspond to a variety of ECG ones., and the
conversion of PPG to ECG can be regarded as an ill-posed inverse problem. This
implies that multiple ECG solutions may correspond to the same input PPG
signal. Nevertheless, previous work in this domain has tended to focus on
providing a single ECG solution to the conversion task, ignoring the inherent
uncertainty in the conversion and thus failing to capture the complete
information contained in the PPG signal. Such studies suggest optimizing
neural models either via regularized regression with an $L_{1}$/$L_{2}$ loss
li2020inferring; chiu2020reconstructing; zhu2021learning; tang2022robust;
ho2022quickly; tian2022cross; lan2023performer; abdelgaber2023subject;
hinatsu2023heart or relying on a generative approach that produces a single
solution that approximates a sample from the posterior distribution of ECG
given the PPG signal sarkar2021cardiogan; vo2021p2e; vo2023ppg;
shome2023region. Here, we argue that both these strategies are lacking, and
the conversion in question calls for a different approach.

In this paper, we introduce _Uncertainty-Aware PPG-2-ECG_ (UA-P2E), a novel
PPG-2-ECG conversion methodology, aimed at enhancing the classification of
cardiovascular conditions using ECG signals derived from PPG data. Unlike
prior work, our approach aims to address both the uncertainty arising from the
conversion task, regarding the spread and variability of the possible ECG
solutions given the PPG angelopoulos2022image; belhasin2023principal, and also
the uncertainty in the classification task, enabling the selection of PPG
samples with low classification risk and high confidence geifman2017selective.

Figure 1: Illustration of our proposed UA-P2E for PPG-2-ECG conversion-
classification framework.

To achieve the above, we leverage a conditional diffusion-based methodology
sohl2015deep; ho2020denoising; song2020denoising. These generative models have
recently emerged as the leading synthesis approach in various tasks
dhariwal2021diffusion; yang2023diffusion, and have been shown to enable high-
fidelity posterior modeling for inverse problems yang2023diffusion;
kawar2021snips; kawar2022denoising; song2022pseudoinverse; chung2022diffusion.
We utilize this capability to produce a multitude of posterior sampled ECG
results that correspond to the input PPG, and provide classification decisions
based on this cluster of outcomes. Additionally, we introduce several methods
for displaying the resulting ECG candidates for better human interpretability.
Figure 1 provides an illustration of our approach to the conversion-
classification process.

Our computational methodology is grounded on a mathematical proof showcasing
the optimality of the expected classification score derived from our proposed
approach for classifying cardiovascular conditions from source PPG signals.
This optimality relies on the assumption that we have access to a perfect
posterior sampler and ECG classification models. This stands in contrast to
the single-solution approach used by prior work, as described above, which
necessarily performs worse.

Our work presents a thorough empirical study demonstrating superior
performance in classification and uncertainty estimation compared to baseline
methods. Our main results are demonstrated on the “Computing in Cardiology”
(CinC) dataset goldberger2000physiobank, which provides the most challenging
classification data in cardiology.

In summary, our contributions are the following: (1) We introduce a state-of-
the-art diffusion-based methodology for PPG-2-ECG conversion-classification
that accounts for the uncertainty in the conversion task, thereby enhancing
performance; (2) We provide a mathematical proof that justifies our proposed
computational process for the classification task; (3) We offer effective
methods for displaying the ECG solutions obtained, enhancing interpretability
while taking into account the uncertainty perspective; and (4) We present an
empirical study that supports our mathematical proof and demonstrates the
superiority of our methodology over baseline strategies.

##  2 Related Work

Naturally, our work is not the first one to consider the conversion of PPG
signals to ECG ones. Below we outline the main trends in this line of work,
and add a discussion on uncertainty quantification, as we aim to practice in
this paper.

Regression: Earlier approaches to the PPG-2-ECG conversion problem primarily
focused on approximating either the Minimum Mean Absolute Error (MMAE)
estimator using $L_{1}$ loss li2020inferring; chiu2020reconstructing;
ho2022quickly; lan2023performer or the Minimum Mean Squared Error (MMSE)
estimator with $L_{2}$ loss zhu2021learning; tang2022robust; tian2022cross;
abdelgaber2023subject; hinatsu2023heart. These works commonly employed
regularization techniques to enhance perceptual outcomes. However, the work
reported in blau2018perception which discovered the distortion-perception
tradeoff has shown that when optimizing for distortion (of any kind),
perceptual quality is necessarily compromised, thus pushing the solutions in
the above methods to signals that are likely to drift out of the valid ECG
manifold, thus weakening their classification performance.

Generation: The first attempt at PPG-2-ECG conversion using generative models
was introduced by the authors of CardioGAN sarkar2021cardiogan. They employed
an adversarial training scheme with an encoder-decoder architecture.
Similarly, P2E-WGAN vo2021p2e utilized conditional Wasserstein GANs for this
task. vo2023ppg introduced a variational inference methodology to approximate
the posterior distribution with a Gaussian prior assumption. The authors of
RDDM shome2023region have first adapted a diffusion-based model that
accelerates a DDPM ho2020denoising diffusion process for this conversion task.
However, despite the success of the generative approach, all the above works
have chosen to use a single (possibly random) ECG solution from the posterior
distribution, thus missing the complete picture of the possibilities of the
different ECGs given the PPG signal.

ECG Domain: Due to the near periodicity of typical ECG signals, and the
tendency in the medical community to encapsulate these signals by identifying
key intervals (e.g., PQRST alghatrif2012brief complexes) as valid
representations for them, previous work debated with the question of the
domain to adopt in handling such data. As opposed to such papers
chiu2020reconstructing; li2020inferring; zhu2021learning; ho2022quickly;
shome2023region, in this work we operate in the raw data domain (sampled
signals), aiming to preserve all the details that may eventually influence
later classification.

Uncertainty Quantification: In the framework of the conversion-classification
explored in this paper, uncertainty quantification serves a dual purpose.
Firstly, it seeks to characterize the range, spread, and variability of
potential solutions for a given input PPG signal angelopoulos2022image;
belhasin2023principal. Secondly, it also aims at assessing the confidence in
the correctness of classification predictions, e.g. via a selective
classification geifman2017selective scheme, which we adopt in this paper.
Building upon principles from these two threads of work, we characterize the
diverse solutions that can potentially describe the input, leading to an
improved methodology within the conversion-classification framework.

##  3 Problem Formulation and Goals

Let $X\in\mathcal{X}\subseteq\mathbb{R}^{d}$ be a random vector representing
an ECG signal that follows the prior distribution $\pi(X)$. We assume access
only to its observation $Y=h(X)\in\mathcal{Y}\subseteq\mathbb{R}^{d}$, where
$h$ is an unknown, possibly stochastic, and non-invertible function. The
observation $Y$ represents a PPG signal that follows the distribution
$\pi(Y|X)$. In this paper, our first and primary objective is to provide a
comprehensive and interpretable estimation of the ECG signal(s) that
corresponds to the given $Y$, focusing on the approximation of the posterior
distribution $\pi(X|Y)$. More specifically, our PPG-2-ECG conversion is
designed to assist medical professionals in understanding and interpreting the
different ECG signal variations obtained through this conversion process,
emphasizing the uncertainty inherent in the spread and variability of these
ECG signals.

Prior work in this field (see Section 2) adopted a _single-solution_ approach
for the conversion task, focusing on estimating a single reconstructed ECG
solution from $\pi(X|Y)$, typically achieved through regression or generation
models. Here, we propose a generalization of this conversion approach using a
conditional stochastic generative model, denoted as
$g:\mathcal{Y}\times\mathcal{Z}\rightarrow\mathcal{X}$ s.t.
$g(Y,Z)=\hat{X}\sim\hat{\pi}(X|Y)$, where $Z\in\mathcal{Z}$ denotes a
stochastic component that typically follows a normal distribution
$\pi(Z):=\mathcal{N}(0,\mathbf{I})$, and $\hat{X}$ represents a potential ECG
solution derived as a sample from the approximate posterior distribution
$\hat{\pi}(X|Y)$. In this context, a single ECG solution may be derived by
fixing $Z=z$ in $g(Y,Z)$, or by marginalizing over the random seed, i.e.,
$\mathbb{E}_{z\sim\pi(Z)}[g(Y,Z)]$. In contrast, allowing $Z$ to be random may
provide a more comprehensive set of solutions to the inverse problem, exposing
the inherent uncertainty within this conversion task. Our prime goal in this
work is therefore to train the stochastic conversion model $g$ using a
training data consisting of many (ECG, PPG) signal pairs, denoted by
$\\{(X_{i},Y_{i})\\}_{i=1}^{n}$.

A second objective in this work is classification, enabling diagnosis of
various cardiovascular conditions via the incoming PPG signals. Consider a
binary classification problem where the input is an observation
$Y\in\mathcal{Y}$ with a corresponding binary label $C\in\\{0,1\\}$ that
follows a conditional distribution $\pi(C|Y)$. In this classification setting,
we aim to model a classifier
$f_{\mathcal{Y}}:\mathcal{Y}\rightarrow\mathbb{R}^{+}$ for approximating the
posterior distribution, denoted as $\hat{\pi}(C|Y)$.

Addressing this classification task, one might be tempted to adopt a
straightforward and naive strategy, of training directly on data that consists
of PPG signals and their associated labels. Nevertheless, appropriate
cardiovascular labels linked with PPG signals are relatively rare and hard to
acquire. An alternative is to use labels that were produced for ECG signals,
$X$, and tie these to their matched PPG, $Y$. However, as the mapping between
PPG and ECG is not injective, this implies that other ECG signals (possibly
with a different label) could have led to the same PPG. As a consequence,
these labels are necessarily noisy, impairing the learning task.

A presumably better alternative is to operate in the reconstructed ECG domain,
hoping for improved performance. Indeed, the classification proposed in prior
work operates on the single estimated ECG $\hat{X}$, approximating this way
the posterior $\pi(C|\hat{X})$. This is achieved using a classifier
$f_{\mathcal{X}}:\mathcal{X}\rightarrow\mathbb{R}^{+}$, trained over data of
the form $\\{(X_{i},C_{i})\\}_{i=1}^{n}$, where $X_{i}$ represents a grounded
ECG signal and $C_{i}$ denotes its associating label. As we rely now on labels
that do match their corresponding signals, this may seem like a better
approach. However, the same flaw as above applies here – the single-produced
ECG, $\hat{X}$, captures a partial truth about the given PPG, thus weakening
the classification. Here is a formal definition of this classification
strategy, as practiced in prior work:

######  Definition 3.1 (Single Score Classifier).

We define a stochastic classifier $f_{\mathcal{Y}}$ as a Single Score
Classifier (SSC) if it is given by

$\displaystyle
f_{\mathcal{Y}}(Y)=f_{\mathcal{X}}(\mathbb{E}_{Z^{\prime}\sim\pi(Z^{\prime})}[g(Y,Z^{\prime})]).$

When choosing $\pi(Z^{\prime})=\pi(Z)$, the term
$\mathbb{E}_{Z^{\prime}\sim\pi(Z^{\prime})}[g(Y,Z^{\prime})]$ becomes the MMSE
estimate of the ECG, and when $\pi(Z^{\prime})=\delta(Z-W)$ for $W\sim\pi(Z)$,
we get a single sample from the approximate posterior distribution,
$\hat{\pi}(X|Y)$.

Following the data processing inequality beaudry2011intuitive, any
transformation
$\hat{X}=\mathbb{E}_{Z^{\prime}\sim\pi(Z^{\prime})}[g(Y,Z^{\prime})]$
satisfies the inequality

$\displaystyle I(Y;C)\geq I(\hat{X};C),$ (1)

where $I(U;V)$ denotes the mutual information between the random variables $U$
and $V$. The above inequality implies that the classification information
contained within a single reconstructed ECG signal $\hat{X}$, as utilized by
prior work, is always weakly lower than the classification information
contained within the original PPG signal $Y$. This suggests that despite the
transformation from PPG to ECG, there may be some loss or reduction in the
discriminative power for classification tasks due to information processing
constraints.

To address this challenge, we adopt a _multi-solution_ formulation approach
that takes into account all possible ECG solutions that can be derived from
the PPG signal $Y$. Below is the formal definition of our classification
strategy.

######  Definition 3.2 (Expected Score Classifier).

We define a classifier $f_{\mathcal{Y}}$ as an Expected Score Classifier (ESC)
if it is given by

$\displaystyle
f_{\mathcal{Y}}(Y)=\mathbb{E}_{Z\sim\pi(Z)}\left[f_{\mathcal{X}}(g(Y,Z))\right].$

Intuitively, this strategy averages the classification scores of posterior
samples that emerge from the posterior distribution $\pi(X|Y)$, rather than
averaging the samples themselves. As the following Theorem claims, this
classification approach is optimal.

######  Theorem 3.1 (Optimality of the Expected Score Classifier).

Consider the Markovian dependency chain $Y\rightarrow X\rightarrow C$,
implying that $\pi(C|X,Y)=\pi(C|X)$, and assume the following:

  1. 1.

The conversion model $g$ is a posterior sampler, i.e.,
$g(Y,Z)=\hat{X}\sim\pi(X|Y)$; and

  2. 2.

The classification model $f_{\mathcal{X}}$ is optimal by satisfying
$f_{\mathcal{X}}(X)=\pi(C=1|X)$,

Then the obtained classification is optimal, satisfying
$f_{\mathcal{Y}}(Y)=\pi(C=1|Y)$.

Supported by Equation (1), Theorem 3.1 implies that our approach utilizes the
full extent of the information within the observation $Y$, and is therefore
optimal. In particular, this demonstrates that our approach is superior to the
single-solution classification strategy (Definition 3.1), which has been
adopted by prior work. See Appendix A for the proof of the above claim, along
with an extension showing the ESC optimality in practical settings.

##  4 UA-P2E: Uncertainty-Aware PPG-2-ECG

We now turn to introduce _Uncertainty-Aware PPG-2-ECG_ (UA-P2E), our
methodology for PPG-2-ECG conversion, aiming to provide a comprehensive
understanding of the various ECG solutions derived from a given PPG signal.
UA-P2E may assist medical professionals in analyzing diverse ECG signal
variations resulting from the conversion process. Additionally, our method can
be leveraged to enhance cardiovascular diagnostic accuracy and confidence
assessments, facilitating more informed clinical decisions.

While our proposed approach is applicable using any conditional stochastic
generative estimator, denoted as
$g:\mathcal{Y}\times\mathcal{Z}\rightarrow\mathcal{X}$, we focus in this work
on conditional diffusion-based models sohl2015deep; ho2020denoising;
song2020denoising, which have emerged as the leading synthesis approach in
various domains dhariwal2021diffusion; yang2023diffusion. Our diffusion
process is trained and evaluated on raw signal data of fixed-length
$Y,X\in\mathbb{R}^{d}$ PPG and ECG signal pairs, sampled at $125$Hz, thus
covering $d/125$ seconds.222in our experiments $d=1024$ and thus the signals
cover $\approx 8$ seconds. The central part of the diffusion process involves
a conditional denoiser, denoted as $\epsilon(X_{t},Y,t)$. This denoiser is
trained to predict the noise in $X_{t}$, which is a combination of $X_{0}$
(representing a clean instance) and white additive Gaussian noise at a noise
level denoted by $t$. In this conditional process, the denoiser leverages the
PPG signal $Y$ and the noise level $t$ as supplementary information. Training
$\epsilon$ is based on pairs of $(X,Y)$ signals - a clean ECG and its
corresponding PPG. After training, we apply multiple diffusion operations on a
given PPG signal $Y\in\mathcal{Y}$ to draw potential ECG solutions
$\\{\hat{X}_{i}\\}_{i=1}^{K}$, where each $\hat{X}_{i}\sim\hat{\pi}(X|Y)$ is a
sample from the approximated posterior distribution of the ECG given the PPG
signal.

In Section 4.1 we describe our enhanced classification procedure that relies
on the above posterior sampler, denoted by $g$. Additionally, in Section 4.2
we discuss how to improve the quantification of uncertainty in classification,
aiming to enhance confidence in predictions. Lastly, in Section 4.3, we offer
methods to visualize the cloud of ECG solutions $\\{\hat{X}_{i}\\}_{i=1}^{K}$.
This visualization aims at enhancing interpretability for medical
professionals, allowing for a better understanding of the various ECG signals
that emerge from a given PPG signal.

###  4.1 UA-P2E: Enhanced Classification

The methodology for UA-P2E’s enhanced classification consists of three phases.
The first involves training the probabilistic classification model, denoted as
$f_{\mathcal{X}}:\mathcal{X}\rightarrow\mathbb{R}^{+}$, using ECG signals and
their associated labels. $f_{\mathcal{X}}$ can be optimized through the
commonly used Cross-Entropy loss function, producing an output that has a
probabilistic interpretation. In the second phase, each potential posterior
sample $\hat{X}_{i}$ from the set $\\{\hat{X}_{i}\\}_{i=1}^{K}$ (derived by
applying $g$ to a given $Y$) is evaluated by this classifier, extracting an
output score $f_{\mathcal{X}}(\hat{X}_{i})$, being an estimated probability of
belonging to the positive class ($C=1$). In the final phase, the average
classification score, $\frac{1}{K}\sum_{i=1}^{K}f_{\mathcal{X}}(\hat{X}_{i})$,
is computed, and compared to a decision threshold $t\in\mathbb{R}^{+}$. The
complete workflow of our proposed method to enhance classification is detailed
in Algorithm 1.

Algorithm 1 UA-P2E for Classification Framework

1:Observation $Y\in\mathcal{Y}$. Pretrained conditional stochastic generative
model $g:\mathcal{Y}\times\mathcal{Z}\rightarrow\mathcal{X}$. Pretrained
classification model $f_{\mathcal{X}}:\mathcal{X}\rightarrow\mathbb{R}^{+}$.
Number of samples $K\in\mathbb{N}$. Decision threshold $t\in\mathbb{R}^{+}$.

2:Classification decision $\hat{C}\in\\{0,1\\}$, and its positive score
$f_{\mathcal{Y}}(Y)\in\mathbb{R}^{+}$.

3:for $i=1$ to $K$ do $\triangleright$ Apply the conversion process

4: Draw $Z_{i}\sim\mathcal{N}(0,1)$ and compute $\hat{X}_{i}\leftarrow
g(Y,Z_{i})$.

5:end for

6:$f_{\mathcal{Y}}(Y)\leftarrow\frac{1}{K}\sum_{i=1}^{K}f_{\mathcal{X}}(\hat{X}_{i})$
$\triangleright$ Approximate the ESC (Definition 3.2)

7:if $f_{\mathcal{Y}}(Y)>t$ then $\hat{C}\leftarrow 1$ else $\hat{C}\leftarrow
0$.

Algorithm 1 seeks to approximate the ESC (Definition 3.2), which has been
proven optimal in Theorem 3.1. The performance of Algorithm 1 in practical
settings is rigorously analyzed in Appendix A.3, demonstrating its reliance on
the information loss inherent to the PPG-2-ECG conversion.

###  4.2 UA-P2E: Uncertainty Quantification

The proposed PPG-2-ECG conversion has two layers of uncertainty awareness, as
described herein. The first takes into account the inherent uncertainty that
emerges from the ill-posed conversion itself, quantifying the spread and
variability of the possible ECG candidates that derive from the PPG inputs.
The second uncertainty layer leverages the above property for providing
classification predictions with a better confidence in their correctness. This
serves a selective classification geifman2017selective scheme, in which a
carefully chosen subset of the test PPG data is selected by their calculated
confidence, exhibiting improved accuracy.

As mentioned in Section 4.1, the posterior sampling interpretation of UA-P2E
inherently accounts for the uncertainty that arises from the variability of
the ECG solutions derived from any given PPG signal. The quantification of
this phenomenon can be evaluated by following closely with previous work
belhasin2023principal. In Section 5.2 we provide empirical evidence that
refers to the above, and more specifically demonstrates that the generated ECG
signals tend to have a misleadingly wide spread and variability in the signal
domain. While this might suggest that the uncertainty volume implied is too
large for practical needs, fusing the classification of these ECG candidates
proves to be highly informative and effective, as we discuss next.

Selective classification geifman2017selective enables handling cases in which
classifying every PPG signal within a test set is challenging due to
uncertainty or data corruption. This technique allows for abstaining from
making predictions for signals with low confidence, thus reducing
classification errors on the remaining data. A confidence function, defined as
$\kappa:\mathcal{Y}\rightarrow\mathbb{R}^{+}$, serves as the engine for this
task by predicting the correctness of predictions. One common choice of this
confidence function $\kappa$ is to use the maximal softmax score of a
classifier. In a similar spirit, in this work we set $\kappa(Y)$ to take the
value $\kappa(Y)=\max\\{1-f_{\mathcal{Y}}(Y),f_{\mathcal{Y}}(Y)\\}$, where
$f_{\mathcal{Y}}(Y)$ approximates the ESC (Definition 3.2), as summarized in
Algorithm 1.

In addition to the above, we may consider applying a calibration scheme on a
heldout set, denoted as $\\{(Y_{i},C_{i})\\}_{i=1}^{m}$, to determine a
threshold $\lambda\in\mathbb{R}^{+}$ for $\kappa(Y)$ that defines the reliable
test PPG signals whose classification error is statistically guaranteed to be
lower than a user-defined risk level $\alpha\in(0,1)\ll 1$. Such a guarantee
has the following form:

$\displaystyle\mathbb{P}\left(\frac{\mathbb{E}_{\pi(Y,C)}\left[\mathbf{1}\\{\hat{C}\neq
C\\}\mathbf{1}\\{\kappa(Y)>\lambda\\}\right]}{\mathbb{E}_{\pi(Y,C)}\left[\mathbf{1}\\{\kappa(Y)>\lambda\\}\right]}<\alpha\right)>1-\delta~{},$
(2)

where $\delta\in(0,1)\ll 1$ is a user-defined calibration error. Intuitively,
by setting $\alpha=\delta=0.1$, the above implies that the classification
error among selected PPG signals, whose confidence scores are greater than
$\lambda$, will be lower than $\alpha$ ($=10\%$) with probability of
$1-\delta$ ($=90\%$). In Appendix B, we describe how to calibrate $\lambda$ to
provide the guarantee stated in Equation (2).

###  4.3 Visualizing UA-P2E’s ECG Output(s)

A primary objective of UA-P2E is the PPG-2-ECG conversion, and a natural
question to pose is what ECG signal to present to medical professionals after
such a conversion. Due to the stochastic nature of our estimation approach,
this question becomes even more intricate, as we have multitude of candidate
ECG signals, all valid from a probabilistic point of view. This question can
be addressed by proposing two distinct and complementary strategies: _static_
ECG, as described hereafter, or _dynamic_ ECG, as discussed in Appendix C. We
emphasize, however, that there is no “correct” answer to the question of the
ECG result to present, and thus the options discussed hereafter can be
considered as possibilities to be chosen by the human observer.

We now turn to discuss several choices under the static approach. Note that
although a single ECG solution represents only a partial truth that refers to
the given PPG signal (see Section 3), here we aim to present such a single ECG
that encapsulates the main information conveyed by the given PPG signal. In
terms of interpretability, we assume that professionals might prefer receiving
a single solution that closely aligns with the underlying medical information
within the PPG signal.

To address this challenge, while acknowledging the inherent uncertainty in
converting PPG-2-ECG from a classification perspective, we first apply our
diagnostic approach as described in Section 4.1, which involves analyzing
multiple ECG solutions. Consequently, we propose presenting a single ECG that
corresponds to the diagnostic findings, specifically tied to the histogram of
ECG solutions whose classification aligns with the classification obtained
using the ESC (Definition 3.2) approximation. Formally, these ECG solutions
are derived from $\\{f_{\mathcal{X}}(\hat{X}_{i})\\}_{i=1}^{k}$, where
$\hat{X}_{i}=g(Y,Z_{i})\sim\hat{\pi}(X|Y)$ and
$\mathbf{1}\\{f_{\mathcal{X}}(\hat{X}_{i})>t\\}=\mathbf{1}\\{f_{\mathcal{Y}}(Y)>t\\}$,
with $t\in\mathbb{R}^{+}$ representing the decision threshold. We may offer to
use one of the following strategies:

Most Likely Score ECG: Present an ECG sample $\hat{X}^{*}$ that matches the
most likely score. i.e.,

$\displaystyle\hat{X}^{*}\in\arg\max_{b\in\\{I_{1},I_{2},\ldots,I_{B}\\}}\text{KDE}_{\\{f_{\mathcal{X}}(\hat{X}_{i})\\}_{i=1}^{k}}(b)~{},$
(3)

where $b\in\\{I_{1},I_{2},\ldots,I_{B}\\}$ denotes the bin intervals of the
KDE function of $\\{f_{\mathcal{X}}(\hat{X}_{i})\\}_{i=1}^{k}$.

Expected Score ECG: Present an ECG sample $\hat{X}^{*}$ that best matches the
average score, i.e.,

$\displaystyle\hat{X}^{*}=\arg\min_{\hat{X}\in\\{\hat{X}_{i}\\}_{i=1}^{k}}\left|f_{\mathcal{X}}(\hat{X})-\frac{1}{k}\sum_{i=1}^{k}f_{\mathcal{X}}(\hat{X}_{i})\right|~{}.$
(4)

Min/Max Score ECG: Present an ECG sample $\hat{X}^{*}$ that best represents
the classification, i.e.,

$\displaystyle\hat{X}^{*}=\arg\max_{\hat{X}\in\\{\hat{X}_{i}\\}_{i=1}^{k}}f_{\mathcal{X}}(\hat{X})\textbf{
~{}~{}if~{}~{} }f_{\mathcal{Y}}(Y)=1\textbf{ ~{}~{}else~{}~{}
}\arg\min_{\hat{X}\in\\{\hat{X}_{i}\\}_{i=1}^{k}}f_{\mathcal{X}}(\hat{X})~{}.$
(5)

##  5 Empirical Study

(a) Low Uncertainty (b) Medium Uncertainty (c) High Uncertainty

Figure 2: Diffusion-based multiple solutions in the temporal domain. Each
temporal point displays an interval containing 95% of our ECG/PPG synthetic
values. The top row depicts the ECG solution cloud (GREEN) resulting from the
PPG-2-ECG conversion, while the bottom row shows the PPG solution cloud (RED)
from the reverse conversion of ECG-2-PPG. The bold curves in GREEN and RED are
the mean of the sampled estimates.

This section presents a comprehensive empirical study of our proposed method,
UA-P2E, for converting PPG signals to ECG ones (lead II), as detailed in
Section 4. We employ this conversion technique with two prime objectives in
mind: (1) Providing an interpretable estimation of the ECG signals, denoted as
${\hat{X}}$, that correspond to the given PPG $Y$, while targeting the
approximation of samples from the posterior distribution $\pi(X|Y)$; and (2)
Providing accurate classification and confidence assessment for a series of
cardiovascular conditions. In both objectives, the presented results showcase
the superiority of the proposed paradigm across all conducted experiments.

Unlike prior work, this paper aims to address the uncertainty in the
conversion of PPG to ECG signals, and this is obtained by leveraging a
diffusion-based conversion methodology. More specifically, we adopt the DDIM
framework song2020denoising, aiming to reduce the number of iterations (in our
experiments we applied $T=100$ iterations) required for the sampling
procedure, and thus increasing the efficiency of our conversion algorithm. The
diffusion denoising model is trained using the MIMIC-III matched waveform
database johnson2016mimic, which is one of the most extensive dataset for
paired PPG and ECG signals comprising 22,317 records from 10,282 distinct
patients, with each record spanning approximately 45 hours. For each PPG
signal, we sample $K=100$ ECG candidates using the diffusion-based model. We
operate on raw signal data with a fixed PPG/ECG length of $d=1024$ temporal
signal values, each sampled at $125$Hz thus the signals cover $\approx 8$
seconds.

Our strategy relies on the core ability of generative models to obtain an
effective sampling from the posterior distribution $\hat{\pi}(X|Y)$, where $Y$
stands for the given PPG and $X$ for its corresponding ECG. Figure 2 (top row)
illustrates typical sampling outcomes of our diffusion-based conversion model,
showcasing the diverse ECG signals generated from different PPG inputs, each
exhibiting varying levels of uncertainty, as expressed by the variability of
the samples obtained.

As for the second objective mentioned above regarding classification, we train
and evaluate the classification model, denoted as $f_{\mathcal{X}}$, using the
‘Computing in Cardiology” (CinC) goldberger2000physiobank dataset, containing
multi-labeled ECG signals of 11 cardiovascular conditions 333The following
cardiovascular conditions are considered: (1) Ventricular premature beats, (2)
Atrial flutter, (3) Sinus rhythm, (4) Atrial fibrillation, (5)
Supraventricular premature beats, (6) Bradycardia, (7) Pacing rhythm, (8)
Sinus tachycardia, (9) 1st degree av block, (10) Sinus bradycardia, and (11)
Sinus arrhythmia. that are detectable by lead II ECG signals. Interestingly,
our experiments rely in part on a reversed diffusion-based conversion model,
designed to sample from the likelihood distribution $\hat{\pi}(Y|X)$, i.e.
converting ECG (lead II) to PPG. This reversed sampling is utilized in order
to complete the CinC goldberger2000physiobank dataset used in our tests,
enabling an analysis of challenging cardiac classification tasks. Figure 2
(bottom row) presents typical sampling outcomes of the reversed conversion
model.

For further details on the datasets used for training the conversion and
classification models, along with details on data pre-processing and training
schemes, we refer the reader to Appendix F.

In Section 5.1, we begin by assessing the quality of our diffusion-based model
for the conversion task, and in Section 5.2, we provide empirical evidence for
the inherent uncertainty in the conversion-classification framework. In
Section 5.3, we leverage this sampling ability to obtain our experimental
results of classifying cardiovascular conditions, showcasing the superiority
of our approach to baseline alternatives, as utilized by prior work. Lastly,
Section 5.4 analyzes the quality of our proposed static visualization
approaches tailored for medical professionals.

###  5.1 Diffusion-Based Conversion: Quality Assessment

|  | Signal Domain | Embedding Domain
---|---|---|---
|  | RMSE ↓ | 1-FD ↓ | 100-FD ↓ | 1-FD ↓ | 100-FD ↓
Baseline Methods |  CardioGAN sarkar2021cardiogan | 0.54 | 99.0 | - | - | -
RDDM shome2023region (T=50) | 0.22 | 6.71 | - | - | -
Our Method: UA-P2E | MMSE-Approx | 0.22 | 11.5 | - | 10.7 | -
DDIM (T=25) | 0.30 | 2.86 | 2.79 | 5.87 | 5.54
DDIM (T=50) | 0.30 | 0.52 | 0.44 | 3.83 | 3.49
DDIM (T=100) | 0.29 | 0.32 | 0.24 | 4.67 | 4.40
Table 1: Quality assessment of our diffusion-based conversion model compared
to state-of-the-art baseline methods: CardioGAN sarkar2021cardiogan and RDDM
shome2023region. The results demonstrate the superiority of our approach in
posterior modeling, as expressed by the much better Perceptual Quality
measures.

We begin by assessing the quality of our diffusion-based PPG-2-ECG conversion
model using the MIMIC-III johnson2016mimic dataset, showcasing its superiority
over state-of-the-art baseline methods. Here, we examine the quality of
$\\{\hat{X}_{i}\\}_{i=1}^{K}$ with $K=100$, sampled from our approximate
posterior distribution $\hat{\pi}(X|Y)$ for each PPG signal $Y$.

Table 1 presents the conversion performance results, evaluating the quality of
the obtained ECG signals across two domains, first, in the signal domain, and
second, in the embedding domain using our classification model, denoted as
$f_{\mathcal{X}}$. we provide a comparison with two state-of-the-art methods,
CardioGAN sarkar2021cardiogan and RDDM shome2023region, using their reported
metrics. More specifically, we assess the conversion quality through RMSE
calculations between the generated ECG(s) and the ground truth signals.
Additionally, we employ the Fréchet Distance (FD) to evaluate the perceptual
quality of the produced signals, considering either a single ECG sample from
each PPG (1-FD) or 100 samples from each (100-FD).

The results brought in Table 1 clearly testify to the superiority of our
diffusion-based conversion model. Observe that by averaging the $K=100$
samples, we obtain an approximation of the MMSE estimator, yielding a
comparable performance to the reported result in shome2023region. When
considering the individual ECG solutions for each PPG, the RMSE values
increases, as expected blau2018perception. In terms of perceptual quality,
both the single-solution and multi-solution approaches show pronounced
improvement in FD compared to the baselines. Note that computing the FD on
multiple solutions helps in improving further the signals’ quality.

Table 1 includes also perceptual quality results that are computed in an
embedding domain. By leveraging our pretrained classifier $f_{\mathcal{X}}$,
the distributions of true and synthesized ECG signals are compared via a
feature layer within this classifier, similarly to the approach taken by the
FID measure heusel2017gans.

###  5.2 Conversion-Classification Uncertainty Evidence

Conversion Uncertainty Classification Uncertainty

(a)

(b)

Figure 3: Empirical evidence of inherent uncertainty in the PPG-2-ECG
conversion-classification framework for the CinC goldberger2000physiobank
dataset. (a) The conversion spread and variability as proposed by
belhasin2023principal. High reconstruction error with many PCs indicates high
uncertainty in terms of spread and variability. (b) Classification uncertainty
shows relatively small interval sizes of ECG probability scores for each PPG
(right), which can be effectively modeled using just 100 ECG samples (left).

Here, we present empirical evidence of the inherent uncertainty in the
proposed PPG-2-ECG conversion, along with the classification uncertainty
implied. Figure 3 summarizes the results that quantify these uncertainty
layers, referring to the CinC goldberger2000physiobank dataset.

In Figure 3(a), we illustrate the uncertainty as proposed by
belhasin2023principal, showing the maximal reconstruction error of the
projected ground truth ECG signals among $90\%$ of the temporal signal values
using a varying number of _Principal Components_ (PCs) of the ECG solutions’
cloud. Intuitively, a high reconstruction error using many PCs indicates high
uncertainty in terms of the spread and variability of possible solutions to
the inverse problem tackled. For additional information and details on this
metric, we refer the reader to previous work belhasin2023principal.

Figure 3(a) compares the reconstruction error for our PPG-2-ECG conversion
with three similar graphs that correspond to image restoration tasks
(colorization, super-resolution and inpainting). This comparison exposes the
tendency of our task to have a much wider uncertainty, as far more than $100$
PCs are required in order to reconstruct the ground truth ECG signals with a
small error. This exposes the fact that PPG-2-ECG conversion is highly ill-
posed. Put bluntly, this means that given a PPG signal, the ECG corresponding
to it could be one of a wide variety of possibilities. This fact undermines
our task, as it suggests that too much information has been lost in migrating
from ECG to PPG, to enable a “safe” come-back. However, we argue that this is
a misleading observation, and, in fact, the true uncertainty is much better
behaved. More specifically, within the variability that our posterior sampler
exposes, many changes manifested in the obtained ECG signals are simply
meaningless and medically transparent. Indeed, if we migrate from the signal
domain to classification, this uncertainty shrinks to a meaningful and crisp
decisions.

The above takes us to the results shown in Figure 3(b), depicting the
uncertainty in classification, as arises from potential variability in the
conversion process. Inspired by angelopoulos2022image, in Figure 3(b) (right)
we present histograms of interval sizes that contain 90% of the probability
scores extracted using the ECG cloud that corresponds to each PPG signal,
denoted as $\\{f_{\mathcal{X}}(\hat{X}_{i})\\}_{i=1}^{K}$. These histograms
refer to the series of 11 cardiovascular conditions mentioned earlier. Small
intervals in this context correspond to a high certainty in classification,
and the results reveal that for most of the conditions assessed this is indeed
the case. We also see that a small fraction of PPG signals may exhibit a wide
range of possible probabilities among ECG candidates, a fact that may motivate
our later selective classification approach.

Figure 3(b) (left) presents the classification accuracy gaps for the 11
cardiac conditions mentioned. These gaps correspond to the difference between
the accuracy of original ECG classification, which serves as an upper bound
for our performance, and the accuracy of the average of ECG classification
scores, as summarized in Algorithm 1. As can be seen, these graphs show a
rapid stabilization, suggesting that using $100$ samples is definitely
sufficient for modeling the uncertainty that arises from the conversion
process in the classification domain.

###  5.3 Classifying Cardiovascular Conditions

To demonstrate the benefits of leveraging our diffusion-based synthesized
ECGs, we present classification experimental results on two datasets, MIMIC-
III johnson2016mimic and CinC goldberger2000physiobank, analyzing the
classification performance across a series of 11 cardiovascular conditions
mentioned earlier. For both datasets, we compare our approach to
classification strategies utilized by prior work, varied by the choice of
$\pi(Z^{\prime})$ in Definition 3.1, which refers to as the SSC. Specifically,
we examine the classification performance of the MMSE estimate and a random
single sample from the approximated posterior distribution, $\hat{\pi}(X|Y)$.
For more details on the classification strategies considered, we refer the
reader to Appendix E.

Note that MIMIC-III lacks cardiovascular conditions labels, and thus cannot be
utilized as-is for classification tasks. As a remedy, we apply our experiments
using the CinC dataset. However, while CinC contains ECG data (all leads) and
their corresponding labels, it does not include PPG signals, and as such it
cannot serve our overall PPG classification goals. We resolve this difficulty
by generating synthetic PPG signals from the given ECG ones, using a reversed
diffusion-based methodology. The idea is to sample valid and random PPG
signals that correspond to their ECG (lead II) counterparts, this way
augmenting CinC to form a complete suite. In Section 5.3.2 we provide
empirical evidence for the validity of our reversal conversion model of
ECG-2-PPG in comparison with the original PPG signal.

####  5.3.1 Cardiovascular Conditions in CinC

We now turn to describe the main classification results of cardiovascular
conditions, referring to the augmented CinC dataset, and using the synthesized
PPG signals. Figure 4 presents ROC curves for the classification strategies
considered, and Figure 5 completes this description by presenting risk-
coverage curves geifman2017selective, which evaluate the performance of
different confidence functions, denoted as $\kappa$, for PPG signals in regard
to uncertainty quantification.

As can be seen from these two figures, our approach (GREEN) demonstrates
superior performance over all the alternatives, apart from the direct ECG
(BLUE) classification, which serves as an upper-bound for the achievable
performance.

Theorem 3.1 may be interpreted as claiming that our approach performance is
expected to align with that of classifying the original PPG signals directly
(ORANGE), under optimal setup conditions. However, we observe that our
approach achieves better performance. We argue that this performance gap is
due to the erroneous labels used in the direct classification approach: Note
that these labels refer to the original ECG signals, and do not necessarily
match the corresponding PPG, which may correspond to other valid ECG signals.

Still referring to the results in these two figures, it is evident that prior
work which takes advantage of either the classification strategies of the mean
estimate (PURPLE) or a random ECG sample (BROWN), exhibits a weaker
performance compared to our approach. These findings strengthen our claim that
ignoring the uncertainty that emerges from the conversion task might impair
performance.

Figure 4:  CinC goldberger2000physiobank Results: ROC curves illustrating the
classification performance of various strategies (see Appendix E), showcasing
the superiority of our proposed approach. Higher curves indicate better
performance. Figure 5:  CinC goldberger2000physiobank Results: Risk-Coverage
curves geifman2017selective illustrating the uncertainty quantification
performance of various strategies (see Appendix E), showcasing the superiority
of our proposed approach. Lower curves indicate better performance.

####  5.3.2 Atrial Fibrillation in MIMIC

(a)

(b)

(c)

Figure 6:  MIMIC-III johnson2016mimic Results: (a) ROC Curves illustrating
classification performance. (b) Risk-Coverage curves geifman2017selective
demonstrating uncertainty quantification performance. (c) Incremental
classification error as a function of the number of ECG solutions utilized for
each PPG signal.

The main question arising from the results in Figures 4 and 5 is whether using
synthesized PPG for drawing the various conclusions is a fair and trust-worthy
strategy. In order to address this question, we return to the MIMIC-III
dataset, this time using a much smaller held-out subset of it that contains
binary labels indicating the presence of Atrial Fibrillation (AFib). Thus, for
this heart condition, we can evaluate classification accuracy for both true
PPG signals and synthesized ones. More details on this dataset are provided in
Appendix F.

In Figure 6 we bring the classification results in an experiment that follows
the ones performed on CinC, with one distinct difference – we assess both real
and synthesized PPG signals. Broadly speaking, all the conclusions from the
previous sub-section generally hold, namely (i) The best performance is
obtained by classifying ECG directly; (ii) Our approach is superior to all
other alternatives, including a direct classification of PPG (be it real or
synthesized).  [Omer: TODO for me: check if the following holds] We do note
that in the risk-coverage curve experiment, our approach delivers better
confidence scores than the classifier tested on the original ECGs, and we
attribute this behavior to the relatively small size of the test-set used in
this experiment.

Focusing on the main theme of this experiment, we see that the performance of
classifying the original PPG (ORANGE) is comparable to the alternative of
operating on the synthetic signals (RED). In terms of AUROC, the performances
are nearly identical. When evaluating the area under the Risk-Coverage curve
geifman2017selective, we observe a similar trend; the uncertainty performance
of both approaches is almost identical when considering around half of the
test samples. These findings offer empirical evidence supporting the validity
of our reversal conversion model.

###  5.4 Which ECG Solution to Deliver to Medical Professionals?

| Signal Domain | Embedding Domain
---|---|---
Static Visualization Strategy | RMSE ↓ | 1-FD ↓ | RMSE ↓ | 1-FD ↓
Min/Max Score ECG, see Equation (5) | 0.31 | 1.31 | 0.31 | 16.7
Expected Score ECG, see Equation (4) | 0.31 | 1.28 | 0.31 | 13.2
Most Likely Score ECG, see Equation (3) | 0.31 | 1.27 | 0.30 | 10.5
Table 2: Comparative quality assessment of the three proposed static ECG
visualization strategies, as described in Section 4.3. The results show that
the Most Likely Score ECG strategy is the best among the three in terms of
perceptual quality.

In this section, we examine the static ECG visualization strategies, as
discussed in Section 4.3. By comparing our proposed visualization strategies,
we aim to provide empirical insights on how to choose between them. As already
mentioned in Section 4.3, there is no “correct” choice for the visualization
strategy, and it should be considered by the medical user. Table 2 presents
the comparative results of the three proposed strategies: Min/Max Score ECG,
Expected Score ECG, and Most Likely Score ECG. This table brings RMSE results
for the signals chosen and FD values for their distributions, all compared
with their ECG origins. As can be seen from the table, while all three
considered approaches are good performing in these measures, the “Most Likely
Score ECG” seem to be the best, both in RMSE and FD.

##  6 Concluding Remarks

This paper presents Uncertainty-Aware PPG-2-ECG (UA-P2E), a novel approach
that addresses the inherent uncertainty in the PPG-2-ECG task. By
incorporating the potential spread and variability of derived ECG signals from
PPG inputs through sampling from a diffusion-based model, we achieve state-of-
the-art performance in the conversion task. In addition, we show that our
approach enhances cardiovascular classification, through the approximation of
our proposed Expected Score Classifier (ESC), which averages classification
scores of the ECG solution cloud for each input PPG signal. This
classification strategy is proven to be optimal, and demonstrated to perform
better than baseline methods in practice. Finally, we present ECG
visualization methods, designed to enhance interpretability for medical
professionals while considering the multitude ECG solutions associated with
each PPG signal.

Our work presents rich empirical results that demonstrate the superior
classification performance of our approach, compared to a direct
classification that operates directly on given PPG signals. In contrast, our
theoretical analysis suggests that the two options should perfectly align. We
argue that the reason behind this phenomenon is the inherent noise in the
labels used, as they match the ECG data but contain misleading guidance for
the corresponding PPG. Further theoretical analysis in this context is left
for future work.

##  Appendix A Optimality of the Expected Score Classifier

This section provides theoretical results proving the optimality of the
Expected Score Classifier (ESC), defined in Definition 3.2.

###  A.1 Optimal Settings

We begin by presenting a proof for Theorem 3.1 in optimal settings:

Consider the Markovian dependency chain $Y\rightarrow X\rightarrow C$,
implying that $\pi(C|X,Y)=\pi(C|X)$, and recalling the assumptions of Theorem
3.1:

  1. 1.

The conversion model $g$ is a posterior sampler, i.e.,
$g(Y,Z)=\hat{X}\sim\pi(X|Y)$; and

  2. 2.

The classification model $f_{\mathcal{X}}$ is optimal by satisfying
$f_{\mathcal{X}}(X)=\pi(C=1|X)$.

By definition, the ESC is written as

$f_{\mathcal{Y}}(Y)=\mathbb{E}_{Z\sim\pi(Z)}\left[f_{\mathcal{X}}(g(Y,Z))\right]=\int
f_{\mathcal{X}}\Big{(}g(Y,Z=z)\Big{)}\pi(Z=z)dz.$

Next, we make a change of variables $\hat{X}=g(Y,Z)$. This gives us

$\displaystyle f_{\mathcal{Y}}(Y)$ $\displaystyle=\int
f_{\mathcal{X}}\Big{(}g(Y,Z=z)\Big{)}\pi(Z=z)dz$ $\displaystyle=\int
f_{\mathcal{X}}(\hat{X}=\hat{x})\pi(\hat{X}=\hat{x}|Y)d\hat{x}$
$\displaystyle=\mathbb{E}_{\hat{X}\sim\pi(X|Y)}\left[f_{\mathcal{X}}(\hat{X})\right],$

where we rely on the assumption that $g$ is a posterior sampler, namely,
$g(Y,Z)\sim\pi(X|Y)$. Therefore, for any $y\in\mathcal{Y}$, it follows from
the optimality of $f_{\mathcal{X}}$ that

$\displaystyle f_{\mathcal{Y}}(Y=y)$
$\displaystyle=\mathbb{E}_{\hat{X}\sim\pi(X|Y=y)}\left[f_{\mathcal{X}}(\hat{X})\right]$
$\displaystyle=\mathbb{E}_{\hat{X}\sim\pi(X|Y=y)}\pi(C=1|\hat{X})$
$\displaystyle=\int_{\mathcal{X}}\pi(C=1|\hat{X}=\hat{x})\pi(\hat{X}=\hat{x}|Y=y)d\hat{x}$
$\displaystyle=\int_{\mathcal{X}}\pi(C=1|\hat{X}=\hat{x},Y=y)\pi(\hat{X}=\hat{x}|Y=y)d\hat{x}$
$\displaystyle=\pi(C=1|Y=y),$

where the second-to-last transition holds due to the Markov chain assumption.
Thus, we obtain $f_{\mathcal{Y}}(Y)=\pi(C=1|Y)$, completing the proof.

###  A.2 Optimality for General Classifiers

Here, we show that the ESC (Definition 3.2), denoted as $f_{\mathcal{Y}}(Y)$,
is an optimal estimator of $f_{\mathcal{X}}(X)$, regardless of the optimality
of $f_{\mathcal{X}}$.

######  Theorem.

Let $X$ be the source ECG signal whose observation is the PPG signal $Y$.
Additionally, assume the conversion model $g$ is a posterior sampler such that
$g(Y,Z)=\hat{X}\sim\pi(X|Y)$ and let $f_{\mathcal{Y}}(Y)$ denote the ESC, as
defined by Definition 3.2. Then, $f_{\mathcal{Y}}(Y)$ constitutes the minimum
mean square error (MMSE) estimator of $f_{\mathcal{X}}(X)$. Namely,

$f_{\mathcal{Y}}(Y)=\underset{\hat{f}(Y)}{\arg\min}\;\mathbb{E}\Big{[}\Big{(}f_{\mathcal{X}}(X)-\hat{f}(Y)\Big{)}^{2}\Big{|}Y\Big{]}$

where the expectation above is taken over $X$ given $Y$.

The proof follows a straightforward approach. We first expand the squared
error term

$\mathbb{E}\Big{[}\Big{(}f_{\mathcal{X}}(X)-\hat{f}(Y)\Big{)}^{2}\Big{|}Y\Big{]}=\mathbb{E}[f_{\mathcal{X}}(X)^{2}|Y]-2\hat{f}(Y)\mathbb{E}[f_{\mathcal{X}}(X)|Y]+\hat{f}(Y)^{2}.$

Differentiating the above expression with respect to $\hat{f}(Y)$ and setting
it to zero yields

$\hat{f}^{*}(Y)=\mathbb{E}[f_{\mathcal{X}}(X)|Y].$

Recognizing that
$f_{\mathcal{Y}}(Y)=\mathbb{E}_{\hat{X}\sim\pi(X|Y)}\left[f_{\mathcal{X}}(\hat{X})\right]=\mathbb{E}[f_{\mathcal{X}}(X)|Y]$,
we conclude that

$f_{\mathcal{Y}}(Y)=\hat{f}^{*}(Y).$

Therefore, $f_{\mathcal{Y}}(Y)$ is the MMSE estimator of $f_{\mathcal{X}}(X)$.

###  A.3 Performance Analysis of the Approximated Expected Score Classifier

While we established the optimality of the ESC (Definition 3.2), practical
implementation often requires approximation. As summarized in Algorithm 1, we
rely on an approximation defined by

$f_{\mathcal{Y}}(Y;K):=\frac{1}{K}\sum_{i=1}^{K}f_{\mathcal{X}}\Big{(}g(Y,Z_{i})\Big{)},$
(6)

where $\\{Z_{i}\\}_{i=1}^{K}$ are i.i.d. samples drawn from the normal
distribution. Notably, this approximation does not assume that the conversion
model $g$ is a posterior sampler, nor that the classifier $f_{\mathcal{X}}$ is
optimal. Additionally, the expectation is replaced with a finite-sample
average, introducing dependence on the number of samples $K>0$.

This section analyzes the asymptotic performance of the approximated ESC as
$K$ tends to infinity. We consider the _Area Under the Receiver Operating
Characteristic_ (AUROC) as our performance metric. Given independent random
negative and positive ECG samples $(X_{0},X_{1})$, and their respective PPG
signals $(Y_{0},Y_{1})$, AUROC represents the probability that the classifier
ranks $Y_{1}$ higher than $Y_{0}$:

$\text{AUROC}(f_{\mathcal{Y}};K)=\mathbb{P}\Big{(}f_{\mathcal{Y}}(Y_{0};K)\leq
f_{\mathcal{Y}}(Y_{1};K)\Big{)}.$ (7)

Since performance depends on $K$, we investigate its asymptotic behavior
$\underset{K\rightarrow\infty}{\lim}\,\text{AUROC}(f_{\mathcal{Y}};K)$. We
begin by stating the following lemma, the proof of which can be found in
Section A.3.1.

######  Lemma 1 (Monotonicity).

Assume there exist constants $\delta>0$ and $\epsilon>0$ such that for any
$Y\in\mathcal{Y}$:

$\displaystyle\Big{\lvert}\mathbb{E}_{Z\sim\pi(Z)}\left[f_{\mathcal{X}}(g(Y,Z))\right]-\mathbb{E}_{X\sim\pi(X|Y)}[f_{\mathcal{X}}(X)]\Big{\rvert}\leq\frac{\delta}{2},$
$\displaystyle\mathbb{E}_{X\sim\pi(X|Y)}\left[\lvert
f_{\mathcal{X}}(X)-\pi(C=1|X)\rvert\right]\leq\frac{\epsilon}{2}.$

Now, consider independent random negative and positive ECG samples
$(X_{0},X_{1})$, where we have access only to their respective PPG signals
$(Y_{0},Y_{1})$. Under the stated assumptions, the following inequality holds,
where $\gamma:=\epsilon+\delta$:

$\bigg{\lvert}\Big{(}\mathbb{E}_{Z\sim\pi(Z)}\left[f_{\mathcal{X}}(g(Y_{1},Z))\right]-\mathbb{E}_{\tilde{Z}\sim\pi(Z)}\left[f_{\mathcal{X}}(g(Y_{0},\tilde{Z}))\right]\Big{)}-\Big{(}\pi(C=1\,|Y_{1})-\pi(C=1\,|Y_{0})\Big{)}\bigg{\rvert}\leq\gamma.$

Furthermore, when $\gamma$ is sufficiently small, the following relationship
holds:

$\Big{(}\mathbb{E}_{Z\sim\pi(Z)}\left[f_{\mathcal{X}}(g(Y_{1},Z))\right]-\mathbb{E}_{\tilde{Z}\sim\pi(Z)}\left[f_{\mathcal{X}}(g(Y_{0},\tilde{Z}))\right]\Big{)}\Big{(}\pi(C=1\,|Y_{1})-\pi(C=1\,|Y_{0})\Big{)}>0.$

This implies that when $Y_{1}$ is more likely to positive than $Y_{0}$, the
classifier ranks $Y_{1}$ higher than $Y_{0}$, and vice versa.

The conditions in the above imply that both the conversion model $g$ and the
classifier $f_{\mathcal{X}}$ are only approximately optimal. This aligns with
practical scenarios where perfect optimality is typically infeasible. With
Lemma 1 established, we now turn to the main result of this section.

######  Theorem.

Assume Lemma 1 below holds for sufficiently small $\gamma$ such that

$\Big{(}\mathbb{E}_{Z\sim\pi(Z)}\left[f_{\mathcal{X}}(g(Y_{1},Z))\right]-\mathbb{E}_{\tilde{Z}\sim\pi(Z)}\left[f_{\mathcal{X}}(g(Y_{0},\tilde{Z}))\right]\Big{)}\Big{(}\pi(C=1\,|Y_{1})-\pi(C=1\,|Y_{0})\Big{)}>0$

Then,

$\underset{K\rightarrow\infty}{\lim}\,\text{{AUROC}}(f_{\mathcal{Y}};K)=\mathbb{P}\Big{(}\pi(C=1\,|Y_{1})\geq\pi(C=1\,|Y_{0})\Big{)},$

where the outer probability is taken over $(Y_{0},Y_{1})$ while the inner
conditional probabilities are taken over the labels given $Y_{0}$ and $Y_{1}$.

The above theorem reveals the asymptotic performance of the proposed
classifier fundamentally relies on the discriminative information within the
observation $Y$ regarding the true label $C$. When $Y$ holds no information
about $C$, the probability $\pi(C=1\,|Y)$ falls to $0.5$, indicating complete
randomness. Conversely, as $Y$ encodes increasingly richer information about
$C$, the classifier displays improved classification performance. Thus, the
information loss, inherent to the PPG-2-ECG inverse problem, dictates the
achievable performance of the classifier.

###### Proof.

We begin by assuming that $\Big{(}\pi(C=1\,|Y_{1})-\pi(C=1\,|Y_{0})\Big{)}>0$
and for a given pair of observations $(Y_{0},Y_{1})$. We define

$\mu:=\mathbb{E}_{Z\sim\pi(Z)}\left[f_{\mathcal{X}}(g(Y_{1},Z))\right]-\mathbb{E}_{\tilde{Z}\sim\pi(Z)}\left[f_{\mathcal{X}}(g(Y_{0},\tilde{Z}))\right].$

It follows that $\mu>0$. Next, we define
$\bar{U}_{K}:=f_{\mathcal{Y}}(Y_{1};K)-f_{\mathcal{Y}}(Y_{0};K)$, which can be
further expressed as

$\bar{U}_{K}=\frac{1}{K}\sum_{i=1}^{K}f_{\mathcal{X}}\Big{(}g(Y_{1},Z_{i})\Big{)}-f_{\mathcal{X}}\Big{(}g(Y_{0},\tilde{Z}_{i})\Big{)},$

where $\\{Z_{i},\tilde{Z}_{i}\\}_{i=1}^{K}$ are i.i.d. samples drawn from the
normal distribution. Notice $\bar{U}_{K}$ can be seen as a sample mean of
i.i.d. samples whose mean is $\mu$. Thus, by the weak law of large numbers

$\underset{K\rightarrow\infty}{\lim}\,\mathbb{P}\big{(}\big{|}\bar{U}_{K}-\mu\big{|}<\gamma\big{)}=1.$

for any $\gamma>0$. Furthermore, define the events $A_{K}=\bar{U}_{K}\geq 0$
and $B_{K}=\lvert\bar{U}_{K}-\mu\rvert<\gamma$. Then, for sufficiently small
$\gamma$ (e.g. $\mu/4$) we have $B_{K}\subset A_{K}$, thus
$\mathbb{P}(A_{K})\geq\mathbb{P}(B_{K})$. This in turn implies that

$\underset{K\rightarrow\infty}{\lim}\,\mathbb{P}\Big{(}f_{\mathcal{Y}}(Y_{0};K)\leq
f_{\mathcal{Y}}(Y_{1};K)\;\Big{\lvert}Y_{0},Y_{1}\Big{)}=\underset{K\rightarrow\infty}{\lim}\,\mathbb{P}(A_{K})\geq\underset{K\rightarrow\infty}{\lim}\,\mathbb{P}(B_{K})=1.$

Thus, when $\Big{(}\pi(C=1\,|Y_{1})-\pi(C=1\,|Y_{0})\Big{)}>0$ we obtain
$\underset{K\rightarrow\infty}{\lim}\,\mathbb{P}\Big{(}f_{\mathcal{Y}}(Y_{0};K)\leq
f_{\mathcal{Y}}(Y_{1};K)\;\Big{\lvert}Y_{0},Y_{1}\Big{)}=1$. Conversely, when
$\Big{(}\pi(C=1\,|Y_{1})-\pi(C=1\,|Y_{0})\Big{)}<0$, a similar analysis
reveals that
$\underset{K\rightarrow\infty}{\lim}\,\mathbb{P}\Big{(}f_{\mathcal{Y}}(Y_{0};K)\leq
f_{\mathcal{Y}}(Y_{1};K)\;\Big{\lvert}Y_{0},Y_{1}\Big{)}=0$ . Thus, in
general, we have

$\underset{K\rightarrow\infty}{\lim}\,\mathbb{P}\Big{(}f_{\mathcal{Y}}(Y_{0};K)\leq
f_{\mathcal{Y}}(Y_{1};K)\;\Big{\lvert}Y_{0},Y_{1}\Big{)}=\mathbb{I}\Big{(}\pi(C=1\,|Y_{1})\geq\pi(C=1\,|Y_{0})\Big{)},$

where $\mathbb{I}$ is the indicator function. Finally, we conclude that

$\displaystyle\underset{K\rightarrow\infty}{\lim}\,\text{{AUROC}}(f_{\mathcal{Y}};K)$
$\displaystyle=\underset{K\rightarrow\infty}{\lim}\,\mathbb{P}\Big{(}f_{\mathcal{Y}}(Y_{0};K)\leq
f_{\mathcal{Y}}(Y_{1};K)\Big{)}$
$\displaystyle=\underset{K\rightarrow\infty}{\lim}\,\mathbb{E}\left[\mathbb{P}\Big{(}f_{\mathcal{Y}}(Y_{0};K)\leq
f_{\mathcal{Y}}(Y_{1};K)\;\Big{\lvert}Y_{0},Y_{1}\Big{)}\right]$
$\displaystyle=\mathbb{E}\left[\underset{K\rightarrow\infty}{\lim}\,\mathbb{P}\Big{(}f_{\mathcal{Y}}(Y_{0};K)\leq
f_{\mathcal{Y}}(Y_{1};K)\;\Big{\lvert}Y_{0},Y_{1}\Big{)}\right]$
$\displaystyle=\mathbb{E}\left[\mathbb{I}\Big{(}\pi(C=1\,|Y_{1})\geq\pi(C=1\,|Y_{0})\Big{)}\right]$
$\displaystyle=\mathbb{P}\Big{(}\pi(C=1\,|Y_{1})\geq\pi(C=1\,|Y_{0})\Big{)},$

where the expectations above are over $(Y_{0},Y_{1})$, and the third equality
is due to Lebesgue’s dominated convergence theorem. ∎

####  A.3.1 Proof Lemma 1

First notice that

$\displaystyle\Big{\lvert}\mathbb{E}_{X\sim\pi(X|Y)}[f_{\mathcal{X}}(X)]-\pi(C=1|Y)\Big{\rvert}$
$\displaystyle=\Big{\lvert}\mathbb{E}_{X\sim\pi(X|Y)}\big{[}f_{\mathcal{X}}(X)\big{]}-\mathbb{E}_{X\sim\pi(X|Y)}\big{[}\pi(C=1|X)\big{]}\Big{\rvert}$
$\displaystyle=\Big{\lvert}\mathbb{E}_{X\sim\pi(X|Y)}\big{[}f_{\mathcal{X}}(X)-\pi(C=1|X)\big{]}\Big{\rvert}$
$\displaystyle\leq\mathbb{E}_{X\sim\pi(X|Y)}\Big{[}\big{\lvert}f_{\mathcal{X}}(X)-\pi(C=1|X)\big{\rvert}\Big{]}\leq\frac{\epsilon}{2}.$

Now, by the triangle inequality we obtain

$\displaystyle\Big{\lvert}\mathbb{E}_{Z\sim\pi(Z)}\left[f_{\mathcal{X}}(g(Y,Z))\right]-\pi(C=1|Y)\Big{\rvert}$
$\displaystyle\leq\Big{\lvert}\mathbb{E}_{Z\sim\pi(Z)}\left[f_{\mathcal{X}}(g(Y,Z))\right]-\mathbb{E}_{X\sim\pi(X|Y)}[f_{\mathcal{X}}(X)]\Big{\rvert}$
$\displaystyle+\Big{\lvert}\mathbb{E}_{X\sim\pi(X|Y)}[f_{\mathcal{X}}(X)]-\pi(C=1|Y)\Big{\rvert}$
$\displaystyle\leq\frac{\delta}{2}+\frac{\epsilon}{2}=\frac{\epsilon+\delta}{2}$

Next, we apply the triangle inequality again to arrive at the desired
inequality:

$\displaystyle\bigg{\lvert}\Big{(}\mathbb{E}_{Z\sim\pi(Z)}\left[f_{\mathcal{X}}(g(Y_{1},Z))\right]-\mathbb{E}_{\tilde{Z}\sim\pi(Z)}\left[f_{\mathcal{X}}(g(Y_{0},\tilde{Z}))\right]\Big{)}-\Big{(}\pi(C=1\,|Y_{1})-\pi(C=1\,|Y_{0})\Big{)}\bigg{\rvert}$
$\displaystyle=\bigg{\lvert}\Big{(}\mathbb{E}_{Z\sim\pi(Z)}\left[f_{\mathcal{X}}(g(Y_{1},Z))\right]-\pi(C=1\,|Y_{1})\Big{)}-\Big{(}\mathbb{E}_{\tilde{Z}\sim\pi(Z)}\left[f_{\mathcal{X}}(g(Y_{0},\tilde{Z}))\right]-\pi(C=1\,|Y_{0})\Big{)}\bigg{\rvert}$
$\displaystyle\leq\bigg{\lvert}\Big{(}\mathbb{E}_{Z\sim\pi(Z)}\left[f_{\mathcal{X}}(g(Y_{1},Z))\right]-\pi(C=1\,|Y_{1})\Big{)}\Big{\rvert}+\Big{\lvert}\Big{(}\mathbb{E}_{\tilde{Z}\sim\pi(Z)}\left[f_{\mathcal{X}}(g(Y_{0},\tilde{Z}))\right]-\pi(C=1\,|Y_{0})\Big{)}\bigg{\rvert}$
$\displaystyle=\epsilon+\delta=\gamma.$

Finally, when $\gamma$ is sufficiently small, the two terms of interest share
the same sign, resulting in their product always being positive.

##  Appendix B Assessing Reliable PPG Signals via Calibration

To ensure the selection of valid PPG signals meeting the guarantee outlined in
Equation (2), a calibration scheme is necessary. This scheme’s goal is to
identify a calibrated parameter $\hat{\lambda}\in\mathbb{R}^{+}$ that
maximizes coverage of selected PPG signals whose classification error is below
a user-specified bound, denoted by $\alpha\in(0,1)$. Then, a new unseen PPG
signal, $Y\sim\pi(Y)$, will be considered reliable if its confidence score
$\kappa(Y)$ surpasses $\hat{\lambda}$.

The calibration algorithm is based on a _conformal prediction_ scheme
[vovk2005algorithmic, papadopoulos2002inductive, lei2014distribution], a
general method for distribution-free risk control. Specifically, we employ the
_Learn Then Test_ (LTT) [angelopoulos2021learn] procedure in this paper. The
conformal prediction’s objective is to bound a user-defined risk function,
denoted as $R(\lambda)\in(0,1)$, by adjusting a calibration parameter
$\lambda\in\mathbb{R}^{+}$ based on empirical data, aiming for the tightest
upper bound feasible [angelopoulos2021gentle]. Through this calibration
process, our aim is to identify $\hat{\lambda}$ that ensures the following
guarantee:

$\displaystyle\mathbb{P}(R(\hat{\lambda})<\alpha)>1-\delta~{},$ (8)

where $\alpha\in(0,1)$ is an upper bound for the risk function, and
$\delta\in(0,1)$ is the calibration error. These parameters are user-defined
and should approach zero for an effective calibration.

Recalling the desired guarantee specified in Equation (2),

$\displaystyle\mathbb{P}\left(\frac{\mathbb{E}_{\pi(Y,C)}\left[\mathbf{1}\\{\hat{C}\neq
C\\}\mathbf{1}\\{\kappa(Y)>\lambda\\}\right]}{\mathbb{E}_{\pi(Y,C)}\left[\mathbf{1}\\{\kappa(Y)>\lambda\\}\right]}<\alpha\right)>1-\delta~{},$

we turn to describe the steps for calibration to achieve this guarantee. To
simplify the above into the form posed in Equation (8), we define the
_selective risk_ as:

$\displaystyle
R(\lambda):=\frac{\mathbb{E}_{\pi(Y,C)}\left[\mathbf{1}\\{\hat{C}\neq
C\\}\mathbf{1}\\{\kappa(Y)>\lambda\\}\right]}{\mathbb{E}_{\pi(Y,C)}\left[\mathbf{1}\\{\kappa(Y)>\lambda\\}\right]}~{}.$
(9)

Intuitively, this measure quantifies the classification error among selected
PPG signals with confidence scores surpassing $\lambda$.

In practical scenarios, computing the above risk function, $R(\lambda)$, is
not feasible due to the unavailability of the joint distribution $\pi(Y,C)$.
However, we do have samples from this distribution within our calibration
data, $\\{(Y_{i},C_{i})\\}_{i=1}^{m}$. Hence, we utilize the _empirical
selective risk_ , denoted as $\hat{R}(\lambda)$, and defined as:

$\displaystyle\hat{R}(\lambda):=\frac{\sum_{i=1}^{m}\mathbf{1}\\{\hat{C}_{i}\neq
C_{i}\\}\mathbf{1}\\{\kappa(Y_{i})>\lambda\\}}{\sum_{i=1}^{m}\mathbf{1}\\{\kappa(Y_{i})>\lambda\\}}~{}.$
(10)

Note that $\hat{C}_{i}=\mathbf{1}\\{f_{\mathcal{Y}}(Y_{i})>t\\}$, where
$t\in\mathbb{R}^{+}$ represents the decision threshold.

For each $\lambda\in\Lambda$, where $\Lambda$ represents a set of calibration
parameter values defined by the user as
$\Lambda:=[0\dots\lambda_{\text{max}}]$, we utilize the empirical risk
function mentioned earlier to establish an _upper-confidence bound_ , denoted
as $\hat{R}^{+}(\lambda)$, and defined as:

$\displaystyle\hat{R}^{+}(\lambda):=\hat{R}(\lambda)+r_{\delta}~{},$ (11)

where $r_{\delta}\in(0,1)$ serves as a concentration bound, such as
Hoeffding’s bound, which is calculated as
$r_{\delta}=\sqrt{\frac{1}{2m}\log\frac{1}{\delta}}$. It is shown in
[hoeffding1994probability] that
$\mathbb{P}(\hat{R}^{+}(\lambda)<R(\lambda))<\delta$.

Finally, to achieve the desired guarantee outlined in Equation (2), we seek
calibration parameters that satisfy Equation (2) while using the upper
confidence bound defined in Equation (11). The goal is to maximize the
coverage of selected PPG signals, thus the smallest $\lambda\in\Lambda$ that
meets Equation (2) is selected as follows:

$\displaystyle\hat{\lambda}:=\min\\{\lambda:\hat{R}^{+}(\lambda)<\alpha,\lambda\in\Lambda\\}~{}.$
(12)

The algorithm below summarizes the steps outlined in this section for applying
calibration.

Algorithm 2 Calibration of a Selective Classification UA-P2E Framework

1:Calibration set $\\{(Y_{i},C_{i})\\}_{i=1}^{m}$. Selective classification
error level $\alpha\in(0,1)$. Calibration error level $\delta\in(0,1)$.
Pretrained conditional stochastic generative model
$g:\mathcal{Y}\times\mathcal{Z}\rightarrow\mathcal{X}$. Pretrained
classification model $f_{\mathcal{X}}:\mathcal{X}\rightarrow\mathbb{R}^{+}$.
Number of samples $K>0$. Decision threshold $t>0$.

2:for $i=1$ to $m$ do

3: $\hat{C}_{i},f_{\mathcal{Y}}(Y_{i})\leftarrow$ Apply Algorithm 1 using
$Y_{i}$, $g$, $f_{\mathcal{X}}$, $K$, and $t$.

4:
$\kappa(Y_{i})\leftarrow\max\\{1-f_{\mathcal{Y}}(Y_{i}),f_{\mathcal{Y}}(Y_{i})\\}$

5:end for

6:for $\lambda\in\Lambda$ do

7:
$\hat{R}^{+}(\lambda)\leftarrow\frac{\sum_{i=1}^{m}\mathbf{1}\\{\hat{C}_{i}\neq
C_{i}\\}\mathbf{1}\\{\kappa(Y_{i})>\lambda\\}}{\sum_{i=1}^{m}\mathbf{1}\\{\kappa(Y_{i})>\lambda\\}}+\sqrt{\frac{1}{2m}\log\frac{1}{\delta}}$

8:end for

9:$S:=\\{\lambda:\hat{R}^{+}(\lambda)<\alpha,\lambda\in\Lambda\\}$

10:if $S\neq\phi$ then $\hat{\lambda}\leftarrow\min S$ else Calibration
failed.

11:Given a new PPG signal $Y\sim\pi(Y)$, if $\kappa(Y)>\hat{\lambda}$ then $Y$
is reliable.

##  Appendix C Visualizing UA-P2E’s ECG Outputs through a Dynamic ECG

The main advantage with the static visualization approaches, outlined in
Section 4.3, is their relative simplicity, providing reasonable, even if
partial, explainability to the conversion performed on the measured PPG data.
In order to go beyond these solutions, we offer a dynamic alternative, in
which a series of ECG signals are presented to medical professionals. This
approach exposes the uncertainty within the given PPG data, even for those ECG
solutions that do not align with the approximation of the ESC (Definition
3.2). Here as well we provide several possible strategies for visualizing the
ECG solutions, aiming to increase interpretability. These strategies are
associated with the histogram obtained from
$\\{f_{\mathcal{X}}(\hat{X}_{i})\\}_{i=1}^{K}$, where
$\hat{X}_{i}=g(Y,Z_{i})\sim\hat{\pi}(X|Y)$.

Navigating via PCA: Present a dynamic ECG by following the order of its
corresponding projected solutions onto the main PCA axis within the solution
cloud.

Navigating via Shortest Path: Present a dynamic ECG by following the shortest
path among the samples within the solution cloud.

Navigating via Classification for Smoothness: Present a dynamic ECG by
following the classification scores order of the samples within the solution
cloud. For each presented ECG, utilize dynamic programming to ensure maximum
smoothness.

##  Appendix D Exploring Optimal Classification Strategies in Mixture of 2D
Gaussians

In this section, we analyze the performance of our proposed ESC (Definition
3.2) in comparison to the approach adopted by prior work, the SSC (Definition
3.1), utilizing a toy 2D dataset that simulates our Markovian setup for
recovering ECG signals from PPG signals in an ideal environment. Our primary
objective here is to explore optimal strategies when we have access to optimal
conversion and classification models. Furthermore, our goal is to support our
theoretical results, as detailed in Appendix A, which assert the optimality of
our proposed approach.

###  D.1 Dataset Description

(a)

(b)

(c)

Figure 7: Optimal Setup Description: (a) Illustration of the instance
distribution in BLUE, observation distribution in ORANGE, and the conversion
distribution in GREEN. (b) Conversion samples and their classification
according to the optimal classifier, demonstrating potential for incorrect
classification when only a single solution is considered. (c) Multi-solution
classification scores revealing uncertainty in the conversion process. The ESC
(Definition 3.2) is presented in GREEN dashed line.

We begin by describing the optimal setup considered in this section. Figure 7
illustrates the dataset utilized in the experiments of this section,
exemplified by an instance of $X=x$ and its corresponding observation $Y=y$.

As depicted in Figure 7(a), we define $\pi(X)$ as a mixture of two 2D
Gaussians, and $\pi(Y|X)$ as a distribution derived from noisy versions of $X$
with a fixed std, denoted by $\sigma\in\mathbb{R}$. Each Gaussian within the
mixture represents a cluster of instances belonging to either $C=0$ or $C=1$.
Consequently, we have the complete knowledge of the joint distribution
$\pi(X,Y)=\pi(Y|X)\pi(X)$, enabling us to analytically derive the optimal
conversion distribution $\pi(X|Y)$ with the exception of the weight factor
within the mixture, which was trained and evaluated using a simple DNN. The
optimal conversion model is defined by
$g:\mathcal{Y}\times\mathcal{Z}\rightarrow\mathcal{X}$. Theoretical details
regarding the analytical derivation of $\pi(X|Y)$, as well as formal details
and hyperparameters concerning the dataset, are provided in Appendix D.2.

Moving to classification, we define the optimal classification model as
$f_{\mathcal{X}}:\mathcal{X}\rightarrow\mathbb{R}^{+}$. In Figure 7(b), we
present samples from the posterior distribution $\pi(X|Y=y)$, along with their
corresponding classifications. Specifically, $X=x$ corresponds to $C=1$,
whereas the conversion of its observation $Y=y$ might lead to samples being
assigned to $C=0$ as well, a fact that motivates our work. Thus, when
providing the user with a single solution, we offer only partial truth
regarding the original $X=x$.

Lastly, in Figure 7(c), we present the histogram of the classification scores
obtained by samples from the posterior distribution $\pi(X|Y=y)$, indicating
their likelihood to belong to $C=1$ with a decision threshold set at $t=0.5$.
By observing multiple solutions and computing the expected score across the
posterior distribution, as proposed in this paper through the ESC (Definition
3.2), we expose the uncertainty in the conversion, as highlighted by the
dashed green line in this figure.

###  D.2 Theoretical Details and Hyperparameters

Here, our objective is to formally define the dataset under consideration in
this section, as illustrated in Figure 7, as well as to provide the analytic
derivations utilized.

Let $X$ be a mixture of Gaussians distribution, i.e.,
$X\sim\sum_{k=1}^{K}w_{k}\mathcal{N}(X;\mu_{X}^{k},{\sigma_{X}^{k}}^{2})$.

and $Y=X+\epsilon$ is a noisy version of $X$ where
$\epsilon\sim\mathcal{N}(0,\sigma^{2})$. Therefore,
$Y\sim\sum_{k=1}^{K}w_{k}\mathcal{N}(Y;\mu_{X}^{k},{\sigma_{X}^{k}}^{2}+\sigma^{2})$.
Then
$\pi(X|Y)=\sum_{k=1}^{K}W_{k}(Y)\mathcal{N}(X;\mu_{X|Y}^{k},{\sigma_{X|Y}^{k}}^{2})$,
and it can be computed analytically as follows:

###  D.3 Performance Analysis

Figure 8 presents the performance of the baseline strategies and ours under
this setup.

(a)

(b)

(c)

Figure 8: Gaussians Results: (a) Text. (b) Text. (c) Text.

As can be seen, the performance of $f^{*}_{X}(X)$ exhibits superiority in all
experiments and serves as an upper bound for all the results as expected. The
gap between the performance of $f^{*}_{X}(X)$ and $f^{*}_{Y}(Y)$ demonstrates
the loss of information that $Y$ exhibits in regard to $X$. While our
framework being given $Y$ instead of $X$, under optimal conditions, our
strategy’s performance aligns with the performance of $f^{*}_{Y}(Y)$,
indicating optimality. This alignment is supported by Theorem LABEL:th.

Figure 8 also highlights another noteworthy aspect; the alignment of the mean
estimate performance, denoted as $f^{*}_{X}(\mathbb{E}[\hat{X}])$, with
optimal performance. However, this alignment is not applicable in general. We
argue that this behavior occurs due to the smooth nature of our optimal
classifier in this simple toy example. Additional experimental evidence
supporting this observation is provided in Appendix
LABEL:app:_The_Misalignment_of_MMSE_Estimates_with_Optimality.

Finally, it is observed that the performance of the classification strategy
based on a single posterior sample, denoted by $f^{*}(g^{*}(Y))$, exhibits
lower performance in all experiments. This indicates that classification based
on a single sample might aggregate errors, especially when there is a non-
negligible probability to sample from opposite classes. This behavior is
supported by Theorem LABEL:th.

##  Appendix E Classification Strategies

In our experiments we assess several classification strategies, all relying on
paired ECG and PPG signals and their corresponding labels, as obtained using
MIMIC-III [johnson2016mimic] and CinC [goldberger2000physiobank]. We assume
the availability of pretrained classifiers,
$f_{\mathcal{X}}:\mathcal{X}\rightarrow\mathbb{R}^{+}$ and
$f_{\mathcal{Y}}:\mathcal{Y}\rightarrow\mathbb{R}^{+}$, for input ECG and PPG
signals, respectively. These classifiers output classification likelihoods
considering cardiovascular conditions.

We emphasize that the baseline approaches, as utilized in prior work, ignore
the uncertainty in the conversion of PPG to ECG by maintaining only a single
ECG solution given the PPG, whereas our approach maintains and manipulates an
aggregation of multiple ECG candidates for the classification. With this in
mind, here are the classification strategies considered in this work:

Original ECG: Classifier performance when tested on original ECG signals,
i.e., the performance of $f_{\mathcal{X}}(X)$, where $X\sim\pi(X)$. Since we
do not have access to the original ECG signal in our PPG-2-ECG setting, this
measure serves as an upper bound for the achievable performance.

Original PPG: Classifier performance when tested on the original PPG signals,
i.e., the performance of $f_{\mathcal{Y}}(Y)$, where $Y\sim\pi(Y)$.

Synthesized PPG: Classifier performance when tested on diffusion-based random
PPG samples, i.e., the performance of $f_{\mathcal{Y}}(\hat{Y})$, where
$\hat{Y}\sim\hat{\pi}(Y|X)$ are outcomes of the reversed diffusion-based
conversion model. See Section 5.3 for detailed explanation on the need for
these signals.

Synthesized ECG: SSC Mean Classifier performance when tested on single-
solution (Definition 3.1) ECG signals referring to the mean ECG signal from
the diffusion-based ECG samples, i.e., the performance of
$f_{\mathcal{X}}(\frac{1}{K}\sum_{i=1}^{K}\hat{X}_{i})$, where
$\hat{X}_{i}\sim\hat{\pi}(X|Y)$.

Synthesized ECG: SSC Random: Classifier performance when tested on single-
solution (Definition 3.1) ECG signals referring to randomly chosen diffusion-
based ECG samples, i.e., the performance of $f_{\mathcal{X}}(\hat{X})$, where
$\hat{X}\sim\hat{\pi}(X|Y)$.

Synthesized ECG: ESC (Ours): Classifier performance when tested on multi-
solution (Definition 3.2 and Algorithm 1) ECG signals referring to randomly
chosen diffusion-based ECG samples, and then averaging their classification
scores, i.e., the performance of
$\frac{1}{K}\sum_{i=1}^{K}f_{\mathcal{X}}(\hat{X}_{i})$, where
$\hat{X}_{i}\sim\hat{\pi}(X|Y)$.

##  Appendix F Experimental Supplementary for Cardiovascular Conditions
Experiments

For both datasets, we employed the same diffusion-based model following the
DDIM [ddim] framework for the conversion task, which we trained on the MIMIC-
III matched waveform database (MIMIC) [mimic]. To the best of our knowledge,
MIMIC is the largest dataset available for pairs of measured PPGs and ECGs,
containing 22,317 records of 10,282 distinct patients, each approximately 45
hours long. For more details on the preprocessing and training schemes, we
refer the reader to Appendix F.1.

The classification models  [Omer: TODO]. Further details are elaborated in
Appendix F.2.

###  F.1 Data and Training Details for the Conversion Model

[Omer: TODO]

###  F.2 Data and Training Details for the Classification Model

[Omer: TODO]

####  F.2.1 MIMIC

In Section 5.3.2, we present experimental results on a held-out subset of the
MIMIC dataset, where we have access to PPGs, ECGs, and binary labels
indicating the presence of atrial fibrillation.

We present experimental results obtained from a held-out subset of the MIMIC
dataset, comprising 400 patients. For these patients, we extracted binary
labels indicating the presence of atrial fibrillation from medical reports
within the MIMIC-III clinical database [mimic_clinical]. This subset was
carefully selected to include an equal number of patients diagnosed with
normal sinus rhythm and those diagnosed with atrial fibrillation, ensuring a
balanced representation of both conditions.

####  F.2.2 CinC

Following this, in Section 5.3.1, we present an analysis of the Computing in
Cardiology (CinC) challenge 2021 dataset [cinc]. To the best of our knowledge,
CinC is the largest classification dataset in cardiology, containing 94,630
10-second ECG recordings, each labeled with various challenging cardiovascular
conditions. However, it lacks PPG measures.

While striving to demonstrate the effectiveness of our approach within a
challenging environment, we adopted a diffusion-based methodology to convert
ECG signals to PPG signals. This approach was motivated by the inherent noise
and reduced information content observed in PPG signals, making synthetic PPG
signals suitable to simulate our setup effectively. The data preprocessing and
training scheme for this task were the same as for the regular PPG-2-ECG
conversion model but applied with a reversal in the input and output of the
model. Further details are elaborated in Appendix F.1.

###  F.3 Understanding the Success of Our Conversion-Classification Framework
Over PPG Signals

[Omer: TODO]

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complemented by formal proofs provided in appendix or supplemental material.

     * •

Theorems and Lemmas that the proof relies upon should be properly referenced.

  16. 4.

Experimental Result Reproducibility

  17. Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?

  18. Answer: [TODO]

  19. Justification:

  20. Guidelines:

     * •

The answer NA means that the paper does not include experiments.

     * •

If the paper includes experiments, a No answer to this question will not be
perceived well by the reviewers: Making the paper reproducible is important,
regardless of whether the code and data are provided or not.

     * •

If the contribution is a dataset and/or model, the authors should describe the
steps taken to make their results reproducible or verifiable.

     * •

Depending on the contribution, reproducibility can be accomplished in various
ways. For example, if the contribution is a novel architecture, describing the
architecture fully might suffice, or if the contribution is a specific model
and empirical evaluation, it may be necessary to either make it possible for
others to replicate the model with the same dataset, or provide access to the
model. In general. releasing code and data is often one good way to accomplish
this, but reproducibility can also be provided via detailed instructions for
how to replicate the results, access to a hosted model (e.g., in the case of a
large language model), releasing of a model checkpoint, or other means that
are appropriate to the research performed.

     * •

While NeurIPS does not require releasing code, the conference does require all
submissions to provide some reasonable avenue for reproducibility, which may
depend on the nature of the contribution. For example

       1. (a)

If the contribution is primarily a new algorithm, the paper should make it
clear how to reproduce that algorithm.

       2. (b)

If the contribution is primarily a new model architecture, the paper should
describe the architecture clearly and fully.

       3. (c)

If the contribution is a new model (e.g., a large language model), then there
should either be a way to access this model for reproducing the results or a
way to reproduce the model (e.g., with an open-source dataset or instructions
for how to construct the dataset).

       4. (d)

We recognize that reproducibility may be tricky in some cases, in which case
authors are welcome to describe the particular way they provide for
reproducibility. In the case of closed-source models, it may be that access to
the model is limited in some way (e.g., to registered users), but it should be
possible for other researchers to have some path to reproducing or verifying
the results.

  21. 5.

Open access to data and code

  22. Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?

  23. Answer: [TODO]

  24. Justification:

  25. Guidelines:

     * •

The answer NA means that paper does not include experiments requiring code.

     * •

Please see the NeurIPS code and data submission guidelines
(https://nips.cc/public/guides/CodeSubmissionPolicy) for more details.

     * •

While we encourage the release of code and data, we understand that this might
not be possible, so “No” is an acceptable answer. Papers cannot be rejected
simply for not including code, unless this is central to the contribution
(e.g., for a new open-source benchmark).

     * •

The instructions should contain the exact command and environment needed to
run to reproduce the results. See the NeurIPS code and data submission
guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more
details.

     * •

The authors should provide instructions on data access and preparation,
including how to access the raw data, preprocessed data, intermediate data,
and generated data, etc.

     * •

The authors should provide scripts to reproduce all experimental results for
the new proposed method and baselines. If only a subset of experiments are
reproducible, they should state which ones are omitted from the script and
why.

     * •

At submission time, to preserve anonymity, the authors should release
anonymized versions (if applicable).

     * •

Providing as much information as possible in supplemental material (appended
to the paper) is recommended, but including URLs to data and code is
permitted.

  26. 6.

Experimental Setting/Details

  27. Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?

  28. Answer: [TODO]

  29. Justification:

  30. Guidelines:

     * •

The answer NA means that the paper does not include experiments.

     * •

The experimental setting should be presented in the core of the paper to a
level of detail that is necessary to appreciate the results and make sense of
them.

     * •

The full details can be provided either with the code, in appendix, or as
supplemental material.

  31. 7.

Experiment Statistical Significance

  32. Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?

  33. Answer: [TODO]

  34. Justification:

  35. Guidelines:

     * •

The answer NA means that the paper does not include experiments.

     * •

The authors should answer "Yes" if the results are accompanied by error bars,
confidence intervals, or statistical significance tests, at least for the
experiments that support the main claims of the paper.

     * •

The factors of variability that the error bars are capturing should be clearly
stated (for example, train/test split, initialization, random drawing of some
parameter, or overall run with given experimental conditions).

     * •

The method for calculating the error bars should be explained (closed form
formula, call to a library function, bootstrap, etc.)

     * •

The assumptions made should be given (e.g., Normally distributed errors).

     * •

It should be clear whether the error bar is the standard deviation or the
standard error of the mean.

     * •

It is OK to report 1-sigma error bars, but one should state it. The authors
should preferably report a 2-sigma error bar than state that they have a 96%
CI, if the hypothesis of Normality of errors is not verified.

     * •

For asymmetric distributions, the authors should be careful not to show in
tables or figures symmetric error bars that would yield results that are out
of range (e.g. negative error rates).

     * •

If error bars are reported in tables or plots, The authors should explain in
the text how they were calculated and reference the corresponding figures or
tables in the text.

  36. 8.

Experiments Compute Resources

  37. Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?

  38. Answer: [TODO]

  39. Justification:

  40. Guidelines:

     * •

The answer NA means that the paper does not include experiments.

     * •

The paper should indicate the type of compute workers CPU or GPU, internal
cluster, or cloud provider, including relevant memory and storage.

     * •

The paper should provide the amount of compute required for each of the
individual experimental runs as well as estimate the total compute.

     * •

The paper should disclose whether the full research project required more
compute than the experiments reported in the paper (e.g., preliminary or
failed experiments that didn’t make it into the paper).

  41. 9.

Code Of Ethics

  42. Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?

  43. Answer: [TODO]

  44. Justification:

  45. Guidelines:

     * •

The answer NA means that the authors have not reviewed the NeurIPS Code of
Ethics.

     * •

If the authors answer No, they should explain the special circumstances that
require a deviation from the Code of Ethics.

     * •

The authors should make sure to preserve anonymity (e.g., if there is a
special consideration due to laws or regulations in their jurisdiction).

  46. 10.

Broader Impacts

  47. Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?

  48. Answer: [TODO]

  49. Justification:

  50. Guidelines:

     * •

The answer NA means that there is no societal impact of the work performed.

     * •

If the authors answer NA or No, they should explain why their work has no
societal impact or why the paper does not address societal impact.

     * •

Examples of negative societal impacts include potential malicious or
unintended uses (e.g., disinformation, generating fake profiles,
surveillance), fairness considerations (e.g., deployment of technologies that
could make decisions that unfairly impact specific groups), privacy
considerations, and security considerations.

     * •

The conference expects that many papers will be foundational research and not
tied to particular applications, let alone deployments. However, if there is a
direct path to any negative applications, the authors should point it out. For
example, it is legitimate to point out that an improvement in the quality of
generative models could be used to generate deepfakes for disinformation. On
the other hand, it is not needed to point out that a generic algorithm for
optimizing neural networks could enable people to train models that generate
Deepfakes faster.

     * •

The authors should consider possible harms that could arise when the
technology is being used as intended and functioning correctly, harms that
could arise when the technology is being used as intended but gives incorrect
results, and harms following from (intentional or unintentional) misuse of the
technology.

     * •

If there are negative societal impacts, the authors could also discuss
possible mitigation strategies (e.g., gated release of models, providing
defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms
to monitor how a system learns from feedback over time, improving the
efficiency and accessibility of ML).

  51. 11.

Safeguards

  52. Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?

  53. Answer: [TODO]

  54. Justification:

  55. Guidelines:

     * •

The answer NA means that the paper poses no such risks.

     * •

Released models that have a high risk for misuse or dual-use should be
released with necessary safeguards to allow for controlled use of the model,
for example by requiring that users adhere to usage guidelines or restrictions
to access the model or implementing safety filters.

     * •

Datasets that have been scraped from the Internet could pose safety risks. The
authors should describe how they avoided releasing unsafe images.

     * •

We recognize that providing effective safeguards is challenging, and many
papers do not require this, but we encourage authors to take this into account
and make a best faith effort.

  56. 12.

Licenses for existing assets

  57. Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?

  58. Answer: [TODO]

  59. Justification:

  60. Guidelines:

     * •

The answer NA means that the paper does not use existing assets.

     * •

The authors should cite the original paper that produced the code package or
dataset.

     * •

The authors should state which version of the asset is used and, if possible,
include a URL.

     * •

The name of the license (e.g., CC-BY 4.0) should be included for each asset.

     * •

For scraped data from a particular source (e.g., website), the copyright and
terms of service of that source should be provided.

     * •

If assets are released, the license, copyright information, and terms of use
in the package should be provided. For popular datasets,
paperswithcode.com/datasets has curated licenses for some datasets. Their
licensing guide can help determine the license of a dataset.

     * •

For existing datasets that are re-packaged, both the original license and the
license of the derived asset (if it has changed) should be provided.

     * •

If this information is not available online, the authors are encouraged to
reach out to the asset’s creators.

  61. 13.

New Assets

  62. Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?

  63. Answer: [TODO]

  64. Justification:

  65. Guidelines:

     * •

The answer NA means that the paper does not release new assets.

     * •

Researchers should communicate the details of the dataset/code/model as part
of their submissions via structured templates. This includes details about
training, license, limitations, etc.

     * •

The paper should discuss whether and how consent was obtained from people
whose asset is used.

     * •

At submission time, remember to anonymize your assets (if applicable). You can
either create an anonymized URL or include an anonymized zip file.

  66. 14.

Crowdsourcing and Research with Human Subjects

  67. Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?

  68. Answer: [TODO]

  69. Justification:

  70. Guidelines:

     * •

The answer NA means that the paper does not involve crowdsourcing nor research
with human subjects.

     * •

Including this information in the supplemental material is fine, but if the
main contribution of the paper involves human subjects, then as much detail as
possible should be included in the main paper.

     * •

According to the NeurIPS Code of Ethics, workers involved in data collection,
curation, or other labor should be paid at least the minimum wage in the
country of the data collector.

  71. 15.

Institutional Review Board (IRB) Approvals or Equivalent for Research with
Human Subjects

  72. Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?

  73. Answer: [TODO]

  74. Justification:

  75. Guidelines:

     * •

The answer NA means that the paper does not involve crowdsourcing nor research
with human subjects.

     * •

Depending on the country in which research is conducted, IRB approval (or
equivalent) may be required for any human subjects research. If you obtained
IRB approval, you should clearly state this in the paper.

     * •

We recognize that the procedures for this may vary significantly between
institutions and locations, and we expect authors to adhere to the NeurIPS
Code of Ethics and the guidelines for their institution.

     * •

For initial submissions, do not include any information that would break
anonymity (if applicable), such as the institution conducting the review.