# Single-frame transmission and phase imaging using off-axis holography with
undetected photons

Emma Pearce1,2,+ Osian Wolley1,+ Simon P. Mekhail1 Thomas Gregory1 Nathan R.
Gemmell2 Rupert F. Oulton2 Alex S. Clark3 Chris C. Phillips1 Miles J.
Padgett1<EMAIL_ADDRESS>1School of Physics and Astronomy,
University of Glasgow, Glasgow, G12 8QQ, United Kingdom  2Blackett Laboratory,
Department of Physics, Imperial College London, London, SW7 2AZ, United
Kingdom  3Quantum Engineering Technology Labs, H. H. Wills Physics Laboratory
and Department of Electrical and Electronic Engineering, University of
Bristol, BS8 1FD, United Kingdom

###### Abstract

Imaging with undetected photons relies upon nonlinear interferometry to
extract the spatial image from an infrared probe beam and reveal it in the
interference pattern of an easier-to-detect visible beam. Typically, the
transmission and phase images are extracted using phase-shifting techniques
and combining interferograms from multiple frames. Here we show that off-axis
digital holography enables reconstruction of both transmission and phase
images at the infrared wavelength from a single interferogram, and hence a
single frame, recorded in the visible. This eliminates the need for phase
stepping and multiple acquisitions, thereby greatly reducing total measurement
time for imaging with long acquisition times at low flux or enabling video-
rate imaging at higher flux. With this single-frame acquisition technique, we
are able to reconstruct transmission images of an object in the infrared beam
with a signal-to-noise ratio of $1.78\,\pm\,0.06$ at 10 frames per second, and
record a dynamic scene in the infrared beam at 33 frames per second.

Imaging and sensing with undetected photons is now a well-established and
actively researched technique for probing samples in the infrared (IR) while
only ever detecting visible light [1]. Motivating this is the ability to
observe highly specific molecular information by using IR light to investigate
a sample’s vibrational and rotational modes, with use in polymer
identification for recycling [2], gas sensing [3], quality control [4, 5], and
diagnostic medicine [6]. These applications are, however, limited by currently
available IR detection technologies; cameras in the IR remain significantly
more expensive, less efficient, and noisier than silicon-based detectors for
visible wavelengths.

To avoid the shortcomings of IR detection, imaging with undetected photons
(IUP) exploits a nonlinear interferometer to decouple the probe and detection
wavelengths. Photon pairs containing a visible photon (signal) and an infrared
photon (idler) are generated, typically by spontaneous parametric down-
conversion (SPDC) in a nonlinear crystal. By passing the pump through the
crystal twice, pairs can be generated in either pass. Pairs generated in the
first pass are precisely overlapped to be indistinguishable from pairs
generated in the second pass, leading to interference in the signal photon
count rate [7]. Adding an object to the path of the infrared photons from the
first pass introduces distinguishability. This manifests as a reduction in the
amplitude of the visible interference fringes, which is proportional to the
object transmissivity. Changes in phase, which are proportional to the
refractive index of the object, are also observed as a phase shift in the
visible interference fringes. Importantly, coincident detection is not
necessary as it is only the potential to distinguish which affects the
interference. The presence of an object in the idler beam can be therefore be
recorded in the signal photon channel without ever detecting the infrared
idler photons which interact with the object [8]. This approach has been
applied to a wide variety of fields such as spectroscopy [9, 10, 11, 12, 13],
optical coherence tomography [14, 15], hyperspectral imaging [16], and
microscopy [17, 18].

The phase and transmission images are typically obtained from multiple frames
acquired at different phase shifts to reconstruct the full oscillation at each
pixel. Most commonly, this is achieved by a small movement of either the
signal or idler mirror using a motorised stage, which we will refer to as
phase-shifting, with an image taken at each position for a minimum of 3
positions [19]. This significantly affects the overall measurement rate,
requiring time for the stage to move and settle as well as multiple exposure
times. Capturing multiple images becomes increasingly time-consuming when long
camera exposures are required due to low SPDC flux. In these types of IUP
systems, it is possible to image the transmission profile directly, and
therefore quickly [20], but the phase image is only obtainable by phase-
shifting. Alternatively, it has been shown that full phase and transmission
information can be obtained in a single camera frame by splitting the detected
beam into 4 regions on the sensor, each with a different phase shift
introduced, although this comes at the expense of reduced intensity per pixel
and many additional optics [21].

Here we demonstrate a recently proposed single-frame approach [1] that uses
off-axis digital holography. In off-axis holography, a spatial carrier
frequency is introduced between the two interfering waves, referred to here as
the image and reference waves [22]. This is achieved by introducing an angle
between the image and reference beams, typically by a combination of tilting
the mirrors or beamsplitter in the interferometer and adjusting the path delay
until straight tilt fringes are detected. The carrier frequency introduced
separates the $+1$, $-1$, and DC diffracted orders from each other in
$\mathbf{k}\text{-space}$, which allows for spatial filtering in order to
select either the $+1$ or $-1$ order [23]. Reconstruction of the object field
is then possible from a single interferogram.

In off-axis holography with a conventional linear interferometer, the
interference pattern detected on the camera takes the form

$I_{tot}=I_{im}+I_{ref}+2\Re\left[E_{im}(\mathbf{r})E_{ref}^{*}(\mathbf{r})\exp\left(i\mathbf{k_{tilt}}\mathbf{r}\right)\right],$
(1)

where $I_{im}$ and $I_{ref}$ are the image and reference intensities, $E_{im}$
and $E_{ref}$ the complex fields of the image and reference (where the object
transmission and phase is contained within the image field) and
$\mathbf{k_{tilt}}$ is the relative wavevector between fields introduced as a
spatial carrier frequency. Typically the non-interfering terms $I_{im}$ and
$I_{ref}$ are subtracted using pre-recorded images and a Fast Fourier
Transform (FFT) of the remaining terms yields the two interference terms which
are separated in $\mathbf{k}\text{-space}$. Spatial filtering allows for the
selection of one of these terms which, when an inverse FFT (IFFT) is
performed, the result corresponds to
$E_{im}(\mathbf{r})E_{ref}^{*}(\mathbf{r})\exp\left(i\mathbf{k_{tilt}}\mathbf{r}\right)$.
The measurement of $I_{ref}$ allows for an estimate of $E_{ref}$ assuming a
flat phase profile, and the linear phase ramp $\mathbf{k_{tilt}}$ can be
calculated from the location of the centre of the term in the Fourier
transform. This allows for a calculation of the field information about the
object contained within $E_{im}$. Similarly, in a nonlinear interferometer for
IUP in the low gain regime, the recorded intensity takes the form

$I_{tot}=I_{sig,1}+I_{sig,2}+2\Re\left[A_{sig,1}(\mathbf{r})A_{sig,2}(\mathbf{r})\exp\left(i\phi+i\mathbf{k_{tilt}}\mathbf{r}\right)\left[T(\mathbf{r})\exp\left(i\phi_{object}(\mathbf{r})\right)\right]^{2}\right],$
(2)

where the object is described by a position-dependent transmission
$T(\mathbf{r})$ and phase $e^{i\phi_{object}(\mathbf{r})}$, which is squared
as the object is passed twice in the Michelson interferometer. Comparing with
Eq. 2, we see that $I_{sig,1}+I_{sig,2}$ can be determined by recording a
frame when the idler beam is blocked. This can then be subtracted from the
total intensity. An FFT of the subtracted image can be taken and spatial
filtering performed to select one of the diffracted orders. This gives
$A_{sig,1}(\mathbf{r})A_{sig,2}(\mathbf{r})T^{2}(\mathbf{r})\exp\left(i\phi+2i\phi_{object}(\mathbf{r})+i\mathbf{k_{tilt}}\mathbf{r}\right)$
after an IFFT. We assume that the amplitude profile of the first and second
pass signal beams, $A_{sig,1}(\mathbf{r})$ and $A_{sig,2}(\mathbf{r})$,
respectively, are approximately equal, and so division by half the measured DC
signal, $(I_{sig,1}+I_{sig,2})$, yields
$T^{2}(\mathbf{r})\exp\left(i\phi+2i\phi_{object}(\mathbf{r})+i\mathbf{k_{tilt}}\mathbf{r}\right)$.
The phase ramp $\mathbf{k_{tilt}}$ is calculated in exactly the same way as in
a conventional off-axis scheme. The global phase ($\phi$) can also be removed
in a pre-calibration step by calculation of the phase without an object
present. This allows for a calculation of the object transmission
$T(\mathbf{r})$ and phase $\phi_{object}(\mathbf{r})$ from a single camera
frame.

## Results

### Imaging System

Figure 1: Schematic of experimental setup for off-axis holography with
undetected photons. Purple indicates the pump beam (UV, 355 nm). Blue and red
represent signal (visible, 460 nm) and idler (SWIR, 1555 nm) beams,
respectively.

Figure 1 shows the experimental setup used to perform off-axis digital
holography with undetected photons. A 275 mW continuous-wave laser at a
wavelength of 355 nm is prepared in polarization by a half-wave plate (HWP)
and a polarizing beam-splitter (PBS) before pumping a 5 mm-long beta barium
borate (BBO) crystal. Signal (460 nm) and idler (1555 nm) pairs are produced
via type-1 SPDC. The idler photons are separated by a long-pass dichroic
mirror, forming the two paths of the interferometer. Signal and pump propagate
together towards the signal mirror, while the idler is sent towards the idler
mirror. The object to be imaged can be optionally inserted in front of the
idler mirror. A lens in each arm ($f$ = 100 mm) means that the mirrors sit in
the Fourier plane of the crystal. All three wavelengths are then reflected and
pass back through the crystal. Upon the pump’s second pass, there is a
probability to generate another signal-idler photon pair. The signal and idler
photons are then separated from the pump at the PBS, with any residual pump
and idler removed by a combination of filters. The camera is placed in a
projected Fourier plane of the crystal. The idler photons which interact with
the object are never detected.

Straight fringes can be introduced by tilting the idler mirror [24, 25],
denoted by the curved yellow arrow in Fig. 1. Achieving this is possible
because the interference pattern depends on the phase of all fields within the
interferometer. This is a notable difference when compared with typical off-
axis digital holography, where it is necessary for the detected beams to be
tilted relative to each other. The desired fringe pattern may also be achieved
by adjusting the signal mirror in this system, however, we choose to optimize
the signal mirror position for maximum SPDC intensity at the camera. We then
adjust the idler mirror, which affects only the interference and not the
intensity (in the low-gain regime) as it is not directly detected. Adjusting
the signal mirror would also affect the pump beam returning to the BBO and
hence alter the phase-matching.

Figure 2: Digital off-axis holography acquisition and analysis. Example frames
showing (a) interference, (b) the reference beam acquired by blocking the
infrared path, and (c) interference with the reference subtracted. Raw camera
frames are shown in Analog-to-Digital Units (A.D.U.) A vertical cut averaged
over the rows (zoomed inset) shows $\sim 25$ fringes with a maximum
$22.8\,\pm\,0.9\%$ contrast. Also shown is (d) the FFT of the subtracted image
and (e) the mask used in the image reconstruction.

In order to perform the spatial filtering required to isolate the +1
diffracted order from the DC and $-1$ orders, enough tilt fringes must be
introduced such that the $+1$ order is sufficiently separated in
$\mathbf{k}\text{-space}$, and that the filtering applied does not compromise
the resolution of the imaging system by removing high spatial frequencies from
the image. However, as with a conventional interferometer, introducing more
tilt fringes worsens the fringe contrast as the beams walk off from each
other. Aligning the system then becomes a balance between a reasonable fringe
contrast such that the signal-to-noise ratio (SNR) of images is preserved and
enough tilt fringes in the field of view to maintain image resolution. Figure.
2 shows the interference pattern and reference subtraction used to perform
off-axis holography. We align the system such that we have $\sim 30$ fringes
in the field of view (FOV), measured to be $12.47\,\pm\,0.03\text{mm}$. This
results in a maximum fringe contrast of $22.8\,\pm\,0.9\%$ as measured on
$1000$ reference-subtracted images, defined as
$(I_{max}-I_{min})/2A_{sig,1}A_{sig,2}$. This gives a maximum measurable
transmission of $0.48\pm 0.01$. A Gaussian filter is then applied, centred
around $+1$ diffraction order. Subtraction of the reference removes the
majority of the DC component, however we apply a mask to the central $\sim\,5$
pixels to remove DC component that remains. Figure 2 shows an example of an
unmasked and masked FFT of the interference pattern used in the image
reconstruction.

### Transmission and Phase Imaging

Figure 3: Single-frame complex image reconstruction of (a) a phase object
(overlapping pieces of cellotape) and (b) a binary transmission object (3D
printed star). Objects measured over a $12.47\,\pm\,0.03\,\text{mm}$ field of
view. Note the phase measured here is twice the optical phase thickness of the
object due to the double pass.

Figure 3 shows transmission and phase images of a binary transmission star-
shaped target placed in the idler arm acquired with the imaging system. Both
the intensity and the phase are reconstructed from a single interference
pattern, which were acquired at a rate of $10\,\text{Hz}$. In order to
properly demonstrate phase imaging, an example of a transmissive ‘phase
object’ made from two pieces of overlapping cellotape is also shown. From the
phase images, variation in the optical thickness in the cellotape can be seen
which is not present in the transmission image. Note that, in the phase
images, a pre-calibrated reference phase has been subtracted in order to
correct for the global phase term present in the object field and no
unwrapping is currently performed. Figure 3 demonstrates that by using off-
axis holography, it is possible to reconstruct transmission and phase images
from a single interference pattern in a nonlinear interferometer where the
light detected has never interacted with the object.

Figure 4: Reconstructed image of a slanted edge placed in the idler arm, with
the region used to measure the edge response of the system indicated. The tilt
of the edge was corrected for and the edge response fitted with a Gaussian
error function (zoomed inset). The derivative of the error function is a
Gaussian, the FWHM of which is used to calculate the point spread function.
This was repeated for 1000 frames with the mean value calculated and error
stated as the standard error on the mean.

In order to assess the performance of the imaging system, we measure the SNR
and resolution using a slanted edge placed in the idler beam. To measure the
resolution of the system, a Gaussian error function is fitted to a section of
the edge response as indicated in Fig. 4. The derivative of the error function
gives a Gaussian function from which the point spread function (PSF) of the
system can be calculated as the full width at half maximum (FWHM). This
calculation was averaged over $1000$ frames giving a PSF of
$287\,\pm\,1\,\mu\text{m}$. With the FOV of the system measured as
$12.47\,\pm\,0.03\,\text{mm}$, this gives an approximation of the total number
of modes for the system as $1890\,\pm\,20$. In the far-field [17], the FOV is
calculated as $\text{FOV}=2f\theta_{i}/M$ where $f$ is the focal length of the
lens matching the far-field of the crystal to the object, $\theta_{i}$ is the
emission angle of the idler photons and $M$ the magnification factor due to
the difference in wavelength of signal and idler photons [26]. The emission
angle depends on the phase-matching conditions in the crystal, which we model
to account for the broadband nature of the down-converted photons [27]. From
this calculation, we would expect a FOV of $\sim$ 12.97 mm, accounting for the
5 nm bandpass filter used in the experiment. The resolution is calculated as

$\delta x_{corr}=\frac{\sqrt{2\ln{2}}f\lambda_{i}}{\pi w_{p}M},$ (3)

where $\lambda_{i}$ is the idler wavelength and $w_{p}$ the pump beam waist.
For the imaging system here, this gives a resolution of $\sim$ 406 $\mu$m. The
total number of spatial modes can then be calculated as

$m_{2D}=\left(\frac{\text{FOV}}{\delta x_{corr}}\right)^{2},$ (4)

giving a total number of modes as $\sim$ 1000\. The difference between the
theoretical and measured values for the resolution are likely due to error in
the value of the pump beam waist, which was taken from the manufacturer
specification sheet as $1.0\,\pm\,0.2\,\text{mm}$. The difference in the FOV
values can be explained by slight discrepancy in the angle of the BBO compared
to the nominal value used in calculations and the broadband nature of the
down-conversion. The angle of emission, and hence the resulting FOV, are
sensitive to the centre wavelength and bandwidth of the bandpass filter placed
before the camera. As we slightly tilt the filter to optimise the detection of
the signal, it is difficult to precisely estimate the centre wavelength and
bandwidth of the down-conversion. Here, the difference between the measured
and theoretical FOV corresponds to only a $0.04^{\circ}$ difference in
emission angle and the difference in measured and theoretical resolution
values corresponds to a difference in pump waist of $0.4\,\text{mm}$. As the
predicted resolution does not assume anything about the method of
reconstruction, we conclude that we see no deterioration of resolution in the
imaging system by using the off-axis approach.

We measure the SNR of the reconstructed transmission images by masking the
bright and dark regions of the image and average over $1000$ images. This
gives a SNR value of $1.78\,\pm\,0.06$. We note that, as mentioned previously,
there is a trade off between SNR and resolution with the size of the mask
applied in Fourier space, with higher SNR attainable with a smaller mask but
at the cost of a reduction in the spatial resolution.

### Video Rate Imaging

The single-frame nature and Fourier transform-based analysis of this approach
means that not only can images be recorded at video rate, but the transmission
and phase information can also be extracted at this speed. The analysis can
therefore be easily integrated into a graphical user interface. Figure 5 shows
a sequence of images from a video of the binary transmission target as it is
moved across the IR path (Supplementary Video), captured and analysed at 33
frames per second (fps).

Figure 5: Reconstructed transmission and phase images from selected frames
from a 33 frames per second video of a moving binary transmission target.

## Discussion

Most IUP implementations use quasi-phase-matching in periodically-poled
nonlinear crystals for photon pair generation, offering strong nonlinearities
and tailored photon pair wavelengths. However, the crystal aperture is often
only 1 - 2 mm due to the poling process. When significant tilt is introduced
by the mirror in far-field in the IR path, the returning light will be
laterally displaced at the crystal and potentially clipped by the crystal
aperture. Here, the bulk BBO crystal’s large transverse dimensions (10 mm
$\times$ 10 mm) allow significant tilt to be introduced in the IR path without
concern. This effect is evident here when tilting the signal mirror instead,
as the first and second pass of the pump are visibly separated at the crystal
when many fringes are present in the interferogram. The geometric constraints
of the crystal aperture should therefore be considered when applying this
technique to IUP systems.

The main drawback in performing off-axis holography for imaging with
undetected photons when compared with phase-shifting is the trade-off between
SNR and resolution introduced by the tilt fringes themselves and the filtering
process. Whilst in principle it is possible to perform off-axis holography
without loss in resolution, obtaining a sufficient angle between interfering
beams is a challenge within conventional holography and can be a further
challenge in a nonlinear interferometer due to the short coherence lengths and
often low visibility. Whilst an off-axis approach is unlikely to produce
images with comparably high SNR and resolution as phase-shifting, it does have
a significant advantage in the speed of both image acquisition and image
processing, as well as experimental complexity.

Digital holography is itself a large field of research, with many approaches
to optimising image reconstruction [28, 29, 30]. While we use a simple
approach to image reconstruction here, there is much scope for future work
exploring more complicated protocols which may address some of the resolution
and SNR issues discussed above [31]. The demonstration of imaging with
undetected photons in an off-axis configuration allows for the extension of
many techniques within digital holography literature to be explored at
previously challenging wavelengths, for example off-axis spatial multiplexing
[32] and quantitative phase imaging techniques for disease identification
[33].

In conclusion, we have applied off-axis digital holography to imaging with
undetected photons for the first time. We are able to reconstruct both
intensity and phase images of binary and transmissive objects from a single
interference image. This facilitates video-rate imaging of dynamic scenes
within the infared path, which we demonstrate at 33 fps. Our demonstration
helps extend the potential applications of imaging with undetected photon
schemes to scenarios where single-shot imaging and frame rate are important
considerations, such as the monitoring dynamic biological processes and rapid
industrial assessments of materials [34, 35].

## Methods

In order to generate signal and idler pairs, a $355\,\text{nm}$ wavelength
continuous-wave laser (Coherent CX355-250 CW) with nominal beam waist
$1.0\,\pm\,0.2\,\text{mm}$ is prepared in polarisation and pumps a
$10\,\times\,10\,\times\,5$ mm3 BBO crystal. The crystal is nominally cut to
produce non-degenerate photon pairs at $460\,\text{nm}$ (signal) and
$1555\,\text{nm}$ (idler) via type-1 SPDC. The idler photons are separated by
a long-pass dichroic mirror (Chroma T425lpxr). Both mirrors are placed in the
far-field of the crystal by use of $f$ = 100 mm lenses. A 5 nm bandpass filter
(Semrock BrightLine FF01-461/5-25) centred at 461 nm is placed after a PBS
which directs photon pairs to the camera. Camera lens 1 ($f$ = 100 mm) also
places the camera in the far-field of the crystal, where we use a relay,
camera lenses $2$ ($f$ = 120 mm) and $3$ ($f$ = 35 mm), to demagnify by a
factor of 3.4 onto the camera. The camera used is a Hamamatsu ORCA-Quest
(C15550-20UP), run in ultra-quiet scan mode with a pixel size of 4.6 $\mu$m.

## Acknowledgements

We acknowledge funding from the UK National Quantum Hub for Imaging (QUANTIC,
No. EP/T00097X/1) and the Royal Society (No. UF160475). This work was funded
by the UK EPSRC (QuantIC EP/M01326X/1). O.W. acknowledges the financial
support from the EPSRC (QuantIC EP/R513222/1). S.P.M. acknowledges the EPSRC
(QuantIC EP/M01326X/1) and the Leverhulme Trust. T.G. acknowledges the
financial support from the EPSRC (EP/SO19472/1). M.J.P. acknowledges the
financial support from the Royal Society (RSRP/R1/211013P).

## Author contributions statement

E.P., O.W., S.P.M., T.G., N.R.G., and M.J.P. devised the project methodology.
E.P. and O.W. conducted the experiments and data analysis. R.F.O., A.S.C.,
C.C.P, and M.J.P. provided supervision. All authors reviewed the manuscript.

## Additional information

Competing interests: The authors declare no competing interests.

## Data Availability

Data Availability Statement: All data used to produce the figures in this work
will be made available upon request.

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On the origin of escort distributions for q-entropies

NIKOLAOS KALOGEROPOULOS †

Center for Research and Applications
of Nonlinear Systems (CRANS),
University of Patras, Patras 26500, Greece.

Abstract

We present an argument about the origin of escort distributions used in
conjunction with the q-entropy in non-additive thermo-statistics. The origin
of the escort distributions is ascribed to the fact that the effective
statistical description of the underlying system is provided by the measured
Gromov-Hausdorff limit of a sequence of manifolds having a warped product
metric structure. We interpret the entropic parameter q appearing in escort
distributions as a “dimension” related to the fibration structure of the
underlying phase spaces.

Keywords: Escort distributions, q-entropy, Tsallis entropy, Metric measure
spaces, Warped products.

† Electronic mail<EMAIL_ADDRESS>

##  Introduction

The q-entropy (also known as “Tsallis entropy”) has occupied the attention of
a part of the Statistical Mechanics commuity during the last three decades
[1]. Despite some progress in developing a thermodynamic formalism based on
it, and the many articles which claim to have detected data which can be
interpreted as its extremizing distributions, called the q-exponentials, under
the familiar constraints from the conventional Statistical Mechanics, it is
probably fair to say that the origin and foundations of q-entropies are still
largely unknown. One of the questions which have arisen related to the
q-entropies is the origin, and the role of the entropic parameter q [2], [3]
and of the escort distributions [4] which appear in several places in the
developed formalism [1], [5], [6], [7], [8].

In this work we attempt to provide an explanation, at the
mesoscopic/macroscopic levels, for the origin and use of the escort
distributions. We rely on the formalism of smooth metric measure spaces [9]
and our approach follows a path similar to that of [10], [11], but is totally
distinct from [12]. We rely on the results of [13], [14] which we interpret
appropriately. One should turn to [15] for the broader context of metric
measure spaces and their Ricci curvature through optimal transport, and to
[16] for an overview of the comparison geometry of the Bakry-Émery-Ricci
tensor on which this work relies and for all mathematical details and proofs
which may be needed.

In Section 2, we explain the meaning and our use of warped products. In
Section 3, we provide a few definitions about the measured Gromov-Hausdorff
convergence to make this work readable to our intended audience. Section 4
explains how the escort distributions arise from the warped product
construction. Section 5 makes some general comments on the present, and
provides an outlook for future work.

##  Warped products in the present context

We will have in mind in this work the frameworks of the micro-canonical and of
the canonical ensemble of equilibrium Statistical Mechanics. To begin with,
let us assume that we have a system under study which is in contact with a set
of thermostats. Let the combined system plus thermostats be described by a
manifold $\mathcal{M}$ of dimension $N$. We can see $\mathcal{M}$ as the
“universe”, although not in a literal sense, and assume that it describes an
isolated system. Someone might also want to assume that $\mathcal{M}$ is the
phase space of a Hamiltonian system. In our metric treatment $\mathcal{M}$ is
endowed with a Riemannian metric $\mathbf{g}_{M}$. Using a Riemannian metric
is quite common in most physical models. The separation between the system
under study and the set of thermostats can be modelled topologically by
assuming that $\mathcal{M}$ is a fibration, which is locally a direct product
$\mathcal{B}\times\mathcal{F}$. In this product, $\mathcal{B}$ is a manifold
of dimension $n$ which describes the system under study, and $\mathcal{F}$ is
a manifold of dimension q which represents each “thermostat” coupled to our
system of study. One could assume $\mathcal{M}$ to have non-trivial algebraic-
topological structure, but we will not need to elaborate at this point
inasmuch it does not provide any obstructions and allows the constructions of
interest to go through.

In our previous work [11], we assumed that such a structure should be a
Riemannian submersion. This assumption turns out to be too general and too
flexible for our purposes. Instead, let us assume something more specific:
that the metric structures inherited by $\mathcal{B}$ which we call it a
“leaf”, and $\mathcal{F}$ which we call a “fiber” in this mathematical
construction, are inherited from $\mathbf{g}_{\mathcal{M}}$ through some kind
of projections. To keep some flexibility and still allow for differences among
the fibers $\mathcal{F}$, let us assume that the metric
$\mathbf{g}_{\mathcal{M}}$ has a warped product form

$\mathbf{g}_{\mathcal{M}}=\mathbf{g}_{\mathcal{B}}+(\varepsilon
e^{-f})^{2}\mathbf{g}_{\mathcal{F}}$ (1)

Here $f:\mathcal{B}\rightarrow\mathbb{R}$ is assumed to be smooth, and let
$\varepsilon\in\mathbb{R}$ be a free parameter which will be used to create a
sequence, when $\varepsilon$ will be taken to approach zero, in what follows.

One can see that topologically $\mathcal{M}=\mathcal{B}\times\mathcal{F}$.
Condition (1) generalizes the direct sum metric
$\mathbf{g}_{\mathcal{M}}=\mathbf{g}_{\mathcal{B}}\oplus\mathbf{g}_{\mathcal{F}}$
through the use of the “warping” function
$e^{-f}:\mathcal{B}\rightarrow\mathbb{R}$ whose role is to “weigh”, metrically
at least, each fiber differently. The “warping” function $e^{-f}$ is taken to
be positive in order to maintain the Riemanian character of the metric
$\mathbf{g}_{\mathcal{M}}$ as well as for the ascertaining that the metrics of
this construction will have negative curvature in the construction of [17].

One can see in (1) that the metric projections onto the leaves $\mathcal{B}$
are isometries and that the metric projections onto the fibers $\mathcal{F}$
are homotheties. Moreover one also observes in (1) that the leaves and fibers
are orthogonal at their intesection points. One can also prove that the leaves
$\mathcal{B}$ are totally geodesic submanifolds of $\mathcal{M}$, and that the
fibers $\mathcal{F}$ are, in turn, totally umbilic submanifolds of
$\mathcal{M}$ [17], [18].

Warped products are extensively studied structures in Geometry over at least
six decades since [17], and a great deal is known about them [18], [19]. What
is more relevant for our work is that such structures appear quite frequently
in Physics, especially in the context of General Relativity and other
geometric theories [18], [20]. One can see warped products as generalizations
of the Euclidean metric of the plane without the origin which is topologically
$\mathbb{R}_{+}\times\mathbb{S}^{1}$, where $\mathbb{S}^{1}$ stands for the
unit radius circle, as can be seen in polar coordinate parametrization
$ds^{2}=dr^{2}+r^{2}d\theta^{2}$. More broadly, warped products can be seen as
generalizations of surfaces of revolution where the fibers are circles. It is
worth noticing that many familiar space-times such as the Schwarzschild and
Friedmann-Lemaitre-Roberston-Walker can also be expressed as, multiple, warped
products [18], [20].

##  Measured Gromov-Hausdoff convergence

We now provide a few definitions needed to formulate our proposal, about the
measured Gromov-Hausdorff distance and its associated convergence. This form
of convergence applies to metric measure spaces rather than to just Riemannian
manifolds. Metric measure spaces have been a very active area of research
during the last forty five years and references abound. We point out [21],
[22], [23] for details and proofs which we omit altogether.

Let $\mathfrak{X}$ be an arbitrary metric space with distance function $d$ and
let $A,\ B$ be arbitrary non-empty subsets of $\mathfrak{X}$. Let
$A\subset\mathfrak{X}$ and let $U_{r}(A)$ indicate an $r$-neighborhood of $A$,
namely

$U_{r}(A)\ =\ \left\\{x\in\mathfrak{X}:d(x,a)<r,\ \ \ \forall\ a\in
A\right\\}$ (2)

The Hausdorff distance $d_{H}(A,B)$ between $A$ and $B$ is defined as

$d_{H}(A,B)\ =\ \inf\left\\{r\in[0,+\infty]:\ A\subset U_{r}(B)\ \
\mathrm{and}\ \ B\subset U_{r}(A)\right\\}$ (3)

It turns out that the Hausdorff distance $d_{H}$ is indeed a metric on the set
of all compact subsets of $\mathfrak{X}$.

A question which arises is how to generalize this distance between two
different metric spaces $\mathfrak{Y},\ \mathfrak{Z}$ which are not subsets of
the same metric space and which, in general, have different distance functions
$d_{\mathfrak{Y}},\ d_{\mathfrak{Z}}$. A way to proceed is to consider the set
$\mathfrak{Y}\sqcup\mathfrak{Z}$ which is the disjoint union of $\mathfrak{Y}$
and $\mathfrak{Z}$ endowed with the distance function
$d_{\mathfrak{Y}\sqcup\mathfrak{Z}}$ which is such that

$d_{\mathfrak{Y}\sqcup\mathfrak{Z}}=\left\\{\begin{array}[]{ll}d_{\mathfrak{Y}}(y_{1},y_{2}),&\mathrm{if}\
\ y_{1},y_{2}\in\mathfrak{Y}\\\ d_{\mathfrak{Z}}(z_{1},z_{2}),&\mathrm{if}\ \
z_{1},z_{2}\in\mathfrak{Z}\\\ \infty,&\mathrm{otherwise}\\\
\end{array}\right.$ (4)

More generally, one can consider a metric space $\mathfrak{X}$ and two
subspaces $\mathfrak{Y}^{\prime}$, $\mathfrak{Z}^{\prime}$ isometric to
$\mathfrak{Y}$, $\mathfrak{Z}$ respectively, where the distance functions of
$\mathfrak{Y}^{\prime}$, $\mathfrak{Z}^{\prime}$ are the restrictions of the
metric of $\mathfrak{X}$ to them. Then the Gromov-Hausdorff distance $d_{GH}$
between $\mathfrak{Y}$ and $\mathfrak{Z}$ is given by

$d_{GH}(\mathfrak{Y},\mathfrak{Z})=d_{H}(\mathfrak{Y}^{\prime},\mathfrak{Z}^{\prime})$
(5)

for all possible isometric embeddings of $\mathfrak{Y}$, $\mathfrak{Z}$ into
all possible ambient spaces $\mathfrak{X}$. İt turns out that it is sufficient
to consider in the definition of the Gromov-Hausdorff distance as ambient
space $\mathfrak{X}$ the disjoint union
$\mathfrak{Y}^{\prime}\sqcup\mathfrak{Z}^{\prime}$ with the distance function
(4).

It turns out that the Gromov-Hausdorff distance is indeed a metric on the
space of isometry classes of compact metric spaces, giventhat the distance
between two such isometric spaces is zero. The above results can be extended
to non-compact metric spaces either through a pointed version of the above
definitions or by use of ultralimits, but we will not expand upon such
definitions as we will only need to consider compact manifolds in this work,
as will be seen below.

Given the Gromov-Hausdorff distance, one can define convergence within the set
of compact metric spaces, following the familiar definition: a sequence of
compact metric spaces $\mathfrak{X}_{n},\ n\in\mathbb{N}$ converges to a
compact metric space $\mathfrak{X}_{\infty}$ if

$d_{GH}(\mathfrak{X}_{n},\mathfrak{X}_{\infty})=0,\ \ \ \mathrm{as}\ \
n\rightarrow\infty$ (6)

The Gromov-Hausdorff convergence is a purely metric concept which does not
involve volumes or any other measures. Let us recall that there is no
“natural” concept of volume in a general metric space. In Riemannian manifolds
however, such a natural choice exists, mainly because Riemannian manifolds are
infinitesimally Euclidean spaces which admit a natural definition of volume.
By contrast, immediate generalizations of Riemannian manifolds, such as the
Finslerian manifolds which substitute the ellipsoids as unit spheres with the
boundaries of convex bodies do not admit a canonical volume function [22].
Since we will be working with Riemannian manifolds, we need a version of the
Gromov-Hausdorff convergence which involves volumes, or more generally
“measures” in metric spaces. The measures $\mu$ we consider are assumed to be
sufficiently “nice” (Borel, regular) as, at this level of generality, they are
adequate for our purposes.

Let us consider a sequence of metric measure spaces
$(\mathfrak{X}_{n},d_{n},\mu_{n}),\ n\in\mathbb{N}$. If one sees functions, as
subsets of distributions, as dual to measures with respect to integration,
then one can consider the sequence of continuous functions
$f_{n}:\mathfrak{X}_{n}\rightarrow\mathbb{R}$ converging to
$f_{\infty}:\mathfrak{X}_{\infty}\rightarrow\mathbb{R}$. For the convergence
of measures $\mu_{n}\rightarrow\mu_{\infty}$ one demands

$\int_{\mathfrak{X}_{n}}f_{n}\ d\mu_{n}\ \rightarrow\
\int_{\mathfrak{X}_{\infty}}f_{\infty}\ d\mu_{\infty},\ \ \ \mathrm{as}\ \ \
n\rightarrow\infty$ (7)

When both (6) and (7) are satisfied then we can state that the metric measure
space $(\mathfrak{X}_{n},d_{n},\mu_{n})$ converges to the metric measure space
$(\mathfrak{X}_{\infty},d_{\infty},\mu_{\infty})$ in the measured Gromov-
Hausdorff sense.

##  Escort probabilities from warped product metrics

We return to the warped product metrics (1) and the main line of our argument.
First of all, we have to state that the manifolds $\mathcal{M},\ \mathcal{B},\
\mathcal{F}$ whose metrics are related by (1) are assumed to be compact. This
is a reasonable requirement since all these three manifolds are essentially
the phase spaces of the “universe” $\mathcal{M}$, of the system under study
$\mathcal{B}$ and of the thermostats $\mathcal{F}$. We are working within the
framework of Hamiltonian systems where the total energy of the “universe”
$\mathcal{M}$ is conserved. This essentially amounts to working within each
constant energy hypersurface of $\mathcal{M}$ which is compact, if we assume
that the Hamiltonian systems we are working with involve only proper
functions. This is usually the case with polynomial, sinusoidal/cosinusoidal,
exponential etc interactions.

One has to be careful at this point not to trivially apply the arguments of
this work to interactions expressed through rational functions, logarithms, or
higher transcendental functions which could give rise to infinites and hence
to non-compact phase spaces or to spaces with singularties. This does not mean
that the final result of the current argument will not be valid, but it just
indicates that one would have to model such systems differently, probably
using the pointed measured Gromov-Hausdorff convergence and pay attention to
how infinites which may emerge should be handled. In our case of compact
$\mathcal{M}$, by projection one gets compact $\mathcal{B}$ and $\mathcal{F}$
and therefore we can apply the above concepts of the measured Gromov-Hausdorff
convergence without any problem.

One wishes to know what happens when we take the limit $\varepsilon\rightarrow
0$ in (1). To be more precise, we consider the measured Gromov-Hausdorff limit
of $\mathcal{M}$ with the Riemannian distance function induced by
$\mathbf{g}_{\mathcal{M}}$ but endowed with the renormalized volume

$\widetilde{dvol_{\mathcal{M}}}\ =\
\frac{dvol_{\mathcal{M}}}{vol_{\mathcal{M}}(B_{\mathcal{M}}(1))}$ (8)

where $B_{\mathcal{M}}(1)$ indicates the ball of unit radius in $\mathcal{M}$.
The reason for this volume renormalization is technical, but need not concern
us since in the definition of probabilities and entropy where this formalism
is applied, one has the freedom to appropriately renormalize these quantities
without changing the physical content of the theory. Given this renormalized
measure of $\mathcal{M}$, one finds that the measured Gromov-Hausdorff limit
is $\mathcal{B}$ endowed with the distance induced by
$\mathbf{g}_{\mathcal{B}}$ and with measure

$d\nu\ =\ (e^{-f})^{q}\ dvol_{\mathcal{M}}$ (9)

A way to understand this relation is as follows: the warped product (1)
“expands” or “contracts” each fiber $\mathcal{F}$ by the metric factor
$e^{-f}$ per dimension of $\mathcal{F}$ when manifold collapse is not
considered. Therefore, upon collapse of the q different dimensions of the
fiber $\mathcal{F}$ the contribution to the resulting measure is a
modification of the initial measure/volume of $\mathcal{M}$ by an overall
factor $(e^{-f})^{q}$.

We would like to mention at this point, so as to also make the connection with
the proposal of [10] regarding the origin of the q-entropy, that this limiting
process gives rise to the Bakry-Émery-Ricci tensor
$\overline{\mathrm{Ric}}_{\mathcal{B}}$ of $\mathcal{B}$ which is given by

$\overline{\mathrm{Ric}}_{\mathcal{B}}(X,Y)\
=\mathrm{Ric}_{\mathcal{B}}(X,Y)+\mathrm{Hess}_{\mathcal{B}}f(X,Y)-\frac{df\otimes
df}{q}(X,Y)$ (10)

where $\mathrm{Ric}_{\mathcal{B}}$ indicates the Ricci tensor of
$\mathbf{g}_{\mathcal{B}}$, $\mathrm{Hess}_{\mathcal{B}}$ is the Hessian of
$f$ using the Levi-Civita connection compatible with
$\mathbf{g}_{\mathcal{B}}$, and $X,Y\in T\mathcal{B}$. As it was stated in
[10] lower bounds of the Bakry-Émery-Ricci tensor on a manifold are directly
related to the convexity properties of the q-entropy functional on the space
of its probability distributions with finite second moments and endowed with
the Wasserstein metric [9], [13], [14], [15].

The physical interpretation of (9) is that when the system under study
$\mathcal{B}$ is coupled to the set of “thermostats” $\mathcal{F}$, then the
combimation forms the “universe” $\mathcal{M}$, to which the microcanonical
distribution with respect to $dvol_{\mathcal{M}}$ is applicable. We also make
the tacit assumption about the Hamiltonian evolution on $\mathcal{M}$ being
ergodic, or even better mixing, in order to follow as closely as we can the
assumptions of Statistical Mechanics. When one performs calculations at the
level of $\mathcal{B}$, then the natural measure to use is $d\nu$ which arises
from the microcanonical measure of $\mathcal{M}$, as seen from the above
arguments and (9), which is exactly the unnormalized escort distribution
$d\nu$ related to $dvol_{\mathcal{M}}$.

The reason that the escort distributions arise in this picture, is the
coupling of the system $\mathcal{B}$ to the environment, which is the union of
the “thermostats” $\mathcal{F}$ as in the canonical ensemble set up in
equilibrium Statistical Mechanics. The entropic parameter q, as can be seen in
(9), is the number of the effective degrees of freedom entering the mesoscopic
or macroscopic description of the system. This number is expressed as the
dimension q of each “fiber”/“thermostat” $\mathcal{F}$. We stress that the
actual phase space volume of these “thermostats” $\mathcal{F}$ is not really
relevant in this description. What is relevant is the number of the effective
degrees of freedom of these “thermostats” which is the dimension q of
$\mathcal{F}$. This conclusion is in agreement with the conclusions of [3].

One can extend the meaning of the entropic parameter q to non-integer values,
by transitioning from Geometry to Analysis through the use of general
functions and real-valued parameters in the place of characteristic functions
of sets and integer-valued parameters [15]. This is a standard procedure in
Analysis which allows the transition from geometric relations to analytical
ones (e.g. the Brascamp-Lieb, Poincaré, Sobolev etc inequalities) which extend
the domain of applicability of geometric inequalities to a larger part of
their parameter space.

##  Conclusions and discussion

In this work we tried to provide an explanation about the origin of the escort
probabilities which are used extensively in the framework of non-additive, the
term non-extensive being a misnomer, Statistical Mechanics based on the
q-entropy. The close relation between the origin of the escort distributions
and the origin of the q-entropy, which was also indicated in [10], becomes
clear. They are both part of the same general viewpoint according to which one
starts with probabilities, the Boltzmann/Gibbs/Shannon entropy, and the
microcanonical distributions of an isolated system, here indicated by
$\mathcal{M}$ and called the “universe”, and through a projection, whose
fibers are here called “thermostats” $\mathcal{F}$, one ends with escort
probabilities, the q-entropy and the canonical description of the system under
study $\mathcal{B}$. Hence, there seems to be a rather intimate, if not
necessarily unique, relation between the q-entropies and the escort
distributions.

We believe that it is worth looking more closely into connecting the value of
the entropic parameter q in q-entropies and escort distributions to that of
the microscopic parameters or the scaling dimensions in the Hamiltonian of the
microscopic system, as was suggested in [3]. Ideally, one should be able to
derive the value of q, if one knows the microscopic dynamics, and maybe the
exact coarse-graining process used to connect the microscopic and
mesoscopic/macroscopic worlds. Whether this is feasible, or to what extent,
remains to be seen. It would be enlightening if we could find ways to apply
the above results to models of interest, to check their actual validity with
concrete calculations, rather than leave the arguments at the current,
admittedly rather abstract/theoretical, level.

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https://anton-petrunin.github.io/metric-geometry/tex/lectures.pdf


# Providing a Philosophical Critique and Guidance of Fairness Metrics

Henry Cerbone1 1 Harvard College<EMAIL_ADDRESS>
###### Abstract

In this project, I seek to present a summarization and unpacking of themes of
fairness both in the field of computer science and philosophy. This is
motivated by an increased dependence on notions of fairness in computer
science and the millennia of thought on the subject in the field of
philosophy. It is my hope that this acts as a crash course in _fairness
philosophy_ for the everyday computer scientist and specifically roboticist.
This paper will consider current state-of-the-art ideas in computer science,
specifically algorithmic fairness, as well as attempt to lay out a rough set
of guidelines for metric fairness . Throughout the discussion of philosophy,
we will return to a thought experiment posed by Cynthia Dwork on the question
of randomness.

##  I INTRODUCTION

As roboticists, we are becoming increasingly concerned with whether the
systems we create are fair. Corresponding to this concern, there is a pressure
to figure out just what it means to be fair. Recent work such as [1] has
tended to reference prevailing work by Dwork et. al. [2] in computer science
on algorithmic fairness. In doing so, researchers inherit the implications for
pure metric fairness regardless of its later application in the particular
robotics context. In this paper, I set out what exactly that baggage looks
like. To do this, work in philosophy is drawn out to lay out not only what
definition of fairness the prevailing technical definition relies on but also
how it _should_ look. Unlike work such as [3], this is not a survey or
ordering of definitions of fairness. I open with the motivation for pursuing a
philosophical critique and guidance to algorithmic fairness before outlining
what the views of fairness are from each field. I close with how to marry
these two views in particular what can be learned from philosophy. This work
serves not as a solution from philosophy for fairness in computer science, but
instead, as a rough road map for researchers hoping to better understand the
work surrounding what it means to be fair. Additionally, it is useful to
inherit the definition of robot from Coglianese et. al. [4] which defines an
algorithm as a “digital robot”. Under this definition, we are able to both
directly apply the discussion of fairness from computer science to robotics as
well as better separate the component parts of a system and reason about them
philosophically.

##  II MOTIVATION

We find ourselves setting out in our design and creation of algorithms with a
sense of duty. This is sourced from the increasing reality that our
algorithms’ decisions affect real peoples’ lives in real ways. One result of
this duty is that we should take into account how these algorithms make these
decisions. In modern society, we are increasingly caught up in questions of
what is fair or what is just. This is largely a result of a basic human drive
for fairness on a personal level and a reaction to centuries of oppression of
people based on race, gender, and social class. As computer scientists, we
have also begun an investigation into what is fair and what is not. This is a
result of the growing field of algorithmic fairness and people designing
algorithms of the aforementioned sort. In our investigation of fairness, it is
important to remember the words of Ludwig Wittgenstein in his Philosophical
Investigations [5].

> We have got on to slippery ice where there is no friction, and so, in a
> certain sense, the conditions are ideal; but also, just because of that, we
> are unable to walk. We want to walk: so we need friction. Back to the rough
> ground!

In other words, we are asking what is fair for real people living real lives.
A definition or notion of fairness that is not contextualized or is overly
abstracted will fail to capture its sole benefactor: people. We are motivated
in our investigation of what is fair to turn to a field that has produced the
greatest volume of material on the topic — philosophy. It is with the hope of
applying some of the philosophical thought of the last century to new areas of
computer science that we arrive at something of use to researchers working in
algorithmic fairness and computer science/robotics more broadly.

##  III Perspective: Algorithmic Fairness

The primary notion of fairness in computer science is from the definition
given by Dwork et. al. in [2] as given below

> We capture fairness by the principle that any two individuals who are
> similar with respect to a particular task should be classified similarly. In
> order to accomplish this individual-based fairness, we assume a distance
> metric that defines the similarity between the individuals. This is the
> source of “awareness” in the title of this paper. We formalize this guiding
> principle as a Lipschitz condition on the classifier. In our approach a
> classifier is a randomized mapping from individuals to outcomes, or
> equivalently, a mapping from individuals to distributions over outcomes. The
> Lipschitz condition requires that any two individuals $x,y$ that are at
> distance $d(x,y)\in[0,1]$ map to distributions $M(x)$ and $M(y)$,
> respectively, such that the statistical distance between $M(x)$ and $M(y)$
> is at most $d(x,y)$. In other words, the distributions over outcomes
> observed by x and y are indistinguishable up to their distance $d(x,y)$.

This definition is certainly appealing. It is intuitive that in a discussion
of fairness, we want people that are similar to be treated in a similar
fashion. The obvious question that arises from this definition, however, is —
What does it mean to be similar? In the context of the paper, how do we come
up with a metric? As mentioned in the paper, it seems that the goal of the
metric definition is to have a Rawlsian style mechanism of social revision.
This is important as otherwise we seem to have a circular notion with a
distinction of good similarities and bad similarities, important ones and non-
important ones. The goal of imposing the metric externally is certainly sound.
An alternative is given by Ilvento in [6] through the inclusion of a human
arbiter. The potential problems with these approaches arise from a closer
examination of the philosophy on which they are based.

##  IV Perspective: Philosophy

###  IV-A A Question of Randomness

Throughout our discussion of the philosophical perspective, we will find it
useful to return to a concrete example, to stay on the rough ground so to
speak. We will rely on an example from Professor Cynthia Dwork which was
brought out in conversation with Deborah Hellman.

> Suppose at interview time the interviewer were to flip a coin to choose a
> letter of the alphabet and to summarily reject all applications with names
> beginning with the chosen letter. This seems “fair” but arbitrary,
> regardless of how it feels to Allen and Anna. If instead the interviewer has
> a fixed antipathy towards “A”, can we view this as having the randomness
> occurring 20 years earlier, at the time the baby was named? Does the timing
> of the randomness matter?

and

> Suppose we have 10 sentencing cases and 5 prison beds. Apparently, it is
> unacceptable to randomly choose a subset of the 5 of the felons for
> incarceration. But it is viewed as OK to randomly order the cases, and for
> the judge to send the first 5 to prison and release the next 5 due to
> overcrowding.

I believe that these examples allow us to think through the various advantages
and disadvantages brought to the table by various philosophical views. They
also contain simplified pieces of key aspects of algorithmic fairness. Namely,
the first example deals with randomness which we often think of as a cure-all
for fair distribution. The second example focuses on the problem of
contextualization and implicit effects.

###  IV-B Rawls

Although the discussion of “What is it to be fair?” is a discussion that has
persisted for millennia in philosophy, we will take the work of John Rawls as
a starting point for the discussion of fairness in philosophy. Rawls’ book,
_Theory of Justice_ [7], was a groundbreaking work in political philosophy. We
will borrow several small parts from this work in our discussion but will not
discuss it in detail. Also, of particular interest to us is a paper published
as a precursor to _Theory of Justice_ entitled _Justice as Fairness_ [8].

####  IV-B1 Justice as Fairness

Rawls gives an account of justice that takes itself as relying on fairness.
Much like notions in computer science, one worries that this is shifting the
question. However, in [8], Rawls gives us a thought experiment in which he
gives us a picture of what fairness might be. If we imagine members of a
society coming together to select the rules on which they should be judged,
knowing that these rules once agreed upon will be used to judge all other
members, this must be regarded as fair. This is an interesting and attractive
notion of what is fair. It seems to be simple and disposes with what is known
as the veil of ignorance from Rawls’ later work. We can also see how this has
direct consequences for metric fairness. If people who are being assigned
positions by an algorithm all mutually agree upon the metric with which they
will be judged, it seems to follow that they have a prima facia right to
follow the rulings.

####  IV-B2 The Veil of Ignorance and The Difference Principle

The primary takeaways from [7] for this context are the veil of ignorance and
the difference principle. The veil of ignorance is Rawls’ requirement that in
order to be fair/just, we must imagine ourselves behind a veil, ignorant of
our own standing and selves. Behind this veil, we are all rational, moral
beings with equal standing. This differs from the aforementioned system in
which people make rules with knowledge of how they themselves will be
affected. It also has interesting implications for computer science. Namely,
we can imagine an algorithm as learning metric fairness such as the methods
found by Gillen et. al in [9]. This is primarily due to algorithms operating
behind a veil of ignorance as they obviously have no self to be concerned
about. Although this claim cannot be taken to be self-evident, various
scenarios of machine learning based solutions built upon many scenarios of
fairness across all levels of society would then be operating behind a veil of
ignorance (or at least more ignorance than Rawls’ law makers). Behind this
veil of ignorance, Rawls’ believes we must maintain two principles, the latter
of which is the difference principle. The difference principle states that
social and economic inequalities must satisfy that they are the greatest
benefit to the least advantaged [7]. We see here a departure from both
“similar people are treated similarly” and from a random approach as
previously presented. What ensues can be seen as a balancing act between
differentials and allowing disadvantaged equal access to social and political
offices.

The importance of the Rawlsian perspective of fairness to this paper is most
obviously in its citation in algorithmic fairness literature. Computer
scientists read Rawls as supporting the notion of metric fairness presented
earlier. Whether or not this is an accurate reading is out of the scope of
this work, however, egalitarianism as a whole is both influenced and in some
ways responding to Rawls. Therefore, presenting Rawls’ views are essential to
any discussion of fairness. The trend toward a more complete Rawlsian approach
to fairness as discussed is starting to be seen such as in work by Hurtado et.
al. [10]. This work appropriately leveraged a relearning step to fix initial
preconceptions of what was fair. This is similar to Rawls’ principle of
reflective equilibrium which serves to correct for decisions initially made
behind the veil of ignorance.

###  IV-C Egalitarianism

It is natural to turn to egalitarianism as the next step in our discussion of
fairness in the field of philosophy. This is a natural point of interest as
the core premise of egalitarianism is that people are equal and deserve rights
and opportunities. Of particular interest is the debate between luck and
relational egalitarians. This debate begin with the 1999 paper by Elizabeth
Anderson entitled _What is the point of equality?_. We will examine both sides
both by presenting them and using their methods to work through our randomness
thought experiments.

####  IV-C1 Luck Egalitarianism

The basic premise of luck egalitarianism is that any inequality due to morally
arbitrary circumstances is unjust. The goal then is at the end of the day to
balance out these lucky advantages to reach equality. A primary figure in luck
egalitarianism, John Roemer, gives us the following review of how one might
equalize these inequalities/what should be viewed as such in [11]

> John Rawls proposes “primary goods”; Ronald Dworkin proposes “resources,” a
> re-source being defined in a comprehensive way to include various talents
> and handicaps; Amartya Sen proposes “capabilities to function”; in two
> recent articles, Richard Arneson has proposed “opportunity for welfare” as
> the appropriate equalisandum; G. A. Cohen has recently proposed “access to
> advantage.” All of these proposals attempt to equalize opportunities, rather
> than outcomes: for Rawls and Dworkin, primary goods and resources,
> respectively, are the wherewithal with which people carry out projects that
> lead to outcomes that have value to them; for Sen, the capabilities to
> function in various ways are the prerequisites for what individuals make of
> themselves; and Cohen’s “access” is similar to Arne- son’s straightforward
> “opportunity.”

Returning to our examples, we see that per a luck egalitarian our first
example is unjust. We are saying that people with a first name beginning with
_A_ , a trait that is morally arbitrary, are to be treated differently. Our
second example gives an interesting distinction to luck egalitarianism. We see
why the two posed scenarios are different. In the first, everyone is treated
justly as we have a random subset. In the second, we have some people being
treated unjustly as a morally arbitrary circumstance has sentenced them a fate
conditioned by overcrowding. If the luck egalitarian breakdown of the second
example leaves a bad taste in our mouth, this is a natural reaction. Luck
egalitarianism suffers from a seemingly absurd distinction. The clearest
example of this is pointed out by Elizabeth Anderson in [12], with the
following example from G. Cohen

> A talented person (call her Sue) who prefers gardening at the average wage
> to being a doctor at the same wage should be motivated by the egalitarian
> ethos to be a doctor because this benefits the least advantaged more than
> being a gardener (Cohen 2008, 184-85). In fact, the difference principle
> does not give anyone the aim of maximizing benefit to the least advantaged.
> It only constrains inequality. If Sue chose gardening, she would not make
> more than others, so she would create no inequality that would have to be
> justified as maximizing the good of the least well-off. Cohen suggests that
> Sue, in choosing to garden, has created an inequality, since, as one who
> loves to garden, she enjoys higher work satisfaction than the least
> advantaged.

We see here how many views of luck egalitarians seem to offer absurd, inhuman
results when applied to seemingly basic scenarios. It is this lack of
intuitiveness that gives rise to the subfield termed relational
egalitarianism.

####  IV-C2 Relational Egalitarianism

As previously mentioned, the obvious problems of luck egalitarianism give rise
to relational egalitarianism. Participants in this field take themselves as
relying on some deeper moral notion of fairness to judge equality and
fairness. This is best illustrated in the coiner of the term _relational
egalitarian_ , Elizabeth Anderson’s closing lines of her paper _The
Fundamental Disagreement between Luck and Relational Egalitarians_

> Once everyone has done everything justice requires of them, the world is
> just, whatever other negative evaluations one might make of it. For justice
> is fundamentally a virtue of agents, not a distributive pattern.

There is not as rigorous or fleshed out a theory amongst relational
egalitarians as to what fairness is. Looking at Anderson’s example from above,
we see that what is required technically is something more than an ML
classification. It is not sufficient to group a number of cases together,
label them as fair, and proceed. In a similar vein, it is also not sufficient
per relational egalitarianism to say that “similar individuals are treated
similarly”. This is because we are creating a distributive pattern for
fairness which empties out the meaning of fairness. What Anderson encourages
us to do is to allow what is fair (justice in this case) to be the result of
the virtue of agents, a result of a moral standing not a statistical one.

A modern relational egalitarian (or something akin to such) can be found in
[13] by Deborah Hellman. Although a full discussion of her book is out of the
scope of this work, her notion of discrimination as compounding previous
wrongs is a good example of a relational egalitarian standpoint. In this view,
we can return to our examples and see that in both scenarios, we cannot say
that we have witnessed a case of injustice or discrimination.

##  V Directions for Robotics

Now that we have established various perspectives of what it means to be fair,
we are at a crossroads as to what to do next. The field of fairness in
robotics is still in its infancy and therefore, a review of technical results
here would be short lived. It is worthwhile to note that work by both Brandao
et. al. [14] and Claure et. al. [15] provide initial technical applications of
various notions of fairness. What this work provides additionally is a
comprehensive explication and outline of a framework to both evaluate and
implement fair systems. Although technical work tends to give a brief
introduction to the particular definition of fairness is used, understandably,
not enough reasoning is given to decide if that was a good decision. In order
for roboticists to make real progress in fair robotics, we must be able to ask
and decide for ourselves which definition of fairness we are implementing.

Two good thought experiments for deciding what kind of framework we might lay
out are loan applications and automated package deliveries. The first takes
the aforementioned definition of an algorithm as a digital robot. Loan
application approval is a common example in algorithmic fairness and modern
egalitarianism such as work by Meyer [16]. When setting up such a system, we
might refer to the various definitions laid out above and ask ourselves,
”Which is most appropriate for this task?” The answer to this should be based
on an interdisciplinary approach, supported by the definitions and
explications laid out here. The question of an automated package delivery
robot provides a more concrete example in terms of physicality, however, we
are still concerned with the algorithm that directs our robot on how to
deliver the packages. Borrowing the rough set-up from [14], we can readily
imagine ways in which historical bias would creep in through spatial data i.e.
areas with histories of over-policing and racial policing would potentially be
avoided. Here we are not directly helped by saying, “Similar individuals
should be treated similarly,” as it is unclear what makes individuals similar.
If we say that they are similar based on the area they live in, then we have
not done anything to remove bias. Here, we might look to the Rawlsian
definition to answer how to incorporate the biased data such that individuals
are treated justly. It might be that the user of this package delivery service
has to agree to some ranking of safety of delivery based on their location.
Per Rawls, this would be fair. Another view, that of relational
egalitarianism, might push us to apply more “feedforward” morality to the
issue and think about the factors that led recipients to live where they do.
This might drive us to substantially devalue the biased data in favor of
collecting new data or taking on a socio-spatial ambiguous navigation metric.

While these examples are simple, they help to outline the reasoning process
that roboticists should adopt going forward. The problem of how to evaluate
the definition of what is fair is one that should be answered not only by
philosophers but also those who use fairness in their work. Although the
technical problems are still at a point of vagueness due to their lack of
development, looking to the future, it is clearer how we might as roboticists
develop systems with fairness (and it’s definition) in mind.

##  VI Conclusion

My hope is that at this point, the implications are clear. In the field of
computer science, we must not rely on a single definition of fairness such as
luck egalitarianism would have us do. Relational egalitarianism seems more
promising. This is the direction that the current _similar treated similarly_
seems to be headed. However, it is also important as the designers of such
algorithms that we not pass off decisions solely on society. We must have some
notion of fairness in our design that can then be leveraged to be
contextualized to setting. It is not sufficient to rely solely on the vanilla
definitions of fairness sourced from computer scientists. As roboticists, it
is essential that we think about what fairness means as we architect a system
not as a post hoc project. The support for this can be directly sourced from
Elizabeth Anderson’s definition. Relying only on society is the result of a
distributive pattern matching. We instead must frame our project as imbuing
our agents (robotic and otherwise) with some sense of virtue. This sets up the
challenge as providing a methodology for robotic morals. This cannot simply be
done via example but also requires a consideration of just what it means to be
moral/ethical. This has begun to be tackled in works such as by [17] but must
become a part of our practice as a whole.

## ACKNOWLEDGMENT

Many thanks to Professor Cynthia Dwork for teaching as well as feedback on
this paper. Additional thanks to Lily Hu for intial discussions surrounding
this work.

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