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

## Abstract

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 [].
The analysis of cardiovascular conditions using Electrocardiography (ECG) signals has emerged as one of the most widely utilized diagnostic tool [], 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 [].

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 [].
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 [], blood oxygen saturation [], blood pressure [], and to assess vascular function [, ].
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 [, , , , , , , , , , , , ].
However, such a mapping
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 [, , , , , , , , ] 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 [, , , ].
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 [, ], and also the uncertainty in the classification task, enabling the selection of PPG samples with low classification risk and high confidence [].

Figure: Figure 1: Illustration of our proposed UA-P2E for PPG-2-ECG conversion-classification framework.
Refer to caption: x1.png

To achieve the above, we leverage a conditional diffusion-based methodology [, , ]. These generative models have recently emerged as the leading synthesis approach in various tasks [, ], and have been shown to enable high-fidelity posterior modeling for inverse problems [, , , , ].
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 [], 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 [, , , ] or the Minimum Mean Squared Error (MMSE) estimator with $L_{2}$ loss [, , , , ].
These works commonly employed regularization techniques to enhance perceptual outcomes.
However, the work reported in [] 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 [].
They employed an adversarial training scheme with an encoder-decoder architecture.
Similarly, P2E-WGAN [] utilized conditional Wasserstein GANs for this task.
[] introduced a variational inference methodology to approximate the posterior distribution with a Gaussian prior assumption.
The authors of RDDM [] have first adapted a diffusion-based model that accelerates a DDPM [] 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.

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 [, ].
Secondly, it also aims at assessing the confidence in the correctness of classification predictions, e.g. via a selective classification [] 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 $\pi(X|Y)$.
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 [], 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.
In Appendix A we prove Theorem A.1, implying that our approach utilizes the full extent of the information within the observation $Y$, and is therefore optimal.

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

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

## 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 [, , ], which have emerged as the leading synthesis approach in various domains [, ].
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.(^1^11in 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.

Figure: Algorithm 1 UA-P2E for Classification Framework

Algorithm 1 seeks to approximate the ESC (Definition 3.2), which has been proven optimal in Theorem A.1.

### 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.
In Appendix C, we present empirical evidence illustrating the uncertainty from this perspective.
The second uncertainty layer leverages the above property for providing classification predictions with a better confidence in their correctness.
This serves a selective classification [] scheme, in which a carefully chosen subset of the test PPG data is selected by their calculated confidence, exhibiting improved accuracy.

Selective classification [] 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.
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.

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.,

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

In Appendix D, we provide empirical results regarding the quality of the three visualization approaches mentioned above. These results suggest that the most likely score ECG, as defined in Equation (3), provides the highest quality.

## 5 Empirical Study

Figure: Figure 2: MIMIC-III Results: 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.
Refer to caption: x2.png

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.
All experiments described hereafter are conducted using three random seeds, and the results shown are the average of their outcomes.
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 [], 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 [], which is one of the most extensive dataset for paired PPG and ECG signals.

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 classification objective, we train and evaluate the classification model, denoted as $f_{\mathcal{X}}$, using the ‘Computing in Cardiology” (CinC) [] dataset, containing multi-labeled ECG signals of 11 cardiovascular conditions
(^2^22The 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 [] 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 experimental details, including the datasets used for training and evaluating the conversion and classification models, along with details on data pre-processing and training schemes, we refer the reader to Appendix E.

### 5.1 Diffusion-Based Conversion: Quality Assessment

**Table 1: MIMIC-III Results: Quality assessment of our diffusion-based conversion model compared to state-of-the-art baseline methods: CardioGAN [] and RDDM []. The results show the mean metric estimates and their standard errors across the three seeds.**
|  |  | RMSE ↓ | 1-FD ↓ | 100-FD ↓ |
| --- | --- | --- | --- | --- |
| Baseline Methods | CardioGAN [] | 0.54 | 99.0 | - |
| RDDM [] (T=50) | 0.22 | 6.71 | - |  |
| Our Method:<br>UA-P2E | MMSE-Approx | 0.2222 $\pm$ 0.0000 | 11.527 $\pm$ 0.0179 | - |
| DDIM (T=25) | 0.3031 $\pm$ 0.0000 | 2.8671 $\pm$ 0.0034 | 2.7873 $\pm$ 0.0007 |  |
| DDIM (T=50) | 0.2956 $\pm$ 0.0000 | 0.5194 $\pm$ 0.0015 | 0.4418 $\pm$ 0.0007 |  |
| DDIM (T=100) | 0.2939 $\pm$ 0.0001 | 0.3198 $\pm$ 0.0020 | 0.2379 $\pm$ 0.0005 |  |

We begin by assessing the quality of our diffusion-based PPG-2-ECG conversion model using the MIMIC-III [] 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 $\hat{\pi}(X|Y)$.

Table 1 presents the conversion performance results, evaluating the quality of the obtained ECG signals. we provide a comparison with two state-of-the-art methods, CardioGAN [] and RDDM [], 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 in Table 1 demonstrate the superiority of our 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 []. When considering the individual ECG solutions for each PPG, the RMSE values increases, as expected []. In terms of perceptual quality, 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.

### 5.2 Classifying Cardiovascular Conditions

To demonstrate the benefits of our diffusion-based synthesized ECGs, we present classification results on two datasets, MIMIC-III [] and CinC [], 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 F.

Note that MIMIC-III lacks cardiovascular labels, and thus cannot be utilized as-is for classification. As a remedy, we apply our experiments using the CinC dataset. However, while CinC contains ECG signals (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 Appendix G we provide empirical evidence for the validity of our reversal conversion model of ECG-2-PPG in comparison with the original PPG signal.

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 3 presents ROC curves for the classification strategies considered, and Figure 4 completes this description by presenting risk-coverage curves [],
which evaluate the performance of different confidence functions, denoted as $\kappa$, for PPG signals in regard to uncertainty quantification.
For detailed numerical results, including standard errors, refer to Appendix H.

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 A.1 may be interpreted as claiming that our approach is expected to align with that of classifying the original PPG signals directly (ORANGE), under optimal conditions.
However, we observe that our approach achieves better performance.
We argue that this gap is due to the
erroneous labels used in the direct classification: 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: Figure 3: CinC Results: ROC curves for the classification performance of various strategies (see Appendix F), showcasing the superiority of our approach. Higher curves indicate better performance.
Refer to caption: x3.png

Figure: Figure 4: CinC Results: Risk-Coverage curves [] for the uncertainty quantification performance of various strategies, showcasing the superiority of our approach. Lower is better performance.
Refer to caption: x4.png

## 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 showcasing the superior classification performance of our approach, compared to direct classification methods operating on given PPG signals.
Nevertheless, these results are partially reliant on synthetic PPG signals.
While we provide empirical validation for these synthetic signals in this paper, further investigation should be dedicated for the study of our approach with original PPG signals, along with further exploration of additional cardiovascular conditions. Future studies in these areas will require additional data collection.

## 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 showing the optimality of the ESC in optimal settings:

###### Theorem A.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.
The conversion model $g$ is a posterior sampler, i.e., $g(Y,Z)=\hat{X}\sim\pi(X|Y)$; and
- 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 $f_{\mathcal{Y}}(Y)=\pi(C=1|Y)$.

The following is the proof:

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 A.1:

- 1.
The conversion model $g$ is a posterior sampler, i.e., $g(Y,Z)=\hat{X}\sim\pi(X|Y)$; and
- 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.

###### Theorem A.1 (Optimality of the Expected Score Classifier) .

### 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 $f_{\mathcal{X}}(X)$.

###### Theorem .

## Appendix B Assessing Reliable PPG Signals via Calibration

Recalling the desired selective guarantee specified in Equation (2),

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

To ensure the selection of valid PPG signals meeting the above guarantee, 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 [, , ], a general method for distribution-free risk control.
Specifically, we employ the *Learn Then Test* (LTT) [] 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 [].
Through this calibration process, our aim is to identify $\hat{\lambda}$ that ensures the following guarantee:

$$ $\displaystyle\mathbb{P}(R(\hat{\lambda})<\alpha)>1-\delta~{},$ (6) $$

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.

We now turn to describe the steps for calibration to achieve the guarantee outlined in Equation (2).
To simplify Equation (2) into the form posed in Equation (6), we define the *selective risk* as:

$$ $\displaystyle R(\lambda):=\frac{\mathbb{E}_{\pi(Y,C)}\left[\mathbf{1}\{\hat{C} \neq C\}\mathbf{1}\{\kappa(Y)>\lambda\}\right]}{\mathbb{E}_{\pi(Y,C)}\left[ \mathbf{1}\{\kappa(Y)>\lambda\}\right]}~{}.$ (7) $$

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:

$$ $\displaystyle\hat{R}(\lambda):=\frac{\sum_{i=1}^{m}\mathbf{1}\{\hat{C}_{i}\neq C _{i}\}\mathbf{1}\{\kappa(Y_{i})>\lambda\}}{\sum_{i=1}^{m}\mathbf{1}\{\kappa(Y_ {i})>\lambda\}}~{}.$ (8) $$

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:

$$ $\displaystyle\hat{R}^{+}(\lambda):=\hat{R}(\lambda)+r_{\delta}~{},$ (9) $$

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 [] 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 (9).
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:

$$ $\displaystyle\hat{\lambda}:=\min\{\lambda:\hat{R}^{+}(\lambda)<\alpha,\lambda \in\Lambda\}~{}.$ (10) $$

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

Figure: Algorithm 2 Calibration of a Selective Classification UA-P2E Framework

## Appendix C Conversion-Classification Uncertainty Evidence

Figure: (a)
Refer to caption: x5.png

Here, we present empirical evidence of the inherent uncertainty in the proposed PPG-2-ECG conversion, along with the classification uncertainty implied. Figure 5 summarizes the results that quantify these uncertainty layers, referring to the CinC [] dataset.

In Figure 5(a), we illustrate the uncertainty as proposed by [], 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 [].

Figure 5(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 5(b), depicting the uncertainty in classification, as arises from potential variability in the conversion process.
Inspired by [], in Figure 5(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.
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 5(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.

## Appendix D Quality Assessment of ECG Visualization Approaches

**Table 2: CinC Results: Quality assessment of the three proposed ECG visualization strategies, as described in Section 4.3. The results show the mean metric estimates and their standard errors across the three seeds and demonstrate that the Most Likely Score ECG strategy is the best among the strategies in terms of perceptual quality.**
|  | Signal Domain | Embedding Domain |  |  |
| --- | --- | --- | --- | --- |
| Visualization Strategy | RMSE ↓ | 1-FD ↓ | RMSE ↓ | 1-FD ↓ |
| Min/Max Score ECG (5) | 0.3130 $\pm$ 0.0004 | 1.3113 $\pm$ 0.0302 | 0.3079 $\pm$ 0.0037 | 16.676 $\pm$ 0.7135 |
| Expected Score ECG (4) | 0.3119 $\pm$ 0.0004 | 1.2820 $\pm$ 0.0144 | 0.3091 $\pm$ 0.0044 | 13.172 $\pm$ 0.2228 |
| Most Likely Score ECG (3) | 0.3105 $\pm$ 0.0004 | 1.2726 $\pm$ 0.0101 | 0.3040 $\pm$ 0.0043 | 10.491 $\pm$ 0.1879 |

In this section, we examine the ECG visualization strategies, as discussed in Section 4.3, over the CinC dataset.
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.
The three strategies are evaluated across two domains, first, in the signal domain, and second, in the embedding domain using our classification model, denoted as $f_{\mathcal{X}}$.
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.

## Appendix E Experimental Details

This section provides a thorough description of the experimental methodology utilized in our paper.
It encompasses the datasets employed for both training and evaluating the conversion and classification models, along with details regarding data pre-processing and training schemes.

### E.1 Datasets and Preprocessing

In this work we rely on two datasets: the MIMIC-III matched waveform database [], which consists of pairs of measured ECG signals, denoted as $X$, and corresponding PPG signals, denoted as $Y$; and the “Computing in Cardiology” dataset (CinC) [], which contains pairs of measured ECG signals along with associated cardiovascular condition labels, referred to as $C$.

MIMIC-III is one of the most extensive datasets for paired PPG and ECG signals.
It comprises 22,317 records from 10,282 distinct patients, with each record spanning approximately 45 hours.

The MIMIC-III dataset includes a substantial portion of noisy signals, which are unsuitable for training models due to various distortions, artifacts, synchronization issues, and other anomalies. Consequently, a comprehensive preprocessing protocol is employed for preparing the data for subsequent analysis. The preprocessing of each record in MIMIC-III is conducted through the following steps:

- 1.
Resampling: Each record is resampled to a uniform sampling rate of 125 Hz.
- 2.
Signal Smoothing: A Butterworth bandpass filter is applied to all signals to reduce noise.
- 3.
Signal Alignment: Ensures that all signals are temporally aligned.
- 4.
Artifact Removal: Signal blocks with unacceptable heart rates, as extracted from the electrocardiogram (ECG), are removed.
- 5.
Consistency Check: Removes signal blocks where the heart rates derived from the photoplethysmogram (PPG) differ by more than 30% from those extracted from the ECG.
- 6.
Trend Removal: The global trend (DC-drift) in the signal is removed to prevent bias.
- 7.
Normalization: All signals are normalized to have zero mean and unit variance.

Moving to CinC, which to the best of our knowledge, is the largest and most challenging classification dataset in cardiology, containing 94,630 of 10-second multi-labeled ECG recordings from distinct patients.
Each recording is labeled with the presence or absence (binary label) of 11 cardiovascular conditions: (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.
We note that the labels considered in CinC exhibit significant imbalance.

The CinC database, noted for its cleaner and more structured data, lacks the pairings and complexities found in the MIMIC-III dataset and only includes ECG samples. For this dataset, the preprocessing steps are slightly modified and include:

- 1.
Resampling: Each record is resampled to 125 Hz to standardize the data input.
- 2.
Signal Smoothing: A Butterworth bandpass filter is employed to smooth the ECG signals.
- 3.
Trend Removal: Global trends are removed.
- 4.
Normalization: Signals are normalized to have zero mean and unit variance.

### E.2 Training Details for the Conversion Model

Our study leverages a diffusion-based conversion methodology.
Specifically, we adopt the DDIM framework [].
All the training details for the reversed conversion model, designed for generating PPG signals given ECG ones, are similar to the training details for the ECG-2-PPG conversion, as described hereafter.

The denoising model employed in our experiments is based on a U-Net [] architecture with an attention mechanism, as proposed in [, ].
Originally introduced in the field of image processing, this architecture comprises a series of residual layers and downsampling convolutions, followed by a subsequent series of residual layers with upsampling convolutions.
Additionally, a global attention layer with a single head is incorporated, along with a projection of the timestep embedding into each residual block.
To adapt the architecture for our purposes, we replace the convolutions and attention layers with 1D counterparts, enabling training and evaluation directly in the raw 1D signal domain.
In our study we use $d=1024$ PPG and ECG signal length.

Regarding hyperparameters for the U-Net architecture, we set 256 dimensions for time embedding.
The downsampling and upsampling is applied by factors of [1, 1, 2, 2, 4, 4, 8], with each factor corresponding to a stack of 2 residual blocks.
Attention is applied after the downsampling/upsampling matching to the factor of 4.
To address overfitting, we incorporate dropout with a ratio of 0.2.

Concerning training, the model architecture is trained on MIMIC-III.
We randomly split the data into 70% for training and 30% for validation after preprocessing.
The model architecture is trained for 10,000 epochs, employing a batch size of 2,048 PPG and ECG recordings across 8$\times$V100 GPUs.
The noise schedule progresses linearly over 1,000 time steps, starting at 1e-6 and ending at 1e-2.
For optimization, we employed the standard Adam optimizer with a fixed learning rate of 1e-4.
An $L_{2}$ loss is utilized to estimate the noise in the input instance.

For evaluation, we sample $K=100$ ECG candidates for each PPG signal using the diffusion-based framework.
We adopt a non-stochastic diffusion process as proposed in [], with skip jumps of 10 iterations, resulting in a total of $T=1000/10=100$ diffusion iterations.

### E.3 Training Details for the Classification Model

Our paper adopts a classification methodology over the CinC dataset, referred to as $f_{\mathcal{X}}$, which classifies ECG signals, whether original or synthesized, into the 11 cardiovascular conditions in CinC.
Additionally, we report the classification performance of a direct approach, denoted as $f_{\mathcal{Y}}$, which directly gets PPG signals instead of ECG ones.
The training scheme outlined hereafter applies to both classifiers.

As discussed earlier, the presence of each cardiovascular condition in the CinC dataset is imbalanced, and thus CinC poses significant challenges for developing a robust classifier.
In Figure 6 (red) we report the frequency of each cardiovascular condition in the dataset.
To address this issue, we implement a novel class-balance sampling technique.
Given the multi-label nature of the classification, where class dependencies exist, our strategy not only ensures a uniform class distribution for positive samples but also carefully selects negative samples from a balanced pool reflecting common class co-occurrences.
This approach significantly aids in achieving a more uniform label distribution, the effectiveness of which is demonstrated in Figure 6 (blue).

The classification architecture utilized in our studies is based on a simplified VGG [] model with 8 layers, incorporating batch normalization modules to promote stability, along with dropout at ratio of 0.2 to address overfitting.
Similar to the conversion model, we modify the VGG architecture for 1D input signals by reconfiguring the convolutional layers for 1D operation.

Optimization is performed on 1$\times$V100 GPU using the Adam optimizer and an $L_{2}$ weight decay of 1e-4. The model is trained over 20 epochs with a batch size of 128, employing a learning rate of 1e-3 with a linear decay and using binary cross-entropy loss as the loss function.

Figure: Figure 6: An illustration of the label histograms for two different sampling methods on the CinC training dataset: vanilla-sampling and balanced-sampling. In the histogram for vanilla sampling, the distribution of labels is as they naturally appear in the dataset. On the other hand, in the histogram for balanced sampling, the labels are intentionally adjusted to achieve a more uniform distribution.

## Appendix F 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 [] and CinC [].
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 5.2 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, as practiced by prior work, 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, as practiced by prior work, 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 the proposed 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 G Exploring the Validity of Synthesized PPG Signals within MIMIC-III

Figure: (a)
Refer to caption: x7.png

The main question arising from the results depicted in Section 5.2 is whether using synthesized PPG signals for drawing the various conclusions is a fair and trust-worthy strategy.
One potential medical rationale for this approach is that PPG signals inherently encompass noise. Consequently, employing synthesized PPG signals, even if noisy themselves, can accurately model real-world scenarios.
Still, in order to address this question in a more direct form, 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 performance for both true PPG signals and synthesized ones.

As previously stated in Section 5.2, the MIMIC-III matched waveform database lacks cardiovascular labels.
However, textual cardiac reports are available for a subset of patients within MIMIC-III, referred to as the MIMIC-III clinical database [].
In order to extract AFib labels from this subset, we filtered 400 patients, with 200 reported cases of AFib and 200 with normal sinus rhythm. The determination of AFib or sinus rhythm presence was made by searching for the terms "atrial fibrillation" and "sinus rhythm" within the cardiac reports.
We note that extracting labels using this searching approach may provide erroneous labels.

In Figure 7 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 Section 5.2 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).

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 [], 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.

## Appendix H CinC Numerical Results

**Table 3: CinC Results: Mean and standard error values of macro-level AUROC and AURC across three random seeds, shown for each cardiovascular condition examined. See Appendix F for the classification strategies considered.**
| Classification Strategy | Macro-AUROC ↑ | Macro-AURC ↓ |
| --- | --- | --- |
| Original ECG | 0.9435 $\pm$ 0.0020 | 0.0048 $\pm$ 0.0000 |
| Synthesized PPG | 0.7857 $\pm$ 0.0021 | 0.0313 $\pm$ 0.0001 |
| Synthesized ECG: SSC Mean | 0.7286 $\pm$ 0.0068 | 0.0405 $\pm$ 0.0017 |
| Synthesized ECG: SSC Random | 0.7458 $\pm$ 0.0007 | 0.0392 $\pm$ 0.0002 |
| Synthesized ECG: ESC (Ours) | 0.8496 $\pm$ 0.0010 | 0.0211 $\pm$ 0.0001 |

**Table 4: CinC Results: Mean and standard error values of AUROC and AURC [] across three random seeds, shown for each cardiovascular condition examined. See Appendix F for the classification strategies considered.**
| Cardiovascular Condition | Classification Strategy | AUROC ↑ | AURC ↓ |
| --- | --- | --- | --- |
| 1) Ventricular premature beat | Original ECG | 0.8953 $\pm$ 0.0081 | 0.0042 $\pm$ 0.0005 |
| Synthesized PPG | 0.6724 $\pm$ 0.0033 | 0.0216 $\pm$ 0.0009 |  |
| Synthesized ECG: SSC Mean | 0.6290 $\pm$ 0.0194 | 0.0171 $\pm$ 0.0006 |  |
| Synthesized ECG: SSC Random | 0.6344 $\pm$ 0.0087 | 0.0178 $\pm$ 0.0014 |  |
| Synthesized ECG: ESC (Ours) | 0.8006 $\pm$ 0.0072 | 0.0077 $\pm$ 0.0006 |  |
| 2) Atrial flutter | Original ECG | 0.9710 $\pm$ 0.0004 | 0.0056 $\pm$ 0.0001 |
| Synthesized PPG | 0.8457 $\pm$ 0.0018 | 0.0289 $\pm$ 0.0006 |  |
| Synthesized ECG: SSC Mean | 0.7648 $\pm$ 0.0170 | 0.0424 $\pm$ 0.0040 |  |
| Synthesized ECG: SSC Random | 0.8018 $\pm$ 0.0035 | 0.0380 $\pm$ 0.0009 |  |
| Synthesized ECG: ESC (Ours) | 0.8816 $\pm$ 0.0009 | 0.0220 $\pm$ 0.0003 |  |
| 3) Sinus rhythm | Original ECG | 0.9728 $\pm$ 0.0004 | 0.0175 $\pm$ 0.0003 |
| Synthesized PPG | 0.8111 $\pm$ 0.0010 | 0.1309 $\pm$ 0.0005 |  |
| Synthesized ECG: SSC Mean | 0.7638 $\pm$ 0.0018 | 0.1700 $\pm$ 0.0012 |  |
| Synthesized ECG: SSC Random | 0.7773 $\pm$ 0.0009 | 0.1618 $\pm$ 0.0010 |  |
| Synthesized ECG: ESC (Ours) | 0.8547 $\pm$ 0.0004 | 0.1114 $\pm$ 0.0013 |  |
| 4) Atrial fibrillation | Original ECG | 0.9489 $\pm$ 0.0008 | 0.0059 $\pm$ 0.0001 |
| Synthesized PPG | 0.8263 $\pm$ 0.0007 | 0.0283 $\pm$ 0.0005 |  |
| Synthesized ECG: SSC Mean | 0.8320 $\pm$ 0.0144 | 0.0262 $\pm$ 0.0004 |  |
| Synthesized ECG: SSC Random | 0.8582 $\pm$ 0.0044 | 0.0317 $\pm$ 0.0010 |  |
| Synthesized ECG: ESC (Ours) | 0.9196 $\pm$ 0.0016 | 0.0147 $\pm$ 0.0003 |  |
| 5) Supraventricular premature<br>beats | Original ECG | 0.9080 $\pm$ 0.0017 | 0.0068 $\pm$ 0.0002 |
| Synthesized PPG | 0.6673 $\pm$ 0.0033 | 0.0302 $\pm$ 0.0011 |  |
| Synthesized ECG: SSC Mean | 0.6176 $\pm$ 0.0223 | 0.0313 $\pm$ 0.0027 |  |
| Synthesized ECG: SSC Random | 0.6404 $\pm$ 0.0054 | 0.0351 $\pm$ 0.0006 |  |
| Synthesized ECG: ESC (Ours) | 0.7739 $\pm$ 0.0069 | 0.0163 $\pm$ 0.0006 |  |
| 6) Bradycardia | Original ECG | 0.8072 $\pm$ 0.0224 | 0.0010 $\pm$ 0.0002 |
| Synthesized PPG | 0.7734 $\pm$ 0.0154 | 0.0023 $\pm$ 0.0001 |  |
| Synthesized ECG: SSC Mean | 0.6405 $\pm$ 0.0284 | 0.0143 $\pm$ 0.0057 |  |
| Synthesized ECG: SSC Random | 0.6927 $\pm$ 0.0098 | 0.0028 $\pm$ 0.0005 |  |
| Synthesized ECG: ESC (Ours) | 0.7853 $\pm$ 0.0113 | 0.0011 $\pm$ 0.0001 |  |
| 7) Pacing rhythm | Original ECG | 0.9734 $\pm$ 0.0048 | 0.0009 $\pm$ 0.0002 |
| Synthesized PPG | 0.7258 $\pm$ 0.0119 | 0.0054 $\pm$ 0.0001 |  |
| Synthesized ECG: SSC Mean | 0.6148 $\pm$ 0.0212 | 0.0139 $\pm$ 0.0010 |  |
| Synthesized ECG: SSC Random | 0.6193 $\pm$ 0.0181 | 0.0137 $\pm$ 0.0008 |  |
| Synthesized ECG: ESC (Ours) | 0.7966 $\pm$ 0.0061 | 0.0055 $\pm$ 0.0002 |  |
| 8) Sinus tachycardia | Original ECG | 0.9932 $\pm$ 0.0005 | 0.0015 $\pm$ 0.0001 |
| Synthesized PPG | 0.9674 $\pm$ 0.0003 | 0.0094 $\pm$ 0.0002 |  |
| Synthesized ECG: SSC Mean | 0.9451 $\pm$ 0.0030 | 0.0136 $\pm$ 0.0010 |  |
| Synthesized ECG: SSC Random | 0.9263 $\pm$ 0.0030 | 0.0184 $\pm$ 0.0009 |  |
| Synthesized ECG: ESC (Ours) | 0.9792 $\pm$ 0.0018 | 0.0050 $\pm$ 0.0005 |  |
| 9) 1st degree AV block | Original ECG | 0.9634 $\pm$ 0.0011 | 0.0029 $\pm$ 0.0001 |
| Synthesized PPG | 0.6350 $\pm$ 0.0043 | 0.0365 $\pm$ 0.0006 |  |
| Synthesized ECG: SSC Mean | 0.6139 $\pm$ 0.0092 | 0.0332 $\pm$ 0.0009 |  |
| Synthesized ECG: SSC Random | 0.6184 $\pm$ 0.0050 | 0.0347 $\pm$ 0.0011 |  |
| Synthesized ECG: ESC (Ours) | 0.7193 $\pm$ 0.0095 | 0.0203 $\pm$ 0.0010 |  |
| 10) Sinus bradycardia | Original ECG | 0.9935 $\pm$ 0.0002 | 0.0025 $\pm$ 0.0001 |
| Synthesized PPG | 0.9438 $\pm$ 0.0008 | 0.0304 $\pm$ 0.0005 |  |
| Synthesized ECG: SSC Mean | 0.8932 $\pm$ 0.0028 | 0.0565 $\pm$ 0.0028 |  |
| Synthesized ECG: SSC Random | 0.9005 $\pm$ 0.0010 | 0.0502 $\pm$ 0.0011 |  |
| Synthesized ECG: ESC (Ours) | 0.9580 $\pm$ 0.0003 | 0.0199 $\pm$ 0.0011 |  |
| 11) Sinus arrhythmia | Original ECG | 0.9512 $\pm$ 0.0003 | 0.0039 $\pm$ 0.0001 |
| Synthesized PPG | 0.7741 $\pm$ 0.0032 | 0.0207 $\pm$ 0.0004 |  |
| Synthesized ECG: SSC Mean | 0.6999 $\pm$ 0.0039 | 0.0272 $\pm$ 0.0019 |  |
| Synthesized ECG: SSC Random | 0.7344 $\pm$ 0.0037 | 0.0268 $\pm$ 0.0006 |  |
| Synthesized ECG: ESC (Ours) | 0.8771 $\pm$ 0.0006 | 0.0086 $\pm$ 0.0001 |  |

This section extends the empirical classification results referring to CinC [], that were outlined in Section 5.2. We provide here the numerical results of the mean metric values along with their corresponding standard errors.

Tables 3 and 4 display the AUROC (Area Under the ROC curve) and AURC (Area Under the Risk-Coverage curve []) metrics.
Specifically, Table 4 outlines AUROC/AURC mean and standard error values associated with each classification strategy per cardiovascular condition, whereas Table 3 provides macro-level metrics derived by averaging over all examined cardiovascular conditions.

## Peer Reviews

confidence: 5
contribution: 3
initial_rating: 5
presentation: 2
questions: Hallucination Analysis: How does the model ensure that synthetic ECG signals do not contain unrealistic artifacts?

Baseline Comparisons: The comparison is limited to CardioGAN and RDDM. Have you considered including additional baseline models from the literature (e.g., ArXiv:2309.15375, 2012.04949, 2204.11795, 2101.02362) to provide a more comprehensive performance evaluation?

Real vs. Synthetic Data Performance: How do the model's performance metrics change when trained on real versus synthetic PPG-ECG pairs? Can you quantify the impact of synthetic data on classification accuracy?

Generalizability Across Datasets: Have you considered applying this approach to unpaired PPG-ECG datasets or datasets from different demographic groups? Would such testing be feasible within your current framework? Have you employed techniques such as cross-validation or tested on external datasets to ensure that your model generalizes well beyond the training data?

Risk of Overfitting: Given the complexity of diffusion models and the size of the datasets used, what measures have you taken to prevent overfitting?
soundness: 4
strengths: Originality and Significance: This paper offers a novel approach to the challenging PPG-to-ECG conversion by applying a diffusion-based, uncertainty-aware model. This method effectively addresses the inherently ill-posed nature of the task, capturing the distribution of possible ECG outputs rather than a single solution. By doing so, it meets a main need in cardiovascular diagnostics, especially where paired data is limited.

Methodological Rigor: The paper demonstrates strong methodological rigor with a solid theoretical foundation, including proofs of the Expected Score Classifier's (ESC) optimality. This rigorous analysis, supported by detailed equations (e.g., Theorem 3.1), supports the model’s reliability, ensuring both empirical soundness and theoretical robustness.

Comprehensive Evaluation: A thorough evaluation across 11 cardiovascular conditions highlights the model's generalizability and robustness. Compared to baseline models, it shows superior performance in signal reconstruction and classification, with added metrics for uncertainty quantification. This detailed analysis strengthens the case for clinical applicability.

Clarity and Presentation: The paper is well-organized, balancing technical details with intuitive explanations, such as figures illustrating model performance and ECG visualization strategies, enhancing clarity. The emphasis on interpretability supports practical use in clinical settings.

Significance for the Field: The integration of uncertainty-aware diffusion models for physiological signal conversion represents a meaningful enhancement in machine learning for healthcare. The interdisciplinary approach bridges machine learning and biomedical engineering with the potential to drive future innovations in cardiovascular diagnostics and medical device development.
summary: This paper presents "Uncertainty-Aware PPG-2-ECG (UA-P2E)," a novel framework that employs diffusion models to convert photoplethysmography (PPG) signals into electrocardiography (ECG) signals for improved cardiovascular disease classification. Recognizing the ill-posed and inherently ambiguous nature of the PPG-to-ECG conversion—stemming from the loss of certain physiological information in PPG measurements—the authors propose a multi-solution approach to address this challenge. By leveraging diffusion models, UA-P2E captures the full distribution of possible ECG signals corresponding to a given PPG input, effectively modeling the uncertainty inherent in this inverse problem. This allows the framework to generate robust ECG signals that account for the variability and ambiguity of the conversion process. The authors validate their approach through experiments across multiple cardiovascular conditions, demonstrating state-of-the-art classification performance. They provide empirical evaluations, including comparisons with two baseline models, to substantiate the effectiveness of UA-P2E in both signal reconstruction and cardiovascular classification tasks.
weaknesses: Dependence on Synthetic Data: The evaluation heavily relies on synthetic PPG data, especially for augmenting the CinC dataset (Section 5.2​). This dependence raises concerns about potential biases, as synthetic data may not fully capture the variability and complexities of real-world PPG signals. Consequently, the model's generalizability to real clinical settings might be limited, potentially affecting its practical applicability.

Limited Baseline Comparisons: The paper compares UA-P2E primarily with two baseline models: CardioGAN and RDDM (Table 1). While these comparisons provide some insight, the limited scope restricts a comprehensive understanding of the model's performance relative to the broader range of existing methodologies. Including additional, especially more recent baseline models, such as those in the referenced ArXiv papers, would strengthen the evaluation and better position UA-P2E within the current state-of-the-art.

Potential for Synthetic Artifacts: The possibility of generating hallucinations or artifacts in the synthetic ECG signals produced by the diffusion models is not thoroughly examined. Since diffusion-based models can introduce unrealistic features, a lack of analysis on this front may raise concerns about the reliability and clinical validity of the generated signals. Addressing this risk through quantitative assessments would enhance the credibility of the proposed approach.

Limited Dataset Diversity: The study focuses on paired PPG-ECG datasets from CinC and MIMIC-III. This narrow dataset selection may not adequately demonstrate the model's flexibility or adaptability to other data sources. Expanding the evaluation to include larger or unpaired datasets would provide a more robust validation of the model's generalizability and its potential utility across diverse cardiovascular data.

---

confidence: 3
contribution: 3
initial_rating: 6
presentation: 3
questions: 1. Given the stochastic nature of the diffusion model, are 3 seeds enough to capture the full range of ECG variability, or to provide significant statistical measures in your work?
2. Although the paper reports the classification of the mean and random ECG diffusion-based solutions, you do not mention the performance of CardioGAN and RDDM directly, which were initially used to benchmark the ECG generation performance. If the end goal is to improve classification with ECG-generated signals, don't you think that it would be appropriate to use the exact implementations of CardioGAN and RDDM to compare the performance?
3. Did you ensure that there are no segments from the same recording in both the training and test datasets? I believe this is not mentioned. If the same recording appears in both datasets, the generalization ability of the approach might be not properly evaluated.

Also some missing information:
- The exact database(s) of CinC that was(were) used, as there are many databases from this source available on PhysioNet
- A data summary with the number of samples used to train and test the PPG conversion and classification models (and the corresponding class distributions)
- Only the AURC is used to assess the classification performance, but since the pathological labels often suffer from class imbalance, more insightful metrics such as sensitivity, specificity and F1-Score could be reported to improve the clinical relevance of the approach and facilitate comparison with other works.
soundness: 3
strengths: 1. This work uses a diffusion model to propose multiple generated ECG waves for a single PPG signal, instead of a single solution as in previous works.
2. The paper provides the mathematical proof of the optimality of the proposed classifier, which reassures the credibility of the approach. Also, multiple performance metrics are used to evaluate the model.
3. In general the paper is well structured, with relevant tables and figures for comparison, and explanations are thorough.
4. The use of ECG-generated waves from PPG and its improved classification accuracy could be extended to the current PPG-based widespread solutions for cardiovascular monitoring.
summary: This paper presents a methodology to convert PPG signals into ECG ones by accounting for the uncertainty of the conversion process using a diffusion-based model. The methodology is novel arguing that by using a multi-solution approach - where multiple ECG signals can generate the same PPG wave - the combined classification of the generated ECGs is more accurate in comparison to using only the PPG or using a single generated ECG from state-of-the-art methods. The authors use two datasets, one with pairs of unlabeled ECG and PPG signals and another with only labeled ECG signals. A reverse ECG-2-PPG model is used to generate synthetic PPG for classification.
weaknesses: Despite the comprehensive methodology and appendix, some points require some clarification:
1. The authors use 3 random seeds to generate the ECG signals from the PPG ones and report the mean and standard deviation for the chosen metrics. Given the stochastic nature of the diffusion model, are 3 seeds enough to capture the full range of ECG variability, or to provide significant statistical measures?
2. Although the motivation for the use of PPG-2-ECG relies on the widespread of wearable devices, no database with signals acquired in wearable settings was used. For the diffusion model, the MIMIC-III data comes from hospital facilities (to my knowledge), and for classification, this is unclear (unless the CinC2017 database for AFib was used, for example). This should be stated as a limitation of the work, or as a pointer for future work.

---

confidence: 5
contribution: 3
initial_rating: 5
presentation: 3
questions: 1. As Figure 3 shows, the performance of the PPG-derived ECG significantly trails that of the original ECG. How do you explain this discrepancy? Is it possible that the required information for detecting certain arrhythmias is inherently absent in the PPG signals?
2. Appendix G focuses on Atrial Fibrillation (AF), which is relatively easier to detect. It would be beneficial to include results for other types of diseases to provide a broader evaluation.
3. How does pacing rhythm manifest in PPG signals? Is it feasible to accurately generate ECG signals with pacing rhythm characteristics from PPG?
soundness: 2
strengths: 1. The theoretical foundation validating the transition from PPG to ECG and then to classification is robust. The paper demonstrates that generating multiple candidate ECG sample sets can mitigate the uncertainty of ECG conversion and eliminate errors due to mismatch.
2. Quantification of uncertainties during the conversion process enhances the interpretability of the results.
3. The methodological design is well-justified through practical experiments showing the reliability of PPG-derived ECGs, surpassing state-of-the-art methods. Ablation studies further substantiate the validity of the proposed approach.
4. The paper is well-written with a clear structure, making it easy to follow.
summary: The paper introduces a novel methodology for synthesizing Electrocardiogram (ECG) signals from Photoplethysmogram (PPG) signals, aimed at enhancing the reliability of cardiovascular disease diagnostics while avoiding the difficulties associated with ECG acquisition. By measuring the uncertainties involved in generating the ECG from PPG signals and in classifying from the generated signals, the authors convincingly demonstrate the feasibility and superiority of their proposed method.
weaknesses: 1. The paper should specify which particular database the CinC dataset is derived from. The rationale behind choosing these 11 types of anomalies should be clarified.
2. Although the authors provide a thorough theoretical foundation and extensive analysis, the task of generating ECG from PPG lacks inherent rationale. It is unclear whether the generated ECG can reliably reflect arrhythmias, making this approach seem like an uncertain application of analytical techniques with limited practical value.
3. The authors should further compare their classification results with state-of-the-art models, as the current performance appears suboptimal for ECG classification tasks.

Meta-review: This paper presents an approach to generate ECG from PPG for cardiovascular disease detection. However, although the PPG and ECG signals are inter-related, there is a lack of strong evidence that we can use PPG to detect cardio diseases. As pointed out by the authors, the conversion from PPG to ECG is ill-posed problem. Then the question comes: how do we know the correct ECG is obtained (even if we assume that PPG contain full information of the heart diseases). The reviewers have the concerns on the motivation of the paper as PPG cannot solve some of problems and therefore PPG generated ECG cannot solve the problem too. There is also a concern on the lack of test in real-world data and/or the reliability of such method. I agree with the reviewers there is a lack of motivation to generate ECG from PPG. Why not use PPG directly?

Overall, I cannot recommend to accept this paper.

Decision: Reject