# Testing the gravitational redshift during VERITAS cruise phase

Fabrizio De Marchi<EMAIL_ADDRESS>Department of Mechanical and
Aerospace Engineering, Sapienza University of Rome, Via Eudossiana, 18, 00184
Rome, Italy  Gael Cascioli  Department of Mechanical and Aerospace
Engineering, Sapienza University of Rome, Via Eudossiana, 18, 00184 Rome,
Italy  Currently at University of Maryland Baltimore County and NASA Goddard
Space Flight Center, Maryland, USA  Todd Ely  Jet Propulsion Laboratory,
California Institute of Technology  Luciano Iess  Department of Mechanical and
Aerospace Engineering, Sapienza University of Rome, Via Eudossiana, 18, 00184
Rome, Italy  Eric A. Burt  Jet Propulsion Laboratory, California Institute of
Technology  Scott Hensley  Jet Propulsion Laboratory, California Institute of
Technology  Erwan Mazarico  NASA Goddard Space Flight Center, Maryland, USA

###### Abstract

The NASA Discovery-class mission VERITAS, selected in June 2021, will be
launched towards Venus in 2027. In addition to the science instrumentation
that will build global foundational geophysical datasets, VERITAS proposed to
conduct a technology demonstration for the Deep Space Atomic Clock (DSAC-2). A
first DSAC successfully operated in low-Earth orbit for more than two years,
demonstrated the trapped ion atomic clock technology, and established a new
level of performance for clocks in space. DSAC-2 would have further
improvements in size, power, and performance. It would host a
$1\times{10}^{-13}$ grade USO to produce a frequency output with short-term
stability of less than $2\times{10}^{-13}/\sqrt{\tau}$ (where $\tau$ is the
averaging time).
In this work, we investigate the possibility of leveraging DSAC-2’s frequency
stability to measure possible discrepancies in the redshift predicted by
General Relativity. With its capability of establishing a one-way downlink
Doppler link with Earth stations, DSAC-2 would enable precise tests of the
dependency of the redshift on the location and velocity of the spacecraft and
the Earth, thus providing improved tests of the Local Lorentz Invariance and
Local Position Invariance. We perform accurate simulations of the experiment
that would take place during the VERITAS cruise phase. We consider different
parametrizations of the possible violations of the General Relativity,
different operational conditions, and a wide spectrum of expected measurement
performance assumptions. We show that DSAC-2 onboard VERITAS would provide new
and improved constraints with respect to the current knowledge.

##  I Introduction

An important goal of modern physics is the unification of the four fundamental
interactions under a theory that will encompass General Relativity (GR) and
Quantum Mechanics. An overlying theory would converge to GR at large
distances, where gravity is the dominant interaction, and to quantum theory at
atomic/subatomic scales. Since the two theories are incompatible with each
other (e.g., GR is a deterministic theory), a violation of GR is expected at a
certain level. The search for possible violations of GR involves checks of the
Equivalence Principle and, in a weak-field regime, measurements of the so-
called post-Newtonian parameters (PPN) Will (2014). A deviation from the
values predicted by GR for one or more of these parameters would indicate that
GR is not the ultimate theory of gravitation.

The validity of GR has been challenged several times by astronomical
measurements on an interstellar scale Archibald  _et al._ (2018) and in the
solar system by radio tracking to interplanetary probes (e.g., Bertotti  _et
al._ (2003); Genova  _et al._ (2018); Bernus  _et al._ (2022); see De Marchi
and Cascioli (2020) for a summary).
Deviations from GR would appear as small deviations of the trajectories of the
celestial bodies (or interplanetary probes) from the predictions. At the
interstellar scale (e.g., stellar systems composed of compact objects)
relativistic effects are strong and possible deviations from GR would be
easier to detect. However, results are limited by uncertainties on masses and
mutual distances of these objects. On the contrary, in the solar system,
despite very small relativistic effects (weak-field regime), the masses and
trajectories of the principal bodies are known with high accuracy.
PPN parameters are measured through the solution of orbit determination
problems based on ephemerides and on precise radiometric data (range and/or
Doppler data) collected by interplanetary missions. Until now, no violations
of GR have been found.
Range and range-rate data that will be acquired by VERITAS in its orbital
phase may be used to perform tests of GR. The expected performances will,
however, not be at the level of BepiColombo De Marchi and Cascioli (2020)
because VERITAS is not endowed with a complete cancellation system for plama
noise, (VERITAS enables only two simultaneous link – X uplink/X downlink and
Ka uplink/Ka downlink, while BepiColombo has an additional X uplink/Ka
downlink that enables full plasma noise cancellation).
However, we will show that a time transfer experiment during the cruise phase
of VERITAS between the proposed onboard deep space atomic clock, DSAC-2 Burt
_et al._ (2021), and the atomic clocks at the ground stations can provide
precise measurements of both the gravitational redshift (i.e., the
decrease/increase of the frequency of an electromagnetic wave as it
exits/enters from/into a gravitational well) and of the Doppler shift due to
the relative velocity between the emitter and the receiver.
This experiment has deep significance as it belongs to a class of tests
involving the Equivalence Principle (EP), one of the two founding principles
of GR. The fact that gravitation is a manifestation of the geometry of
spacetime and not a “real” force, as the other three fundamental interactions,
is an outcome of the universality of free fall, which is in turn a direct
consequence of the EP. The Einstein Equivalence Principle (EEP) is usually
divided into three sub-principles:

  * •

Universality of Free Fall (UFF), which states that a test particle (i.e., an
object with a negligible self-gravity) follows a trajectory in an external
gravitational field independent of its mass or structure (i.e., trajectories
depend only on the initial state). This sub-principle is often called the Weak
Equivalence Principle (WEP).

  * •

Local Lorentz Invariance (LLI), which states that the result of a local and
non-gravitational experiment is independent of the orientation or velocity of
the free-falling laboratory.

  * •

Local Position Invariance (LPI), which states that a non-gravitational
experiment is independent of where it is performed in space and in time.

By “non-gravitational experiments” we mean that in the laboratory the
gravitational effects are negligible. Otherwise, the principle is called the
Strong Equivalence Principle (SEP).
In other words, when comparing the free fall of two objects in the same
laboratory (i.e., a “null test”), preferred-location and/or preferred-frame
effects occur if the outcome depends on the location of the laboratory in an
external gravitational field and/or its velocity in an inertial frame. If this
occurs, the EEP is violated.
Conversely, if the EEP is valid, then the Doppler effect between two static
clocks (or “gravitational redshift”) in the weak-field approximation is
“universal,” independent of the nature of the clocks, and always given by
$\Delta U/c^{2}$ Turneaure  _et al._ (1983) where $\Delta U$ is the difference
between the Newtonian potentials at the clock locations and c is the speed of
light.
While tests of the UFF/WEP are performed by Eötvös-like experiments, LLI and
LPI are usually verified by comparing the frequency of two clocks in motion at
different locations (i.e., gravitational potential) or times. The two clocks
can be different (e.g., different materials and/or based on different physical
processes to generate time) or identical. In this latter case, the experiment
automatically excludes the coupling between gravitation and the interactions
involved in clock function.
The sub-principles are expected to be connected: the so-called Schiff’s
conjecture states that the validity of WEP guarantees the validity of LLI and
LPI Schiff (1960).
The violation of the EP can be parametrized by assuming that the
inertia/energy of a particle is position- and velocity-dependent. Given a
particle with rest mass $m_{0}$, position $\mathbf{X}$, velocity $\mathbf{V}$
in a gravitational potential $U$, the Lagrangian expanded to first order in
$U$ and $V^{2}$ is given by Wolf and Blanchet (2016)

$L=-m_{0}c^{2}+\frac{1}{2}m_{0}\left(1+\frac{\delta
m_{I}}{m_{0}}\right)V^{2}-m_{0}\left(1+\frac{\delta
m_{G}}{m_{0}}\right)U\left(\mathbf{X}\right)$ (1)

where $\delta m_{G}$, $\delta m_{I}$ represent the coupling of the (passive)
gravitational and inertial masses/energies to the position and velocity of the
frame, respectively. Using the Euler-Lagrange equations, the equations of
motion are

$\mathbf{a}=\mathbf{g}\left(1+\frac{\delta g}{g}\right)$ (2)

where

$\mathbf{g}=-\frac{dU}{d\mathbf{X}}$ (3)

and

$\frac{\delta g}{g}=\frac{\delta m_{G}}{m_{0}}-\frac{\delta m_{I}}{m_{0}}$ (4)

is the Eötvös parameter that can be measured in a dedicated experiment (e.g.,
the MICROSCOPE mission; Touboul  _et al._ (2017)).
Nordtvedt, 1975 (Nordtvedt (1975)) showed that the conservation of energy, in
the presence of a WEP violation, would imply a small supplementary redshift in
photons exchanged between two static identical clocks. The gravitational
redshift between identical receiver “r” and emitter “e” in case of a WEP
violation being given by (weak-field approximation)

$\frac{\nu_{r}-\nu_{e}}{\nu_{e}}=\left(1+\alpha\right)\frac{U_{e}-U_{r}}{c^{2}}$
(5)

where $U_{e}$ and $U_{r}$ are the gravitational potentials at the emitter and
the receiver, $\alpha=\left(\delta m_{G}/m_{0}\right)\left(m_{0}/\delta
m\right)$, and $\nu_{r}$, $\nu_{e}$ are the receiver and emitter frequencies,
respectively. The first factor in $\alpha$ is related to the WEP violation
contribution (Eq. (4)), while the latter is the ratio of the atom rest mass-
energy to its variation due to the specific quantum process involved in the
generation of photons. In practice, for a given atomic clock, the factor
$m_{0}/\delta m$ depends on the type of atoms/molecules and on the type of
transitions used to measure time. It is an “amplification factor” Nobili  _et
al._ (2013) of the LPI violation parameters derived from the WEP.
In principle, different clocks have different “amplification factors” as the
coupling with gravitation could be different. An analog factor is present in
the second-order Doppler shift leading to the general formula for identical
clocks

$\frac{\nu_{r}-\nu_{e}}{\nu_{e}}=\left(1+\varepsilon\right)\frac{V_{r}^{2}-V_{e}^{2}}{{2c}^{2}}+\left(1+\alpha\right)\frac{U_{e}-U_{r}}{c^{2}}$
(6)

where $\varepsilon$ is the LLI violation parameter and $V_{r}$, $V_{e}$ are
the velocities of the receiver and of the emitter with respect to an inertial
frame. The validity of LLI and LPI entails $\varepsilon=\alpha=0$.
To conclude, gravitational redshift and/or Doppler shift experiments would be
more affected than Eötvös-type experiments in the case of WEP violations.
These latter are easier to carry out as they are null-tests ($\delta g/g$ is
currently known at a ${10}^{-15}$ level; Touboul  _et al._ (2017)), while the
“amplification factor” can magnify the signal by 3 orders of magnitude.
In 1976, Gravity Probe A (GP-A) was the first satellite to perform a redshift
test in the gravitational field of the Earth by comparing two hydrogen masers.
It confirmed the validity of LPI with an accuracy
$\sigma\left(\alpha\right)=1.4\times{10}^{-4}$ Vessot (1989).
The satellites GSAT0201 and GSAT0202, were launched together in 2014 as part
of the ESA Galileo constellation. Because of a technical problem, they were
placed on an elliptic ($e$=0.162) orbit with a period of 13h. Due to the
eccentricity, this configuration was ideal for a gravitational redshift
experiment. By comparing the two onboard passive hydrogen-masers, the
experiment led to a factor 6 improvement
$(\sigma\left(\alpha\right)=2.49\times{10}^{-5}$; Delva  _et al._ (2018)) with
respect to the GP-A results.
A further improvement of about a factor 10 is expected from the upcoming
Atomic Clock Ensemble in Space (ACES) mission Savalle  _et al._ (2019). The
NASA Galileo interplanetary mission to Jupiter (launched in late 1989 with an
USO onboard) performed the first redshift test based on the Sun’s
gravitational field. The precision was
$\sigma\left(\alpha\right)=5.0\times{10}^{-3}$ Krisher  _et al._ (1993).
Regarding the LLI violation, the current uncertainty on $\varepsilon$ is at
the ${10}^{-8}$ level, obtained by ground laboratory experiments Tobar  _et
al._ (2010).
In this work we analyze the opportunity to perform at the same time a
gravitational redshift (LPI violation) and Doppler shift (LLI violation)
experiment, during the cruise phase of VERITAS. Our simulations show that the
exquisite stability of an onboard technology demonstrator such as DSAC-2 would
enable a factor of 10 improvement in the uncertainty of $\alpha$. Conversely,
the LLI violation test cannot improve the current accuracy at the ${10}^{-8}$
level.
This paper is structured as follows. Sec. II describes the VERITAS mission and
the atomic clock DSAC-2, and in Sec. III we outline the theoretical framework
to be used in the experiment. In Sec. IV we describe our simulations and we
discuss the results. Finally, in Sec. V we draw the implications and
conclusions of the outcomes of this work.

##  II Experiment concept

###  II.1 VERITAS mission

VERITAS (for Venus Emissivity, Radio Science, InSAR, Topography, and
Spectroscopy) is a NASA Discovery mission planned for launch in late 2027 (to
be confirmed, Smrekar  _et al._ (2020)). VERITAS would host a set of
instruments designed to advance our understanding of Venus, namely a synthetic
aperture radar (VISAR; Hensley  _et al._ (2020)), an infrared spectrometer
(VEM; Helbert  _et al._ (2020)) and the Integrated Deep Space Transponder
(IDST) enabling the gravity science investigation (Cascioli  _et al._ (2021);
Cascioli  _et al._ (2022); Mazarico  _et al._ (2019); Iess  _et al._ (2021a)).
The nominal mission scenario consists of a 7-month cruise phase, followed by
planetary capture in a highly eccentric orbit with period 13h. An aerobraking
campaign then reduces the orbital period and eccentricity. This first
aerobraking campaign is followed by an initial science phase in an elliptical
orbit (science phase 1, SP1) mainly designed for VEM observations. SP1 is
followed by another 5 months of aerobraking, used to circularize the orbit,
and finally the main observation campaign covering 4 Venus sidereal rotation
periods (or cycles, 243 days each) in which VISAR, VEM, and gravity science
measurements will be collected (science phase 2, SP2).
In this work we are mainly interested in the cruise phase. Fig. 1 shows the
path of the spacecraft during the transfer from Earth to Venus. The spacecraft
will be inserted on a direct transfer orbit inclined on the ecliptic plane, to
match the Venus $~{}4^{\circ}$ orbit inclination. During the cruise there will
be no superior solar conjunctions (SSCs). This implies that the radiometric
observables collected in the context of the redshift experiment would not be
substantially affected by solar plasma noise. This fact, coupled with the
planning of a very limited amount of orbit adjustment maneuvers, allows for a
very quiet dynamical and observational environment of the probe, ideal for
testing faint perturbations on the propagation of radio waves related to
possible LLI/LPI violations.

Figure 1: VERITAS cruise trajectory in the ecliptic plane (top panel) and in
the X-Z plane (lower panel). Markers indicate ten-day periods. The black
segments show the Earth-VERITAS distance during the cruise.

The radiometric tracking of the probe will rely on the onboard tracking
system, close heritage of ESA’s BepiColombo (Iess  _et al._ (2021b); Iess  _et
al._ (2009); Cascioli  _et al._ (2021)). The VERITAS tracking system is able
to establish simultaneous two-way tracking links with Earth stations in the X-
and Ka-bands (8 and 32 GHz respectively). Operational tests of the tracking
equipment onboard BepiColombo allowed verification of its excellent
performance at Ka-bands, which are on the order of 0.02 mm/s at 10s
integration time for range rate and 1-2cm for range (2s sampling rate)
Cappuccio  _et al._ (2020). During the VERITAS cruise, however, the Ka-band
would not be baselined, as routine tracking operations will be conducted
through single X-band links with reduced performance. Operational assessments
during the BepiColombo cruise phase have shown an X-band accuracy on the order
of 0.1 mm/s and 0.04 mm/s at 10s and 60s integration times, respectively
(Paolo Cappuccio, private communication).

###  II.2 DSAC-2

NASA’s Deep Space Atomic Clock (DSAC) Technology Demonstration Mission
recently completed its two-year mission in low-Earth orbit (operations ended
on September 18, 2021). DSAC successfully showed the technology’s viability
for sustained, reliable operations and for providing the most stable frequency
ever demonstrated in space ($\approx 3\times 10^{-15}$ at one-day and a drift
of $<3\times 10^{-16}$/day; Burt  _et al._ (2021)). This success warranted
development of a next-generation DSAC, called DSAC-2, that originally was to
be hosted on VERITAS through as a Technology Demonstration Opportunity (TDO)
but, because of funding shortfalls, had to be canceled. The DSAC-2 conceptual
design intended to use less power, to be smaller, and to be longer-lived than
DSAC-1, while maintaining excellent performance. With DSAC-2 operating as an
external reference to VERITAS’ IDST, it would have been possible to collect
X-band, Ka-band, and combined X/Ka one-way downlink Doppler to support this
proposed GR test. This study examines the GR tests and outcomes that would
have been possible if funding for the DSAC-2 TDO were maintained.
An important aspect of the DSAC-2 project is to advance the trapped ion clock
technology beyond DSAC-1 to include: 1) superior stability, 2) increased
lifetime, 3) reduced size, weight, and power (SWaP), and 4) improve the
fabrication yield percentage and robustness. DSAC-2 would have a stability of
$2\times 10^{-13}/\sqrt{\tau}$ in the short term and $3\times 10^{-15}$ at one
day – similar to the DSAC-1 stability achieved on the ground. To address the
DSAC-2 SWaP requirements of 13 L, 13 kg, and 42 W, DSAC-2 would simplify the
ion trap and frequency chain architectures used in DSAC-1, thereby reducing
components and size as well as power, with current design estimates falling
well below these requirements. Fig. 2 shows a scale comparison of DSAC-1 and
DSAC 2. In addition to design simplification, use of commercial off-the-shelf
(COTS) parts and removal of high tolerances where possible will increase
instrument yield and reliability.

Figure 2: On the left is a photo of the DSAC-1 instrument (silver box) during
space craft integration (a GPS receiver, shown in the foreground, will not be
part of DSAC-2). On the right the DSAC-2 system concept is shown at the same
scale.

##  III Mathematical framework

The relativistic Doppler effect is a combination of the classical Doppler
effect and of time dilation.
Given a station “st” (the “receiver”) and a spacecraft “sc” (the “emitter”),
the relativistic Doppler shift is (Bertotti  _et al._ (2003); Wolf and Petit
(1995))

$\frac{\nu_{st}}{\nu_{sc}}=\left(\frac{1-\mathbf{n}\cdot\mathbf{v}_{st}/c}{1-\mathbf{n}\cdot\mathbf{v}_{sc}/c}\right)\
\ \frac{d\tau_{sc}}{d\tau_{st}}$ (7)

where $\mathbf{n}$ is the unit vector along the propagation path (i.e., from
the spacecraft to the station)

$\mathbf{n}=\frac{\mathbf{r}_{st}-\mathbf{r}_{sc}}{\left|\mathbf{r}_{st}-\mathbf{r}_{sc}\right|}$
(8)

and $\mathbf{r}_{st},\mathbf{r}_{sc},\mathbf{v}_{st},\mathbf{v}_{sc}$ are
their respective positions and velocities in an inertial frame.
The first term on the right-hand side in Eq. (7) represents the “classical”
Doppler effect while the second is the time dilation factor (or “transverse
Doppler effect”). Referring to Barycentric Coordinate Time (TCB), the time
coordinate for the Barycentric Celestial Reference System (BCRS), we estimate
the time dilation factor as

$\frac{d\tau_{st}}{d\tau_{sc}}=\frac{d\tau_{sc}}{dt_{TCB}}/\frac{d\tau_{st}}{dt_{TCB}}$
(9)

$t_{TCB}$ represents the time of a clock moving as the Solar System barycenter
but not influenced by the gravitational potential caused by the Solar System.
The Schwarzschild metric in isotropic coordinates and in the weak field
approximation, reads

$ds^{2}=\left(c\ d\tau_{sc}\right)^{2}\approx\left(1+2U_{sc}\right)\left(c\
dt\right)^{2}-\left(dx^{2}+dy^{2}+dz^{2}\right)$ (10)

where $U_{sc}$ is the gravitational potential at the spacecraft. From Eqs. (9)
and (10), we obtain

$\frac{d\tau_{sc}}{dt_{TCB}}=\sqrt{1+\frac{2U_{sc}-v_{sc}^{2}}{c^{2}}}$ (11)

with
$v_{sc}^{2}=\left|\mathbf{v}_{sc}\right|^{2}=\left(dx^{2}+dy^{2}+dz^{2}\right)/dt^{2}$.
Applying this method to both station and spacecraft, the time dilation reads

$\frac{d\tau_{sc}}{d\tau_{st}}=\sqrt{\frac{1+\left(2U_{sc}-v_{sc}^{2}\right)/c^{2}}{1+\left(2U_{st}-v_{st}^{2}\right)/c^{2}}}.$
(12)

The relativistic Doppler formula is therefore (in the weak-field
approximation)

$\frac{\nu_{st}}{\nu_{sc}}=\sqrt{\frac{1+\left(2U_{sc}-v_{sc}^{2}\right)/c^{2}}{1+\left(2U_{st}-v_{st}^{2}\right)/c^{2}}}\left(\frac{1-\mathbf{n}\cdot\mathbf{v}_{st}/c}{1-\mathbf{n}\cdot\mathbf{v}_{sc}/c}\right)$
(13)

Finally, the expansion up to $c^{-2}$ yields Krisher  _et al._ (1993) (Krisher
_et al._ (1993) assumes positively defined $U_{sc}$ and $U_{st}$, while we
assume that gravitational potential terms are negative)

$\frac{\Delta\nu}{\nu_{sc}}=\frac{\nu_{st}-\nu_{sc}}{\nu_{sc}}\approx-\frac{\mathbf{n}\cdot\left(\mathbf{v}_{st}-\mathbf{v}_{sc}\right)}{c}-\frac{\left(\mathbf{n}\cdot\mathbf{v}_{sc}\right)\left(\mathbf{n}\cdot\mathbf{v}_{st}\right)}{c^{2}}+\frac{\left(\mathbf{n}\cdot\mathbf{v}_{st}\right)^{2}}{c^{2}}+\frac{v_{st}^{2}-v_{sc}^{2}}{2c^{2}}+\frac{U_{sc}-U_{st}}{c^{2}}.$
(14)

Two-way Doppler measurements do not sense (at the $c^{-2}$ level) the
relativistic contributions (i.e., the last two terms in Eq. (14)) due to
mutual cancellation. To detect the signals from the last two terms, two clocks
at both ends of a one-way radio link are necessary.
In order to account for the most general case with different clocks, we adopt
the following parametrization for the redshift signal due to LPI and LLI
violations

$\frac{\Delta\nu}{\nu_{sc}}\left(\alpha_{sc},\alpha_{st},\varepsilon_{sc},\varepsilon_{st}\right)=\left(1+\alpha_{sc}\right)\frac{U_{sc}}{c^{2}}-\left(1+\alpha_{st}\right)\frac{U_{st}}{c^{2}}+\left(1+\varepsilon_{st}\right)\frac{v_{st}^{2}}{2c^{2}}-\left(1+\varepsilon_{sc}\right)\frac{v_{sc}^{2}}{2c^{2}}.$
(15)

The parameters $\alpha_{sc},\alpha_{st},\varepsilon_{sc},\varepsilon_{st}$ are
by definition zero in GR.
We will consider 4 cases: 1) LPI and LLI test with different clocks: the most
general case 2) LPI test only based on different clocks. 3) LPI and LLI test
based on identical clocks and 4) LPI test only and based on identical clocks
(hereafter, “Delva-like” configuration).

##  IV Simulations and results

We conducted an extensive set of numerical simulations to assess the expected
performance of the proposed experiment. The simulations have been conducted
using the NASA-JPL MONTE orbit determination software library Evans  _et al._
(2018). We assume spacecraft tracking from Earth for 8 hours/day - 7
days/week. Four days a week are reserved for standard 8 hours/day DSN two-way
X-band tracking, while three days a week for 3 hours of DSN two-way tracking
followed by 5 hours of DSAC-2 one-way downlink tracking (see Fig. 3). We
assume the Doppler tracking to be supported by the station DSS-25 of the NASA
DSN located in Goldstone (CA). For the two-way X-band tracking data noise we
assume an RMS of 0.04 mm/s at 60s integration time in accordance with
operational observations on BepiColombo (see Sec. II). Note that Ka band
tracking would reduce the Doppler noise to 0.01-0.02 mms/s at the same
integration time. An estimate of the noise level of the DSAC-2 observables
(one-way downlink Doppler) has been obtained by performing a detailed noise
budget, using the same assumptions for noise sources (other than the intrinsic
stability of the DSAC-2 clock) used for the VERITAS error budget (Smrekar  _et
al._ (2020), Sec. IV).
For the intrinsic stability of the DSAC-2 frequency standard we assumed a
white noise scaling on short integration timescales (under a day), closely in
accord with measured performances of DSAC. The expected noise characteristics
for timescale from about 30 seconds to a day are

$\sigma_{y}=\frac{2\times{10}^{-13}}{\sqrt{\tau}}$ (16)

where $\sigma_{y}$ is the Allan standard deviation (a figure of merit for the
fractional frequency stability $y=\frac{\Delta f}{f})$ and $\tau$ is the
integration time. At longer integration times, we expect the clock to
‘flicker’ at a level of about ${10}^{-15}$, where the flickering is likely due
to very long period environmental systematics.

In our simulations we assume an integration time of 60 seconds for both the
DSAC-2 one-way Doppler and the two-way Doppler observables, both types at
X-band. The sum of the noise sources considered, for the DSAC-2 error budget,
is on the order of ${\sigma_{y}\approx 3\times 10}^{-14}$. The tropospheric
contribution, given the level of performance demonstrated at DSS-25 during
Cassini data analysis using the DSN media calibration system (MCS, Keihm  _et
al._ (2004)), is expected to be $<20$% of the total. The plasma noise
contribution is $<15$% of the total (no superior conjunctions are expected).

Figure 3: Simulated tracking schedule.

The numerical simulations are conducted by simulating the tracking data and
the DSAC-2 data based on the current design trajectory accounting for the
above-mentioned assumptions, computing the partial derivatives of the
observations with respect to the parameters of interest and by inverting the
classical orbit determination normal equations (see e.g., Tapley  _et al._
(2004)).
We simulate the experiment during the 15-week period free of maneuvers to
minimize the number of dynamical discontinuities in the trajectory. We use the
ORACLE orbit determination filter developed at Sapienza University which has
been successfully validated and applied to many data analyses and simulations
(e.g., Iess  _et al._ (2018); Durante  _et al._ (2020); Cascioli  _et al._
(2020)). The dynamical model employed to simulate the trajectory of the probe
accounts for the gravitational interaction of the main solar system bodies and
the radiation pressure imparted by the solar photons on the probe. To simulate
the real data analysis approach, we have run two distinct sets of simulations
employing two different orbit determination frameworks. We have conducted a
multi-arc analysis and a single arc analysis. In the multi-arc analysis, we
divide the trajectory into 10-days arcs and treat each period as a separate
data arc. This method entails two separate sets of solve-for parameters: local
parameters and global parameters. The local parameters are the ones affecting
only one arc, while the global ones are common to all the arcs.
The set of estimated local parameters include the state of the probe (position
and velocity at the beginning of the arc) a solar radiation pressure scale
factor (a priori uncertainty 25%). The global parameters include the
gravitational redshift-related parameters $(\alpha,\varepsilon)$ and the
position and velocity of the Earth (with a large, unconstraining, a priori
uncertainty of 50km and ${10}^{-8}$ km/s, respectively, derived from De Marchi
and Cascioli (2020)).
To introduce the methodology used to account for the possible fluctuations in
the frequency stability of the DSAC-2, a discussion on its behavior during the
experiment is required. At the beginning of the experiment DSAC-2 will be
switched on; the reference frequency of the DSAC-2, after a calibration phase,
will be known with a certain precision and accuracy, depending on the internal
calibration hardware. Indeed, the actual DSAC reference frequency ($f_{0}$)
will differ from the expected one ($f_{0}^{e}$) by a frequency bias ($B_{0}$)
such that

$f_{0}=f_{0}^{e}+B_{0}$ (17)

After the calibration phase, DSAC-2 will continue running with a stability as
discussed in Sec. II.2. Indeed, at the beginning of the i-th arc we can expect
the reference frequency to be into the interval

$f_{i-1}-f_{i-1}\sigma_{y}<f_{i}<f_{i-1}+f_{i-1}\sigma_{y}$ (18)

Where $f_{i-1}$ is the reference frequency of the previous arc and
$\sigma_{y}$ is the fractional frequency stability (in this case corresponding
to $\tau$=10 days). In terms of biases:

$B_{i-1}-f_{i-1}\sigma_{y}<B_{i}<B_{i-1}+f_{i-1}\sigma_{y}$ (19)

In the orbit determination filter, then, we solve for the initial bias with an
a priori uncertainty $\sigma_{0}$ in line with the predicted calibration
capability and in each arc, we solve for the local frequency bias. The biases
of each arc, however, are not unrelated, thus in each arc we estimate the
local frequency bias subject to the inter-arc constraint of Eq. (19) with an a
priori uncertainty $(\sigma_{N})$ in line with the predicted 10-day frequency
stability.
The single-arc analysis reproduces the multi-arc frequency bias management
approach by modeling the frequency bias as a stochastic process. We model the
frequency bias as a white noise uncorrelated stochastic process with an update
time of 10 days. The a priori uncertainty on the initial value of the
stochastic parameter is set to $\sigma_{init}$ and the stochastic process
uncertainty is set to $\sigma_{c}$ (i.e., the inter-arc constraint reads
$\left|\left|B_{i}-B_{i-1}\right|\right|<\sigma_{c}f_{i-1})$.
The simulations presented below are performed to explore different assumptions
regarding the Doppler shift parametrization (accordingly to Eq. (15)) while
estimating only $\alpha$ or both $\alpha,\varepsilon$ together, with different
assumptions regarding the type of clock equipped at the DSN receiver (i.e., an
atomic clock based on the same atom and on the same internal atomic transition
as the DSAC). We also explore the redshift-related parameter retrieval varying
the stability assumptions on DSAC ($\sigma_{init}$ and $\sigma_{c}$).
The range of selected values of $\sigma_{init}$ is based on the DSAC-2 retrace
value and characterized on the ground. It is expected to be between
${10}^{-14}$ and ${10}^{-13}$ (this latter is a very conservative assumption).
The simulations presented here were performed on an even broader spectrum,
ranging from ${10}^{-14}$ to ${10}^{-11}$.
The parameter $\sigma_{c}$, corresponds to the 10-day ADEV value, an
integration time where the clock is expected to be largely limited by its
noise floor. As explained above, this number could be ${10}^{-15}$ but further
characterization is needed. The DSAC-1 flight experiment achieved a
$\sigma_{c}<{3\ \times 10}^{-15}$. This value was mainly driven by several
issues that will be resolved for the DSAC-2 development.
For the simulations we assumed ${10}^{-15}<\sigma_{c}<{10}^{-14}$ (i.e., a
very conservative interval).
The results by Krisher  _et al._ (1993) reflect the capabilities enabled by
the previous generation of tracking systems. We explore the expected
uncertainty retrieved by VERITAS under four different parametrization
assumptions to provide a wide array of possible outcomes based on the real
data analysis strategy employed.
The first case, Case I, we considered is the most general one: LPI and LLI
violation with different clocks (4 parameters to be estimated). The results
are reported in Fig. 4.
In the second case, Case II, we considered the LPI violation only, tested with
different clocks (Fig. 5). The uncertainties are about 2.5 smaller than the
current uncertainty (Delva  _et al._ (2018)).
As a third case, Case III, we considered a parametrization employing both
$\alpha$ and $\varepsilon$ but assuming that atoms and atomic transitions
employed in the ground receiving clock are the same as that used by DSAC. In
this case we do not show a “state-of-the-art line” as, to our knowledge, no
previous experiment has met these assumptions: the expected uncertainties are
about ${3\ \times 10}^{-5}$ in both cases (and shown in Fig. 6). Regarding the
LLI, this result is about 3 orders of magnitude worst that the current
uncertainty Tobar  _et al._ (2010).
The last case, Case IV, explores a Delva-like configuration (see Sect. 3) in
which a single $\alpha$ is estimated (i.e., a test of the LPI violation with
identical clocks). Results are plotted in Fig. 7. The expected formal
uncertainty of the parameter $\alpha$, compared with the results of Delva  _et
al._ (2018), shows an improvement of a factor 2.5-5, a quite significant
amelioration.

Figure 4: Case I. Simulation results in terms of formal uncertainty on the
parameter of interest (y-axis), initial frequency bias uncertainty
$\sigma_{0}$ (x-axis) with varying assumptions for the 10-day frequency
stability of DSAC-2, $\sigma_{N}$ (colored markers). Top (bottom): expected
formal uncertainties for LPI (LLI) violation parameters. All results are
compared with the Galileo probe experiments outputs. The dotted vertical line
represents the conservative upper limit ($5\times{10}^{-15}$) for the
$\sigma_{0}$ constraint. The dotted horizontal lines report the results by
Krisher  _et al._ (1993). Figure 5: Case II. LPI violation test with different
clocks. Left (right): expected formal uncertainty for the spacecraft (ground
station) clocks. Figure 6: Case III. LPI and LLI violation test with identical
clocks. Left (right): expected formal uncertainty for LPI (LLI) violation
parameter. Figure 7: – Case IV. Simulation of the LPI violation test with
identical clocks. The dotted horizontal line reports the current knowledge of
the $\alpha$ parameter by Delva  _et al._ (2018).

##  V Conclusions

In this work we described the possibility to perform an advanced test of Local
Lorentz and Local Position invariance during the cruise phase of the VERITAS
probe to Venus, planned in 2027-2028. The scarcity of planned maneuvers and
gravity assists makes this phase particularly suited for this kind of
measurement. As originally proposed, our simulations show that these
gravitational redshift and transverse Doppler (time dilation) experiments
would have benefitted from the excellent stability of an onboard atomic clock
DSAC-2 and would have served as a test of General Relativity predictions.
In our simulations we considered the possibility to test both LLI and LPI
violation. The LLI is currently verified to a ${10}^{-8}$-level Tobar  _et
al._ (2010). An experiment with VERITAS would not provide any improvement
since the expected accuracy is about 3 orders of magnitude bigger. For the
LPI, our simulations show that the experiment with VERITAS could improve the
current accuracy on LPI by a factor between 2.5 and 5, depending on the
assumptions on the DSAC-2 stability.

###### Acknowledgements.

FDM, GC and LI acknowledge support from the Italian Space agency under
contracts 2020-15-HH.0 and 2022-15-HH.0. A portion of the work described in
this paper was carried out at the Jet Propulsion Laboratory, California
Institute of Technology, under a contract with the National Aeronautics and
Space Administration (80NM0018D0004). The authors are thankful to S. Smrekar
for useful discussions. FDM would like to thank N. Bartel for the fruitful
discussion at the COSPAR 2022 Assembly.

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# Implicit ZCA Whitening Effects of Linear Autoencoders for Recommendation

Katsuhiko Hayashi  Faculty of IST, Hokkaido University  0000-0002-3240-4697
Kazuma Onishi  School of Engineering, Hokkaido University

###### Abstract

Recently, in the field of recommendation systems, linear regression
(autoencoder) models have been investigated as a way to learn item similarity.
In this paper, we show a connection between a linear autoencoder model and ZCA
whitening for recommendation data. In particular, we show that the dual form
solution of a linear autoencoder model actually has ZCA whitening effects on
feature vectors of items, while items are considered as input features in the
primal problem of the autoencoder/regression model. We also show the
correctness of applying a linear autoencoder to low-dimensional item vectors
obtained using embedding methods such as Item2vec to estimate item-item
similarities. Our experiments provide preliminary results indicating the
effectiveness of whitening low-dimensional item embeddings.

##  1 Introduction

E-commerce has become an indispensable part of our daily lives. Recommender
systems help users find products they want to buy on e-commerce sites and have
a wide range of applications, such as in movie recommendations and product
recommendations. Collaborative filtering (CF) is one of the most widely used
approaches in recommender systems. Nearest neighbor CF approaches are divided
into user-based and item-based ones. In item-based collaborative filtering
(ICF), recommendations to users are made by finding items that are similar to
other items that a given user has already had an interaction with. Therefore,
the similarity measure between two items plays an important role in the ICF
approach. While early ICF models used such statistical measures as Pearson
correlation and cosine similarity [21, 11, 5], model-based methods [15, 9, 18]
have recently been investigated as a way to learn item similarity. In
particular, ICF models based on linear regressions (autoencoders) [15, 18, 8,
14] have achieved current state-of-the-art performances on several benchmark
datasets for implicit recommendation.

As to the reason for the empirical success of linear autoencoders, in this
study, we reveal that they actually have ZCA whitening [10] effects on
recommendation data. The whitening transformation removes correlations between
the feature dimensions of the item vectors, and the item-item similarity
estimated from the transformed vectors improves the quality of the
recommendations. Our finding also has the following empirical contribution:

  * •

Whitening Item Embeddings: In the field of natural language processing, it is
known that whitening word embeddings improves the performance of various
retrieval and similarity tasks [19, 7]. Analogously, embedding methods such as
Item2vec [1], which learns a latent semantic feature vector representation of
items, are useful in the field of information retrieval. Thus, whitening item
embeddings is expected to improve recommendation performance. Our experiments
provide preliminary results that show the effectiveness of whitening low-
dimensional item embeddings.

#### Notation and Preliminaries

Vectors are represented by boldface lowercase letters, e.g., a.
$\textbf{0}_{D}$ and $\textbf{1}_{D}$ are $D$-dimensional vectors of zeros and
ones, respectively. Matrices are represented by boldface capital letters,
e.g., A. The $i$-th row of a matrix A is represented by ai:, and the $j$-th
column of A is represented by a:j. The element $(i,j)$ of a matrix A is
denoted by $a_{ij}$. $\textbf{A}^{\mathrm{T}}$ and $\textbf{A}^{-1}$ denote
the transpose and inverse of a matrix A, respectively. $\textbf{I}_{D}$
denotes the $D$-dimensional identity matrix.
$\mathop{\text{diag}}(\textbf{A})$ is the diagonal of a square matrix A.
$\mathop{\text{diagMat}}(\textbf{a})$ denotes the diagonal matrix whose
diagonal is the vector a.

##  2 Item-based Neighborhood Model

Let $U$ and $I$ be sets of users and items, respectively. Like in many papers
on recommender systems [15, 18, 5], we consider implicit feedback data. Here,
the user-item interaction matrix X can be considered to be a binary one:

$\textbf{X}=\left(\begin{array}[]{cccc}x_{11}&x_{12}&\cdots&x_{1|I|}\\\
x_{21}&x_{22}&\cdots&x_{2|I|}\\\ \vdots&\vdots&\ddots&\vdots\\\
x_{|U|1}&x_{|U|2}&\cdots&x_{|U||I|}\end{array}\right)\in\\{0,1\\}^{|U|\times|I|}.$

where $x_{ui}=1$ represents that there is an interaction between user $u$ and
item $i$. If there is no interaction between $u$ and $i$, then $x_{ui}=0$.

To make recommendations for a user $u$, item-based neighborhood collaborative
filtering (ICF) models [11] require pre-computed similarities associated with
each item-item pair from X. We denote an item-item similarity matrix as
$\textbf{B}\in\mathbb{R}^{|I|\times|I|}$, where $b_{ij}$ is the similarity
between two items $i$ and $j$. The item set that the user $u$ has interacted
with is represented by $\textbf{y}_{(u)}\in\\{0,1\\}^{|I|}$, where the $j$-th
element is 1 if there is an interaction between $u$ and $j$, otherwise 0.
Accordingly, ICF models simply compute user $u$’s preference score $s_{ui}$ on
item $i$ as the following dot-product
$s_{ui}=\textbf{y}_{(u)}^{\mathrm{T}}\textbf{b}_{:i}$, where $\textbf{b}_{:i}$
is the $i$-th column of B.

##  3 Shallow Linear Autoencoders

As the previous section shows, a key step in ICF methods is to estimate the
item-item similarity matrix $\textbf{B}\in\mathbb{R}^{|I|\times|I|}$ from
$\textbf{X}\in\\{0,1\\}^{|U|\times|I|}$. While early ICF approaches used such
statistical measures as Pearson correlation and cosine similarity [21, 11, 5],
model-based methods [15, 9, 18] have recently been investigated as a way to
learn item similarity. In this section, we introduce shallow linear
autoencoder models [15, 18] that learn the item-item similarity matrix as a
regression problem.

###  3.1 Linear Autoencoder with L2 Regularization

Linear autoencoders can be trained with the multivariate least squares fitting
approach. The linear algebraic formulation can be represented as a ridge
linear regression problem:

$\widehat{\textbf{B}}=\operatorname*{arg\,min}_{\textbf{B}}\Bigl{\\{}||\textbf{X}-\textbf{X}\textbf{B}||_{F}^{2}+\lambda||\textbf{B}||_{F}^{2}\Bigr{\\}}$
(1)

where $\lambda>0$ is a regularization parameter. The closed form solution of
the above equation is given as

$\widehat{\textbf{B}}=(\textbf{X}^{\mathrm{T}}\textbf{X}+\lambda\textbf{I}_{|I|})^{-1}\textbf{X}^{\mathrm{T}}\textbf{X}$
(2)

or, equivalently [17],

$\widehat{\textbf{B}}=\textbf{X}^{\mathrm{T}}(\textbf{X}\textbf{X}^{\mathrm{T}}+\lambda\textbf{I}_{|U|})^{-1}\textbf{X}.$
(3)

For a proof of the equivalence between Eqs.(2) and (3), we refer the read to
the Appendix A of the paper [3] (in the case of $|I|>|U|$) and our Appendix A
(in the case of $|U|>|I|$). Note that though the minimization of Eq.(1) is
achieved in an obvious way ($\textbf{B}=\textbf{I}_{|I|}$),
$\mathop{\text{diag}}{(\textbf{B})}=\textbf{0}_{|I|}$ is imposed as a
constraint condition in practical linear autoencoder models like SLIM [15] and
EASE [18].

###  3.2 EASE: Linear Autoencoder with
Diagonal Constraints [18]

Steck [18] introduced a linear autoencoder model, called EASE. The objective
function of EASE for learning the item-item similarity matrix B is:

$\displaystyle\widehat{\textbf{B}}_{\text{EASE}}=\operatorname*{arg\,min}_{\textbf{B}}\Bigl{\\{}||\textbf{X}-\textbf{X}\textbf{B}||_{F}^{2}+\lambda||\textbf{B}||_{F}^{2}\Bigr{\\}}$
$\displaystyle\text{s.t.}\ \ \
\mathop{\text{diag}}{(\textbf{B})}=\textbf{0}_{|I|}.$

The optimization problem can be solved with the following Lagrangian:

$L=||\textbf{X}-\textbf{X}\textbf{B}||_{F}^{2}+\lambda||\textbf{B}||_{F}^{2}+2\bm{\alpha}^{\mathrm{T}}\mathop{\text{diag}}{(\textbf{B})}$

where $\bm{\alpha}=[\alpha_{1},\dots,\alpha_{|I|}]^{\mathrm{T}}$ is the vector
of Lagrange multipliers. By setting the derivative to zero, we derive the
estimate of the similarity matrix B:

$\widehat{\textbf{B}}_{\text{EASE}}=(\textbf{X}^{\mathrm{T}}\textbf{X}+\lambda\textbf{I}_{|I|})^{-1}(\textbf{X}^{\mathrm{T}}\textbf{X}-\mathop{\text{diagMat}}{(\bm{\alpha})}).$

By imposing the constraint
$\mathop{\text{diag}}(\widehat{\textbf{B}}_{\text{EASE}})=0$ and defining
$\widehat{\textbf{P}}=(\textbf{X}^{\mathrm{T}}\textbf{X}+\lambda\textbf{I}_{|I|})^{-1}$,
$\bm{\alpha}$ is determined as

$\bm{\alpha}=\textbf{1}_{|I|}\oslash\mathop{\text{diag}}(\widehat{\textbf{P}})-\lambda\textbf{1}_{|I|}.$

Steck [18] showed that the solution can be derived in the following closed
form:

$\widehat{\textbf{B}}_{\text{EASE}}=\textbf{I}_{|I|}-\widehat{\textbf{P}}\mathop{\text{diagMat}}{(\textbf{1}_{|I|}\oslash\mathop{\text{diag}}(\widehat{\textbf{P}}))}$

where $\oslash$ denotes elementwise division.

##  4 Relationship between Linear Autoencoder
and ZCA Whitening

###  4.1 ZCA Whitening (Zero-phase Component Analysis)

Whitening is an operation that eliminates correlations between features in a
data sample. That is, correlation between any two components $x_{i}$ and
$x_{j}$ of a sample vector $\textbf{x}=[x_{1},\dots,x_{D}]$ in the
$D$-dimensional feature space is reduced to zero. In the following, we assume
that a feature vector of each sample is centered; i.e.,
$\frac{1}{D}\sum_{i=1}^{D}x_{i}=0$. Given $N$ data samples
$\textbf{X}=[\textbf{x}_{1},\dots,\textbf{x}_{N}]\in\mathbb{R}^{D\times N}$,
the correlation coefficients between features are represented by the following
covariance matrix:

$\textbf{$\Phi$}_{\textbf{X}}=\frac{1}{N}\textbf{X}\textbf{X}^{\mathrm{T}}.$

We now consider a linear transformation $\textbf{P}\in\mathbb{R}^{D\times D}$
of feature vectors:

$\textbf{w}_{n}=\textbf{P}\textbf{x}_{n}\ \ \ (n=1,\dots,N).$

Accordingly, the covariance matrix of the transformed vectors
$\textbf{W}=[\textbf{w}_{1},\dots,\textbf{w}_{N}]$ is defined as:

$\textbf{$\Phi$}_{\textbf{W}}=\frac{1}{N}\textbf{W}\textbf{W}^{\mathrm{T}}.$

The purpose of whitening is to find a projection matrix P that makes the
covariance matrix $\textbf{$\Phi$}_{\textbf{W}}$ be the identity matrix
$\textbf{I}_{D}$. If $\textbf{$\Phi$}_{\textbf{W}}=\textbf{I}_{D}$, P must
satisfy the following equation:

$\textbf{P}^{\mathrm{T}}\textbf{P}=\textbf{$\Phi$}_{\textbf{X}}^{-1}.$ (4)

P that satisfies Eq.(4) can be represented by using the eigenvectors U of
$\textbf{X}\textbf{X}^{\mathrm{T}}$.

$\textbf{X}\textbf{X}^{\mathrm{T}}=\textbf{U}\bm{\Sigma}\textbf{U}^{\mathrm{T}}$

where U is a square $D\times D$ matrix whose $i$-th column is the $i$-th eigen
vector of $\textbf{X}\textbf{X}^{\mathrm{T}}$, and $\bm{\Sigma}$ is a diagonal
matrix whose diagonal elements are the corresponding eigenvalues,
$d_{ii}=\lambda_{i}$. Since $\bm{\Sigma}$ is an orthogonal matrix
($\textbf{U}^{\mathrm{T}}\textbf{U}=\textbf{U}\textbf{U}^{\mathrm{T}}=\textbf{I}_{D}$),
$\textbf{$\Phi$}_{\textbf{X}}^{-1}$ can be denoted by
$\textbf{U}\bm{\Sigma}^{-1}\textbf{U}^{\mathrm{T}}$. In zero-phase component
Analysis (ZCA) or ZCA whitening [10], we use

$\textbf{P}_{\text{ZCA}}=\textbf{U}\bm{\Sigma}^{-1/2}\textbf{U}^{\mathrm{T}}$

as a transformation matrix P. In practice, noise $\epsilon$ is often added to
the diagonal of $\bm{\Sigma}$:

$\textbf{P}_{\text{ZCA}}=\textbf{U}(\bm{\Sigma}+\epsilon\textbf{I}_{D})^{-1/2}\textbf{U}^{\mathrm{T}}.$

###  4.2 Whitening Effects of Linear Autoencoders

#### Linear Autoencoder with L2 Regularization

First, we consider the ZCA whitening transformation of the user-item
interaction matrix $\textbf{X}\in\\{0,1\\}^{|U|\times|I|}$ (here, we assume
$|U|>|I|$):

$\textbf{W}=\textbf{P}_{\text{ZCA}}\textbf{X}$

where
$\textbf{P}_{\text{ZCA}}=\textbf{U}(\bm{\Sigma}+\epsilon\textbf{I}_{|U|})^{-1/2}\textbf{U}^{\mathrm{T}}$.
Note that U and $\bm{\Sigma}$ are the eigen vector and eigenvalue matrices of
$\textbf{X}\textbf{X}^{\mathrm{T}}$. After this whitening transformation, we
simply consider

$\widehat{\textbf{B}}_{\text{ZCA}}=\textbf{W}^{\mathrm{T}}\textbf{W}$

as an item-item similarity matrix for ICF-based recommender systems.

Now let us rewrite $\widehat{\textbf{B}}_{\text{ZCA}}$:

$\displaystyle\widehat{\textbf{B}}_{\text{ZCA}}$ $\displaystyle=$
$\displaystyle\textbf{W}^{\mathrm{T}}\textbf{W}=(\textbf{P}_{\text{ZCA}}\textbf{X})^{\mathrm{T}}(\textbf{P}_{\text{ZCA}}\textbf{X})$
$\displaystyle=$
$\displaystyle\textbf{X}^{\mathrm{T}}\bigl{(}\textbf{U}(\bm{\Sigma}+\epsilon\textbf{I}_{|U|})^{-1/2}\textbf{U}^{\mathrm{T}}\bigr{)}\bigl{(}\textbf{U}(\bm{\Sigma}+\epsilon\textbf{I}_{|U|})^{-1/2}\textbf{U}^{\mathrm{T}}\bigr{)}\textbf{X}$
$\displaystyle=$
$\displaystyle\textbf{X}^{\mathrm{T}}\bigl{(}\textbf{U}(\bm{\Sigma}+\epsilon\textbf{I}_{|U|})^{-1}\textbf{U}^{\mathrm{T}}\bigr{)}\textbf{X}$
$\displaystyle=$
$\displaystyle\textbf{X}^{\mathrm{T}}\bigl{(}\textbf{U}\bm{\Sigma}\textbf{U}^{\mathrm{T}}+\epsilon\textbf{U}\textbf{U}^{\mathrm{T}}\textbf{U}\textbf{U}^{\mathrm{T}}\bigr{)}^{-1}\textbf{X}$
$\displaystyle=$
$\displaystyle\textbf{X}^{\mathrm{T}}\bigl{(}\textbf{X}\textbf{X}^{\mathrm{T}}+\epsilon\textbf{I}_{|U|}\bigr{)}^{-1}\textbf{X}\
\ \ \ \ \ (=\text{Eq.(\ref{eq:solution2})})$ $\displaystyle=$
$\displaystyle\bigl{(}\textbf{X}^{\mathrm{T}}\textbf{X}+\epsilon\textbf{I}_{|I|}\bigr{)}^{-1}\textbf{X}^{\mathrm{T}}\textbf{X}\
\ \ \ \ \ (=\text{Eq.(\ref{eq:solution1})}).$

Refer to our Appendix A for details on the final transformation. This result
clearly shows a connection between the shallow linear autoencoder and ZCA
whitening. We can see that linear autoencoders actually have implicit ZCA
whitening-like effects on feature vectors of items111In general, linear
autoencoder models for recommendation data do not assume that input data are
centered., while items are considered as input features in the primal problem
of the autoencoder models.

#### Linear Autoencoder with Diagonal Constraints

As pointed out in the paper [14], the solution of EASE can be divided into two
terms: regularization and diagonal constraints. The former regularization part
is equivalent to the solution of linear autoencoder with L2 regularization:

$\widehat{\textbf{B}}_{\text{EASE}}=\underbrace{\bigl{(}\textbf{X}^{\mathrm{T}}\textbf{X}+\lambda\textbf{I}_{|I|}\bigr{)}^{-1}\textbf{X}^{\mathrm{T}}\textbf{X}}_{\widehat{\textbf{B}}_{\text{ZCA}}}-\bigl{(}\textbf{X}^{\mathrm{T}}\textbf{X}+\lambda\textbf{I}_{|I|}\bigr{)}^{-1}\mathop{\text{diagMat}}(\bm{\alpha}).$

This result shows that EASE also has implicit whitening effects on item
vectors. From [14], it is known that the latter diagonal constraint part plays
a role that penalizes the impact of unpopular items. In future work, we will
further investigate the role of the diagonal constraint part from the view
point of the data preprocessing stage.

#### Whitening Item Embeddings

Several studies [1, 6, 8] have shown that a latent semantic representation of
items is useful for estimating item similarity. We consider item embeddings
$\textbf{E}\in\mathbb{R}^{D\times|I|}$, where $D<|I|$ is the dimension size of
the item embedding. Our finding on a relationship between the linear
autoencoder and ZCA whitening shows the correctness of using a linear
autoencoder on item embeddings:

$\widehat{\textbf{B}}=\operatorname*{arg\,min}_{\textbf{B}}\Bigl{\\{}||\textbf{E}-\textbf{E}\textbf{B}||_{F}^{2}+\lambda||\textbf{B}||_{F}^{2}\Bigr{\\}}.$

The final solution is

$\widehat{\textbf{B}}=(\textbf{E}^{\mathrm{T}}\textbf{E}+\lambda\textbf{I}_{|I|})^{-1}\textbf{E}^{\mathrm{T}}\textbf{E}.$

In the case of $D<|I|$, by using Lemma 9 of the paper [3], we can transform
the above equation into

$\widehat{\textbf{B}}=\textbf{E}^{\mathrm{T}}(\textbf{E}\textbf{E}^{\mathrm{T}}+\lambda\textbf{I}_{D})^{-1}\textbf{E}.$

Moreover, we can further transform this equation as follows:

$\displaystyle\widehat{\textbf{B}}$ $\displaystyle=$
$\displaystyle\textbf{E}^{\mathrm{T}}(\textbf{E}\textbf{E}^{\mathrm{T}}+\lambda\textbf{I}_{D})^{-1}\textbf{E}$
$\displaystyle=$
$\displaystyle\textbf{E}^{\mathrm{T}}(\textbf{U}\bm{\Sigma}\textbf{U}^{\mathrm{T}}+\lambda\textbf{U}\textbf{I}_{D}\textbf{U}^{\mathrm{T}})^{-1}\textbf{E}$
$\displaystyle=$
$\displaystyle\textbf{E}^{\mathrm{T}}\textbf{U}(\bm{\Sigma}+\lambda\textbf{I}_{D})^{-1}\textbf{U}^{\mathrm{T}}\textbf{E}$
$\displaystyle=$
$\displaystyle\textbf{E}^{\mathrm{T}}\textbf{U}(\bm{\Sigma}+\lambda\textbf{I}_{D})^{-1/2}\textbf{U}^{\mathrm{T}}\textbf{U}(\bm{\Sigma}+\lambda\textbf{I}_{D})^{-1/2}\textbf{U}^{\mathrm{T}}\textbf{E}$
$\displaystyle=$
$\displaystyle(\textbf{P}_{\text{ZCA}}\textbf{E})^{\mathrm{T}}(\textbf{P}_{\text{ZCA}}\textbf{E})$

where $\textbf{U}\bm{\Sigma}\textbf{U}^{\mathrm{T}}$ is the eigenvalue
decomposition of $\textbf{E}\textbf{E}^{\mathrm{T}}$ and
$\textbf{P}_{\text{ZCA}}$ denotes
$\textbf{U}(\bm{\Sigma}+\lambda\textbf{I}_{D})^{-1/2}\textbf{U}^{\mathrm{T}}$.
The result shows that the linear autoencoder implicitly decorrelates latent
features of item embeddings through the ZCA whitening process.

##  5 Experiments

###  5.1 Datasets and Evaluation Metrics

We conducted experiments on two publicly available datasets:

  * •

MovieLens 20 Million (ML-20M) [4]: 136,677 users and 20,108 movies with about
10.0 million interactions,

  * •

Netflix Prize (Netflix) [2]: 463,435 users and 17,769 movies with about 56.9
million interactions,

For a fair comparison, we followed the experimental settings used in [12] and
kept the same pre-processing steps222The program code is provided by the
authors of [13]:
https://github.com/samlobel/RaCT_CF/blob/master/setup_data.py..

The experiments considered two ranking metrics, Recall$@R$ and the truncated
NDCG (NDCG$@R$), where $R$ is a cut-off hyper-parameter [12]. Here,
$\omega(r)$ is defined as the item at rank $r$, $\mathbbm{I}[\cdot]$ as the
indicator function, and $\mathcal{I}_{u}$ as the held-out unobserved items
that a user $u$ will interact. Recall$@R$ for user $u$ is

$\text{Recall}@R:=\sum_{r=1}^{R}\frac{\mathbbm{I}[\omega(r)\in\mathcal{I}_{u}]}{\min(R,|\mathcal{I}_{u}|)}.$

The truncated discounted cumulative gain (DCG$@R$) is

$\text{DCG}@R:=\sum_{r=1}^{R}\frac{2^{\mathbbm{I}[\omega(r)\in\mathcal{I}_{u}]}-1}{\log{(r+1)}}.$

NDCG$@R$ is obtained by dividing DCG$@R$ by its best possible value where all
the held-out items are ranked at the top.

###  5.2 Results

Table 1: Comparison of recommendation performances: * and ** denote results transcribed from [12] and [18], respectively. | ML-20M | Netflix
---|---|---
Model | Recall@20 | NDCG@100 | Recall@20 | NDCG@100
Embed (Inner Product) | 0.279 | 0.317 | 0.246 | 0.279
Embed+AE | 0.372 | 0.402 | 0.353 | 0.385
Embed+EASE | 0.361 | 0.394 | 0.341 | 0.375
*SLIM [15] | 0.370 | 0.401 | 0.347 | 0.379
*EASE [18] | 0.391 | 0.420 | 0.362 | 0.393
WMF [6] | 0.360 | 0.386 | 0.316 | 0.351
CDAE [20] | 0.391 | 0.418 | 0.343 | 0.376
Mult-VAE [12] | 0.395 | 0.426 | 0.351 | 0.386
RaCT [13] | 0.403 | 0.434 | 0.357 | 0.392

In section 4.2, we showed that linear autoencoders have ZCA whitening-like
effects on item feature vectors. In our experiments, we investigated how
linear autoencoders improve the quality of item embeddings. Using a singular
value decomposition $\textbf{X}=\textbf{U}\bm{\Sigma}\textbf{V}^{\mathrm{T}}$,
we computed item embeddings
$\textbf{E}=\bm{\Sigma}^{1/2}\textbf{V}^{\mathrm{T}}\in\mathbb{R}^{D\times|I|}$.
To estimate item similarity, we applied a linear autoencoder with L2
regularization (AE) and EASE to item embeddings, respectively. We also tried a
simple inner product $\textbf{E}^{\mathrm{T}}\textbf{E}$ as a baseline item-
item similarity matrix.

The number of dimensions $D$ was set to 800 and 4,000 for the ML-20M and
Netflix datasets, respectively. The hyperparameter $\lambda$ of the linear
autoencoders was fixed to 200 in all settings. Note that an advantage of using
low-dimensional vectors is the computational savings regarding the calculation
of the inverse matrix; i.e.,
$(\textbf{E}\textbf{E}^{\mathrm{T}}+\textbf{I}_{D})^{-1}\in\mathbb{R}^{D\times
D}$ is much more efficient than
$(\textbf{X}^{\mathrm{T}}\textbf{X}+\lambda\textbf{I}_{|I|})^{-1}\in\mathbb{R}^{|I|\times|I|}$.

Tab.1 compares recommendation performances. Our results clearly show that
linear autoencoders improve the quality of the item embeddings. In particular,
Embed+AE outperformed Embed+EASE. We think this is because a low-rank
approximation (item embeddings) eliminates the effects of unpopular items,
while the diagonal constraint part in the solution of EASE also plays the same
role, as noted in Section 4.2. Therefore, we think that the normal AE already
has sufficient ability to improve the quality of the item embeddings.

##  6 Conclusion

In this paper, we have shown that linear autoencoder models have ZCA
whitening-like effects on recommendation data. This finding ensures the
correctness of applying a linear autoencoder model to low-dimensional item
embedding vectors. Our initial experiments also reveal the effectiveness of
whitening low-dimensional item embeddings.

In future work, we will try other methods to make item embeddings, such as
Item2vec [1] and GloVe [16].

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##  Appendix A Proof of the Equivalence of Eqs.(2) and (3) when $|U|>|I|$

###### Proof.

Given a matrix $\textbf{X}\in\mathbb{R}^{|U|\times|I|}$, we assume $|U|>|I|$.

Let $\textbf{X}=\textbf{U}_{f}\bm{\Sigma}_{f}\textbf{V}^{T}$ be the full SVD
representation of X with
$\textbf{U}_{f}^{\mathrm{T}}\textbf{U}_{f}=\textbf{U}_{f}\textbf{U}_{f}^{\mathrm{T}}=\textbf{I}_{|U|}$
and
$\textbf{V}^{\mathrm{T}}\textbf{V}=\textbf{V}\textbf{V}^{\mathrm{T}}=\textbf{I}_{|I|}$.
Further, $\bm{\Sigma}_{f}=\begin{pmatrix}\bm{\Sigma}\\\
\textbf{0}\end{pmatrix}\in\mathbb{R}^{|U|\times|I|}$ and
$\textbf{U}_{f}=(\textbf{U}\ \textbf{U}_{\perp})\in\mathbb{R}^{|U|\times|U|}$
where $\bm{\Sigma}\in\mathbb{R}^{|I|\times|I|}$ is a diagonal matrix whose
diagonal entries are the singular values of X and
$\textbf{U}_{\perp}\in\mathbb{R}^{|U|\times(|U|-|I|)}$ denotes the matrix of
the bottom $|U|-|I|$ left singular vectors.

We can rewrite:

$\textbf{X}\textbf{X}^{\mathrm{T}}+\lambda\textbf{I}_{|U|}=\textbf{U}_{f}(\bm{\Sigma}_{f}\bm{\Sigma}_{f}^{\mathrm{T}}+\lambda\textbf{I}_{|U|})\textbf{U}_{f}^{\mathrm{T}}.$

Then, we have:

$\displaystyle\widehat{\textbf{B}}$ $\displaystyle=$
$\displaystyle\textbf{X}^{\mathrm{T}}(\textbf{X}\textbf{X}^{\mathrm{T}}+\lambda\textbf{I}_{|U|})^{-1}\textbf{X}$
(5) $\displaystyle=$
$\displaystyle\textbf{X}^{\mathrm{T}}(\textbf{U}_{f}(\bm{\Sigma}_{f}\bm{\Sigma}_{f}^{\mathrm{T}}+\lambda\textbf{I}_{|U|})^{-1}\textbf{U}_{f}^{\mathrm{T}})\textbf{X}$
$\displaystyle=$
$\displaystyle\textbf{V}\bm{\Sigma}_{f}^{\mathrm{T}}(\bm{\Sigma}_{f}\bm{\Sigma}_{f}^{\mathrm{T}}+\lambda\textbf{I}_{|U|})^{-1}\textbf{U}_{f}^{\mathrm{T}}\textbf{X}.$

Here, we rewrite:

$\displaystyle\bm{\Sigma}_{f}^{\mathrm{T}}(\bm{\Sigma}_{f}\bm{\Sigma}_{f}^{\mathrm{T}}+\lambda\textbf{I}_{|U|})^{-1}$
$\displaystyle=$ $\displaystyle(\bm{\Sigma}\
\textbf{0})\begin{pmatrix}(\bm{\Sigma}^{2}+\lambda\textbf{1}_{|I|})^{-1}&\textbf{0}\\\
\textbf{0}&\frac{1}{\lambda}\textbf{I}_{|U|-|I|}\end{pmatrix}$ (6)
$\displaystyle=$
$\displaystyle(\bm{\Sigma}(\bm{\Sigma}^{2}+\lambda\textbf{1}_{|I|})^{-1}\
\textbf{0})\in\mathbb{R}^{|I|\times|U|}.$

We combine Eqs.(5) and (6):

$\displaystyle\widehat{\textbf{B}}$ $\displaystyle=$
$\displaystyle\textbf{V}(\bm{\Sigma}(\bm{\Sigma}^{2}+\lambda\textbf{1}_{|I|})^{-1}\
\textbf{0})\textbf{U}_{f}^{\mathrm{T}}\textbf{X}$ $\displaystyle=$
$\displaystyle\textbf{V}(\bm{\Sigma}(\bm{\Sigma}^{2}+\lambda\textbf{1}_{|I|})^{-1}\
\textbf{0})(\textbf{U}\ \textbf{U}_{\perp})^{\mathrm{T}}\textbf{X}$
$\displaystyle=$
$\displaystyle\textbf{V}\bm{\Sigma}(\bm{\Sigma}^{2}+\lambda\textbf{1}_{|I|})^{-1}\textbf{U}^{\mathrm{T}}\textbf{X}$
$\displaystyle=$
$\displaystyle\textbf{V}\bm{\Sigma}(\bm{\Sigma}^{2}+\lambda\textbf{1}_{|I|})^{-1}(\bm{\Sigma}^{-1}\textbf{V}^{\mathrm{T}})(\textbf{V}\bm{\Sigma}\textbf{U}^{\mathrm{T}})\textbf{X}$
$\displaystyle=$
$\displaystyle\textbf{V}(\bm{\Sigma}^{2}+\lambda\textbf{1}_{|I|})^{-1}\textbf{V}^{\mathrm{T}}\textbf{X}^{\mathrm{T}}\textbf{X}$
$\displaystyle=$
$\displaystyle(\textbf{X}^{\mathrm{T}}\textbf{X}+\lambda\textbf{I}_{|I|})^{-1}\textbf{X}^{\mathrm{T}}\textbf{X}\
\ \ \ \ (=\text{Eq.(\ref{eq:solution1})}).$

This shows the equivalence of Eqs.(2) and (3) when $|U|>|I|$. ∎