# Improving Generative Model-based Unfolding with Schrödinger Bridges

Sascha Diefenbacher<EMAIL_ADDRESS>Physics Division, Lawrence Berkeley
National Laboratory, Berkeley, CA 94720, USA  Guan-Horng Liu<EMAIL_ADDRESS>Autonomous Control and Decision Systems Laboratory, Georgia Institute of
Technology, Atlanta, GA 30332, USA  Vinicius Mikuni<EMAIL_ADDRESS>National
Energy Research Scientific Computing Center, Berkeley Lab, Berkeley, CA 94720,
USA  Benjamin Nachman<EMAIL_ADDRESS>Physics Division, Lawrence Berkeley
National Laboratory, Berkeley, CA 94720, USA  Berkeley Institute for Data
Science, University of California, Berkeley, CA 94720, USA  Weili Nie
<EMAIL_ADDRESS>Machine Learning Research Group, NVIDIA Research

###### Abstract

Machine learning-based unfolding has enabled unbinned and high-dimensional
differential cross section measurements. Two main approaches have emerged in
this research area: one based on discriminative models and one based on
generative models. The main advantage of discriminative models is that they
learn a small correction to a starting simulation while generative models
scale better to regions of phase space with little data. We propose to use
Schrödinger Bridges and diffusion models to create SBUnfold, an unfolding
approach that combines the strengths of both discriminative and generative
models. The key feature of SBUnfold is that its generative model maps one set
of events into another without having to go through a known probability
density as is the case for normalizing flows and standard diffusion models. We
show that SBUnfold achieves excellent performance compared to state of the art
methods on a synthetic $Z$+jets dataset.

##  I Introduction

Correcting detector effects – called deconvolution or unfolding – is the
central statistical task in differential cross section measurements in
particle, nuclear, and astrophysics. Classical unfolding methods are based on
histograms, which result in binned measurements in a small number of
dimensions. Machine learning has the potential to revolutionize differential
cross section measurements by enabling unbinned and high-dimensional
measurements. A number of machine learning-based unfolding techniques have
been proposed [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15] (see also
Ref. [16] for an overview) and the OmniFold method [5, 9] has recently been
applied to studies of hadronic final states with data from H1 [17, 18, 19,
20], LHCb [21], CMS [22], and STAR [23].

Let $X$ represent111Upper case letters denote random variables and lower case
letters correspond to realizations of those random variables. an event at
detector-level and $Z$ represent the same event at particle-level. The goal of
unfolding is to infer the most likely density $p(z)$ using simulated pairs
$(Z,X)$ and observations $x$ from data. In classical approaches, $X$ and $Z$
are discretized and unfolding proceeds via regularized matrix inversion to
approximate the maximum likelihood solution. The likelihood is a product of
Poisson probability mass functions, although most measurements do not directly
maximize this likelihood. For example, one of the most common classical
unfolding algorithms is called Lucy-Richardson deconvolution [24, 25] (also
known as Iterative Bayesian Unfolding [26]) which uses an Expectation-
Maximization (EM) algorithm to converge to a maximum likelihood estimator.

In the unbinned case, the likelihood is not known. One solution is to use the
EM algorithm, which is at the core of two maximum likelihood machine learning
approaches. The first is OmniFold, which proceeds as follows222This integral
format is a continuum limit representation. In practice, the integrals are
replaces with sums over examples.:

E step

$\omega_{i+1}(x)=p_{\text{data}}(x)/\tilde{p}_{\text{sim.}}(x)$

$\tilde{p}_{\text{sim.}}(x)\equiv\int\text{d}z\,p_{\text{sim.}}(x,z)\,\nu_{i}(z)$

M step

$\nu_{i+1}(z)=\bar{p}_{\text{sim.}}(z)/p_{\text{sim.}}(z)$

$\bar{p}_{\text{sim.}}(z)\equiv\int\text{d}x\,p_{\text{sim.}}(x,z)\,\omega_{i+1}(x)$

Both the Expectation (E) and Maximization (M) steps are achieved in practice
by training classifiers and interpreting the resulting score as the target
likelihood ratio, e.g. take samples from $p_{\text{data}}$ and from
$\tilde{p}_{\text{sim.}}$ (samples from $p_{\text{sim.}}$ weighted by $\nu$)
and train a classifier to distinguish them for the E step. The final result is
the set of simulated events weighted by $\nu$ with probability density
$p_{\text{sim.}}(z)\,\nu(z)$ in the continuum limit. An alternative approach
called IcINN [11] is instead based on generative models:

E step

$p_{i}(z|x)\propto p_{\text{sim.}}(x|z)\,\nu_{i}(z)$

M step

$\nu_{i+1}(z)=\check{p}_{\text{sim.}}(z)/p_{\text{sim.}}(z)$

$\check{p}_{\text{sim.}}(z)\equiv\int\text{d}x\,p_{i}(z|x)\,p_{\text{data}}(x)$

The E step of the IcINN is achieved by training a generative model (a
normalizing flow [27]) to emulate $p_{\text{sim.}}(x|z)\,\nu_{i}(z)$ while the
M step uses a classifier as in OmniFold. The groundwork for the IcINN was laid
in Ref. [7], which focused on the E step, as have other papers using
normalizing flows, diffusion models, and generative adversarial networks [2,
6, 14]. Comparing both the E and M steps of OmniFold and the IcINN, one can
see that they are formally the same and thus the EM proof in Ref. [5] applies
to both approaches.

While formally both algorithms achieve the same EM algorithm step by step,
they have complementary strengths in practice. One of OmniFold’s strengths is
that it starts from an existing simulation and if that simulation is close to
nature, then the neural networks only have to learn a small correction. In
contrast333This is true for normalizing flows and traditional diffusion
models. Generative Adversarial Networks (GANs) [28] do not require a tractable
noise distribution, but are less stable. Reference [10] used a physical latent
space for a Varitional Autoencoder (VAE), but VAEs are not state of the art in
generative modeling., the normalizing flows in IcINN have to map a known
probability density (e.g. a multidimensional Gaussian) to the data and these
can be quite different. However, the E step of OmniFold trains directly on
data and thus its performance degrades when there are not many events. In
contrast, the E step of the IcINN is trained using simulation and so can
better handle the case of fewer events.

We propose an approach called SBUnfold that incorporates strengths of both
OmniFold and the IcINN. We employ a technique called a Schrödinger Bridge (SB)
using a diffusion model to learn the generative model in the IcINN workflow.
In contrast to a normalizing flow or standard diffusion model, a Schrödinger
Bridge learns to map one dataset into another without needing to know the
probability density of one of the datasets. Thus, the Schrödinger Bridge
should ideally learn a small correction while also preserving the E step
learning with simulation and not data. The latter property means that SBUnfold
should outperform OmniFold when there are few events in data. Since OmniFold
and the IcINN differ only in the E step, we focus exclusively on the E step of
the first iteration for all methods and will refer to IcINN as cINN for
disambiguation.

This paper is organized as follows. Section II introduces Schrödinger Bridges
and Sec. III describes how this can be used for unfolding. Numerical results
from the OmniFold public $Z$+jets dataset are presented in Sec. IV. The paper
ends with conclusions and outlook in Sec. V.

##  II Schrodinger Bridge and the connection with Diffusion Models

Diffusion models have become popular choices for generative modeling due to
their large degree of flexibility, stable training, and often competitive
results compared to other approaches. The core idea is to design a time-
dependent perturbation process that slowly perturbs data towards a tractable
noise distribution. The goal of the generative model is to then reverse this
process, starting from a noise sample and denoising towards new data
observations. From a fixed choice of stochastic differential equation (SDE)

$\mathrm{d}\textbf{x}=f(\textbf{x},t)\mathrm{d}t+g(t)\mathrm{dw},$ (1)

described by parameters $f(\textbf{x},t)\in\mathbb{R}^{d}$ and
$g(t)\in\mathbb{R}$, the evolution over time of the data observation
$\textbf{x}\in\mathbb{R}^{d}$ is determined. The same initial data point can
undergo different paths due to the additional stochastic term identified by
the Wiener process, or Brownian motion, w$(t)\in\mathbb{R}^{d}$, often sampled
from a normal distribution with the same dimension as the data. The reverse
process follows the reverse SDE equation that reads:

$\mathrm{d}\textbf{x}=[f(\textbf{x},t)-g(t)^{2}\nabla\log
p(\textbf{x},t)]\mathrm{d}t+g(t)\mathrm{d\bar{w}}.$ (2)

The only unknown term in the Eq. 2 is the score function, $\nabla\log
p(\textbf{x},t)$, which can be approximated using denoising score matching
[29] and minimizing the loss function

$\mathcal{L}_{SGM}=\frac{1}{2}\mathbb{E}_{\textbf{x}_{t},t}\left\|\epsilon_{\theta}(\textbf{x}_{t},t)-\sigma_{t}\nabla_{\textbf{x}_{t}}\log
q(\textbf{x}_{t}|\textbf{x})\right\|^{2}_{2},$ (3)

where a neural network with trainable parameters $\theta$ is optimized to
approximate the score function of data that have been perturbed by a Gaussian
distribution with time-dependent parameters $\alpha_{t}$ and $\sigma_{t}$. For
this Gaussian perturbation,
$q(\textbf{x}_{t}|\textbf{x})=\mathcal{N}(\textbf{x}_{t},\alpha_{t}\textbf{x},\sigma_{t}^{2}\textbf{I})$,
requiring $f(\textbf{x},t)$ to be affine with respect x and resulting in a
Gaussian noise at the end of the diffusion process. For an arbitrary noise
distribution, the corresponding choice of $f(\textbf{x},t)$ cannot be easily
identified, often restricting the family of distributions one is allowed to
choose for the diffusion process. A more general framework that allows the
mapping between general distributions corresponds to the solution of the
Schrödinger Bridge (SB) problem. Initially proposed by Erwin Schrödinger [30],
the problem concerned the inference of the trajectories from particles
undergoing a diffusion process where experimental observations of the
particle’s trajectory are available only at specific time values, fixing the
boundary conditions of the problem. The connection with diffusion generative
models becomes more clear by considering the following forward and backward
SDEs:

$\displaystyle\mathrm{d}\textbf{x}$
$\displaystyle=[f(\textbf{x},t)+g(t)^{2}\nabla\log{\Psi}(\textbf{x},t)]\mathrm{d}t+g(t)\mathrm{dw},$
(4a) $\displaystyle\mathrm{d}\textbf{x}$
$\displaystyle=[f(\textbf{x},t)-g(t)^{2}\nabla\log\hat{\Psi}(\textbf{x},t)]\mathrm{d}t+g(t)\mathrm{d\bar{w}}.$
(4b)

The wave-functions $\Psi$ and $\hat{\Psi}$ satisfy

$\displaystyle\frac{\partial{\Psi(\textbf{x},t)}}{\partial t}$
$\displaystyle=-\nabla\Psi(\textbf{x},t)^{\intercal}f(\textbf{x},t)-\frac{1}{2}g(t)^{2}\Delta\Psi(\textbf{x},t)$
$\displaystyle\frac{\partial{\hat{\Psi}(\textbf{x},t)}}{\partial t}$
$\displaystyle=-\nabla\cdot(\hat{\Psi}(\textbf{x},t)f(\textbf{x},t))+\frac{1}{2}g(t)^{2}\Delta\hat{\Psi}(\textbf{x},t)$

with $\Psi(\textbf{x},0)\hat{\Psi}(\textbf{x},0)=p_{A}(\textbf{x})$ and
$\Psi(\textbf{x},1)\hat{\Psi}(\textbf{x},1)=p_{B}(\textbf{x})$ for densities
$p_{A}(\textbf{x})$ and $p_{B}(\textbf{x})$. Compared to Eq. 2, the presence
of an additional non-linear term to the drift function,
$g(t)^{2}\nabla\log\hat{\Psi}(\textbf{x},t)$, enables the diffusion between
densities that are not necessarily represented by a standard normal
distributions at the end of the diffusion process. Additionally,
$\nabla\log\hat{\Psi}(\textbf{x},t)$ no longer represents the score function
of the perturbed data, but is related to it since

$\Psi(\textbf{x},t)\hat{\Psi}(\textbf{x},t)=q(\textbf{x},t),$ (5)

hence

$\nabla\log\Psi(\textbf{x},t)+\nabla\log\hat{\Psi}(\textbf{x},t)=\nabla\log
q(\textbf{x},t).$ (6)

While the equations describing the general SB problem have similarities with
the standard framework for diffusion generative models, a general strategy to
solve the problem is not immediately obvious. The authors of Ref. [31] propose
a tractable solution named I2SB, by considering the presence of pairs of
observations in the dataset such that
$p(\textbf{x}_{a},\textbf{x}_{b})=p_{A}(\textbf{x}_{a})p_{B}(\textbf{x}_{b}|\textbf{x}_{a})$.
This assumption also holds for the unfolding methodology where pairs of
particle collisions before and after detector interactions are always
available in the simulation. With this approximation, the authors of Ref. [31]
have shown that by setting the linear drift term $f(\textbf{x},t){:=}0$, the
posterior of Eq. 4, $q(\textbf{x}|\textbf{x}_{a},\textbf{x}_{b})$, has the
analytic form:

$\displaystyle
q(\textbf{x}|\textbf{x}_{a},\textbf{x}_{b})=\mathcal{N}(\textbf{x}_{t};\mu_{t}(\textbf{x}_{a},\textbf{x}_{b}),\Sigma_{t}),$
(7) $\displaystyle\mu_{t}$
$\displaystyle=\frac{\bar{\sigma}_{t}^{2}}{\bar{\sigma}_{t}^{2}+\sigma^{2}_{t}}\textbf{x}_{a}+\frac{\sigma^{2}_{t}}{\bar{\sigma}_{t}^{2}+\sigma^{2}_{t}}\textbf{x}_{b},\quad\Sigma_{t}=\frac{\sigma_{t}^{2}\bar{\sigma}_{t}^{2}}{\bar{\sigma}_{t}^{2}+\sigma^{2}_{t}}\cdot
I,$ (8)

with $\sigma^{2}_{t}{:=}\int_{0}^{t}g^{2}(\tau)\mathrm{d}\tau$ and
$\bar{\sigma}^{2}_{t}{:=}\int_{t}^{1}g^{2}(\tau)\mathrm{d}\tau$. From this
expression, given pairs $(\textbf{x}_{a},\textbf{x}_{b})$, we can directly
determine $\textbf{x}_{t}=\mu_{t}+\Sigma_{t}\epsilon$,
$\epsilon\sim\mathcal{N}(0,1)^{d}$ for any time step $t$. The loss function is
then identified similar to Eq. 3 as:

$\mathcal{L}_{\mathrm{I^{2}SB}}=\frac{1}{2}\mathbb{E}_{\textbf{x}_{t},t}\left\|\epsilon_{\theta}(\textbf{x}_{t},t)-\frac{\textbf{x}_{t}-\textbf{x}_{0}}{\sigma_{t}}\right\|^{2}_{2},$
(9)

where the right term of the loss function approximates the score function of
the backward drift $\nabla\log\hat{\Psi}(\textbf{x},t)$, which can then be
used during sampling to transport samples from $p_{B}$ to $p_{A}$.

During sampling, standard recursive samplers like DDPM [32] can be used with
the prediction of the denoised data
$\textbf{x}_{0}^{\theta}=\textbf{x}_{n+1}-\sigma_{n+1}\epsilon_{\theta}(\textbf{x}_{n+1},n+1)$
at time step $n<N$ defined as:

$\textbf{x}_{n}\sim
q(\textbf{x}_{n}|\textbf{x}_{0}^{\theta},\textbf{x}_{n+1}),\textbf{x}_{N}\sim
p_{B}.$ (10)

The stochastic solver is able to produce different observations given the same
exact inputs, which can then be used for different coverage tests regarding
the validity of the generated outputs. A second option is to consider the
deterministic case where the posterior distributions are effectively replaced
by their means. This particular case is described by the solution of the
following ODE:

$\mathrm{d}\textbf{x}_{t}=\mathbf{v}_{t}(\textbf{x}_{t}|\textbf{x}_{0})\mathrm{d}t=\frac{\beta_{t}}{\sigma_{t}^{2}}(\textbf{x}_{t}-\textbf{x}_{0})\mathrm{d}t\,,$
(11)

which describes an optimal transport plan [33]. In our studies, we observe
similar sample quality between the stochastic and deterministic settings and
report the results based on the deterministic case for simplicity.

We set $f(\textbf{x},t){:=}0$ and $g(t)=\sqrt{\beta(t)}$, with $\beta(t)$ the
triangular function:

$\beta(t)=\begin{cases}\beta_{0}+2(\beta_{1}-\beta_{0})t,&0\leq
t<\frac{1}{2},\\\
\beta_{1}-2(\beta_{1}-\beta_{0})(t-\frac{1}{2}),&\frac{1}{2}\leq t\leq
1.\end{cases}$ (12)

with $\beta_{0}=10^{-5}$ and $\beta_{1}=10^{-4}$.

##  III Unfolding Methodology

We test SBUnfold using the public dataset from Ref. [5], available on Zenodo
[34] and briefly summarized in the following. Proton-proton collisions
producing a $Z$ boson are generated at a center-of-mass energy of
$\sqrt{s}=14$ TeV. A non-trivial test of unfolding requires at least two
datasets, one that acts as the ‘data’ and one that is the ‘simulation’. For
the ‘data’, collisions are simulated with the default tune of Herwig 7.1.5
[35, 36, 37]. We only make use of the reconstructed events from Herwig as
would be the case with real data. The Herwig particle-level events are only
used when evaluating the performance of different methods. For the
‘simulation’, events are simulated with Tune 26 [38] of Pythia 8.243 [39, 40,
41]. Detector distortions are simulated with Delphes 3.4.2 [42] and the CMS
tune that uses a particle flow reconstruction. In future work, we will
investigate the performance of SBUnfold on the full phase space of particles
(particle-level) and particle flow objects (detector-level). For this study,
we focus on a fixed set of six observables that serve as the benchmark for
fixed-dimension unfolding. These observables are computed from the
substructure of the leading jet. The jets are clustered using all particle
flow objects at detector level and all stable non-neutrino truth particles at
particle level. They are defined by the anti-$k_{T}$ algorithm [43] with
radius parameter $R=0.4$ as implemented in FastJet 3.3.2 [44, 45]. The first
four observables are the jet mass $m$, constituent multiplicity $M$, the
$N$-subjettiness ratio $\tau_{21}=\tau_{2}^{(\beta=1)}/\tau_{1}^{(\beta=1)}$
[46, 47], and the jet width $w$ (implemented as $\tau_{1}^{(\beta=1)}$). The
remaining two observables are the jet mass $\ln\rho=\ln
m_{\text{SD}}^{2}/p_{T}^{2}$ and momentum fraction $z_{g}$ after Soft Drop
grooming [48, 49] with $z_{\text{cut}}=0.1$ and $\beta=0$. Many of the
observables are computed with FastJet Contrib 1.042 [50].

The $Z$ bosons are required to have $p_{T}>200$ GeV in order to render
acceptance effects negligible444See Ref. [9] for how to include acceptance
effects.. Each dataset consists of about 1.6 millon events.

The noise prediction model of SBUnfold is implemented using a fully-connected
architecture incorporating multiple skip connections. Specifically, the model
employs four ResNet [51] blocks, where each residual layer is connected to the
output of a single-layer network through a skip connection. The activation
function used is LeakyRelu [52] with a slope of $\alpha=0.01$, and all layer
sizes are set to 32. The implementation of the model is carried out using
Pytorch [53]. The generation is carried out by starding from data observations
at reconstruction level and using the SB to move towards the generator level
events using 1000 time steps to improve accuracy. The cINN implementation uses
the normalizing flow model presented in Ref. [7] while the first step of the
first iteration of OmniFold uses a fully-connected model with three hidden
layers, ReLu activation function, and node sizes set to 50, 150, 50, before
the output layer with a single node and sigmoid activation function. Both cINN
and OmniFold are implemented in TensorFlow [54]. For all models, the number of
trainable parameters are kept similar at around 13k. The initial learning rate
is set to $10^{-3}$ and all models are trained for 30 epochs with batch size
of 128. A total of 1M training events are used to train all models taken from
the Pythia simulations. The pseudo-data used to evaluate the performance of
the unfolding is taken from Herwig. Both SBUnfold and cINN do not have access
to the data during training while OmniFold uses a total of 2M training events
(1M from Pythia and 1M from Herwig). We investigate the impact of the pseudo-
data training size in Sec. IV when determining the unfolded results. All
trainings are carried out using the Perlmutter supercomputer with a single
NVIDIA A100 GPU.

Figure 1: Comparison between different unfolding algorithms for six different
physics observables unfolded. All observables are unfolded simultaneously
without binning, with histograms shown only for evaluation. Results are
evaluated over 600’000 pseudo-data points. Statistical uncertainties are shown
only in the ratio panel. Pseudo-data and simulation are described by Pythia.

##  IV Results

We evaluate the performance of SBUnfold compared to other unfolding
methodologies by first looking at unfolded distributions where both pseudo-
data and samples used to train the different unfolding methods originate from
the Pythia simulation. The results showing the unfolded distributions are
shown in Fig. 1. In App. A, we also provide the comparison of the results
obtained by a standard diffusion model.

Figure 2: Pairwise correlation plot between all unfolded variables using the
cINN (left) or SBUnfold (right) algorithms.

We observe a good agreement with the expected distributions from Pythia at
generator level and the responses of the generative models for unfolding. We
omit the OmniFold results since the weighting function becomes trivial when
both data and simulation are statistically identical. We also investigate the
distribution of the pairwise correlation between each pair of unfolded
features to verify that SBUnfold is also capable of learning the correct
correlations between distributions. The results are shown in Fig. 2.

Table 1: Comparison of the earth mover’s distance (EMD) and triangular discriminator between different algorithms. EMD is calculated over unbinned distributions while triangular discriminator uses histograms as inputs. Uncertainties from EMD are derived using 100 bootstraps with replacement taken from the unfolded data. Results are evaluated using 600’000 pseudo-data points sampled from Pythia. Quantities in bold represent the method with best performance. Model |  EMD($\times 10$)/Triangular Discriminator($\times 10^{3}$)
---|---
| cINN | SBUnfold
Jet mass |  1.4$\pm$0.2/0.29 |  0.70$\pm$0.06/0.30
Jet Width |  0.013$\pm$0.002/0.25 |  0.0029$\pm$0.0005/0.06
N |  2.3$\pm$0.8/0.09 |  0.57$\pm$0.04/0.9
$\log\rho$ |  1.1$\pm$0.3/0.64 |  0.27$\pm$0.01/0.68
$z_{g}$ |  0.095$\pm$0.003/10.9 |  0.009$\pm$0.001/3.1
$\tau_{21}$ |  0.2$\pm$0.1/0.6 |  0.016$\pm$0.001/0.2

We again observe a good agreement between the generated distributions and
expected distributions from the simulation for both generative models, with
SBUnfold showing improved description compared to the cINN for a few
distributions such as $z_{g}$ vs. $\log\rho$ and $z_{g}$ vs $\tau_{21}$. The
distribution of $z_{g}$ shows a sharp cutoff at $z_{g}=0$ which is hard to
reproduce with generative models. Since the same sharp distribution is
observed at reconstructed level events, SBUnfold can take advantage of the
reconstructed prior when starting the diffusion process, contrary to the cINN
that always requires samples from the standard normal distribution for the
prior transformation. We also quantify the agreement between unfolded
distributions with expected values derived from Pythia by calculating both the
triangular discriminator metric [55, 56, 57] over histograms as presented in
Fig. 1, as well as using the Earth mover’s distance (EMD) directly on the
marginalized one-dimensional features generated by the unfolding methods. The
results are listed in Table 1.

Figure 3: Comparison between different unfolding algorithms for six different
physics observables unfolded. All observables are unfolded simultaneously
without binning, with histograms shown only for evaluation. Results are
evaluated over 600’000 pseudo-data points. Statistical uncertainties are shown
only in the ratio panel. Pseudo-data is taken from Herwig while Pythia is
taken as the main simulator.

For all distributions we observe significantly lower EMD values for SBUnfold
compared to cINN with four out of six also showing lower values of the
triangular discriminator. Uncertainties from the EMD calculation are derived
from the standard deviation of 100 bootstraps with replacement taken from the
unfolded data. We observe the EMD uncertainties from cINN to be often higher
than SBUnfold, which points to improved stability from SBUnfold due the more
informative prior.

Next, we keep the same unfolding methodology from SBUnfold and cINN trained
over Pythia samples but instead use reconstructed events from Herwig as the
data representative. We compare the unfolded results between the different
distributions in Fig. 3.

We observe improved agreement in all unfolded distributions compared to the
distributions of reconstructed events, while systematic shifts are observed in
all unfolding methodologies for the jet mass and $\tau_{21}$ distributions.
These features highlight the prior dependence of all models which cannot be
excluded without the presence of the M step and additional iterations. In
Table 2 we calculate the EMD and triangular discriminator using the unfolded
distributions with Herwig as pseudo-data.

Table 2: Comparison of the earth mover’s distance (EMD) and triangular discriminator between different algorithms. EMD is calculated over unbinned distributions while triangular discriminator uses histograms as inputs. Uncertainties from EMD are derived using 100 bootstraps with replacement taken from the unfolded data. Results are evaluated using 600,000 pseudo-data points sampled from Herwig. Quantities in bold represent the method with best performance. Model |  EMD($\times 10$)/Triangular Discriminator($\times 10^{3}$)
---|---
| OmniFold Step 1 | cINN | SBUnfold
Jet mass |  6.1$\pm$0.1/1.5 |  10.1$\pm$1.2/2.4 |  9.0$\pm$0.1/3.1
Jet Width |  0.06$\pm$0.001/1.1 |  0.05$\pm$0.003/0.7 |  0.02$\pm$0.001/0.2
N |  1.7$\pm$0.1/0.2 |  6.1$\pm$4.0/0.2 |  3.0$\pm$0.1/0.6
$\log\rho$ |  1.35$\pm$0.03/1.1 |  3.1$\pm$2.1/0.8 |  0.4$\pm$0.1/0.7
$z_{g}$ |  0.086$\pm$0.001/1.2 |  0.3$\pm$0.1/12.7 |  0.049$\pm$0.001/3.5
$\tau_{21}$ |  0.23$\pm$0.02/4.6 |  0.7$\pm$0.4/3.5 |  0.12$\pm$0.02/1.4

Once again we observe a good performance of SBUnfold, achieving the lowest EMD
values for four out of six distributions and lowest triangular discriminator
values for three of the six. We also observe OmniFold achieving similar
performance for all observables, in particular OmniFold shows improved
performance for the particle multiplicity, which is not a continuous
distribution and hence harder for generative models to determine precisely. We
also study how SBUnfold corrects physics observables back to generator level
events by calculating the migration matrix between reconstructed and unfolded
observables. Results showing that SBUnfold learns to apply small corrections
to reconstructed events are shown in App. B.

OmniFold is also the only algorithm that uses the statistical power of the
data to determine the unfolded distributions. We investigate how the unfolding
results change when the available number of data entries is reduced to only
1,000 instead of the 600,000 thousands we used previously. For both SBUnfold
and cINN this change only modifies the number of generated samples from the
trained model while OmniFold only has access to the 1,000 data examples during
training while the number of Pythia samples is not changed and kept at
1,000,000. The unfolded results are shown in Fig. 4.

Figure 4: Comparison between different unfolding algorithms for six different
physics observables unfolded. All observables are unfolded simultaneously
without binning, with histograms shown only for evaluation. Results are
evaluated over 1’000 pseudo-data points, the same number was used to train
OmniFold while other algorithms only rely on simulation. Statistical
uncertainties are shown only in the ratio panel. Pseudo-data is taken from
Herwig while Pythia is taken as the main simulator.

Due to the lower number of data entries we observe larger statistical
fluctuations from the unfolded results which still shows a good agreement
compared to the expected quantities. We evaluate further the unfolded results
by calculating the EMD and triangular discriminator, listed in Table 3.

Table 3: Comparison of the earth mover’s distance (EMD) and triangular discriminator between different algorithms. EMD is calculated over unbinned distributions while triangular discriminator uses histograms as inputs. Uncertainties from EMD are derived using 100 bootstraps with replacement taken from the unfolded data. Results are evaluated using 1’000 pseudo-data points sampled from Herwig. Quantities in bold represent the method with best performance. Model |  EMD($\times 10$)/Triangular Discriminator($\times 10^{3}$)
---|---
| OmniFold Step 1 | cINN | SBUnfold
Jet mass |  8.7$\pm$1.8/13.6 |  9.2$\pm$3.0/8.4 |  7.7$\pm$2.5/6.9
Jet Width |  0.14$\pm$0.02/18 |  0.07$\pm$0.02/5.7 |  0.05$\pm$0.02/4.6
N |  12$\pm$3/10.9 |  5.4$\pm$1.3/3.8 |  5.8 $\pm$1.6/3.7
$\log\rho$ |  4.0$\pm$0.8/11 |  1.6$\pm$0.5/6.2 |  1.2$\pm$0.3/4.4
$z_{g}$ |  0.08$\pm$0.02/1.5 |  0.08$\pm$0.03/7.2 |  0.06$\pm$0.01/7.1
$\tau_{21}$ |  0.4$\pm$0.07/16 |  0.2$\pm$0.05/12 |  0.1$\pm$0.04/8

The reported results for SBUnfold and cINN are less affected by the reduced
data sizes, even though the uncertainties from EMD have greatly increased due
to limited pseudo-data examples. On the other hand, OmniFold shows a worse
degradation compared to previous results, with EMD values often disagreeing by
more than two standard deviations.

##  V Conclusion and Outlook

In this paper we presented SBUnfold, an unfolding algorithm that uses a
Schrödinger bridge (SB) to design a stochastic mapping between distributions.
Since the SB allows the transport between arbitrary distributions555Our
Schrödinger bridge is not a universal function approximator; it will map the
distributions to each other, but may not preserve the conditional probability
density [31]. Our numerical results indicate that this is not an issue and we
expect that monotonic detector distortions should be in the class of functions
that SBs are able to accommodate, but we leave further investigations to
future work. We thank Jesse Thaler for useful discussions on this point.,
SBUnfold does not require a tractable prior similar to other generative
models, but instead uses a diffusion process to directly denoise physics
observables from detector effects. We have demonstrated the performance of
SBUnfold using synthetic $Z$+jets samples and compared with both state-of-the-
art methods using generative models (cINN) and reweighting (OmniFold). Using
different metrics, we observe an excellent performance of SBUnfold. Compared
to cINN, SBUnfold shows improved fidelity for distributions with sharp
features and smaller uncertainties overall likely due to more informative
priors from reconstructed level events. Compared to OmniFold, SBUnfold shows a
more robust performance against variations of the number of available data
observations, showing promising results for cases where the amount of
available data is reduced.

We have only investigated the expectation step of different unfolding
algorithms, leaving the maximization step for future study. We notice that the
maximization step based on classifiers used by IcINN is also applicable for
SBUnfold, and would be interesting to compare how the unfolding results
improve with subsequent iterations between both algorithms.

Finally, similar to OmniFold, SBUnfold can be readily adapted to different
data structures such as image-like datasets. On the other hand, the diffusion
process currently relies on the specific ordering of the features used during
unfolding (see Eq. 8), making the application of SBUnfold with low-level
features, such as particles, not trivial. Following studies will investigate
how to accommodate permutation invariance within SBUnfold.

## Code Availability

The code for this paper can be found at
https://github.com/ViniciusMikuni/SBUnfold.

## Acknowledgments

We thank Daniel Whiteson, Tilman Plehn, and Jesse Thaler for thoughtful
discussions and feedback on the manuscript. VM, SD, and BN are supported by
the U.S. Department of Energy (DOE), Office of Science under contract DE-
AC02-05CH11231. GHL would like to thank Evangelos Theodorou for helpful
discussions. This research used resources of the National Energy Research
Scientific Computing Center, a DOE Office of Science User Facility supported
by the Office of Science of the U.S. Department of Energy under Contract No.
DE-AC02-05CH11231 using NERSC award HEP-ERCAP0021099.

##  Appendix A Comparison with standard diffusion models

We investigate the benefits of using SBUnfold as opposed to a standard
diffusion model. The diffusion model implementation follows the FPCD model
proposed in [58] for jet kinematic generation. The backbone neural network
architecture used is the same as the one used in SBUnfold. Similarly, the
learning rate, number of training epochs, and training samples are kept the
same between the two models. The main conceptual difference between the
implementations is that SBUnfold starts from the reconstruction level events
instead of a Gaussian prior used in FPCD. Similarly to the cINN
implementation, the FPCD model is only conditioned on reconstruction level
events to determine the unfolded response. Results of the unfolded
distributions using Pythia both as pseudo-data and simulation are shown in
Fig. 5.

Figure 5: Comparison between different unfolding algorithms for six different
physics observables unfolded. Results are evaluated over 600’000 pseudo-data
points. Statistical uncertainties are shown only in the ratio panel. Pseudo-
data and simulation are described by Pythia.

For all distributions we observe a good agreement between the unfolded results
and the pseudo-data. In Table 4 we calculate the EMD and triangular
discriminator using the unfolded distributions obtained by the different
methodologies.

Table 4: Comparison of the earth mover’s distance (EMD) and triangular discriminator between different unfolding methodologies. EMD is calculated over unbinned distributions while triangular discriminator uses histograms as inputs. Uncertainties from EMD are derived using 100 bootstraps with replacement taken from the unfolded data. Results are evaluated using 600’000 pseudo-data points sampled from Pythia. Quantities in bold represent the method with best performance. Model |  EMD($\times 10$)/Triangular Discriminator($\times 10^{3}$)
---|---
| FPCD | cINN | SBUnfold
Jet mass |  0.74$\pm$0.08/0.19 |  1.4$\pm$0.2/0.29 |  0.70$\pm$0.06/0.30
Jet Width |  0.0087$\pm$0.0006/0.9 |  0.013$\pm$0.002/0.25 |  0.0029$\pm$0.0005/0.04
N |  0.81$\pm$0.06/0.1 |  2.3$\pm$0.8/0.09 |  0.57$\pm$0.04/0.9
$\log\rho$ |  0.34$\pm$0.01/0.77 |  1.1$\pm$0.3/0.64 |  0.27$\pm$0.01/0.68
$z_{g}$ |  0.035$\pm$0.007/12.4 |  0.095$\pm$0.003/10.9 |  0.009$\pm$0.001/3.1
$\tau_{21}$ |  0.024$\pm$0.002/0.3 |  0.2$\pm$0.1/0.6 |  0.016$\pm$0.001/0.2

From the results we observe that FPCD often outperforms the cINN in the EMD
metric while still not achieving the same level of performance as SBUnfold.
These results indicate that standard diffusion models may be more expressive
compared to the normalizing flow model implemented in the original cINN, but
still not as precise as SBUnfold which leverages the more informative prior
distribution from reconstruction level objects rather than the Gaussian prior
required by both cINN and FPCD.

##  Appendix B Migration matrix of unfolded events

In this appendix we investigate how SBUnfold maps reconstructed level
distributions back to generator level samples by calculating the corresponding
migration matrix using Herwig as the pseudo-data while the detector response
is derived from the Pythia simulation. The distributions are shown in Fig. 6.

Figure 6: Migration distribution between reconstructed level events and
generator level distribution after unfolding with SBUnfold. Results are
evaluated over 600’000 pseudo-data points. Pseudo-data is taken from Herwig
while Pythia is taken as the main simulator.

The migration matrix for all observables has a high fraction of events present
close to the diagonal, consistent with SBunfold learning to apply only a small
correction to the inputs.

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# Survey: Vitals Screening Techniques for a Safer Environment

Sarah Baig, Mohammed Elbadry

###### Abstract

With COVID-19 disrupting operations across various sectors of the workforce
(e.g., offices, airports, libraries, schools), preventative measures enabling
resumption of work are quickly becoming a necessity. In this paper, we present
the need for Vitals Screening Techniques (VIST) where more than one vital is
screened to ensure the safety of the population (e.g. temperature, heart rate,
and blood oxygen levels). VIST can be deployed in crowded environments to
provide the new necessary layer of safety. We provide extensive coverage of
state-of-art technology that can assist in tackling this emerging problem, and
evaluate one of the existing products on the market that employ VIST.

##  1 Introduction

The COVID-19 Pandemic has fundamentally changed society’s perspective [1] on
common spaces (e.g., airports, schools, commercial establishments). The ’new
normal‘ has heralded a new age of technological innovations, from contact
tracing, telemedicine to newer screening methods[2]. Although temperature
screening has been widely implemented [3, 4] as a way to safely reopen, multi-
vital screening provides greater insight into the health of an individual.
Vital signs, including temperature, blood oxygen levels and heart rate, are
the easiest and most critical data points gathered from an individual to
assess their general health. [5].

In emergency settings, patients have to be prioritized and guided to the
correct place of treatment (“triage”) largely based on their vital signs [6,
7, 8, 9]. Vitals screening methods are applied to high throughput areas (e.g.,
airports, businesses, warehouses, factories), especially in the case of a
global pandemic [10]. One of the vitals that is important during screening is
heart rate. Elevated heart rate (Tachycardia) is a telltale sign associated
with fever in the case of viral respiratory infections [11]. Another vital
that can be helpful is blood oxygen levels (SpO2). Low blood oxygen levels
were seen before the onset of a fever in many COVID-19 patients [12] and can
indicate underlying signs of other viral diseases [13]. Body temperature,
heart rate, and blood oxygen levels together provide further insight into the
health of an individual.

Human body temperature is well established as one of the key vital signs [14].
The accepted mean value for normal human body temperature measured orally is
37°C (98.6°F). However newer research indicates that the average may actually
be closer to 36.6°C (97.9°F) [15]. Each individual has their normal body
temperature, which varies slightly from the ideal value. Human body
temperature constantly adapts to its environmental conditions. A body
temperature of 38°C (100.4°F) or more is considered to be a fever [16]. The
most recent viral epidemics have had fever as the most common symptom (e.g.,
Ebola, SARS, H1N1 [17], and COVID-19 [18]). It is evident that fever detection
is one of the key components in screening.

All current screening techniques rely on remote body temperature measurements.
There are two problems that must be acknowledged when measuring the body
temperature: i) normal body temperature variation; ii) infrared thermal
imaging limitations. Body temperature can fluctuate based on the region
selected for measurement [19, 20]. Furthermore, research has shown that body
temperature is a nonlinear function of several variables such as age, state of
health, gender, environmental temperature, time of the diurnal cycle, and
among many others [14]. On average, healthy elderly people have lower body
temperature compared to younger adults [14]. The human body is constantly
adapting its temperature to environmental conditions (e.g., goes up in the
afternoon and lower at night). Despite these minor variations, elevated body
temperature is still a universally accepted indicator of fever.

Remote body temperature screening is a fast, non-invasive alternative to
conventional clinical thermometers for monitoring body temperature [21, 20].
Average external body temperature (peripheral skin temperature) is 2-4°C
(3.6-7.2°F) less than the core temperature [19]. Therefore, mean body
temperature must be calculated from external (or skin) temperature using an
estimation algorithm. Infrared radiation emitted by a surface depends on the
environmental conditions such as moisture, airflow and surrounding temperature
[22, 23]. Other factors that impact temperature sensing are ambient
temperature drift and aging of the sensor. An individual’s thermal state also
affects the radiated heat (e.g., running, coming from cold environments, etc).
Further, the distance and angle of the thermal camera relative to the subject
plays a critical role in the sensor’s fidelity. Blackbody devices (temperature
references) are known to solve the issues related to ambient temperature and
sensor aging, improving the accuracy of the sensor. However, they are often
forgone due to the cost or complexity of deployment [24]. Remote body
temperature sensing is an ideal alternative to clinical thermometers that are
sometimes cumbersome and often require an attendant [21, 25].

Beyond temperature sensing, blood oxygen levels are used to infer any
impairment in lung function [26]. Blood oxygen level is usually measured with
a pulse oximeter (finger clip). A resting oxygen saturation level between 95%
and 100% is regarded as normal for a healthy person at sea level, and below
95% is considered abnormal [27, 28, 29, 30]. Low blood oxygen can serve as an
indicator to many different viral pneumonias [12]. The recent global pandemic
(i.e. COVID-19) has demonstrated that many people can have dangerously low
oxygen levels, without showing any other symptoms (’silent hypoxemia’) [31].
The detection of low oxygen levels in asymptomatic individuals can facilitate
early diagnosis of an underlying illness. Blood oxygen measurement serves as a
key component of Vital Screening.

Heart rate is measured through pulse oximeters, in addition to blood oxygen
levels. The American Heart Association defines the normal sinus heart rate as
between 60 and 100 bpm at rest [32, 33, 11] (it is important to note that
athletes often have heart rates below 60 bpm at rest). Tachycardia is observed
in case of anemia, intake of caffeine, and exercise [34]. Tachycardia is seen
concomitantly with fever due to an increase in the Basal Metabolic Rate and
cardiac output. In one study, when the temperature rose by 1°C (1.8°F) due to
fever, the heart rate increased on the average by 8.5 beats per minute [11].
Thus, tachycardia, when seen along with fever, can point to possible
infection.

Pulse oximetry technology involves shining light at specific wavelengths
through tissue (most commonly the fingernail bed) and using a detector to
determine the amount of light that passes through [35, 36]. There are several
inherent limitations to pulse oximetry. One of the common examples of
interfering factors is poor signal due to certain nail polish and artificial
fingernails [37]. Poor peripheral perfusion because of cold, hypotension, or
Raynaud’s disease is the principal cause for failure to obtain a satisfactory
signal, mainly because of an inadequate pulse wave [29, 35]. Motion artifacts
can interfere with signal detection because of an unstable waveform.
Improperly seated sensors, shivering, seizures, or tremors can cause movement
leading to inaccurate readings. Pulse oximetry, despite its limitations, is
universally recognized as an essential vital measurement tool.

The need to take preventative measures to prepare for inevitable future
outbreaks is apparent. We present a solution using existing sensors: Vital
Screening Techniques (VIST). VIST involves scanning individual’s multiple
vitals within seconds using robust sensors. It provides the added layer of
safety needed to move past COVID-19 pandemic.

##  2 Vitals Screening Techniques

VIST encompasses any device that measures more than one independent vital(
e.g., body temperature, heart rate, and blood oxygen level) of an individual.
VIST devices usually have built in sensors in the form of thermal cameras and
pulse oximeters. Most of the products display the readings on a user-friendly
interface. A live video feed usually allows the user to adjust their
positioning. VIST allows for the rapid mass screening of individuals to ensure
a safe environment.

For external temperature sensing, thermal cameras are used to obtain targets’
radiated temperature. The thermal camera should ideally be set up in a room
temperature environment [38, 39, 40]. Measurements should be made only at a
fixed distance, with the subject directly facing the camera. The incorporation
of a blackbody device allows for higher accuracy in the narrow range of normal
human body temperature. The subject should acclimatize to room temperature for
a few minutes prior to measurement. Per the FDA [25], it is recommended to
measure one person at a time.

In pulse oximetry, blood oxygen levels and heart rate are measured through the
process of photoplethysmography (PPG). PPG is a non-invasive technology that
uses a light source and a photodetector at the surface of the skin (finger-
tip) to measure the volumetric variations of blood circulation with a
resulting waveform. Pulse oximeters can incorporate either the transmissive or
reflective mode. In the transmissive mode, the light sources and the
photodiode are opposite to each other with the measurement site between them
[41]. In the reflective mode, the light sources and photodiode are on the same
side, and light is reflected to the photodiode across the measurement site.
The transmissive mode is not only the most commonly used method, it is the
only clinically approved one, because of its high accuracy and stability. The
clip-style of the pulse oximeter probe eliminates some of the errors due to
finger movement. The only drawback of the clip style is that it is difficult
to clean and cannot incorporate UV-C disinfection.

A pulse oximeter used in mass screening is a high-touch surface with the
potential for disease transmission. Ultraviolet (UV-C) sterilization is one
method that may be incorporated to address this concern. UV-C’s effectiveness
against different strains of viruses has long been established [42]. Studies
show that UV-C light at 267 and 279 nm was very effective at inactivating the
Coronavirus [43].Recent studies have shown the chance of transmission of SARS-
CoV-2 (through inanimate surfaces) is less frequent than previously recognised
[44]. The improper exposure to UV-C radiation poses risks to human’s skin and
eyes. UV-C can be offered as an optional feature to alleviate concerns amongst
the general public about touching this high-contact surface.

Figure 1:  Sample Interface from Symptomsense

##  3 Evaluation

In this section, we evaluate Soter Technologies’ sensors that are available
for licensing, as well as full-integrated in products of various form factors
(e.g., kiosk, handheld, gateway, etc.).

###  3.1 Sensors Fidelity

We compare Soter’s sensors against other sensors on the market. The thermal
sensor is compared against Braun thermometer [45] and Famidoc [46]. Braun has
FDA 510(K) Pre-market Notification, and Famidoc passed the EU standards for
infrared thermometers accuracy. To the best of our knowledge, there is no
existing remote thermal scanner that is FDA approved. The experiments were
done over 20 different individuals with age ranges from 23 to 65 and various
skin pigments. For each test, 10 samples were collected leading to total of
250 scans per each sensor.

Temperature Sensing. Soter’s thermal camera warms within 2 minutes after
start-up. With an incorporated blackbody, ambient temperature does not affect
readings. There is no drift or aging of the sensor due to the built-in
blackbody technology. Multi-point temperature readings are made from different
regions of the face. A unique weighted estimation algorithm is used adjust
skin temperature to body temperature. Only one exposed region is necessary to
obtain a reading, allowing for accurate temperature readings from individuals
with head coverings, masks, etc.

We find that Famidoc has ±0.09°C (0.17°F) precision and Braun has ±0.12°C
(0.21°F) precision. On average, Famidoc and Braun are about 0.40°C (0.72°F)
from each others measurements. We find that Braun cannot measure people of
lower than average body temperature (e.g. body temperature of 96°F or less).
For lower body temperature individuals, we find that Famidoc performs better
than Braun, however with degraded accuracy and precision performance. Soter’s
temperature sensor retains precision of ±0.28°C (0.51°F) and accuracy of
±0.15°C (0.27°F) compared to Braun. Compared to Famidoc, Soter’s thermal
sensor has an accuracy of ±0.18°C (0.33°F) Further, Soter’s sensor is able to
robustly detect those individual’s with generally lower than average body
temperature with high precision and accuracy relative to Braun and Famidoc.
The average scan time of Soter’s temperature sensor is 4.76 seconds. Further,
we find that Soter’s sensor yielded 100% completed scans, unlike Braun, 97.5%
(Famidoc had 100% completed scans). This shows that Soter’s sensor can detect
larger population than the Braun sensors.

Figure 2:  Accuracy of Temperature Sensors

Available | ✓ |  SymptomSense Kiosk |  SymptomSense Gateway | TAVIS | Olea Irvine | Olea Austin + | Ksubaka | Rapidscreen
---|---|---|---|---|---|---|---|---
Unavailable | X
Temperature | Thermal Sensor | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
Blackbody | ✓ | ✓ | X | X | X | X | X
Live Video | ✓ | X | ✓ | ✓ | ✓ | ✓ | ✓
Facial Detection | ✓ | ✓ | X | ✓ | ✓ | ✓ | ✓
Pulse Oximeter | Blood Oxygen | ✓ | ✓ | ✓ | X | X | X | X
Heart Rate | ✓ | ✓ | ✓ | X | X | X | X
| Configurations | ✓ | ✓ | ✓ | X | ✓ | X | ✓
Integration Potential | ✓ | ✓ | ✓ | X | ✓ | X | X

Table 1: In-depth comparison between existing products in the market [47, 48,
49, 50, 51]. The table shows existing sensors that provide reliable
temperature sensing by combining known techniques, thus enabling the highest
sensing accuracy (e.g., facial detection, blackbody usage, etc.). We also show
existing products in the market that support pulse oximetry sensor to provide
oxygen levels and heart rate readings.

Heart Rate and Blood Oxygen Soter’s product line offers various fully-
integrated pulse oximeters available in different form factors (open design,
clip etc.). All of Soter’s pulse oximeters use transmissive PPG technology.
Here, we evaluate the off-the-shelf, medical-grade, FDA approved pulse
oximeter (Nonin 3231)111Test subjects included those that traditionally may
have difficulty obtaining pulse oximetry readings, including those with powder
coated nails, poor perfusion, as well as cold hands.. We find the average
pulse oximetry scan time is 7.7 seconds from our testing. Nonin 3231 yielded
100% successfully completed scans. This pulse oximeter is FDA approved and has
undergone a clinical study with a stated accuracy of ±1.31% between 70-100%
blood oxygen [52]. Heart rate accuracy is ±3 digits between 20-250 bpm.

The Nonin 3231 clip pulse oximeter measures a wide range of heart rates: 18 –
321 beats per minute. It can measure blood oxygen levels between 0-100%. The
ratio between the amplitude of the red light at 660nm and infrared light at
910nm wavelength is used to determine oxygen saturation. The pulse oximeter
requires only 2 beats and uses an averaging algorithm to obtain an accurate
heart rate. Longer scan times (with more readings taken) will result in more
accurate pulse oximetry values.

###  3.2 User Experience

As an example of existing products’ user interfaces, SymptomSense provides a
user friendly interface (Figure 1) where users can read their results (either
vital values or pass/fail) right after screening. Figure 1 shows a completed
scan with temperature, heart rate and blood oxygen readings (on the left of
the interface), and pulse oximetry scan complete (on the right). Once readings
are obtained, visual indicators are used to indicate if the individual has
passed the screening or not per the ranges of acceptable pass defined by the
system operator.

###  3.3 Existing Products

Table 1 shows in-depth comparison between existing products that leverage both
single and multi-vital sensing. Configurations include the ability to set
vital ranges, pass or fail vs numeric readings. Integration features include
an optional battery, printer, barcode system, etc.

##  4 Discussion

UV-C and standard medical pulse oximeter. There is a shift to incorporate UV-C
into pulse oximeters. Soter’s proprietary pulse oximeter with optional UV-C is
currently in the final phase of development. This pulse oximeter utilizes a
double-fail safe to automatically turn off when a finger is detected ensuring
limited exposure to harmful UV-C rays. 275 nm of UV-C light (limited to the
contact area) disinfects the surface for a 30 second duration after every use.
To our knowledge, there is currently no FDA approved medical finger pulse
oximeter with UV-C disinfection capability.

Remote respiratory rate is in development. There is no established technology
that can remotely measure the respiratory rate with high confidence.
Nonetheless, there is recent work with various technology to estimate and
measure the respiratory rate (thermal sensing [53], mmwave [54]). Given the
current issues remote respiratory rate technology is facing (e.g., random
individual vibrations, and better sensors required), it is not mature yet for
the consumer market. However, to obtain a valid respiratory rate reading with
high confidence, the individual has to get scanned for a long duration (e.g.,
> 10 seconds) to obtain a valid reading, since respiratory rate is much lower
than heart rate (5-20).

Screening best practices. The FDA [55, 56, 38, 39, 40, 25] has provided proper
guidelines on screening best practices (including thermal camera environment
and calibration). Soter’s sensors make the process of following the FDA
procedure easier as they have self-calibrated blackbody based thermal system
that shows final output to the user. Soter does not claim to do multi-person
detection which has been shown to be ineffective. The Nonin 3231 pulse
oximeter is already classified as a medical device and is FDA approved.

Further screening. The entity using our kiosks may choose to do further
screening/testing before rejecting the individual that our system flags [39].
We leave the guidelines of handling the user to the entity as it may differ
and vary given location’s conditions. We recommend following CDC guidelines.

HIPAA compliance. Most of the screening systems do not store or retain any
information about the user and their vitals, it only shows the vital results
for a brief moment. Facial recognition technology is currently not implemented
in the SymptomSense product-line, Thus, there is no need for HIPAA clearance,
hence anonymized vital screening [57].

##  5 Acknowledgement

We thank Asheik Hussain and Cary Chu for their invaluable feedback and
continuous assistance of this work.

##  6 Conclusion

In this paper, we introduce VIST, vitals screening techniques. We cover state
of the art vitals scanners that exist in the market, with in depth comparison
between them. We further cover details about how the underlying sensing
technologies work and their drawbacks.

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# A method of defining central and Gibbs measures and the ergodic method

A. M. Vershik111St. Petersburg Department of Steklov Institute of Mathematics,
St. Petersburg State University, and Institute for Information Transmission
Problems.

###### Abstract

We formulate a general statement of the problem of defining invariant measures
with certain properties and suggest an ergodic method of perturbations for
describing such measures.

##  1 Defining probability measures via projective conditional measures

In the second half of the last century, a new method of defining probability
measures in infinite-dimensional systems, alternative to the classical
(Kolmogorov’s) one, was gradually developed. Instead of a system of consistent
finite-dimensional distributions, which uniquely defines a measure via
projections, the new method involves another data system, roughly speaking, a
consistent system of conditional measures. The method appeared independently
in the theory of Markov processes (E. B. Dynkin), in statistical physics (R.
L. Dobrushin), etc. We present an abstract version of the method, regarding it
simultaneously as a far-reaching generalization of the theory of measurable
partitions of Lebesgue spaces and Rokhlin’s systems of conditional measures
and as the problem of describing invariant measures in the theory of dynamical
systems and Gibbs measures. The presentation is in terms of equipped
equivalence relations (e.e.r.), or, in other words, Borel partitions in a
standard Borel space and “projective conditional measures” on elements of
these partitions. We could also use the language of groupoids or the language
of the theory of extensions of measures from special algebras (but not
$\sigma$-algebras) of sets to a $\sigma$-algebra (not the whole
$\sigma$-algebra) of measurable sets. Similar considerations in smaller
generality can be found in [2, 6].

The new method has essentially introduced a large number of combinatorial,
analytic, and algebraic problems: about central measures on path spaces of
graded graphs, about Markov measures with given cotransitions, etc.

Consider a standard Borel space $(X,\mathfrak{A})$ (isomorphic to the interval
$[0,1]$ regarded as a Borel space) with the $\sigma$-algebra ${\mathfrak{A}}$
of all Borel sets. Let $\tau$ be a Borel equivalence relation (e.r.) (i.e.,
partition) on this space with countable classes and $\rho$ be a Borel
$2$-cocycle with nonnegative real values on this relation, i.e., a Borel
function on pairs of equivalent points satisfying the conditions
$\rho(x,x)=\rho(x,y)\rho(y,x)=1$, $\rho(x,y)\rho(y,z)=\rho(x,z)$. The cocycle
$\rho$ uniquely (up to a positive coefficient) defines a finite or
$\sigma$-finite nonnegative measure (in short, “conditional projective
measure”) on each equivalence class. The pair $(\tau,\rho)$ will be called an
equipped equivalence relation on the space $(X,\mathfrak{A})$.222It would be
natural, extending Rokhlin’s terminology, to introduce the term
“semimeasurable partition with a system of conditional projective measures.”
If all classes are finite, then the e.e.r. is nothing else than a Borel
measurable partition, and the cocycle defines conditional probability measures
on all classes.

On the other hand, it is well known that if $\mu$ is a probability measure on
$X$ (i.e., on a Lebesgue space) and $\tau$ is an e.r., then $\mu$ uniquely
defines an equipment of this e.r., that is, a $2$-cocycle, or a conditional
projective measure on almost every equivalence class. In the case where the
e.r. is the partition into the orbits of an action of a countable group with a
quasi-invariant measure, this cocycle is the so-called Radon–Nikodym cocycle
$RN_{\mu}$. If the cocycle is identically equal to $1$, then the measure is
said to be invariant. If the cocycle is not indicated, it is assumed to be
identically equal to $1$. We now formulate the main problem.

######  Problem 1.

Let $(\tau,\rho)$ be an e.e.r. on a standard Borel space $(X,\mathfrak{A})$;
find all Borel probability measures $\mu$ for which the Radon–Nikodym cocycle
$RN_{\mu}$ coincides with $\rho$ almost everywhere with respect to the measure
$\mu$, or, in other words, find all probability measures with given
conditional projective measures on equivalence classes.

In the case where such a measure is unique (it is this case that corresponds
to Kolmogorov’s system of finite-dimensional distributions), we may speak of
extending the measure to the $\sigma$-algebra of all measurable sets; in the
general case, there can be no uniqueness.

The family of all measures on $(X,\mathfrak{A})$ determined by Problem 1 is
well defined and forms, in a natural way, a Choquet simplex. The set of its
extreme points (Choquet boundary) is called the absolute of the e.e.r.
$(\tau,\rho)$ and denoted by $\operatorname{Ab}(X,\tau,\rho)$. Any two
different measures from $\operatorname{Ab}(X,\tau,\rho)$ are mutually singular
and defined on different complete $\sigma$-algebras.

The traditional construction of Gibbs measures, as well as the problem of
describing invariant measures of group actions in dynamics, obviously, fits in
this scheme. The notion of absolute is closely related to various notions of
boundary.

If all classes of an e.e.r. are finite, then we have a well-defined Borel
quotient space $X/{\tau}$, which coincides with
$\operatorname{Ab}(X,\tau,\rho)$, and describing all (not ergodic) measures
from ${\cal M}_{(\tau,\rho)}$ reduces to indicating a measure on this quotient
space. But if the e.r. does not define a Borel quotient space (space of
classes), then studying the structure of the absolute is a difficult problem
and depends significantly on the geometry of classes. The solution of Problem
1 can be “wild,” i.e., it may happen that the absolute has no reasonable
parametrization, but in many (for example, combinatorial) problems, a
parametrization can be found.

It is not difficult to extend all these definitions to e.r.’s $\tau$ whose
equivalence classes are not countable, but endowed with a well-defined locally
compact topology.

It is important to emphasize that an e.r. can be studied only together with a
cocycle, i.e., a system of conditional projective measures (even if the
cocycle is identically equal to $1$). The main role in the further analysis of
the subject (uniqueness, special properties of measures, etc.) must be played
by the geometry of classes of e.e.r.’s, but it is still poorly studied.

We now state the inverse problem.

######  Problem 2.

Let $M$ be a family of probability measures defined on the $\sigma$-algebra
$\mathfrak{A}$ of a standard Borel space $X$. Find the minimal e.r. $\tau$ for
which all measures $\mu\in M$ define the same cocycle $\rho\equiv RN_{\mu}$.

This problem is a generalization of the traditional problem related to
sufficient statistics (cf. [1]), in which one usually seeks only measurable
(for example, finite) equivalence relations. In the above setting, there are
no restrictions on the e.r. For example, consider the set of all Bernoulli
measures $\prod_{1}^{\infty}(p,1-p)$, $p\in[0,1]$, on the space of sequences
$\prod_{1}^{\infty}\\{0;1\\}$. The required e.e.r. is the de Finetti partition
with cocycle identically equal to $1$: two sequences are equivalent if they
coincide from some index $n$ on and have the same number of zeros among the
first $n$ coordinates.

##  2 Hyperfinite and tame e.e.r.’s, the universal Markov model

We now consider Problem 1 for a special case whose importance is due to a
large number of applications. Namely, it includes the problem of describing
characters of locally finite groups, or, more generally, describing traces on
AF-algebras, as well as the problem of describing central measures on path
spaces of graded graphs.

An equipped equivalence relation $\tau$ is said to be hyperfinite if it is a
monotonely increasing limit of a sequence of finite equivalence relations:
$\tau=\bigcup_{n}\xi_{n}$. Thus, a hyperfinite e.e.r. can be defined by a
sequence of its finite approximations, i.e., a decreasing sequence of
measurable partitions $\\{\xi_{n}\\}_{n}$ with finite elements and conditional
measures on them. Such sequences are called filtrations. For details, see [8].

By a number of well-known theorems, the orbit partition for a group action
with an invariant measure is hyperfinite if and only if the group is amenable.
However, orbit partitions with nontrivial cocycles can be hyperfinite also for
nonamenable groups. Note that the hyperfiniteness condition for an e.e.r. is a
condition on the cocycle, i.e., on the conditional projective measures, but
apparently it has not been stated in this form. For Lebesgue spaces, a
hyperfinite e.e.r. is unique up to isomorphism (generalized H. Dye’s theorem).

We impose a slightly stronger (than hyperfiniteness) condition on the
approximating sequence of measurable partitions $\\{\xi_{n}\\}$: an e.r. is
said to be tame, or locally hyperfinite, if for every $n$ the number of types
of conditional measures of the partition $\xi_{n}$ is finite. This condition
singles out a class of hyperfinite e.e.r.’s that is of most interest for
applications. Now we describe a universal model of a tame e.e.r.

######  Definition 1.

Let $X_{n}$ be a finite or compact space and $\\{\pi_{n}\\}$ be a set of
“transition operators” that send a point $x\in X_{n}$ to a subset
$\pi_{n}(x)\subset X_{n+1}$. The corresponding Markov (nonstationary)
compactum $\operatorname{Mar}$ is the space of sequences

$\operatorname{Mar}\subset\\{\\{x_{n}\\}_{n=1}^{\infty}:x_{n}\in
X_{n},\;n=1,2,\dots\\}$

where

$\\{x_{n}\\}\in\operatorname{Mar}\iff x_{n+1}\in\pi_{n}(x_{n})\quad\text{for
every $n\geq 1$}.$

Elements of $\operatorname{Mar}$ are called trajectories, or paths. The tail
equivalence relation $\tau$ on the Markov compactum $\operatorname{Mar}$ is
the following relation on trajectories:

$\\{x_{n}\\}\sim_{\tau}\\{y_{n}\\}\iff\quad\text{there exists $N$ such that
$x_{n}=y_{n}$ for every $n>N$}.$

The Markov compactum $\operatorname{Mar}$ is endowed with the weak topology
and Borel structure. A Markov Borel measure $P$ is defined by an initial
distribution $\mu_{1}(\cdot)$ of the coordinate $x_{1}$ and a collection of
transition probabilities, i.e., a family of measures $\\{P_{n,x}\\}$,
$n=1,2,\dots$, $x\in X_{n}$, where
$P_{n,x}(y)=\operatorname{Prob}(x_{n+1}=y|x_{n}=x)$.

But we will need another data system on a Markov compactum, a system of
cotransition probabilities. It is a family of measures $\\{P^{n,x}\\}$ on
$X_{n}$, ${n=1,2,\dots}$, $x\in X_{n+1}$, where
$P^{n,x}(y)=\operatorname{Prob}(x_{n}=y|x_{n+1}=x)$. Such a system does not
yet define a global measure on the entire Markov compactum.

######  Lemma 1.

Every system of cotransition measures defines a cocycle on the tail e.r. of
the Markov compactum: the quotient of the conditional measures of two paths
$\\{x_{n}\\}$ and $\\{y_{n}\\}$ coinciding for $n>N$ is equal to the quotient
of the products of the corresponding transition probabilities

$\prod_{1\leq i\leq
N}\frac{\operatorname{Prob}(x_{i}|x_{i+1})}{\operatorname{Prob}(y_{i}|y_{i+1})}.$

Such cocycles will be called Markov cocycles; a Markov compactum equipped with
a Markov cocycle, i.e., a system of cotransitions, will be called an equipped
Markov compactum.

Obviously, every Markov measure on $\operatorname{Mar}$ uniquely determines a
Markov cocycle, but, in general, a system of contransitions, i.e., a Markov
cocycle, does not uniquely determine a Markov measure. It is equally clear
that on a Markov compactum there can exist non-Markov cocycles.

######  Theorem 1 (universal model).

For every standard Borel space $X$ and a tame equipped equivalence relation
$\tau$ on $X$ with cocycle $\rho$ there exists an equipped Markov compactum
$\operatorname{Mar}$ and a Borel isomorphism
$T:X\rightarrow\operatorname{Mar}$ that sends $\tau$ to the tail e.r. on
$\operatorname{Mar}$ and sends $\rho$ to a Markov cocycle.

Thus, Problem 1 of finding all invariant measures for a tame e.e.r. reduces to
the problem of finding all Markov probability measures $P$ on a compactum
$\operatorname{Mar}$ with a given system of cotransition probabilities. In
other words, to the problem of describing Markov chains with given
cotransitions.

If the cocycle is identically equal to $1$, i.e., all conditional measures of
all orders are uniform, then we obtain the problem of describing all measures
of maximal entropy on a given Markov compactum.

The absolute of a Markov compactum $\operatorname{Mar}$ is denoted by
$\operatorname{Ab}(\operatorname{Mar})$. The proof of the theorem essentially
follows from the results of [8].

Instead of the language of Markov compacta, one can use the language of
$\mathbb{N}$-graded graphs (Bratteli diagrams): the path space of such a graph
is a Markov compactum, which ensures the parallelism. In many situations
(mainly of combinatorial nature), the language of graphs is more preferable.
The author does not know models similar to the universal model for general
e.e.r.’s.

##  3 The ergodic method and the method of perturbations

By the ergodic method of solving Problem 1 about invariant measures for
hyperfinite equivalence relations we mean the method of finding invariant
distributions and invariant measures based on the pointwise ergodic theorem
or, more exactly, on the pointwise martingale convergence theorem, applied to
the characteristic functions of sets from a basis of the $\sigma$-algebra on
which the required measure is defined. In this meaning, the term is used in
the author’s paper [7] and in earlier papers (see, e.g., [4]). But the
practical task of finding invariant measures, i.e., probabilities of
cylinders, or transition probabilities, as limits of some conditional
expectations can be quite difficult. A very important factor is the choice of
a basis of sets whose measures are being calculated. But, on the other hand,
the problem of finding all ergodic measures can be “wild,” so complete
calculations can be essentially infeasible. A reasonable classification of
hyperfinite absolutes (i.e., systems of conditional projective measures) is
hardly possible; while the Borel classification of e.e.r.’s is, on the
contrary, too rough (see [2]); the author does not know any other,
intermediate, classification criteria. That is why, it is important to have
feasible solvability criteria for the problem of describing invariant
measures, as well as methods of reduction of problems to a few canonical
problems.

One of these fundamental problems, whose solution is obtained by a canonical
application of the ergodic method, is the problem of describing all ergodic
measures on the infinite product $X^{\infty}=\prod_{1}^{\infty}X$ (where $X$
is a Borel space) invariant under the group $S_{\infty}$ of all finite
permutations of coordinates. Denote by $\tau^{F}$ the e.r. on $X^{\infty}$
generated by the partition into the orbits of the action of $S_{\infty}$. The
answer to Problem 1 is given by de Finetti’s theorem and says that every
ergodic invariant measure is a Bernoulli measure with an arbitrary one-
dimensional distribution (= measure on $X$). Thus,
$\operatorname{Ab}(X^{\infty},\tau^{F})=\operatorname{Meas}(X)$.

If we regard this answer up to a measure-theoretic isomorphism, it turns out
that the absolute consists of a unique purely continuous measure on $X$, a
continuum of discrete measures, and their mixtures, i.e.,
$\operatorname{Ab}(X^{\infty},\tau^{F})=\\{\\{\alpha_{n}\\}:\alpha_{n}\geq
0,\,\sum_{n}\alpha_{n}\leq 1\\}$.

In many recent examples of problems related to absolute in combinatorial and
algebraic situations, the answer (hypothetical or otherwise) has a similar
structure: the absolute is a Choquet simplex (which can be called the
secondary simplex), i.e., the ergodic measures themselves also admit a
decomposition. That is why, it is natural to conjecture that a proof of this
fact should be sought not in a direct calculation but in analyzing the
reduction to the fundamental de Finetti problem described above. We suggest
the following method, which can be called the method of perturbations. For an
unperturbed problem, we take the de Finetti problem about $\tau^{F}$. The
first step is to construct a homomorphism $T$ of the space where we seek the
absolute for some e.r. $\tau$ to the space $X^{\infty}$ such that $T(\tau)$ is
a subpartition of $\tau^{F}$. At the second step, we must verify that the
absolute of $T(\tau)$ is similar to, or even coincides with, the absolute of
$\tau^{F}$. To find $T$, if it exists, is the most nontrivial part of the
method.

The second step is related to a problem about the infinite product
$X^{\infty}$ which is of independent interest.

######  Problem 3.

For what e.r.’s $\tau$ satisfying the condition $\tau\succ\tau^{F}$ does the
absolute $\operatorname{Ab}(X^{\infty},\tau)$ consist of all Bernoulli
measures?

The following partial answer is useful:

######  Lemma 2.

Let $\tau,\tau^{\prime}$ be two e.e.r.’s on a Borel space $X$ with cocycles
identically equal to $1$, and let
$\operatorname{Ab}(X,\tau^{\prime})\subset\operatorname{Ab}(X,\tau)$. These
absolutes coincide if and only if the following condition is satisfied: for
every ergodic measure ${\mu\in\operatorname{Ab}(X,\tau^{\prime})}$, the e.e.r.
$\tau$ is ergodic with respect to $\mu$.

In turn, proving ergodicity reduces to verifying that a certain sequence of
functionals converges to a constant in measure, and not to the more difficult
problem of finding weak limits, as in the general ergodic method.

An illustrative example of the usefulness of the method of perturbations is
given by the problem about central measures on the Young graph. Thoma’s
theorem about characters, more exactly, a rephrasing of this theorem as an
assertion about the absolute of the Young graph, leaves no doubt that this
problem should be regarded in connection with de Finetti’s theorem. The fact
is that the answers to these problems are remarkably similar. Namely, the
absolute is stratified: a stratum of discrete measures parametrized by one-
dimensional frequencies with sum $1$ and a stratum of measures with zero
frequencies. However, all proofs known to date (see [9]) are not elementary
and do not reveal the closeness of these problems. This relation is indeed
nontrivial, and the main role here is played by the dynamic properties of the
RSK algorithm, which makes it possible to construct a required lifting of the
graph of $Q$-tableaux to the Schur–Weyl graph.

Using $Q$-tableaux arising in the RSK algorithm to cover central measures on
the Young graph by Bernoulli measures was first suggested in [3]; the fact
that this correspondence is an isomorphism was proved in [5]. But the
embedding discussed above has not been noted; at the same time, a careful
analysis shows that the method of perturbations allows one also to prove the
theorem about the absolute, i.e., to prove that in this way we obtain all
ergodic central measures. For Young tableaux with finitely many rows, this
result is implicitly contained in [11].

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