# Analysis of Light Attenuation Length Measurement of a
High Quality Linear Alkylbenzene for the JUNO Experiment

Guojun Yu<EMAIL_ADDRESS>National Laboratory of Solid State
Microstructure, Nanjing University  School of Physics, Nanjing University,
Hankou Road 22, 210093, Nanjing, China  Jialiang Zhang  National Laboratory of
Solid State Microstructure, Nanjing University  School of Physics, Nanjing
University, Hankou Road 22, 210093, Nanjing, China  Shuo Li  National
Laboratory of Solid State Microstructure, Nanjing University  School of
Physics, Nanjing University, Hankou Road 22, 210093, Nanjing, China  Zifeng Xu
National Laboratory of Solid State Microstructure, Nanjing University  School
of Physics, Nanjing University, Hankou Road 22, 210093, Nanjing, China  Lei
Zhang  School of Physics, Nanjing University, Hankou Road 22, 210093, Nanjing,
China  Aizhong Huang  Jinling Petrochemical Corporation Ltd, 210033, Nanjing,
China  Ming Qi<EMAIL_ADDRESS>National Laboratory of Solid State
Microstructure, Nanjing University  School of Physics, Nanjing University,
Hankou Road 22, 210093, Nanjing, China

###### Abstract

Jiangmen Underground Neutrino Observatory (JUNO) is the next generation
neutrino experiment which aims at neutrino mass hierarchy problem along with
many other cutting-edge studies concerning neutrinos. Located $700\mathrm{m}$
underground in Jiangmen China, JUNO’s central detector is an acrylic sphere
filled with $20\mathrm{k}\mathrm{t}$ liquid scintillator with linear
alkylbenzene(LAB) as scintillator solvent. To ensure that an unprecedented
energy resolution of $\sigma_{E}/E\leqslant 3\%$ can be reached, LAB used in
JUNO must have excellent transparency at the wavelength ranging from
$350\mathrm{n}\mathrm{m}$ to $450\mathrm{n}\mathrm{m}$.

In the past decade much effort has been devoted to the development of high
transparency LAB based on the measurement of light attenuation length. Through
a close cooperation with Jingling Petrochemical Corporation in Nanjing,
transparency of LAB samples prepared for JUNO has been improved progressively.
However, this improvement is also pushing our apparatus towards approaching
its measuring limit, undermining the credibility of our measurement. In order
to get a result accurate and precise, an apparatus upgrading and a more
detailed error analysis is inevitable.

In this article, we present an analysis of how apparatus upgrading helps with
decreasing measuring errors, and we conducted measurements using the new
apparatus on several samples. A detailed error analysis is followed to
validate the results. We propose to apply statistical methods featuring Monte
Carlo simulation to estimate systematic uncertainties. Deviations caused by
fit models is also considered and the overall uncertainty is obtained by
combining two independent measurements. We finally report the light
attenuation length of a newly improved LAB sample to be $29.90\pm
0.95\mathrm{m}$, which gives a new high of all the preceding samples we
tested. This study may provide a strong evidence of JUNO’s feasibility to
reach its energy resolution.

##  I Introduction

Following the Daya Bay’s success of measuring neutrino oscillation parameter
$\theta_{13}$,[1, 2, 3, 4, 5] Jiangmen Underground Neutrino Observatory (JUNO)
is designed to tackle the neutrino mass hierarchy problem and many other
exciting frontiers of neutrino research.[6, 7, 8, 9] Located $700\mathrm{m}$
underground in Jiangmen with two nuclear power plants in Yangjiang and Taishan
both $53\mathrm{k}\mathrm{m}$ away, JUNO has a baseline that is optimal for
mass hierarchy experiments.[6, 7] JUNO’s central detector is an arcrylic
sphere with a diameter of $35.4\mathrm{m}$ immersed in a cylinder water tank.
Roughly $18,000$ microchannel plate photomultiplier tubes(PMT) and more than
$20,000$ other PMTs are equipped surrounding the sphere to detect and magnify
weak $\gamma$ signals produced via inverse beta decay
$\mathrm{bar}{\nu}_{e}+p\rightarrow e^{+}+n$ process.[6, 7] To be able to
manifest the extremely small difference of different mass hierarchy models,
JUNO’s energy resolution should at least be at a level of
$\sigma_{E}/E\leqslant 3\%$ at $1\mathrm{M}\mathrm{eV}$, which amounts to a
photon yield of $1,100/\mathrm{M}\mathrm{eV}$.

As a consequence, optical transparency of liquid scintillator(LS) plays a key
role in reaching such a high standard. JUNO uses an organic liquid
scintillator with linear alkylbenzene(LAB) as solvent,
PPO(2,5-diphenyloxazole) as fluor and bis-MSB(1,4-bis(2-methylstyryl)-benzene)
as wavelength shifter. LAB as solvent constitutes the majority of
scintillator’s mass so it dominates the light attenuation performance. LAB is
an ideal organic target material. It is a hydrocarbon-rich compound with a
high quantum yield, high flash point and many other outstanding features such
as low toxicity, biological degradability, and relatively low price.
Unfortunately, industrially produced LAB usually contains various impurities
which greatly weaken its light penetrability. Thus, it cannot be emphasized
how crucial it is to develop an LAB sample with a high transparency quality.

Numerous experiments as well as theoretical studies have been put into
investigating light attenuation in fluid. Some focus on directly measuring
light attenuation length,[10, 11, 12, 13, 14] which is a comprehensive
manifestation of absorption, scattering and even re-emission.[15, 16, 17] Some
focus on scattering process in specific, as a consequence of the fact that
absorption and re-emission is weak in LAB and it is Rayleigh scattering that
dominates the process.[18, 19] Thus it would suffice to measure Rayleigh
scattering length and use it as an indication of material’s transparency.[18,
19, 20, 21] Note that since Rayleigh scattering is an elastic process it
simply turns photons into different directions, making no contributions to
decrease of transparency. Thus a lower scattering length is preferred.[18] The
experiment presented in this article belongs to the former kind.

Through a close cooperation of more than 10 years with Jingling Petrochemical
Corporation in Nanjing, LAB samples has been improved to show an increase of
attenuatioan length from $14\mathrm{m}$ to $25\mathrm{m}$.[10, 12] Going
further, in this article we report a result of $29.90\pm 0.95\mathrm{m}$
attenuation length of a newly improved LAB sample obtained after an apparatus
upgrade. A detailed error analysis featuring Monte Carlo method is also given
to validate this result.

##  II Experiment

###  II.1 Attenuation length

Light attenuation length $L$ of some substance is defined as the longest
length of a beam can travel in this substance until its intensity drops to
$1/\mathrm{e}$ of the incident. The basic method of measuring the attenuation
length of a liquid substance is fitting intensity data collected at different
liquid levels to the Beer–Lambert law:[22]

$I=I_{0}\mathrm{e}^{-\frac{x}{L}}$ (1)

where $I_{0}$ is the incident intensity, $x$ is the path beam travels.

Obviously if a set of liquid levels and intensity data is provided, $L$ can be
obtained by fitting. Designed based on this, our apparatus comprises two major
parts: optical measuring system and data acquisition system(DAQ). As is shown
in figure 1, the optical measuring system is all contained in a dark closet.
At the top of the closet an LED source with a $430\mathrm{n}\mathrm{m}$ peak
wavelength is installed. Aligned below is an adjustable lens that parallelizes
beam, a plano-convex spherical lens with focus $0.15\mathrm{m}$, a diaphragm,
a one-meter-tube used as a sample container, and a PMT(CR135 from Hamamatsu)
that detects light intensity and outputs electronic signals for later
processing. Note that attenuation length varies with incident light with
different wavelengths so that LED source must have an appropriate wavelength
range. In JUNO experiment, photons arriving at PMTs will be at a wavelength of
roughly $430\mathrm{n}\mathrm{m}$ so this is set to be the LED’s peak
wavelength. At the bottom side of the tube there is a conduit in which a
pressure transmitter and an electromagnetic valve are plugged. By controlling
the valve, sample in the tube can be set to different liquid levels. The
intensity collected by PMT and the level data provided by the transmitter form
a dataset to be fit to finally get the attenuation length.

Figure 1: The whole experiment apparatus is placed in a dark room with
electromagnetism screening to make sure its stability

However, considering the randomness in evaluation, a single measurement of
intensity is far from enough. Through a PCI board, the LED source is
controlled to pulse with an $800\mathrm{Hz}$ on-off signal generated by a
signal generator so that in one minute a dataset of $48,000$ data points are
generated. A trigger signal at the same frequency is also generated to
accommodate the DAQ. The original trigger will be processed and sent as a
square wave to a charge-to-amplitude converter(V965 from CAEN). Finally, the
analog intensity signal will be transformed to a $12$ bit digital ADC channel
and then saved. Eventually around $19,000$ data can be recorded in one minute
due to the memory limit and the intensity is determined to be the mean of
these $19,000$ data through a Gaussian fit.

Inasmuch as the analog signal is processed by an ADC, it is not the absolute
intensity but a digital ADC channel number ranging from $0$ to $2^{12}$ that
is obtained. However, since the value of ADC channel is proved to be linearly
dependent on intensity with an intercept resulted from electronical
pedestal(abbreviated as ped below), fitting ADC channel leads to no difference
in the final result than that obtained by fitting absolute intensity directly.
Thus, for the balance of this article $I$ will be used to denote ADC channel
but still be referred to as intensity just for later convenience.

###  II.2 Preparations

Before the experiment, a full check on the equipment is always indispensable.
It must be ensured that one-meter-tube is not contaminated, optical system is
tuned to its best state, and every electronical device works properly.
Apparatus calibration is done by performing a real measuring using branded
purified water as sample. Taking advantage of the stability of purified water
production, deviations of apparatus’ performance can be detected.

The stability of lab environment is important as well. Figure 2 shows a
$7\mathrm{h}$ surveillance of lab temperature and humidity with AC turned on
using a thermohygrometer, which has a resolution of
$0.01\mathrm{\SIUnitSymbolCelsius}$ and $0.024\%$RH. It can be seen that under
such circumstances the fluctuation is small. Though it is impossible to keep
temperature and humidity a constant, it is always possible to perform
experiment in a relatively short time so that the fluctuations of the
environment is of little importance.

(a) Temperature and relative humidity in the lab from 17:00 to 24:00. The
maximal difference of temperature and that of humidity are respectively
$0.8\mathrm{\SIUnitSymbolCelsius}$ and $1\%$.

(b) Intensity in the same time period. Fluctuation is relatively small when
temperature and humidity is relatively stable.

Figure 2: Environment’s influence on intensity.

To guarantee the reliability of the experiment, it would only start after all
the preparations are done. In this article 4 batches of newly developed LAB
samples tagged as NJ64, NJ66, NJ67, and NJ68 are tested. Besides, a previously
tested batch NJ44 is also used as a cross-check.

##  III Apparatus Upgrading

###  III.1 Necessity of upgrading

For JUNO the longer attenuation length observed in some new sample the better.
However, this increasing of attenuation length also creates a challenge for
the measurement and pushing our apparatus towards its measuring limit,
affecting both accuracy and precision of the result. The apparatus is claimed
to reach its measuring limit in the following sense: collect a dataset and
graph it, if there are two neighboring points coming too close to each other
that their error bars overlap in the ordinate, it is believed that
statistically they cannot be clearly distinguished from each other since this
introduces ambiguity and makes the fitting result unreliable. To show this
quantitatively, suppose $n$ data points $\\{(x_{i},I_{i})\\}$ are collected in
a measurement, we define the relative intensity resolution $\Delta I/I_{0}$ of
a fit model $I=f(x;I_{0})$ as the minimal relative difference $\Delta
I/I_{0}=\displaystyle\min_{|i-j|=1}|I_{i}-I_{j}|$ of two neighboring points
$(x_{i},I_{i})$ and $(x_{j},I_{j})$. In our case the fit model is the
Beer–Lambert law (1), and since $\Delta x=|x_{i}-x_{j}|$ is two orders of
magnitude smaller than $L$, it is natural to write

$\frac{\Delta
I}{I_{0}}=\left|\mathrm{e}^{-\frac{x_{i}}{L}}-\mathrm{e}^{-\frac{x_{j}}{L}}\right|\approx\frac{\Delta
x}{L}$ (2)

In order to ensure that each data point is statistically distinguishable from
the others, it is required that

$\Delta I\geqslant 2\varepsilon_{I}$ (3)

where $\varepsilon_{I}$ is the uncertainty of intensity. This gives the
maximal attenuation length the apparatus can give with accuracy and precision
to some extent

$L_{\max}\approx\frac{\max I_{0}}{2\varepsilon_{I}}\Delta x$ (4)

Note that this limit can always be avoided by simply extending the tube. So it
has to be pointed out that this analysis only works when an extension is
impossible and reduction of the number of data points is unacceptable, as is
our case.

Usually intensity data will be collected at $10$ different levels uniformly
distributed in a distance of $1\mathrm{m}$ so $\Delta x=0.1\mathrm{m}$. In
order to get the uncertainty $\varepsilon_{I}$ and the maximal
$I_{\max}=\max\,I_{0}$ in theory, a stability test involving large number of
repeated measurements at some specific liquid level has been conducted.
Histograms in figure 5(a) and 5(c) show the distribution of estimated means
and standard deviations of roughly 200 repeating measurements at the same
liquid level. The standard deviation of the mean values of intensity speaks
for $\varepsilon_{I}$, and the mean of the standard deviations of intensity
gives information about $I_{\max}$. Fit results show that $I_{\max}=2,900$ and
$\varepsilon_{I}=4.5$. Note that $\varepsilon_{I}$ provided here considers
only the fluctuations of intensity itself. In reality $\varepsilon_{I}$ would
be slightly larger because of other errors such as error from fitting. Here a
conservative estimation setting $\varepsilon_{I}=5$ is applied. With all
parameters provided, the extreme attenuation length under the present
circumstance is roughly

$L_{\max}\approx 29\mathrm{m}$ (5)

From equation (4) it can be easily seen that $L_{\max}$ increases when
$I_{0}$, $\Delta x$ increases or $\varepsilon_{I}$ decreases. Since the total
length of the tube cannot be altered, increasing $\Delta x$ is tantamount to
lowering the number of data point $n$, which would drastically undermines
goodness-of-fit. Thus the only two choices left are raising $I_{0}$ and
reducing $\varepsilon_{I}$. Next we investigate two upgrades that both
increase $I_{0}$ and decrease $\varepsilon_{I}$ at the same time. With these
two upgrades not only would the maximal attenuation length $L_{\max}$ this
apparatus can measure be lifted up but also the uncertainties would be
lowered.

###  III.2 Collimatoed fibre-coupled source

In the original apparatus the light source is simply an LED mounted at the top
of the closet. The half divergence angle is up to
$6\mathrm{\SIUnitSymbolDegree}$ and a diaphragm is needed to confine and block
stray light. Considering this along with other causes such as unavoidable
aberrations resulted from spherical lens, it is nearly impossible for light
beam to propagate from the LED source to PMT in a straight line. This
undermines the validity of fit using the ideal Beer–Lambert law, and at the
same time enhances the fluctuations of intensity and brings down the $I_{0}$.
Thus, we come up with a new collimated fibre-coupled LED source design that
aims at concentrating light and keep it non-divergent as much as possible.

(a) Fiber held by a mount which can be conveniently adjusted by the screws,
helping to align the optical path.

(b) A comparison of two designs. The new collimated fibre source renders
diaphragm unnecessary.

Figure 3: The newly designed fibre-coupled LED source.

In the new apparatus, LED is coupled to an optical fibre. When it comes to
curtailing divergence angle, usually the smaller the numerical aperture(NA) of
a fibre the better, but a too small NA means a too small fibre diameter,
leading to a relatively low light flux. Bound by this trade-off, we tested
various optical fibres and decided to use a multi-mode fibre with an NA of
$0.22$ and a diameter of $300\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ for an
optimal performance. A tailor-made collimator is attached to the other end of
the fibre. The collimator is held by a three-dimensionally adjustable mount as
is shown in figure 3(a). A comparison diagram of the source before and after
is drawn in 3(b).

The direct effects of this optical upgrade is that the size of the spot on
PMT’s receiving window is shrunk tremendously, as is shown in figure 4. A
smaller light spot indicates a more parallel beam, hence less scattered light
which gives more validation of applying Beer–Lambert law and save lots of
effort for fine tuning optical system into an ideal condition. A smaller light
spot also means more concentrated light energy, which allows a higher $I_{0}$,
and less perturbations, which reduces $\varepsilon_{I}$.

(a) before

(b) after

Figure 4: Spot size( c m ) before and after the upgrade. Note that in figure
4(a) only the central image contributes to intensity value, the outer ring
caused by diaphragm is excluded in real measurement.

###  III.3 PMT upgrading

An obvious reason that $I_{0}$ cannot get higher lies in the fact that DAQ
device has a limit in the maximal intensity it can measure. At each liquid
level intensity $I$ observes a normal distribution with a mean of $\mu$ and
standard deviation $\sigma$, which are estimated from $19,000$ ADC data. A fit
with less than $1$ in $19,000$ data is lost in an estimation indicates that
every data within the range of $\mu\pm 4\sigma(P(|x-\mu|\leqslant
4\sigma)=99.99266\%)$ must be included ($1-99.99266\%\approx 1/15787$). Thus
to avoid any possible bias because of loss in data it is required that
$I_{\max}\leqslant 4096-4\sigma-\text{ped}$. For our apparatus
$\text{ped}=273$. Thus we have $I_{\max}=3823-4\sigma$.

PMT plays a key role in collecting ADC channel data and its performance
determines how well ADC data’s distribution can be. A better PMT will have a
faster time response, finer time resolution, and a more sensitive and uniform
cathode, which means a higher robustness against different perturbations that
causes ADC channels to show a wider spread. Naturally this would decrease the
distribution’s $\sigma$ so it increases $I_{\max}$. Moreover, a smaller
$\sigma$ also means a smaller $\varepsilon_{I}$.

Table 1: Key parameters of CR135 and R7724

| spectral response($\mathrm{n}\mathrm{m}$) |  | supply voltage ($\mathrm{V}$) |  | luminous ($\mathrm{\SIUnitSymbolMicro}\mathrm{A}/\textrm{lm}$)
---|---|---|---|---|---
type | range | maximum | rise time($\mathrm{n}\mathrm{s}$) | operational | maximal | dark current($\mathrm{n}\mathrm{A}$) | cathode | anode
CR135 | 300$\sim$650 | 420 | 7 | 1,250 | 1,500 | 30 | 60 | 25$\times 10^{8}$
R7724 | 300$\sim$650 | 420 | 2.1 | 1,750 | 2,000 | 6 | 90 | $3\times 10^{8}$

To achieve this we replace the old-fashioned CR135 from Hamamatsu to R7724
with a faster time response and lower noise. Both PMT’s peak cathode
sensitivity is arrived at $430\mathrm{n}\mathrm{m}$. Other parameters are
shown in Table 1. A new voltage divider is also designed to accommodate the
new PMT. This voltage divider applies an iterative voltage distribution of
resistors with $4$ decoupling capacitors in series so that it has the biggest
gain and shows the best performance when conducted in counting mode. A total
resistance of $3.5\mathrm{M}\mathrm{\SIUnitSymbolOhm}$ gives a current of
$500\mathrm{\SIUnitSymbolMicro}\mathrm{A}$ operated under a voltage of
$1,750\mathrm{V}$.

(a) CR135 without fibre and collimator

(b) R7724 with fibre and collimator

(c) CR135 without fibre and collimator

(d) R7724 with fibre and collimator

Figure 5: Distributions of mean values(upper) and standard deviations(lower)
provided by Guassian fit of intensities. 5(a) and 5(c) are tested using CR135
and no fibre and collimator. 5(b) and 5(d) are tested using R7724 with the new
collimated fibre-coupled source.

Similarly, a stability test has been performed with every parameter fixed
except for PMT and light source. Histogram in figure 5(b) clearly shows a
smaller variance of intensity than in figure 5(a). Fit results present that
$\varepsilon_{I}\approx 1.5$ and $\sigma\approx 141$, which means
$I_{\max}=3,200$. Again a more conservative estimation is adopted by setting
$\varepsilon_{I}=2$. with these and $\Delta x=0.1\mathrm{m}$ all inserted into
equation (4), the limit attenuation length under the present circumstance is

$L_{\max}\approx 80\mathrm{m}$ (6)

After upgrading $L_{\max}$ is lengthened high enough that little influence is
exerted on the accuracy of the measurement. The following section would show
how these upgrades also improve the precision of the measurement.

##  IV Results and Error Analysis

With the aid of upgraded apparatus, 4 batches of newly developed LAB samples,
namely NJ64, NJ66, NJ67 and NJ68 have been measured. Each sample differs from
each other in respect of impurities. To prove the robustness of the
experiment, an old batch of NJ44 has been re-measured as well. Figure 6
summarizes results for each sample.

(a) NJ64

(b) NJ66 test 1

(c) NJ66 test 2

(d) NJ67

(e) NJ68

(f) NJ44

Figure 6: Fit results of LAB samples. Reduced chi-square $\chi_{\nu}^{2}$ is
chi-square over degrees of freedom. Note that in fit plots error bars are too
small to be seen but can be seen in residual plots.

Compared to the best results we get before,[12] it is implied in the figures
that every new sample evidently shows a better transparency, especially NJ66,
whose attenuation length in the test 1 exceeds $30\mathrm{m}$ for the first
time. Re-measurement using the former NJ44 sample seems to further mutually
confirms the reliability of the results. Meanwhile, the residuals appear to
show a pattern of high ends and low abdomen. To figure out how this phenomenon
affects the result a physical analysis of the fit model along with a detailed
error analysis is also given.

Researches of attenuation length measurement usually report errors of the
Beer–Lambert fit as the final result without further dissecting it into
systematic and statistical errors. Errors from the abscissa are rarely
considered either. To tackle these problems, the error analysis in this
article pays more attention to statistical methods such as Monte Carlo
simulation, rendering the results more informative.

The uncertainty of attenuation length $\varepsilon_{L}$ in our experiment
comprises three parts: systematic uncertainty $\varepsilon_{L}^{\text{sys}}$,
statistical uncertainty $\varepsilon_{L}^{\text{sta}}$, and model uncertainty
$\varepsilon_{L}^{\text{model}}$. Systematic and statistical uncertainty will
both be reflected in the fit error if systematic errors of intensity and
liquid level are contained in the error bars when fitting. In the following
section uncertainties of intensity and liquid level will be examined
separately in detail first and then combined in the Monte Carlo simulation to
give the systematic uncertainty. Model uncertainty is introduced to account
for the residual pattern and will be discussed later.

###  IV.1 Uncertainties of intensity

Obtained by fitting $19,000$ data to the Gaussian model, the uncertainty of
intensity $\varepsilon_{I}$ has two sources: statistical
$\varepsilon_{I}^{\text{sta}}$ and systematic $\varepsilon_{I}^{\text{sys}}$.
$\varepsilon_{I}^{\text{sta}}$ is given by Gaussian fitting straight away
while $\varepsilon_{I}^{\text{sys}}$ has to be determined by experiment since
it results from nuisance parameters reflecting effects of some inevitable
interferences, such as fluctuation of LED’s emission power, oscillation of
building, stray light background, and performance of electronical device, etc.
These factors can all be taken into consideration by collecting intensity data
at some fixed level $x_{i}$ in a fairly long testing cycle. In figure 5, the
two upper plots present distributions of mean and standard deviation of
intensities in NJ66 sample at some liquid level. It is clearly manifested in
the figures that after the upgrades that $\varepsilon_{I}^{\text{sys}}$ drops
drastically, which means precision of $I_{i}$ gets better. Note that in
principle this measurement should be done for every single sample at every
liquid level. Considering the obvious inconvenience and bother this test has
only been done once for each of the two representative samples NJ66 and
purified water. The test confirms that $\varepsilon_{I}^{\text{sys}}$ varies
little with liquid levels or samples and is a good indication of systematic
influence. The uncertainty of intensity at level $x_{i}$ is denoted as

$\varepsilon_{I}(x_{i})=\sqrt{\varepsilon_{I}^{\text{sys}}(x_{i})^{2}+\varepsilon_{I}^{\text{sta}}(x_{i})^{2}}$
(7)

###  IV.2 Uncertainties of liquid levels

Liquid levels are calculated from voltage data provided by a pressure
transmitter. The transmitter responses with hydrostatic pressure $p$, which is
given by the Bernoulli’s equation

$\frac{p}{\gamma}+\frac{v^{2}}{2g}+z=0$ (8)

where $z$ is depth and $x=-z+h_{0}$ with $h_{0}$ the extra length below the
conduit and $\gamma=\rho g$ with $\rho$ the mass density of liquid and $g$
gravity acceleration. For the hydrostatic case here liquid velocity $v=0$.

Figure 7: Linearity test of pressure transmitter. Two methods both give a
$\chi^{2}_{\nu}<2$, thus the linearity is proved.

Usually a qualified transmitter’s response should be linear. If it is not,
substantial errors would appear. We tested the linearity of the transmitter by
adding and draining water in the tube successively and fit its responses to a
straight line. Fit results in figure 7 shows estimated intercepts of both
tests are close to $0$ and $\chi^{2}_{\nu}$ close to $1$, indicating that
linearity of the transmitter is guaranteed. Thus the error of liquid level
$\varepsilon_{x}$ is simply

$\varepsilon_{x}=\frac{1}{\gamma}\varepsilon_{p}$ (9)

PCM300’s data sheet reports a relative error $0.5\%$FS, so that
$\varepsilon_{x}=0.5\%x$.

###  IV.3 Monte Carlo simulation

In principle, if errors exist for both coordinates, the function to be
minimized is generalized to be a weighted sum of residuals’ squares of both
variables.[23] Under some circumstances this sum does not even have a chi-
square distribution, introducing so much trouble in specifying confidence
limits. Fortunately, taking advantage of the approximate linear behavior of
the exponential model when the argument $\Delta x/L$ is small, the chi-square
can be written as[24]

$\chi^{2}=\sum_{i=1}^{n}\frac{\left(I_{i}-f(x_{i};L,I_{0})\right)^{2}}{\varepsilon_{I_{i}}^{2}+k_{i}^{2}\varepsilon_{x_{i}}^{2}}$
(10)

where $k_{i}=-I_{0}/L\mathrm{e}^{-x_{i}/L}$ is the slope of the model at
$x_{i}$. Though simplified, this chi-square is still complicated to optimize
due to the non-linear dependence of $\chi^{2}$ on parameters $L$.

A Monte Carlo simulation is introduced to overcome this problem. In one
measurement a dataset of several liquid levels, intensities and corresponding
uncertainties $\\{(x_{i},I_{i},\varepsilon_{I}(x_{i})\\}$ is obtained and
gives only one result of $L$. In order to exploit the most of this dataset, we
generate a new dataset
$\\{(x_{i},I^{\prime}_{i},\varepsilon_{I^{\prime}}(x_{i})\\}$ from the
original data by sampling from a set of Gaussian distributions
$I^{\prime}_{i}\sim\mathcal{N}(I_{i},\varepsilon_{I}^{\prime}(x_{i}))$ for
each $x_{i}$. Here
$\varepsilon_{I^{\prime}}(x_{i})=\sqrt{\varepsilon_{I_{i}}^{2}+k_{i}^{2}\varepsilon_{x_{i}}^{2}}$.
The probability density function of $I^{\prime}_{i}$ is thus

$p(y)=\frac{1}{\sqrt{2\pi}\varepsilon_{I^{\prime}}(x_{i})}\mathrm{e}^{-\frac{(y-I_{i})^{2}}{2\varepsilon^{2}_{I^{\prime}}(x_{i})}}$
(11)

Fitting this generated dataset provides a new estimated $L$. Repeat this
process enough times so a substantial number of $L$ is obtained. The central
limit theorem guarantees that this distribution is almost Gaussian. Thus it is
reasonable to fit this Gaussian distribution and claim the standard deviation
as an estimation of the systematic uncertainty $\varepsilon_{L}^{\text{sys}}$.

Now that $\varepsilon_{I_{i}}$ and $\varepsilon_{x_{i}}$ are known,
$\varepsilon_{L}^{\text{sys}}$ can be determined. Figure 8 shows results of
Monte Carlo simulations with a sample size of $100,000$. Figure 8(a) and 8(b)
are distributions of NJ44 MC simulated data before and after upgrades. A
reduce in the standard deviation after the upgrade is impossible to ignore.
Figure 8(c) and 8(d) are distributions of two measurements of NJ66. A
convergence test has also been conducted for NJ66 after upgrades, showing the
limit of $\varepsilon_{L}^{\text{sys}}$ in figure 9. Here we claim the
systematic uncertainty of each measurement of NJ66 to be $0.66\mathrm{m}$ and
$0.59\mathrm{m}$ respectively. The overall uncertainty of both systematic and
statistical uncertainty is shown in figure 6 so the statistical uncertainty
can be calculated through the square sum relation of errors. We claim the
statistical uncertainty of each measurement of NJ66 to be $1.28\mathrm{m}$ and
$0.63\mathrm{m}$ respectively.

(a) NJ44 before upgrades

(b) NJ44 after upgrade

(c) NJ66 test 1

(d) NJ66 test 2

Figure 8: Distributions of $L$ simulated by Monte Carlo method. The upper two
plots presents a comparison of systematic uncertainties of NJ44 before and
after the upgrades. The lower two are simulations run for two independent
tests of the same sample NJ66.

(a) NJ44 before upgrades

(b) NJ66 after upgrades

Figure 9: The standard deviation of simulated $L$ converges when the number of
$L$ is large enough. Convergence limit has been set to be the average of the
last 50 values.

###  IV.4 Errors of modeling

In light of complex interactions of photons and sample substance, Beer–Lambert
law is only valid when some strict conditions are met. Any imperfection of
apparatus would result deviations from the ideal exponential law. This
inevitable deviation becomes more significant when other errors get suppressed
through the apparatus upgrade. In this experiment, the fact that residuals
show a common pattern and reduced chi-squares of fit is higher than usuals
clearly demonstrates the existence of higher order terms, implying a deviation
from the fit model. As is mentioned earlier the absorption in LAB is mainly
caused by impurities and it is Rayleigh scattering that dominates the
attenuation process,[16, 17] we propose a modified model giving special
concern on the Rayleigh scattering.

Rayleigh scattering, as is first proposed in 1871 by Lord Rayleigh,[25, 26,
27, 28] is an elastic process, which means it only changes photons’ direction
without changing its frequency. Thus it is natural to assume that the high on
both ends shown in residual plots originates from the downward part of
scattering light that can be received by the PMT. To testify this hypotheses
let us trace back to the grounding assumption of deriving Beer–Lambert law
that for a thin layer the ratio of the attenuated to the incident is a
constant $\mu_{\text{att}}$, which is exactly the inverse of attenuation
length $\mu_{\text{att}}=1/L$. We correct this by simply adding a term
$\mu_{\text{mod}}$ representing the not lost part of the scattering light,

$\frac{1}{I}\frac{\mathrm{d}I}{\mathrm{d}s}=-\mu_{\text{att}}+\mu_{\text{mod}}$
(12)

where $I$ is intensity and $s$ is the length light already traveled in sample.
$\mu_{\text{mod}}$ is determined by the geometry of the tube shown in figure
10. For tubes in a cylindrical shape as used in this experiment, only the part
of scattering light whose scattering angle is within the solid angle subtended
by the tube’s bottom end window can be received. Thus, the not lost part takes
a proportion of

$\eta(s)=\int_{0}^{\alpha(s)}\mathrm{d}\theta\int_{0}^{2\pi}\mathrm{d}\varphi\,f(\lambda,\theta,r)\,r^{2}\sin\theta$
(13)

in which $\varphi,\,\theta,\,r$ is local spherical coordinate, $l$ the length
from scattering vertex to the bottom end of the tube and $s=x-l$ with $x$
being the liquid level, $\alpha(s)=\arctan(R/l)$ is the corresponding maximal
azimuthal angle. $R$ the radius of the bottom window. Integrand
$f(\lambda,\theta,r)$ is the Rayleigh formula for unpolarized incident
beams[29]

$f(\lambda,\theta,r)=\frac{2\pi^{2}(n-1)^{2}}{N\lambda^{4}r^{2}}(1+\cos^{2}\theta)$
(14)

where $n$ is the refractive index, $N$ the number density of the sample, $r$
the distance from the scattering vertex and $\lambda$ the wavelength of the
incident beam. Recently some researches shows that organic liquid such as LAB
is not isotropic and depolarized part also contributes to the scattering.[21,
18, 19, 14] Thus, a depolarized parameter $\rho$ should be introduced. J.
Cabannes and L.V. King generalized Lord Rayleigh’s formula to the anisotropic
molecules of gases and liquids,[30, 31] where

$f(\lambda,\theta,r)=\frac{\pi^{2}(n-1)^{2}}{2N\lambda^{4}r^{2}}\cdot\frac{6+6\rho}{6-7\rho}\left(1+\frac{1-\rho}{1+\rho}\cos^{2}\theta\right)$
(15)

Here $b=\pi^{2}(n-1)^{2}/(2N\lambda^{4})$ is introduced to function as the new
fit parameter and $\rho=0.31\pm 0.02$ is adopted in the new model.[21]

Figure 10: Geometry of light passing through sample in the one-metre-tube. The
blue cone shows the solid angle in which the scattering light can be received.

This part of scattering light attenuates as well going through rest of the
sample, an damping term is inevitable.

$\mu_{\text{mod}}(s)=\mathrm{e}^{-\mu_{\text{att}}(x-s)}\eta(s)$ (16)

To simplify the tedious integration the exponential is firstly expanded and
then cut off. Since higher order terms in the expansion stands for the
strongly attenuated part it has little impact on fit and the first two terms
would suffice to show the physics. The corrected model is finally obtained and
denoted as

$I(x)=I_{0}\mathrm{e}^{-\mu_{\text{att}}x+\int_{0}^{x}\mu_{\text{mod}}(s)\,\mathrm{d}s}$
(17)

This new model is fit to the same datasets of samples like the old model. Here
results of two independent tests of NJ66 are presented in figure 11. As is
drawn from the plots, the $\varepsilon_{L}^{\text{model}}$ for these two
measurements are respectively $1.43\mathrm{m}$ and $0.65\mathrm{m}$.
Quantitatively, $\chi^{2}_{\nu}$ decreasing to near $1$ and $p$ value close to
$0.5$ imply a better goodness-of-fit of the new model than that of
Beer–Lambert law which works only in an ideal situation. However, it has to be
pointed out that for residuals there still seems to be a pattern when
$x>0.4\mathrm{m}$ and the seemingly soaring behavior near $x\rightarrow
0\mathrm{m}$ awaits for deeper scrutiny as well. This may be compensated by
involving more terms when cutting off the exponential but may also indicate
some other factors not taken into account in this physical analysis such as
contributions of diffuse reflection and integrand’s dependence on the
wavelength.

(a) NJ66 test 1

(b) NJ66 test 2

Figure 11: Fits using the new model. Reduced chi-square $\chi_{\nu}^{2}$ has
seen a decrease to close to 1 and most of the data points are located to the
model expectations within 1 error bar.

###  IV.5 Combining

Since we have conducted two independent measurements of NJ66 sample, a
combined result considering both measurements can be given. The two
independent results are presented as follows:

$\begin{split}&L_{1}=30.73\pm 0.66^{\text{sys}}\pm 1.28^{\text{sta}}\pm
1.43^{\text{model}}\mathrm{m}\\\ &L_{2}=29.66\pm 0.59^{\text{sys}}\pm
0.63^{\text{sta}}\pm 0.65^{\text{model}}\mathrm{m}\\\ \end{split}$ (18)

And the the combined result is given as the weighted sum

$L=\frac{L_{1}/\varepsilon_{1}^{2}+L_{2}/\varepsilon_{2}^{2}}{1/\varepsilon_{1}^{2}+1/\varepsilon_{2}^{2}}=29.90\mathrm{m}$
(19)

with an uncertainty of

$\varepsilon_{L}=\frac{1}{\sqrt{1/\varepsilon_{1}^{2}+1/\varepsilon_{2}^{2}}}=0.95\mathrm{m}$
(20)

Finally, the light attenuation length of sample NJ66 is reported to be
$29.90\pm 0.95\mathrm{m}$.

##  V Conclusions

In this article the attenuation length of 4 batches of LAB samples has been
measured and analysed in details. It is explained why an upgrade of apparatus
is inevitable at present. The collimated fibre coupled light source that
shrinks the size of spot together with a PMT with better time performance
increases measuring limit of attenuation length and decreases systematic error
of intensity to a large extent. Fit data show that after the upgrade the
apparatus has received a substantial improvement in both accuracy and
precision. We also proposed an elaborated uncertainty approximation method
featuring Monte Carlo simulation to estimate systematic uncertainty, saving us
from complicated calculations. By proposing a new model, deviations of
theoretical models caused by Rayleigh scattering has also been considered. We
eventually report a result of an attenuation length of the most updated LAB
sample NJ66 to be $29.90\pm 0.95\mathrm{m}$. We hope that not only these
results help to give more confidence on JUNO reaching its energy resolution
but also the statistical method and the theoretical model exhibited in this
article could be inspiring for similar experiments.

###### Acknowledgements.

This work was supported by the National 973 Project Foundation of the Ministry
of Science and Technology of China (Contract No.2013CB834300), the Strategic
Pilot Science and Technology Project of the CAS (Contract No.XDA10010000), and
the National Natural Science Foundation of China (Contract No.11620101004).

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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# When the Music Stops: Tip-of-the-Tongue Retrieval for Music

Samarth Bhargav  University of AmsterdamAmsterdamNetherlands<EMAIL_ADDRESS>0000-0001-5204-8514 ,  Anne Schuth  SpotifyAmsterdamNetherlands
<EMAIL_ADDRESS>0000-0002-5841-2134 and  Claudia Hauff
SpotifyAmsterdamNetherlands<EMAIL_ADDRESS>0000-0001-9879-6470

(2023)

###### Abstract.

We present a study of _ToT_ retrieval for music, where a searcher is trying to
find an existing music entity, but is unable to succeed as they cannot
accurately recall important identifying information. tip-of-the-tongue (ToT)
information needs are characterized by complexity, verbosity, uncertainty, and
possible false memories. We make four contributions. _(1)_ We collect a
dataset—ToTMusic—of 2,278 information needs and ground truth answers. _(2)_ We
introduce a schema for these information needs and show that they often
involve multiple modalities encompassing several Music IR sub-tasks such as
lyric search, audio-based search, audio fingerprinting, and text search. _(3)_
We underscore the difficulty of this task by benchmarking a standard text
retrieval approach on this dataset. _(4)_ We investigate the efficacy of query
reformulations generated by a large language model (LLM), and show that they
are not as effective as simply employing the entire information need as a
query–leaving several open questions for future research.

Music Retrieval; Tip-of-the-Tongue Retrieval; Cross Modal Retrieval

††copyright: acmcopyright††journalyear: 2023††copyright:
rightsretained††conference: Proceedings of the 46th International ACM SIGIR
Conference on Research and Development in Information Retrieval; July 23–27,
2023; Taipei, Taiwan††booktitle: Proceedings of the 46th International ACM
SIGIR Conference on Research and Development in Information Retrieval (SIGIR
’23), July 23–27, 2023, Taipei, Taiwan††doi: 10.1145/3539618.3592086††isbn:
978-1-4503-9408-6/23/07††submissionid: 123-A56-BU3††ccs: Information systems
Music retrieval††ccs: Information systems Retrieval models and ranking††ccs:
Human-centered computing User studies††ccs: Information systems Document
filtering††ccs: Information systems Multimedia and multimodal retrieval††ccs:
Information systems Query reformulation

##  1\. Introduction

The _ToT_ retrieval task involves identifying a previously encountered item
for which a searcher was unable to recall a reliable identifier. ToT
information needs are characterized by verbosity, use of hedging language, and
false memories, making retrieval challenging (Arguello et al., 2021; Bhargav
et al., 2022). As a consequence, searchers resort to communities like
r/TipOfMyTongue and WatzatSong, where they can post descriptions of items that
they know exist but cannot find, relying on other users for help. Recent
research of ToT information needs explored how searchers pose these requests
in specific domains like movies (Bhargav et al., 2022; Arguello et al., 2021),
or games (Jørgensen and Bogers, 2020). Music-ToT, however, is under-explored
despite being frequent: it represents 18% of all posts made in a five-year
period in the r/TipOfMyTongue community (cf. section 3.1). Our work is
motivated by the need to understand how such requests are expressed in the
music domain.

We examined the r/TipOfMyTongue community, focusing on requests looking for
musical entities like albums, artists or songs. We show that these requests
often refer to multiple modalities (cf. section 4) and thus encompass a broad
set of retrieval tasks—audio fingerprinting, audio-as-a-query, lyric search,
etc. In our work, we focus on song search. We create ToTMusic111ToTMusic
(along with annotations) will be made available here:
https://github.com/spotify-research/tot: the dataset consists of 2,278
_solved_ information needs pertaining to a song, each of which is linked to
the corresponding correct answer in the publicly available Wasabi Corpus
(Buffa et al., 2021). Using ToTMusic, we develop a schema for Music-ToT
information needs to reveal what information is contained in them (cf. section
3.2). In addition, we are interested in the extent to which standard text
retrieval approaches are able to deal with ToT queries. To this end, we
benchmark a subset of ToTMusic information needs222Concretely, 1.2K
descriptive information needs not containing hyperlinks – we aimed to exclude
posts where important information is not encoded in the text of the post
itself. on the Wasabi corpus, as well as Spotify search. Across both settings,
the low effectiveness—compared to non-ToT queries—of our evaluated retrieval
methods underscores the necessity of novel methods to tackle this task.
Lastly, we conduct a preliminary study on reformulating Music-ToT queries
using GPT-3 (Brown et al., 2020); we find that the task remains very
challenging.

##  2\. Background

ToT retrieval is related to known-item retrieval (KIR) or item-re-finding
(Sadeghi et al., 2014), however ToT queries are typically issued only once—not
multiple times—and importantly, lack concrete identifiers, instead relying on
verbose descriptions, frequently expressed uncertainty and possible false
memories (Arguello et al., 2021; Bhargav et al., 2022; Hagen et al., 2015;
Jørgensen and Bogers, 2020). Approaches for simulating such queries (Azzopardi
et al., 2007; Elsweiler et al., 2011; Kim and Croft, 2009) may lack realistic
phenomena like false memories (Hauff et al., 2012; Hauff and Houben, 2011),
necessitating the collection of real world data. Data on a large scale is
available for only one domain, movies (Bhargav et al., 2022); smaller scale
datasets are available for games (Jørgensen and Bogers, 2020) and movies
(Arguello et al., 2021). Hagen et al. (2015) collect a corpus of general
known-item queries, including music; however their focus was on general known-
item queries and false-memories, and lacked retrieval experiments. Our focus
is on the music domain, examining modalities employed by searchers and how
they express Music-ToT queries. We build upon Arguello et al. (2021) and
Bhargav et al. (2022), with key differences in (1) the domain—music, (2) the
corpus size—millions of items instead of thousands, and, (3) reformulation
experiments utilizing an LLM. Music-ToT relates to several research areas in
Music IR (MIR).

Lyric- and text-based retrieval involves retrieving a song using lyrics or
text (Downie and Cunningham, 2002; Müller et al., 2007). Techniques to handle
misheard lyrics are common (Ring and Uitdenbogerd, 2009; Xu and Kato, 2012; Xu
et al., 2009; Ye et al., 2020), including modeling speech sounds (Hirjee and
Brown, 2010), which may be insufficient, since ToT queries can contain
descriptions of lyrics, requiring semantic methods (Sasaki et al., 2014), or
utilizing the audio itself (Yu et al., 2019). Apart from lyrics, Music-ToT
queries are frequently free-form natural language queries (cf. section 4),
requiring methods that can retrieve audio using text, as well as tags, genre
or human-generated descriptions (Oncescu et al., 2021; Koepke et al., 2022;
Elizalde et al., 2019; Zhu et al., 2022; Doh et al., 2022).

Content-based audio retrieval (Foote, 1997) includes query-by-example (QBE)
(Helén and Virtanen, 2007), where the audio is being queried as-is, e.g. audio
fingerprinting (Haitsma and Kalker, 2002). Alternatively, users can imitate
the wanted audio by vocalizing it, termed query-by-vocal-imitation (QBV)
(Zhang and Duan, 2018; Kim and Pardo, 2019), which includes query-by-humming
(QBH) (Ghias et al., 1995). ToT queries frequently contain references to user
created audio-clips as well as existing media like audio contained in videos
(cf. section 4).

Other modalities like videos may need to be handled as well, necessitating
multi-modal or cross-modal (retrieving one modality using another) methods
(Simonetta et al., 2019), e.g. retrieving audio using video (Wang et al.,
2016; Hong et al., 2017). Approaches to solve Music-ToT have to account for
multiple modalities and free-form natural language including noise, e.g.,
uncertainty (Arguello et al., 2021) and/or false memories (Arguello et al.,
2021; Jørgensen and Bogers, 2020).

##  3\. Methodology

###  3.1. Data Collection

Gathering ToTAll. We gathered posts made across 2017-2021 in the
r/TipOfMyTongue community, yielding 503,770 posts (after filtering out posts
not marked Solved or Open), each containing two fields: title and description.
We extracted text categories from the title, e.g. SONG from _”[SONG] Slow
dance song about the moon?”_. We manually identified a set of 11 overarching
_music-focused_ categories (e.g. Music Video, Band, Rap Music). We discarded
the remaining non-music posts, resulting in ToTAll: 94,363 (60,870 solved and
33,493 unsolved) Music-ToT posts. These posts form a large
proportion—18.73%—of the 503K posts we started out with.

Extracting ToTMusic. We extracted answers from Solved posts following Bhargav
et al. (2022), retaining Solved posts which have a URL as an answer. If the
URL points to a track on Spotify, obtaining the answer was trivial. Otherwise,
the title portion of the markdown inline URLs, formatted as [title](url) (with
title often formatted as ‘Artist-Song’) was used as a query to the Spotify
search API. Since the API returns multiple results, we created a
classifier333Random Forest classifier, parameters selected with grid search on
{10, 20, 30, 40, 50} estimators, max depth {2, 3, 4} and min/max scaled
features. with 31 features based on the scores of the retriever, the edit
distances between title and artist name, song title, etc. We used the
classifier to predict if a title matches the track and artist, scoring 100% on
precision on a held out set of 100 samples. Low-confidence candidates were
filtered out. This left us with a set of 4,342 posts with Spotify tracks as
answers. Lastly, we only retained those posts where the ISRC444The
international standard recording code (ISRC) is a standardized code for
uniquely identifying recordings. of the answer track is also present in the
Wasabi Corpus (Buffa et al., 2021): a total of 2,278 posts. We call this
collection ToTMusic.

Gathering reformulations. We gathered reformulations for all posts in ToTMusic
by prompting GPT-3 (Brown et al., 2020)555Model: text-davinci-003, with
temperature 0.7 with the respective post description and a word count limit:
_< description> Summarize the query above to <N> words, focusing on musical
elements_. We used $N=\\{10,25,50\\}.$666Based on manual inspection, we
discarded $N=5$ (too few words for a cohesive query, leading to crucial
information being left out) and $N=100$ (model hallucinations). We also
employed a prompt without a specific word limit: _< post description> Shorten
the query above, focusing on musical elements._

###  3.2. Music-ToT Schema

Our annotation process involved three steps. We first developed and then
refined a schema to describe Music-ToT information needs; in the final step,
we annotated 100 samples from ToTMusic.

Developing the schema in 2 steps. A preliminary study conducted with one
author (self-rated music expertise 7 out of 10) and two volunteers (music
expertise 8/10 and 7/10 respectively) involved assigning one or more labels to
78 sentences from 25 randomly sampled posts from ToTMusic. We focused on
developing new labels specific to Music-ToT, while also re-using labels from
Arguello et al. (2021): specifically the Context labels, pertaining to the
context an item was encountered in (_Temporal Context_ , _Physical Medium_ ,
_Cross Media_ , _Contextual Witness_ , _Physical Location_ , _Concurrent
Events_), and Other annotations (_Previous Search_ , _Social_ , _Opinion_ ,
_Emotion_ , _Relative Comparison_). The latter are generally applicable across
ToT information needs. This preliminary study revealed 25 new music labels, in
addition to 11 labels from prior work ($6\times{}$Context and
$5\times{}$Other). In the second step, the three authors (self-rated musical
expertise 7, 6 and 5 respectively) of this paper labeled 110 sentences (20
posts from ToTMusic) to validate the schema. Based on our results and
discussions, we combined a few finer-grained categories with low support into
more general categories, e.g. specific musical elements like _Rhythm_ /
_Repetition_ , _Melody_ , _Tempo_ , etc., were combined to _Composition_ ,
resulting in 28 labels in total.

Annotating. Lastly, in step 3, two authors employed the final schema to
annotate $536$ sentences corresponding to $100$ posts. The resulting labels,
their frequency, category, inter-rater agreement (Cohen’s $\kappa$ (Cohen,
1960; Artstein and Poesio, 2008)) along with their description and an example,
are presented in Table 1.

Table 1. Annotation Schema: Label, frequency of occurrence in 100 submissions / 536 sentences (F), annotator agreement ($\kappa$) and description of label, along with an example for each label. | Label | F | $\mathbf{\kappa}$ |  Description |  Example
---|---|---|---|---|---
MUSIC ANNOTATIONS | Composition | 87 | 0.74 |  Describes (part of) the composition of a piece of music including rhythm, melody, tempo, pitch, chords, notes, and keys; or how they are composed into a cohesive piece of music. |  …playing the same major-key pattern over each chord in a fairly simple repeating loop.
Genre | 77 | 0.92 |  References a genre. |  It sounded like a reggae/ska type beat
Music Video Description | 75 | 0.89 |  Describes a music video associated with a song. |  However, once the music starts, the store is lit up and the tone shifts completely as everything in that store has a pastel colour scheme.
Lyric Quote | 65 | 0.89 |  Directly quotes lyrics that the user overheard, not including sounds / vocalizations |  …it wasn’t until he said something about the “just somebody that I used to know” song that I …
Story/Lyric Description | 60 | 0.71 |  Describes either the story conveyed by the lyrics, or the gist of the lyrics instead of directly quoting it. |  The song is a woman singing to/about a man that she was in love with and died, I think he was in the military and got killed and she had a baby at home?
Artist Description | 54 | 0.92 |  Describes the artist. |  He was maybe a tad overweight, shaggy hair, maybe curly.
Time Period / Recency | 49 | 0.89 |  References the time period the user thought the music was produced. |  Late 90s-early 2000s hip hop song that sounds similar to clip
Instrument | 30 | 0.86 |  Mentions instruments that were overheard. |  The guy performing was at a keyboard/piano …
Vocals | 28 | 0.69 |  Describes the voice or vocal type. |  High pitched but kind of floaty female vocals, a bit …
Name | 23 | 0.81 |  Describes a song/artist/album name, what it resembles/contains, or what the searcher remembers of it. |  …the name of the song was brief, one nordic word.
Popularity | 18 | 0.83 |  Describes the popularity of the music, artist, album or music video. |  I’m surprised I can’t find it since I can remember many specific lyrics, I guess it’s more obscure
Recording | 15 | 0.80 |  A description or reference to user-created content |  I did a vocaroo of the tune, sorry about my voice and any possible background guinea pig noises: URL
Language / Region | 14 | 0.92 |  Either mentions the language of the piece of music and/or references a particular region like state, country, etc. |  A Japanese song that I don’t remember any words to or how the tune goes at all,
Album Cover | 5 | 1.00 |  Describes the album cover. |  On the cover there was also a cyan teal line going along the bottom with white text in it.
Song Quality / Type | 4 | 0.00 |  Describes the type of music (original/cover, live/recorded) or the production quality (professional, amateur, etc.) |  Live Cover of All Along the Watchtower where …
CONTEXT et al. ANNOTATIONS | Uncertainty | 162 | 0.79 |  Conveys uncertainty about information described. |  I don’t know what genre the song was, it was fairly calming and I feel like it couldve been on tiktok but I don’t really know.
Social | 54 | 0.77 |  Communicates a social nicety. |  Any help appreciated!
Opinion | 43 | 0.44 |  Conveys an opinion or judgment about some aspect of the music. |  I don’t remember the lyrics or title, only that it was a kind of angsty teen “I want to set the world on fire”
Temporal Context | 36 | 0.87 |  Describes when the music was heard, either in absolute terms or relative terms. |  …I heard like in a billion videos 6 years ago.
Listening Medium | 26 | 0.75 |  References the medium associated with the item. (e.g., radio, streaming service, etc) |  I heard it on the radio a couple of times in …
Embedded Music | 26 | 0.58 |  References or describes extant media (e.g., Youtube / Twitch URL), including timestamps. |  I do have a video with the song (this video at around minute 4:21: URL)
Other Cross Media | 26 | 0.19 |  Describes exposure to the piece of music through different media, excluding other Cross Modal labels |  …I’m pretty sure was performed on one of the early seasons of Glee or maybe Smash.
Previous Search | 25 | 0.67 |  Describes a previous attempt to find the item, including negative results (i.e., it is not song X). |  I’ve tried humming it into shazam and other sites, looking up the two generic lyrics I remember, even doing those rhythm tapping things and nada
Relative Comparison | 25 | 0.77 |  Describes a characteristic of the music in relative (vs. absolute) terms, by explicitly comparing it with another song / artist / album. |  The melody I remember resembles the beginning of the song ”Run to the hills” by Metallica
Emotion | 25 | 0.05 |  Conveys or describes how a piece of music made the viewer feel |  Even talking about it makes me tear up.
Concurrent Events | 18 | 0.09 |  Describes events relevant to the time period when music was encountered, but excluding descriptions of the music itself. |  …when I was driving down the country but for the life of me can’t remember the name.
Physical Location | 9 | 0.61 |  Describes physical location where music was encountered. |  …record a 9 second portion of this song at a Marriott hotel bar in downtown Chicago …
Contextual Witness | 9 | 0.49 |  Describes other people involved in the listening experience. |  A few years back, a friend of mine showed me an …

##  4\. Data Analysis

We now first discuss Table 1, followed by a brief discussion about the
modalities present in the whole collection, ToTAll.

Annotation results. Among the music-focused annotations, Genre and
Composition, a description of musical elements and how they fit together, are
the two most frequent labels. This is followed by Music Video Description, and
either direct quotes (Lyric Quote) or a description of the lyrics (Story/Lyric
Description) further highlighting the different information needs that need to
be addressed i.e., lyric search, text search and multi-modal search. However,
a simple extraction of Genre and metadata such as Time Period/Recency,
Instrument, etc., may not be useful without considering the most frequent
label, Uncertainty. Search systems therefore would have to handle these
elements, as well as consider potential false memories. Furthermore,
annotations like Social, Opinion are also fairly common occurrences in our
data, which may have limited utility for retrieval (Arguello et al., 2021),
motivating reformulations (cf. section 3.1). Searchers also express their
queries in terms of other music entities in a Relative Comparison, and
describe Previous Search attempts, explicitly ruling out certain candidates.
References to other modalities like user created clips (Recording) or existing
media (Embedded Music) also pose a challenge. We now explore this challenge
with a brief study of references to external content in the entire collection,
ToTAll.

Cross-modal references Music-ToT, like other ToT domains, contains cross-modal
and media references (Arguello et al., 2021), where a searcher refers to
external content. We here show that Music-ToT posts in particular contain such
references frequently. To this end, we gathered frequent websites that appear
in ToTAll. One author manually labeled these as one of:  (1) User Created: a
clip uploaded by a user, e.g., Vocaroo, Clyp.it, Google Drive, Dropbox,
Instaudio, musiclab, Onlinesequencer, Streamable, Speakpipe.  (2) Extant
Media: a clip unlikely to be uploaded by a user, e.g. an existing clip,
corresponding to content/social media websites like Spotify, Twitch, Tiktok,
or YouTube.  (3) _Other URL_ : Not belonging to the previous two categories.
We find that Extant Media forms a larger proportion of queries (19K, 20.9%)
compared to User Created queries (14K, 15.3%), with a small number of posts
containing references to both types (1.1%). Therefore, Music-ToT information
needs are inherently multi-modal. We characterize the remaining 57.7% of
queries as descriptive queries, which include references to lyrics, or story
descriptions (cf. section 3.2). In summary, Music-ToT information needs are
characterized by uncertainty and multi-modality, requiring methods like text-
based audio retrieval, content based audio retrieval/fingerprinting and multi-
or cross-modal retrieval.

##  5\. Benchmarks

###  5.1. Experimental Setup

Corpora. We run experiments on two corpora. The first is the Wasabi 2.0 Corpus
(Buffa et al., 2021, 2020). It consists of 2M commercial songs from 77K
artists and 200K albums. Crucially,  (1) songs have the ISRC linked, enabling
linking to data in Spotify;  (2) it is an open dataset, consisting of rich
information that includes lyrics, extensive metadata, and music snippets  . We
index the Song Name, Artist Name and Lyrics777We also experimented with other
fields like Album Title, but saw no improvement in retrieval effectiveness. of
all songs using Elasticsearch (BM25 with default parameters). The second
corpus corresponds to the Spotify US catalog, consisting of hundreds of
millions of tracks. The Spotify search system (Hashemi et al., 2021) utilizes
multiple retrieval stages (including lexical- and semantic search) and
incorporates historic log data for retrieval purposes.

Queries. We conducted experiments on the 1,256 posts (849 train, 191
validation, and 216 test) from ToTMusic that contain no URLs in the post title
or post text; we make this choice as in the most extreme case, the entire post
may contain just a URL, requiring audio-based search while we focus on text-
based methods. From each post, we create different _queries_ and label them as
follows:  (1) Title: using the post title only;  (2) Text: post text;  (3)
Title+Text: title & text concatenated; and finally,  (4) Keywords: extracting
up to ten keywords from the post text888Keywords were deduplicated with
threshold = $0.2$ and algorithm =seqm. with Yake (Campos et al., 2018);  (5)
$\texttt{Reform}_{N}$: reformulations with $N=\\{10,25,50,\infty{}\\}$.

Evaluation. We report Recall@K, equivalent to Success@K (i.e., one correct
answer) for $K=\\{10,100,1000\\}$ on Wasabi. All reported results are on the
test set. For Spotify search we describe the observed trends (due to the
proprietary nature of the system).

Table 2. Overview of retrieval experiments on Wasabi, using Elasticsearch (BM25). Query | S@10 | S@100 | S@1000
---|---|---|---
Title | 0.0370 | 0.0833 | 0.1389
Keywords | 0.0231 | 0.0463 | 0.0787
Text | 0.0139 | 0.0648 | 0.0926
Title+Text | 0.0324 | 0.0833 | 0.1713
$\texttt{Reform}_{10}$ | 0.0139 | 0.0509 | 0.1204
$\texttt{Reform}_{25}$ | 0.0278 | 0.0602 | 0.1389
$\texttt{Reform}_{50}$ | 0.0185 | 0.0741 | 0.1389
$\texttt{Reform}_{\infty{}}$ | 0.0139 | 0.0741 | 0.1574

###  5.2. Results

Table 2 provides an overview of our Wasabi results.

Post parts as query. The low success across queries and $K$ underscores the
difficulty of the task. On Wasabi, Title queries are more effective than Text
queries—increased verbosity leads to retrieval failure. However, the text may
indeed contain data useful in retrieval, with comparable or higher
effectiveness scores for Title+Text over Title at $K=\\{100,1000\\}$,
motivating keyword extraction: crucial details might be present in the text,
but including the entire need as a query might harm effectiveness. Our keyword
selection method though fails to outperform other queries except for Text on
S@10.

On Spotify search we observe a different trend: Title+Text is the most
effective query followed by Title.

LLM reformulations as query. Examining Table 2, reformulations have limited
success compared to Title queries. $\texttt{Reform}_{25}$ and
$\texttt{Reform}_{50}$ perform as well as Title on S@1000, with
$\texttt{Reform}_{\infty{}}$ outperforming it. While Keywords beat all but
$\texttt{Reform}_{25}$ on S@10, it is outperformed by reformulations on S@100
and S@1000. On Spotify search, we find that reformulations fare worse than
Title queries for S@10, but see limited success on S@100, with
$\texttt{Reform}_{25}$ and $\texttt{Reform}_{50}$ achieving higher
effectiveness. Most importantly, there is no ideal $N$ on either index, with
varying success across metrics. We thus conclude that in our study,
reformulations generated using state-of-the-art LLMs have only mixed success.

##  6\. Conclusions

We explored Tip-of-the-Tongue retrieval for music. Of the 94K posts
corresponding to Music-ToT information needs from an online community for ToT
requests, we linked 2,278 posts to the corresponding answers in the Wasabi
corpus, resulting in ToTMusic, thus enabling further research for this
challenging task.

We iteratively developed and refined a Music-ToT schema that contains 28 fine-
grained labels as shown in Table 1. Labeling 100 posts using this schema, we
showed that users express uncertainty frequently, and almost as often refer to
other modalities. We benchmarked a subset of 1.2K descriptive queries from
ToTMusic, and highlight the difficulty of the task. Future work should
leverage cross- and multi-modal retrieval as well as better approaches for
reformulations.

###### Acknowledgements.

The authors would like to thank Gulfaraz Rahman and Ruben van Heusden for
helping with the preliminary annotation work. The authors also thank Daniel
Lazarovski and Humberto Corona Pampín for their input. Part of this research
was supported by the NWO Innovational Research Incentives Scheme Vidi
(016.Vidi.189.039). All content represents the opinion of the authors, which
is not necessarily shared or endorsed by their respective employers and/or
sponsors.

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  *[ToT]: tip-of-the-tongue
  *[LLM]: large language model
  *[KIR]: known-item retrieval
  *[QBE]: query-by-example
  *[QBV]: query-by-vocal-imitation
  *[QBH]: query-by-humming
  *[ISRC]: international standard recording code


# Harmonic Interpolation and a Brunn-Minkowski Theorem for Random Determinants

Julius Ross  and  David Witt Nyström  Mathematics Statistics and Computer
Science, University of Illinois at Chicago, Chicago IL, USA<EMAIL_ADDRESS>Department of Mathematical Sciences, Chalmers University of Technology and the
University of Gothenburg, Sweden<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We describe the harmonic interpolation of convex bodies, and prove a strong
form of the Brunn-Minkowski inequality and characterize its equality case. As
an application we improve a theorem of Berndtsson on the volume of slices of a
pseudoconvex domain. We furthermore apply this to prove subharmonicity of the
expected absolute value of the determinant of a matrix of random vectors
through the connection with zonoids.

###### 2020 Mathematics Subject Classification:

32J27, 52A40, 52A21 (Primary) 32U05, 14C17, 52A40 (Secondary)

##  1\. Introduction

Let $A$ and $B$ be convex subsets of $\mathbb{R}^{n}$. The Minkowski sum of
$A$ and $B$ is defined as

$A+B:=\\{a+b:a\in A,b\in B\\},$

and the famous Brunn-Minkowski inequality says that

$|A+B|^{1/n}\geq|A|^{1/n}+|B|^{1/n},$

where $|\cdot|$ denotes the Euclidean volume.

We wish to consider the interpolation of convex sets. Given convex $A$ and $B$
there is a natural interpolating family $A_{t}:=(1-t)A+tB$, $t\in[0,1]$, and
it follows from the Brunn-Minkowski inequality that the map

$t\mapsto|A_{t}|^{1/n}$

is concave in $t\in[0,1]$.

For an infinite family of convex sets, there are many possible interpolations.
To consider this in more detail, suppose $\Omega$ is a smoothly bounded domain
in $\mathbb{R}^{m}$ and that we have a continuous family of convex bodies
(i.e. compact convex sets) $A_{\tau}\subset\mathbb{R}^{n}$ parametrized by
$\tau\in\partial\Omega$. If $\Omega$ is itself convex, a natural interpolation
can be obtained by considering

$A=\operatorname{Convexhull}\left(\bigcup_{\tau\in\partial\Omega}A_{\tau}\times\\{\tau\\}\right)\subseteq\mathbb{R}^{n+m}$

and letting $A_{x}$ be the fiber of $A$ over $x\in\Omega$. We call this the
_convex interpolation_ of $\\{A_{\tau}\\}$. Then directly from the Brunn-
Minkowski inequality it follows that the map $x\mapsto|A_{x}|^{1/n}$ is
concave in $x\in\Omega$.

If $\Omega$ is not convex, the convex interpolation is not suitable since it
will not necessarily agree with the given boundary data $\\{A_{\tau}\\}$ on
$\partial\Omega$. For general $\Omega$ a natural interpolation was proposed in
our recent paper [3] that we now describe.

First note that if $A_{y}$ is a continuous family of convex bodies in
$\mathbb{R}^{n}$ over some parameter set $D\subseteq\mathbb{R}^{m}$ and $\mu$
is a Radon measure on $D$, then there is a set-integral

$\int_{D}A_{y}d\mu(y)$

which is itself a subset of $\mathbb{R}^{n}$. To define this precisely recall
that the support function of a convex set $A$ is given by

$h_{A}(\xi):=\sup_{\zeta\in A}(\zeta\cdot\xi)$

with and has the property that

$h_{A}\text{ is convex and }h_{A}(t\xi)=|t|h_{A}(\xi)\text{ for
}t\in\mathbb{R}.$ (1)

On the other hand, if $h$ is a function with those two properties then $h$ is
the support function of a unique closed convex set, which we denote by $A(h)$.

It is an elementary exercise to see that $h_{A+B}=h_{A}+h_{B}$ and more
generally

$h_{t_{1}A_{1}+...+t_{k}A_{k}}=t_{1}h_{A_{1}}+...+t_{k}h_{A_{k}}.$

Furthermore if $A_{t}$ are convex sets such that $A_{t}\to A$ in the Hausdorff
topology, then for each $\xi$, $h_{A_{t}}(\xi)\to h_{A}(\xi)$.

######  Definition 1.1.

Let $d\mu$ be a measure on a measurable set $D$ in $\mathbb{R}^{m}$, and
$A_{y}$ be a convex set for each $y\in D$. We define the _Minkowski integral_
$\int_{D}A_{y}d\mu(y)$ as

$\int_{D}A_{y}d\mu(y):=A\left(\int_{D}h_{A_{y}}d\mu(y)\right).$

Such set-valued integrals have been considered in various places, for example
[1, 5]. As one would expect, some conditions are needed to ensure that the
Minkowski integral is well-defined. For our purpose the following is
sufficient: assume $D$ is compact, $d\mu$ is a Radon measure and $y\mapsto
A_{y}$ is a continuous family of convex bodies. Then for each $\zeta$ the map
$y\mapsto h_{A_{y}}(\zeta)$ is continuous, so $\int_{D}h_{A_{y}}d\mu(y)$
exists and has properties (1), and thus $\int_{D}A_{y}d\mu(y)$ exists.

######  Definition 1.2.

Let $\Omega\subset\mathbb{R}^{m}$ be a smoothly bounded domain. The _harmonic
interpolation_ of a continuous family $\\{A_{\tau}\\}_{\tau\in\partial\Omega}$
of convex bodies is defined as

$A_{x}:=\int_{\partial\Omega}A_{\tau}d\mu_{x}(\tau),$

where $d\mu_{x}$ is the harmonic measure on $\partial\Omega$ with respect to
$x\in\Omega$.

The harmonic interpolation and convex interpolation may differ, even when
$\Omega$ is convex. We argue that the former is better suited in some
contexts, one of which is the theory of zonoids.

A _zonotope_ is a convex set that can be written as the Minkowski sum of line
segments. Clearly any zonotope is a convex polytope, but it is easy to see
that not all convex polytopes are zonotopes. A _zonoid_ is a convex set which
can be approximated arbitrarily well (in the Hausdorff topology) by zonotopes,
or equivalently a convex set that can be written as the Minkowski integral of
line segments (see for example [4] for a introduction to zonoids). The
harmonic interpolation has the property that it preserves zonoids; i.e. if
each boundary set $A_{\tau}$ is a zonoid then each member of the interpolating
family $A_{x}$ will also be a zonoid (and this is not true for the convex
interpolation, even when $\Omega$ is convex).

### Acknowledgements

This material is based upon work supported by the National Science Foundation
under Grant No. DMS-1749447. The second named author is supported by the
Swedish Research Council and the Göran Gustafsson Foundation for Research in
Natural Sciences and Medicine. The authors thank Bo Berndtsson and Dario
Cordero-Erausquin for conversations on this topic.

##  2\. A Brunn-Minkowski Inequality for Harmonic Interpolation

We continue to assume $\Omega\subset\mathbb{R}^{m}$ is a smoothly bounded
domain (which by convention is also bounded), and $A_{\tau}$ for
$\tau\in\partial\Omega$ is a continuous family of convex bodies in
$\mathbb{R}^{n}$. In [3] we proved the following weak version of a Brunn-
Minkowski inequality for the harmonic interpolation.

######  Theorem 2.1.

If $A_{x}$ is the harmonic interpolation of $\\{A_{\tau}\\}$ then
$x\mapsto\log|A_{x}|$ is superharmonic in $x$.

Our main result in this short note is a direct proof of the following stronger
version:

######  Theorem 2.2.

If $A_{x}$ is the harmonic interpolation of $\\{A_{\tau}\\}$ then
$x\mapsto|A_{x}|^{1/n}$ is superharmonic in $x$.

###### Proof.

Let $B_{\epsilon}(x)$ denote the Euclidean ball of radius $\epsilon$ centered
at $x$. We need to show that if $B_{\epsilon}(x)\subseteq\Omega$ then

$|A_{x}|^{1/n}\geq\int_{\partial B_{\epsilon}(x)}|A_{y}|^{1/n}dS(y),$

where $dS$ denotes the normalized Euclidean surface measure on $\partial
B_{\epsilon}(x)$.

A standard property of harmonic measures is that

$\mu_{x}=\int_{\partial B_{\epsilon}(x)}\mu_{y}dS(y),$

and this clearly implies that

$A_{x}=\int_{\partial B_{\epsilon}(x)}A_{y}dS(y).$

Now we approximate the surface measure $dS$ with a sequence of atomic measure
$\nu_{k}=\sum_{i=1}^{N_{k}}\lambda_{i,k}\delta_{y_{i,k}}$ chosen so
$\nu_{k}\to dS$ weakly as $k\to\infty$. Then for $k$ sufficiently large
$\sum_{i=1}^{N_{k}}\lambda_{i,k}A_{y_{i}}$ is arbitrarily close (in the
Hausdorff distance) to $A_{x}$. Thus for any $\epsilon>0$ and $k$ sufficiently
large we have

$\displaystyle|A_{x}|^{1/n}+\epsilon\geq|\sum_{i=1}^{N_{k}}\lambda_{i,k}A_{y_{i,k}}|^{1/n}\geq\sum_{i=1}^{N_{k}}\lambda_{i,k}|A_{y_{i,k}}|^{1/n}\geq\int_{\partial
B_{\epsilon}(x)}|A_{y}|^{1/n}dS(y)-\epsilon,$

where the second follows from the classical Brunn-Minkowski inequality.
Letting $\epsilon\to 0$ we have

$|A_{x}|^{1/n}\geq\int_{\partial B_{\epsilon}(x)}|A_{y}|^{1/n}dS(y),$

which completes the proof. ∎

######  Definition 2.3.

We say that a continuous family of convex sets $A_{x}\subseteq\mathbb{R}^{n}$
over some domain $\Omega\subseteq\mathbb{R}^{m}$ is _subharmonic_ over
$\Omega$ if whenever $B_{\epsilon}(x)\subseteq\Omega$ we have that

$A_{x}\supseteq\int_{\partial B_{\epsilon}(x)}A_{y}dS(y).$

The then get the following Corollary of Theorem 2.2.

######  Corollary 2.4.

If $A_{x}$ is subharmonic over $\Omega$ then $x\mapsto|A_{x}|^{1/n}$ is
superharmonic.

As an application we can give a strengthening of the following theorem of
Berndtsson [2]

######  Theorem 2.5.

Let $U\subseteq\mathbb{C}^{n+m}$ be a pseudoconvex domain with the property
that if $(x_{1}+iy_{1},...,x_{n}+iy_{n},w)\in U$ then
$(x_{1}+iy^{\prime}_{1},...,x_{n}+iy^{\prime}_{n},w)\in U$ for all
$y_{1}^{\prime},\ldots,y_{n}^{\prime}$, and let
$U_{w}:=\\{x\in\mathbb{R}^{n}:(x,w)\in U\\}$.

Then the map $w\mapsto-\log|U_{w}|$ is plurisubharmonic in $w$.

######  Theorem 2.6.

In the same setting as Theorem 2.5, the map $w\mapsto-|U_{w}|^{1/n}$ is
plurisubharmonic.

###### Proof.

Without loss of generality we can assume that $m=1$. Note that the
pseudoconvexity and symmetry of $U$ implies that $U_{w}$ is convex for all
$w$. By approximation we can also without loss of generality assume that the
family $U_{w}$ is bounded and continuous. We claim that the family $U_{w}$ is
subharmonic. Note that for two closed convex sets $A$ and $B$ we have that
$A\supseteq B$ if and only if $h_{A}\geq h_{B}$, so $U_{w}$ is subharmonic if
and only if for any $\xi\in\mathbb{R}^{n}$, $h_{U_{w}}(\xi)=\sup_{x\in
U_{w}}(x\cdot\xi)$ is superharmonic in $w$.

Let $\phi$ be a plurisubharmonic exhaustion function for $U$ which we can
assume to be independent of $\operatorname{Im}(\mathbb{C}^{n})$, just as $U$
itself. Note that $\phi_{R}:=\max(\phi-R,0)$ is also plurisubharmonic and
independent of $\operatorname{Im}(\mathbb{C}^{n})$, and that the same is true
for $\psi_{R}(x+iy,w):=\phi_{R}(x+iy,w)-x\cdot\xi$. Thus by Kiselman’s minimum
principle $\inf_{x\in U_{w}}\psi_{R}(x,w)$ is subharmonic in $w$. We now note
that

$\sup_{x\in U_{w}}(x\cdot\xi)=-\lim_{R\to\infty}\inf_{x\in
U_{w}}\psi_{R}(x,w),$

and hence it follows that $h_{U_{w}}(\xi)$ is superharmonic and thus $U_{w}$
is subharmonic. That $-|U_{w}|^{1/n}$ is subharmonic now follows from
Corollary 2.4. ∎

##  3\. Characterization of the extremal case

By Corollary 2.4 we know that if $\\{A_{x}\\}_{x\in\Omega}$ is a subharmonic
family of convex bodies in $\mathbb{R}^{n}$ over a domain $\Omega$ then
$x\mapsto|A_{x}|^{1/n}$ is superharmonic. Our next result characterizes when
this map is in fact harmonic.

######  Theorem 3.1.

The map $x\mapsto|A_{x}|^{1/n}$ is harmonic if and only if we can write
$A_{x}=c_{x}B+d_{x}$ where $B\subset\mathbb{R}^{n}$ is a fixed convex body,
and $c_{x}$ and $d_{x}$ are harmonic functions on $\Omega$ taking values in
$\mathbb{R}_{+}$ and $\mathbb{R}^{n}$ respectively.

###### Proof.

Let $\Omega^{\prime}$ be a relatively compact subdomain of $\Omega$ with
smooth boundary. Since $A_{x}$ is subharmonic it must dominate the harmonic
interpolation of $A_{y}$ restricted to $\partial\Omega^{\prime}$, but since
$|A_{x}|^{1/n}$ is assumed to be harmonic we must have that $A_{x}$ is equal
to the harmonic interpolation.

Write $\partial\Omega^{\prime}$ as the disjoint union of a finite number of
measurable subsets $D_{i}$ and let

$B_{i}:=\int_{D_{i}}A_{y}d\mu_{x}(y),$

where $\mu_{x}$ is the harmonic measure on $\partial\Omega^{\prime}$ with
respect to $x$. Then $A_{x}=\sum_{i}B_{i}$, and by the Brunn-Minkowski
inequality we have

$|A_{x}|^{1/n}\geq\sum_{i}|B_{i}|^{1/n}.$

On the other hand, as in the proof of Theorem 2.2 one sees that

$|B_{i}|^{1/n}\geq\int_{D_{i}}|A_{y}|^{1/n}d\mu_{x}(y).$

But $|A_{x}|^{1/n}$ being harmonic then implies the equality

$|A_{x}|^{1/n}=\sum_{i}|B_{i}|^{1/n}.$

The well-known characterization of the equality case of the Brunn-Minkowski
inequality then says that we can write $B_{i}=c_{i}B+d_{i}$, where $B$ is some
fixed convex set, and some $c_{i}\in\mathbb{R}_{+}$ and
$d_{i}\in\mathbb{R}^{n}$. We may normalize $B$ to have volume one and center
of gravity at the origin. We thus also see that $A_{x}=c_{x}B+d_{x}$ where
$c_{x}=\sum_{i}c_{i}$ and $d_{x}=\sum_{i}d_{i}$.

Now if we decompose a fixed $D_{i}$ further into disjoint pieces $E_{j}$ the
same argument yields that for each $j$ there are
$c^{\prime}_{j}\in\mathbb{R}_{+}$ and $d^{\prime}_{j}\in\mathbb{R}^{n}$ such
that $\int_{E_{j}}A_{y}dS(y)=c^{\prime}_{j}B+d^{\prime}_{j}$. As we can make
the decomposition arbitrarily fine the continuity of $A_{y}$ implies that
there are continuous functions $c_{y}$ and $d_{y}$ on
$\partial\Omega^{\prime}$ such that $A_{y}=c_{y}B+d_{y}$. It follows that
$A_{x}=c_{x}B+d_{x}$ where $c_{x}$ is the harmonic extension of $c_{y}$ and
$d_{x}$ is the harmonic extension of $d_{y}$ to $\Omega^{\prime}$.

As this can be done for any for relatively compact subdomain of $\Omega$ with
smooth boundary, the result follows. ∎

##  4\. A Brunn-Minkowski theorem for expected absolute random determinants

Consider now a random $(n,n)$ matrix $M_{Y}$ whose columns are iid copies of a
random vector $Y$, corresponding to a Borel probability measure $\nu_{v}$ on
$\mathbb{R}^{n}$. We are then interested in the expected absolute value of the
determinant (ead) $E|\det M_{Y}|$ of $M_{Y}$.

Suppose $Y_{\tau}$ is a family of random vectors parametrized by the boundary
of a smoothly bounded domain $\Omega\subseteq\mathbb{R}^{m}$. We assume that
each $Y_{\tau}$ has finite expectation. Then a natural interpolating family
$Y_{x}$ over $\Omega$ is given by letting

$\nu_{Y_{x}}:=\int_{\partial\Omega}\nu_{Y_{\tau}}d\mu_{x}(\tau),$

where as before $d\mu_{x}$ denotes the harmonic measure with respect to $x$.

######  Theorem 4.1.

The map $x\mapsto(E|\det M_{Y_{x}}|)^{1/n}$ is superharmonic in $x$.

Our proof relies on the connection between eads and a special class of convex
sets called zonoids which was established in [5, Thm 3.1]: to any random
vector $Y$ with finite expectation we may associate a zonoid

$Z(Y):=\int_{\mathbb{R}^{n}}[0,y]d\nu_{v}(y).$

Then the main result [5, Thm. 3.2] says that

$E|\det M_{Y}|=n!|Z(Y)|.$ (2)

###### Proof of Theorem 4.1.

Note that

$\displaystyle
Z(Y_{x})=\int_{\mathbb{R}^{n}}[0,y]d\nu_{Y_{x}}(y)=\int_{\mathbb{R}^{n}}\int_{\partial{\Omega}}[0,y]d\mu_{x}(\tau)d\nu_{Y_{\tau}}(y)=$
$\displaystyle=\int_{\partial{\Omega}}\int_{\mathbb{R}^{n}}[0,y]d\nu_{Y_{\tau}}(y)d\mu_{x}(\tau)=\int_{\partial{\Omega}}Z(Y_{\tau})d\mu_{x}(\tau),$

i.e. $Z(Y_{x})$ is the harmonic interpolation of $Z(Y_{\tau})$. Thanks to the
volume equality (2) the result follows immediately from Theorem 2.2. ∎

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