#  Seeing Unseen: Discover Novel Biomedical Concepts via Geometry-
Constrained Probabilistic Modeling

Jianan Fan  University of Sydney  Dongnan Liu  University of Sydney  Hang
Chang  Lawrence Berkeley National Laboratory  Heng Huang  University of
Maryland at College Park  Mei Chen  Microsoft  Weidong Cai  University of
Sydney

###### Abstract

Machine learning holds tremendous promise for transforming the fundamental
practice of scientific discovery by virtue of its data-driven nature. With the
ever-increasing stream of research data collection, it would be appealing to
autonomously explore patterns and insights from observational data for
discovering novel classes of phenotypes and concepts. However, in the
biomedical domain, there are several challenges inherently presented in the
cumulated data which hamper the progress of novel class discovery. The
non-i.i.d. data distribution accompanied by the severe imbalance among
different groups of classes essentially leads to ambiguous and biased semantic
representations. In this work, we present a geometry-constrained probabilistic
modeling treatment to resolve the identified issues. First, we propose to
parameterize the approximated posterior of instance embedding as a marginal
von Mises-Fisher distribution to account for the interference of
distributional latent bias. Then, we incorporate a suite of critical geometric
properties to impose proper constraints on the layout of constructed embedding
space, which in turn minimizes the uncontrollable risk for unknown class
learning and structuring. Furthermore, a spectral graph-theoretic method is
devised to estimate the number of potential novel classes. It inherits two
intriguing merits compared to existent approaches, namely high computational
efficiency and flexibility for taxonomy-adaptive estimation. Extensive
experiments across various biomedical scenarios substantiate the effectiveness
and general applicability of our method.

##  1 Introduction

The rapid progress in deep learning (DL) has invigorated enormous interests
for automating essential stages in biomedical image analysis through data-
driven methodologies [7, 42]. However, the transformative potential of DL to
facilitate scientific discovery at the fundamental level remains largely
untapped.

Figure 1: Conceptual illustration of the main insight. In the biomedical
domain, violation of the i.i.d. assumption incurred by inconsistent imaging
protocols across cohorts and non-uniform class distributions due to scarcity
of rare classes could deteriorate the generalizability of learned
representations for novel class discovery. We propose to address those issues
via probabilistic modeling on a hyperspherical manifold and incorporation of
geometrical inductive biases for countering semantic ambiguity and open space
risk.

Specifically, the success of existing works hinges on expert-curated data
annotations in which associated biomedical structures (e.g., cells, tissues)
and disease categories are defined a priori [30, 13]. They have little to
offer for exploring unknown concepts and discovering expressive profiles for
novel/undefined biomedical phenotypes from observational data alone. For
instance, cell, the basic unit of human biological system, exhibits vast
diversity with hundreds of different types and states [33]. Compared with
cumbersome manual pattern identification and grouping of cell populations,
autonomously mining new classes of cells with their morphological and
functional characteristics could greatly speed up the scientific discovery
progress of biomedical research.

To this end, we study the problem of discovering novel concepts from
biomedical imaging data, under the _Generalized Novel Class Discovery_ setting
[44]. Given a reference dataset within a set of well-defined base categories,
it aims at identifying and clustering unseen classes in an external collection
of image instances without supervision [22, 20]. The technical challenge of
this task lies in how to leverage the structured prior knowledge secured from
previously known biomedical object categories to guide the unsupervised
discovery of new concepts. Recently, a stream of methods based on
representation learning and clustering [16, 21, 44] have been proposed to
detect unseen objects in generic visual perception scenes. However, the basic
assumption those works built upon may not hold in the biomedical domain from
two aspects: i) It is generally assumed that the labeled and unlabeled data
are independent and identically distributed (i.i.d.) [53, 23], whereas for
biomedical images, data distribution shifts inevitably arise from the
variations in targeted biology and image acquisition among multiple cohorts
[32, 14]. The statistical discrepancy between the reference set of data and
the external collection for discovery would consequently result in ambiguous
semantic representations given the interference of task-irrelevant attributes
[31]. ii) Existing works hypothesize a balanced class distribution a priori
which expects that the quantity and occurrence frequency of unseen classes
should be on par with the base classes [8, 52]. They dismiss the long-tail
nature of class distribution in scientific discovery that instances of classes
never seen before are essentially rare and hard to collect. For instance, rare
diseases are inherently atypical and much fewer in caseload than their common
counterpart [9, 34]. The formulated embedding space is consequently prone to
be biased towards the base classes with dominatedly copious training samples.
Despite the latest efforts delivering explicit modeling of the distribution
biases [50, 48], their self-training founded solutions intrinsically fall
short due to erratic representation space structuring. Without consideration
of the dark information [6] regarding unknown class representations, there is
no guarantee that a structured open space can be preserved for learning new
classes. It deteriorates the generalizability of those methods to uncover and
group the implicit patterns of unseen phenotypes.

In this work, we present a geometry-constrained probabilistic treatment for
discovering novel biomedical classes which resolves the above-mentioned
challenges from a new perspective, as illustrated in Fig. 1. First, to
counteract the semantic ambiguities induced by data distribution shifts, we
suggest to circumvent deterministic image encoding and instead model latent
representations as directional distributions. Specifically, for each instance,
we parameterize the approximated posterior of its semantic embedding with a
marginal von Mises-Fisher (vMF) distribution on a unit hyperspherical
manifold. The probabilistic modeling presents an elegant solution to decouple
latent bias incurred by inconsistent imaging protocols from informative
semantic context as an implicit uncertainty measure. Second, we pinpoint the
necessity to incorporate _inductive biases_ for imposing proper constraints on
the geometric layout of learned embedding space and thereby minimizing the
uncontrollable risk for unknown classes. Aside from the _hyperspherical
manifold_ prior, we identify two critical geometric properties for embedding
space structuring, i.e., _boundness_ and _uniformity_. Here, _boundness_
implies that, in the latent space, both base and novel classes should be
limited to their certain bounded ranges with a large margin separation between
the two groups. _Uniformity_ refers to the degree of distributional diversity
for the set of semantic proxies on the hypersphere. These principles
cooperatively shape the layout of embedding space towards risk-bounded novel
class discovery and perspicuous geometric interpretation. With explicit
constraints to integrate the underlined inductive biases into representation
learning, our proposed method inherits guarantees to attain maximal inter-
class separability and intra-class compactness, which in turn addresses the
inordinate dominance of base classes and favors the generalization of learned
embeddings to recognize unseen concepts.

Additionally, with respect to the practice of existing works assuming the
number of novel classes are known a priori [54, 29], we argue that such
supposition is over-optimistic and unrealistic in the open world of scientific
discovery. We therefore propose a versatile and plug-and-play solution to
estimate the number of categories in unlabeled data. The spectral graph
theory-based approach is computationally efficient and does not introduce
recursive clustering as in [44]. It also holds an appealing merit that it
supports taxonomy-adaptive estimation, which means the users can decide the
granularity level (e.g., coarse or fine-grained) in class hierarchy with
regards to their specific needs and get the corresponding outcome.

Contributions. Our contributions are four-fold: (i) We introduce a novel
paradigm to unleash the tremendous promise of DL to facilitate data-driven
biomedical discovery and account for the distribution biases inherently
presented in the domain. (ii) We propose geometry-constrained probabilistic
modeling to facilitate risk-bounded novel class discovery by exploiting the
_boundness_ and _uniformity_ geometrical inductive biases for representation
space structuring. We perform theoretical justifications from two perspectives
to illustrate the merits of our method. (iii) We devise a spectral graph-
theoretic method to estimate the potential number of novel classes in
unlabeled observational data. (iv) The proposed method is extensively
validated on a diverse suite of challenging scenarios and attains superior
performance compared to existing solutions.

##  2 Related Works

Novel Class Discovery. Inspired by the observation that human visual system
can effortlessly recognize an unseen class of objects based on previously
learnt category concepts, a stream of methods have been proposed for novel
class discovery (NCD) in an unsupervised manner by transferring prior
knowledge gathered from a set of seen categories with annotations [22, 20,
21]. Follow-up works further identify the limitations of NCD in that it
disregards the compositional nature of real-world upcoming data and supposes
encountered classes will not appear again in the unlabeled set. To this end,
they formalize the _Generalized Novel Class Discovery_ (GNCD) setting which
assumes both base and novel classes could simultaneously appear in the
unlabeled data [44]. ComEx [49] and DPN [1] propose to decouple the two sets
of categories for capturing discriminative representations. In a similar
spirit, PromptCAL [51] and DCCL [36] resort to synergistic learning and seek
to improve the constructed representations via auxiliary training objectives.
Nevertheless, those works are rooted in over-optimistic assumptions of i.i.d.
data and balanced class distribution [44] and thereby tend to suffer from
severe learning bias in real-world scientific discovery with intricate
circumstances. Despite the recent efforts to account for non-i.i.d. and non-
uniform data distribution in consonance with NCD [50, 48], their solutions are
limited to the commonly adopted self-training scheme. Without explicit
regularizations imposed on the geometric layout structure of the embedding
space, their performance could fluctuate vastly given the volatile quality of
pseudo-labels [46].

Open-World Semi-Supervised Learning. The technical challenge we intend to
unravel is in line with an arising topic, i.e., open-world semi-supervised
learning [3]. It takes a step further than the closed-world setting held in
standard semi-supervised learning and presumes the emergence of new classes in
unlabeled training and test dataset. Following the insight, several recent
works suggest the usage of pseudo-labels and pair-wise similarity to group and
recognize unseen categories [37, 4, 38, 19]. However, their promise in
biomedical scientific discovery is unfavorably undermined by the challenges
posed by distribution bias.

von Mises-Fisher Distribution. Probabilistic representation learning provides
an elegant viewpoint to measure predictive uncertainty and mitigate feature
ambiguity arising from deterministic mappings [41]. The multivariate Gaussian
distribution is commonly employed as a prominent choice for probabilistic
modeling in Euclidean space, whilst its applicability on spherical embeddings
is limited due to inherent conflicts with the underlying manifold geometry
[11]. On the contrary, the von Mises-Fisher (vMF) distribution is introduced
as a directional statistical model that is defined on the hypersphere. It has
been applied in tasks including facial recognition [27, 47], out-of-
distribution detection [35], and long-tailed learning [45], leading to
remarkable advancements in these areas. Nevertheless, the significance of
incorporating crucial inductive biases pertaining to geometric properties
within the embedding space, such as uniformity, has been overlooked in the
literature. Distinguished from existing works, to the best of our knowledge,
our study is the first to explicitly structure the geometric layout of learned
representations and account for the statistical discrepancy across data with
distributional concentration modeling.

##  3 Methodology

###  3.1 Problem Statement

The objective of this study is to achieve automated discovery of clusters that
correspond to novel concepts using observational data, which is accomplished
by transferring the knowledge acquired from a set of base classes with labeled
image samples. Following the practical setting in [44], we assume that there
exists a labeled base set
$\mathcal{D_{B}}=\\{(\bm{x}_{i},\bm{y}_{i})\\}_{i=1}^{N_{\mathcal{B}}}\in\mathcal{X_{B}}\times\mathcal{Y_{B}}$,
and an unlabeled set
$\mathcal{D_{U}}=\\{(\bm{x}_{i},\bm{y}_{i})\\}_{i=1}^{N_{\mathcal{U}}}\in\mathcal{X_{U}}\times\mathcal{Y_{U}}$
with potential novel classes, where $\mathcal{Y_{U}}$ are not available. All
instances in $\mathcal{D_{B}}$ are of class set $\mathcal{S_{B}}$. As base
classes also appear in the unlabeled set, i.e.,
$\mathcal{Y_{B}}\subset\mathcal{Y_{U}}$, the set of novel classes to be
discovered can be formulated as
$\mathcal{S}_{N}=\mathcal{Y_{U}}\backslash\mathcal{Y_{B}}$, with the total
number $|\mathcal{S}_{N}|$ unknown a priori. Notably, distinguished from [44],
the two sets of data are drawn from different distributions due to
inconsistent biology processing and imaging protocols, i.e.,
$\mathcal{X_{B}}\sim\mathcal{P_{B}},\mathcal{X_{U}}\sim\mathcal{P_{U}},\mathcal{P_{B}}\neq\mathcal{P_{U}}$.
The class distribution of $\mathcal{D_{U}}$ is innately non-uniform as well.
Given a new instance from $\mathcal{P_{U}}$, the goal is to either recognize
it as one of the base classes or find it is closest to which identified novel
class amongst $\mathcal{S}_{N}$.

###  3.2 Geometry-Constrained Probabilistic Modeling

Overview. The technical challenge to adapt base knowledge for novel biomedical
concept discovery lies in two aspects: semantic ambiguity caused by data
distribution shifts and uncontrollable open space risk as a result of major
class dominance. To this end, we propose a novel geometry-constrained
probabilistic treatment. The framework is illustrated in Fig. 2. It explicitly
decouples the underlying distributional bias from informative semantic
representations and structures the geometric layout of the formulated
embedding space based upon inductive biases on latent manifold. First, we
incorporate the _uniformity_ inductive bias by positioning semantic proxies a
priori with uniform distribution guarantees. It helps to mitigate the
representation collapse incurred by the inherent imbalance between base and
novel classes. Subsequently, we leverage vMF probabilistic modeling to promote
latent space _boundness_ by firstly establishing a compact manifold for each
base class and then regularizing the open space to be separated from the base
manifold by a large angular margin. At last, to demystify the dark information
regarding unknown class representations, we propose to structure the embedding
space and minimize the open space risk by imposing statistical discrepancy
constraints on unlabeled instances with semantic consensus.

Figure 2: Overview of the proposed method. We propose to incorporate the
_uniformity_ and _boundness_ geometrical inductive biases by establishing
preorganized proxies as anchors and then structuring the geometric layout of
learned embedding space successively with hyperspherical probabilistic
modeling.

Uniform Proxies Founded A Priori with Energy Minimization. Prior to
representation learning, we propose to at first organize the embedding space
with pre-defined proxies. By disentangling proxy foundation as an individual
step, we seamlessly integrate the _uniformity_ inductive bias to shape model
output space without disturbing discriminative class learning. Here,
uniformity indicates the degree of distributional diversity for the set of
proxies. With the proxies distributed uniformly throughout the entire
manifold, we minimize the open space risk by explicitly structuring the
feature space for unknown classes. The equidistributional organization also
facilitates inter-class separation and intra-class compactness. Specifically,
given a _d_ -dimensional embedding space, we define a set of prior proxies
$\bm{\hat{\Upsilon}}=\\{\bm{\hat{\upsilon}}_{1},...,\bm{\hat{\upsilon}}_{n}\\}\in\mathbb{S}^{d-1}$
on a unit hyperspherical manifold. The number of proxies $n$ should be
generally larger than the total number of classes, for which sensitivity
analysis is conducted in the supplementary. We then propose to characterize
the distributional uniformity of those proxies with their hyperspherical
potential energy [39]. Energy behaves as a measure of distribution redundancy
where lower energy implies the proxies are evenly positioned with large margin
separation. We thereby suggest to minimize the potential energy with the
following objective:

$\mathop{\mathtt{arg\;min}}\limits_{\\{\bm{\hat{\upsilon}}_{1},...,\bm{\hat{\upsilon}}_{n}\\}\in\mathbb{S}^{d-1}}\,\bm{E}_{s,d}(\bm{\hat{\upsilon}}_{i}|_{i=1}^{n})=\sum_{i=1}^{n}\sum_{j=1,j\neq
i}^{n}\xi_{s,d}(\bm{\hat{\upsilon}}_{i},\,\bm{\hat{\upsilon}}_{j}),$ (1)

where the unit hypersphere
$\mathbb{S}^{d-1}=\\{\bm{\hat{\upsilon}}\in\mathbb{R}^{d}|\,\lVert\bm{\hat{\upsilon}}\rVert=1\\}$.
$\xi_{s,d}(\cdot,\,\cdot)$ represents the measurement of the correlation
between proxies, which is typically implemented with the following Riesz _s_
-kernel function:

$\xi_{s,d}(\bm{\hat{\upsilon}}_{i},\,\bm{\hat{\upsilon}}_{j}):=\left\\{\begin{aligned}
\Gamma(\bm{\hat{\upsilon}}_{i},\,\bm{\hat{\upsilon}}_{j})^{-s}&,\ \ s>0,\\\
\log(\Gamma(\bm{\hat{\upsilon}}_{i},\,\bm{\hat{\upsilon}}_{j})&{}^{-1}),\ \
s=0,\end{aligned}\right.$ (2)

where $\Gamma(\cdot,\,\cdot)$ is defined as the hyperspherical geodesic
distance.

Distributional Characterization on Hypersphere. To account for the semantic
ambiguities induced by data distribution shifts, we propose probabilistic
modeling to parameterize latent embeddings as directional distributions on a
hyperspherical manifold. Specifically, we first let $f_{\theta}$ represent a
projection network that maps input images to their hyperspherical embedding,
i.e., $f_{\theta}:\bm{x}\in\mathcal{X}\mapsto\bm{z}\in\mathbb{S}^{d-1}$. Then,
we characterize the approximated posterior of each instance embedding as a
marginal von Mises-Fisher (vMF) distribution, i.e.,
$q_{\theta}(\bm{z}|\bm{x})\sim\mathbf{vMF}(\bm{\tilde{\mu}_{x},\tilde{\kappa}_{x}})$.
Formally, a vMF distribution is defined as:

$q(\bm{z}|\bm{\tilde{\mu}_{x},\tilde{\kappa}_{x}})=C_{d}(\bm{\tilde{\kappa}_{x}})\exp(\bm{\tilde{\kappa}_{x}\tilde{\mu}_{x}}^{T}\bm{z}),$
(3)
$C_{d}(\bm{\tilde{\kappa}_{x}})=\frac{\bm{\tilde{\kappa}_{x}}^{d/2-1}}{(2\pi)^{d/2}\mathcal{I}_{d/2-1}(\bm{\tilde{\kappa}_{x}})},$
(4)

where $\mathcal{I}_{d/2-1}(\cdot)$ denotes the modified Bessel function of the
first kind at order $(d/2-1)$. $\bm{\tilde{\mu}_{x}}$ and
$\bm{\tilde{\kappa}_{x}}$ are statistical parameters representing the
direction and concentration of the distribution. With this formulation, the
discriminative semantic context can be derived from the directional placement
on the hypersphere and hence get decoupled from task-irrelevant variational
attributes. Besides, the concentration parameter $\bm{\tilde{\kappa}_{x}}$
provides an intuitive measurement of semantic embedding ambiguity [40] and can
therefore act as an instance-wise temperature factor. It adaptively scales the
contributions of different image samples based on their representation
uncertainty to deliver dynamic rectifications on the latent space structuring
procedure, which is theoretically analyzed afterwards.

Base Space Bounding and Open Space Dispersion. With the modeled probabilistic
representations, the subsequent objective is to organically structure the
geometric layout of embedding space for discriminative novel class discovery.
Ideally, the space covered by well-defined base categories should be bounded
within a certain range while the unknown classes corresponding to the open
space should be modeled apart from the periphery of base space and dispersed
over the entire manifold [6]. We therefore propose to leverage the pre-defined
proxies to bound the base space and keep the open space isolated. For each
base class, we select one proxy as the class anchor and reformulate it with
Dirac delta $\delta$ to derive a statistical modeling over its deterministic
representation:

$\Delta(\bm{z}):=\left\\{\begin{aligned}
\int_{\mathbb{S}^{d-1}}\delta(\bm{z}-\bm{\hat{\upsilon}}_{\mathcal{B}})d\bm{z}&=1;\\\
\int_{\mathbb{S}^{d-1}}\delta(\bm{z}-\bm{\hat{\upsilon}}_{\mathcal{B}})\phi(\bm{z})d\bm{z}&=\phi(\bm{\hat{\upsilon}}_{\mathcal{B}}).\end{aligned}\right.$
(5)

Here $\bm{\hat{\upsilon}}_{\mathcal{B}}$ denotes the fixed positional values
of the proxies for base classes. It is noted that the selection of proxies is
random since they have been forced to be uniformly distributed on a symmetric
manifold and all proxy pairs have the same distance. Then, the proxies can
behave as desired latent priors to regularize the distribution of base
classes. Specifically, the optimization objective is to enforce the instance-
wise vMF distribution $q(\bm{z}|\bm{\tilde{\mu}_{x},\tilde{\kappa}_{x}})$
approximating the latent prior $\Delta(\bm{z})$ in terms of KL divergence:

$\mathop{\mathtt{min}}\limits_{q}\mathbb{E}_{\mathcal{D_{B}}}[-(\int_{\mathbb{S}^{d-1}}\Delta(\bm{z})\log
q(\bm{z}|\bm{\tilde{\mu}_{x},\tilde{\kappa}_{x}})d\bm{z})-\mathcal{H}_{\Delta}(\bm{z})],$
(6)

where $\mathcal{H}_{\Delta}(\bm{z})$ denotes the differential entropy over
$\Delta(\bm{z})$, which is a constant as $\Delta(\bm{z})$ is fixed a priori.
Combined with Eq. (3)-(5), the objective is equivalent to minimizing the
following loss function:

$\displaystyle\mathcal{L_{B}}(\bm{\tilde{\mu}_{x},\tilde{\kappa}_{x}})=-\int_{\mathbb{S}^{d-1}}[\bm{\tilde{\kappa}_{x}\tilde{\mu}_{x}}^{T}\bm{z}+\log
C_{d}(\bm{\tilde{\kappa}_{x}})]\Delta(\bm{z})d\bm{z}$ (7)
$\displaystyle=-\bm{\tilde{\kappa}_{x}\tilde{\mu}_{x}}^{T}\bm{\hat{\upsilon}}_{\mathcal{B}}-(d/2-1)\log\bm{\tilde{\kappa}_{x}}+\log\mathcal{I}_{d/2-1}(\bm{\tilde{\kappa}_{x}})+\check{\gamma},$

where $\check{\gamma}=(d/2)\log 2\pi$ is a constant constituent. The
regularization term constrains the intra-class variability and shrinks the
base space within a bounded radius. Thereafter, we propose open space
dispersion aimed at separating the space reserved for novel classes away from
the base one. The challenging aspect is how to recognize the unseen concepts
from the hybrid unlabeled set of data where base and novel classes
concurrently exist. In this regard, we first derive the magnitude of
probabilistic deviation as an indication of open-set instances and hereby
enforce large angular margins between those embeddings and the base space.
Specifically, we measure the probability density of each instance-wise vMF
distribution with respect to base class proxies and estimate the ranking over
$\mathcal{X_{U}}$:

$\mathop{argsort}\limits_{\\{\bm{x}_{1},...,\bm{x}_{N_{\mathcal{U}}}\\}\in\mathcal{X_{U}}}\Big{\\{}\mathop{\mathtt{max}}\limits_{\\{\bm{\hat{\upsilon}}_{\mathcal{B}_{i}},\forall
i\in\mathcal{S_{B}}\\}}\,C_{d}(\bm{\tilde{\kappa}_{x}})\exp(\bm{\tilde{\kappa}_{x}\tilde{\mu}_{x}}^{T}\bm{\hat{\upsilon}}_{\mathcal{B}_{i}})\Big{\\}}.$
(8)

The batch-wise instances with maximal divergence from the base class proxies
are hereafter pushed towards the open space:

$\mathcal{L}_{dis}(\bm{\tilde{\mu}_{x},\tilde{\kappa}_{x}})=-\log\frac{\exp(\bm{\tilde{\kappa}_{x}\tilde{\mu}_{x}}^{T}\bm{\hat{\upsilon}}_{\mathcal{R}})}{\exp(\bm{\tilde{\kappa}_{x}\tilde{\mu}_{x}}^{T}\bm{\hat{\upsilon}}_{\mathcal{R}})+\exp(\bm{\tilde{\kappa}_{x}\tilde{\mu}_{x}}^{T}\bm{\hat{\upsilon}}_{\mathcal{B}})},$
(9)

where
$\bm{\hat{\upsilon}}_{\mathcal{R}}=\bm{\hat{\Upsilon}}\setminus\bm{\hat{\upsilon}}_{\mathcal{B}}$
denotes the remaining proxies located outside of the base space. The
optimization objectives of Eq. (7) and (9) cooperatively promote _boundness_
on the underlying manifold. We then propose to explicitly introduce the
$uniformity$ inductive bias and accordingly structure the open space to
control its associated risk, which in turn appreciates the generalizability of
learned representations for recognizing unseen concepts.

Open Space Structuring. The dark information in the unknown space which has
not been properly characterized raises uncontrollable risks for novel classes
learning and discrimination [6]. To this end, we propose to structure the open
space organically with proxy-driven semantic consensus, which encourages the
constructed embeddings to be uniformly distributed over the manifold with
semantically-meaningful clusters such that the representations of different
novel classes can be spaced apart with large angular distance. Our intuition
is that instances with adjacent geometric placements on the underlying
manifold should possess analogous semantics and exhibit marginal
distributional discrepancy. Specifically, given each pair of unlabeled
instances $(\bm{x}_{\alpha},\bm{x}_{\beta})$ and corresponding probabilistic
modeling
$q_{\theta}(\bm{z}_{\alpha}|\bm{x}_{\alpha})\sim\mathbf{vMF}(\bm{\tilde{\mu}}_{\bm{x}_{\alpha}},\bm{\tilde{\kappa}}_{\bm{x}_{\alpha}}),q_{\theta}(\bm{z}_{\beta}|\bm{x}_{\beta})\sim\mathbf{vMF}(\bm{\tilde{\mu}}_{\bm{x}_{\beta}},\bm{\tilde{\kappa}}_{\bm{x}_{\beta}})$,
we measure their distributional overlaps with all pre-set proxies as
$\mathcal{O}_{(\alpha,\beta)}^{\bm{\hat{\Upsilon}}}=\\{C_{d}(\bm{\tilde{\kappa}}_{\bm{x}})\exp(\bm{\tilde{\kappa}}_{\bm{x}}\bm{\tilde{\mu}}_{\bm{x}}^{T}\bm{\hat{\upsilon}}_{i}),\forall\bm{\hat{\upsilon}}_{i}\in\bm{\hat{\Upsilon}},\bm{x}\in(\bm{x}_{\alpha},\bm{x}_{\beta})\\}$,
which indicate the statistical distances between instance embeddings and
proxies. Since the proxies have been priorly positioned covering the entire
manifold with uniformity guarantees, the derived
$\mathcal{O}_{\alpha}^{\bm{\hat{\Upsilon}}}$ and
$\mathcal{O}_{\beta}^{\bm{\hat{\Upsilon}}}$ can comprehensively characterize
the geometric location of instance embeddings. As such, we suggest
$\bm{x}_{\alpha}$ is semantically correlated with $\bm{x}_{\beta}$ and their
distributional discrepancy should be minor if
$\mathcal{O}_{\alpha}^{\bm{\hat{\Upsilon}}}\doteq\mathcal{O}_{\beta}^{\bm{\hat{\Upsilon}}}$.
Formally, the condition holds true if
$\mathtt{arg\,max_{topk}}(\mathcal{O}_{\alpha}^{\bm{\hat{\Upsilon}}})=\mathtt{arg\,max_{topk}}(\mathcal{O}_{\beta}^{\bm{\hat{\Upsilon}}})$.
Let $\mathcal{G}$ denote the collection of instance pairs satisfying the above
condition. In these cases, we aim at minimizing the discrepancy between the
two vMF distributions to escalate their semantic affinities. We resort to
modeling the distributional discrepancy with KL divergence and accordingly
derive the following optimization objective:

$\mathop{\mathtt{min}}\limits_{q}\mathbb{E}_{\mathcal{G}}[-\int_{\mathbb{S}^{d-1}}q(\bm{z}|\bm{\tilde{\mu}}_{\bm{x}_{\alpha}},\bm{\tilde{\kappa}}_{\bm{x}_{\alpha}})\log(\frac{q(\bm{z}|\bm{\tilde{\mu}}_{\bm{x}_{\beta}},\bm{\tilde{\kappa}}_{\bm{x}_{\beta}})}{q(\bm{z}|\bm{\tilde{\mu}}_{\bm{x}_{\alpha}},\bm{\tilde{\kappa}}_{\bm{x}_{\alpha}})})d\bm{z}].$
(10)

Combined with Eq. (3) and (4), we can derive the corresponding regularization
term as follows:

$\displaystyle\mathcal{L}_{str}(\bm{\tilde{\mu}}_{\bm{x}_{\alpha}}$
$\displaystyle,\bm{\tilde{\kappa}}_{\bm{x}_{\alpha}},\bm{\tilde{\mu}}_{\bm{x}_{\beta}},\bm{\tilde{\kappa}}_{\bm{x}_{\beta}})=-[\frac{\mathcal{I}_{d/2}(\bm{\tilde{\kappa}}_{\bm{x}_{\alpha}})}{\mathcal{I}_{d/2-1}(\bm{\tilde{\kappa}}_{\bm{x}_{\alpha}})}\cdot$
(11) $\displaystyle(\bm{\tilde{\kappa}}_{\bm{x}_{\alpha}}-$
$\displaystyle\bm{\tilde{\kappa}}_{\bm{x}_{\beta}}\bm{\tilde{\mu}}_{\bm{x}_{\alpha}}\bm{\tilde{\mu}}_{\bm{x}_{\beta}}^{T})+\log(\frac{C_{d}(\bm{\tilde{\kappa}}_{\bm{x}_{\alpha}})}{C_{d}(\bm{\tilde{\kappa}}_{\bm{x}_{\beta}})})+1]^{-1}.$

The constraints advocate explicit associations between the geometric layout of
embedding manifold and the semantic representations of instances, which in
turn systematically structure the open space. By integrating the optimization
objective along with the ones in Eq. (7) and (9), we jointly incorporate the
_boundness_ and _uniformity_ inductive biases to shape the learned embedding
space and thereby favor its generalization to unseen biomedical concepts. The
overall training procedure is summarized in the supplementary.

###  3.3 Spectral-Based Class Number Estimation

With the well-structured embedding space, the subsequent target is to estimate
the number of classes in unlabeled data and accordingly perform clustering to
assign class label for each instance in $\mathcal{D_{U}}$. Different from
previous methods counting on the results of empirical evaluations and
introducing costly recursive clustering [44], we propose a spectral graph-
theoretic approach based upon the finding that the eigenvalues of graph
Laplacians inherently characterize the underlying connectivity of the latent
manifold [15]. Specifically, we construct a weighted graph over all instances
of $\mathcal{D_{U}}$, in which each node corresponds to the embedding
$\bm{z}_{i}$ of a data point, and the edge weights $\mathbf{W}$ model the
correlations between pairs of instances as $\\{\bm{z}_{i}\bm{z}_{j}^{T}\\}$.
Let $\mathbf{\Lambda}$ denote the diagonal matrix with the row-wise sum of
$\mathbf{W}$ as its entries, the optimization objective for graph partitioning
with minimized energy is:

$\mathop{\mathtt{min}}\limits_{\bm{h}}\,\frac{\bm{h}^{T}\mathbf{\Lambda}^{-1/2}(\mathbf{\Lambda}-\mathbf{W})\mathbf{\Lambda}^{-1/2}\bm{h}}{\bm{h}^{T}\bm{h}},$
(12)

which is equivalent to solve
$\mathbf{\Lambda}^{-1/2}(\mathbf{\Lambda}-\mathbf{W})\mathbf{\Lambda}^{-1/2}\bm{h}=\lambda\bm{h}$.
The results can be intrinsically depicted by the eigenvectors of the
normalized graph Laplacian
$\mathbf{\Lambda}^{-1/2}(\mathbf{\Lambda}-\mathbf{W})\mathbf{\Lambda}^{-1/2}$
[2]. In this regard, the eigenvectors can behave as soft segments to partition
the overall graph into a set of semantically-meaningful splits, while the
corresponding eigenvalues can be considered as indicators for the inner-
connectivity of potential clusters. In other words, instances in splits with
higher eigenvalues exhibit stronger internal semantic affinities and are more
plausible to be assigned into the same class. We therefore propose to estimate
the number of latent classes according to the spectral property of constructed
graph that we rank all possible graph splits with their associated eigenvalues
and identify the split index which corresponds to the largest gap within the
ordering of eigenvalue differences
$\\{argsort\;\mathtt{diff}(eigenvalues)\\}$. The identified index suggests the
start of graph splits which contain limited inner-connectivity and should not
be considered as individual classes. The index is thereby modeled as the
estimated number of classes. Illustrative examples are presented in the
supplementary material. We then perform k-means clustering over
$\mathcal{D_{U}}$ in the hyperspherical space with the estimated class number
for label assignment. Notably, taxonomy-adaptive estimation could be in demand
that it is required to gauge the number of potential fine-grained subclasses.
Our method is well suited for this scenario as it can be adapted to estimate
the inherent subclasses by loosening the above condition and instead indexing
the subordinately largest eigenvalue gap according to the specific need in
class hierarchy.

###  3.4 Theoretical Analysis

In this section, we motivate our method theoretically and provide analytical
insights to validate its merits.

Monotonicity between Distributional Concentration and Semantic Ambiguity. With
the intrinsic semantic ambiguity present in biomedical visual data due to
disparate imaging protocols across cohorts, our probabilistic treatment, which
advocates for vMF statistical modeling, adeptly characterizes the ambiguous
attributes with distributional concentration $\tilde{\kappa}$:

###### Proposition 1

Let $\zeta_{\bm{x}}$ be the continuous entropy of the posterior vMF
distribution parametrized by $\bm{\tilde{\mu}_{x}}\in\mathbb{S}^{d-1}$ and
$\bm{\tilde{\kappa}_{x}}\in\mathbb{R}^{d}_{>0}$. We have
$\zeta_{\bm{x}}(\bm{\tilde{\kappa}_{x}})$ behave as a monotonically decreasing
function in the interval $(0,+\infty)$.

It suggests the statistical parameter $\bm{\tilde{\kappa}_{x}}$ essentially
measures the aleatoric uncertainty of the semantic context in data point
$\bm{x}$. Combined with the analysis of $\bm{\tilde{\kappa}_{x}}$ on
distributional regularization terms, as presented in the supplementary
material, it is demonstrated the probabilistic modeling dynamically calibrates
the biased learning procedure towards discriminative contextual information
modeling.

Towards Bounded Open Space Risk with Uniform Proxies. The most challenging
aspect in the GNCD problem is how to demystify the dark information regarding
unknown class representations in order to restrain the associated open space
risk [36]. We hereby prove the effectiveness of our proposed method towards a
tighter error bound for open space risk characterization and regularization.
Details can be found in the supplemental material.

Table 1: Comparison results of our proposed method against other state-of-the-art methods for discovering novel types of pneumonia infectious organisms and cell nuclei. The subscripts denote the standard deviations. The best and second-best results are highlighted in bold and brown, respectively.  Methods | Pneumonia Infectious Organisms | Cell Nuclei
---|---|---
$\mathsf{Acc}$-all |  $\mathsf{Acc}$-known |  $\mathsf{Acc}$-novel |  $\mathsf{F_{1}}$-all |  $\mathsf{F_{1}}$-known |  $\mathsf{F_{1}}$-novel |  $\mathsf{Acc}$-all |  $\mathsf{Acc}$-known |  $\mathsf{Acc}$-novel |  $\mathsf{F_{1}}$-all |  $\mathsf{F_{1}}$-known |  $\mathsf{F_{1}}$-novel
RankStats+ |  41.320.90 |  40.570.60 | 46.672.37 |  28.961.56 |  33.451.13 | 24.483.13 |  36.250.80 |  36.170.82 | 39.840.65 |  27.170.68 |  42.450.74 | 4.260.41
UNO+ |  49.591.48 |  50.060.92 | 46.311.49 |  31.241.26 |  36.352.11 | 26.142.70 |  41.830.92 |  41.590.74 | 47.831.22 |  33.040.58 | 45.790.69 | 13.540.52
GCD |  67.770.83 |  69.811.45 | 53.340.67 |  38.061.04 |  41.261.30 | 32.871.95 |  45.431.78 |  45.571.26 | 43.732.43 |  38.191.47 | 44.521.05 | 29.642.84
DCCL | 72.732.14 | 74.482.28 | 60.322.82 | 44.821.63 | 49.851.95 | 39.794.08 |  43.931.34 |  43.591.86 | 54.490.53 |  37.931.14 |  40.162.04 | 36.570.77
SLF |  43.810.67 |  42.310.44 | 54.261.12 |  30.721.39 |  32.860.97 | 28.572.86 |  39.150.86 |  39.071.04 | 43.060.69 |  28.950.95 |  44.061.02 | 18.420.81
BYOP |  65.520.95 |  68.871.14 | 58.952.54 |  40.011.02 |  43.121.30 | 38.523.72 | 46.280.68 | 46.020.57 | 55.821.44 | 39.060.59 |  41.010.30 | 35.631.86
Ours | 78.511.05 | 80.191.02 | 66.672.70 | 46.731.47 | 44.270.31 | 49.183.87 | 47.781.28 | 47.411.53 | 57.850.42 | 42.640.36 |  41.480.25 | 43.272.29
Table 2: Comparison results for discovering novel types of skin cancer lesions and degrees of diabetic retinopathy severity.  Methods | Skin Cancer Lesions | Diabetic Retinopathy Severity
---|---|---
$\mathsf{Acc}$-all |  $\mathsf{Acc}$-known |  $\mathsf{Acc}$-novel |  $\mathsf{F_{1}}$-all |  $\mathsf{F_{1}}$-known |  $\mathsf{F_{1}}$-novel |  $\mathsf{Acc}$-all |  $\mathsf{Acc}$-known |  $\mathsf{Acc}$-novel |  $\mathsf{F_{1}}$-all |  $\mathsf{F_{1}}$-known |  $\mathsf{F_{1}}$-novel
RankStats+ |  29.191.03 |  28.650.79 | 33.392.04 |  21.061.60 |  36.430.95 | 9.532.84 |  47.210.94 |  47.900.58 | 41.641.50 |  40.820.69 |  54.780.46 | 19.970.82
UNO+ |  37.802.01 |  37.431.54 | 40.742.38 |  29.161.49 |  40.681.41 | 20.371.82 |  50.430.85 |  51.070.97 | 39.260.68 |  36.570.84 |  52.450.57 | 16.091.23
GCD |  31.581.44 |  49.461.29 | 39.581.06 |  27.180.82 |  39.170.53 | 18.202.25 |  48.432.36 |  49.052.04 | 44.331.22 |  41.691.06 |  53.390.85 | 21.130.93
DCCL |  56.450.78 |  58.920.43 | 37.841.37 |  33.461.14 |  53.460.59 | 18.481.50 |  44.351.58 |  44.931.44 | 46.072.27 |  43.231.55 |  52.161.40 | 27.511.89
SLF |  44.561.29 |  45.140.73 | 39.020.67 |  30.781.77 |  49.750.84 | 16.551.96 |  46.290.99 |  46.930.65 | 44.401.40 |  40.411.13 |  50.920.54 | 20.101.72
BYOP | 65.790.62 | 69.460.44 | 36.651.51 | 35.941.14 | 54.190.71 | 23.041.70 | 51.671.24 | 52.311.27 | 43.630.93 | 44.850.82 | 55.681.23 | 25.560.90
Ours | 69.620.96 | 73.240.32 | 41.670.55 | 41.390.29 | 60.410.40 | 27.120.48 | 58.411.35 | 59.621.60 | 50.520.51 | 47.250.77 | 55.940.56 | 34.220.64

##  4 Experiments

###  4.1 Experimental Setup

Datasets. To evaluate the effectiveness and general applicability of our
method under miscellaneous biomedical discovery scenarios, we conduct
experiments on a diverse suite of challenging tasks encompassing various
biomedical concepts (cell, lesion, disease), imaging modalities (X-ray,
microscopy, dermatoscopy, fundus photography), and pathologies (lung
infection, skin cancer, eye complication). For each task, we adopt two
different datasets with evident data distribution shifts to serve as the
labeled base set and the unlabeled set comprising novel classes, respectively.
In specific, we consider the discovery of novel types or degrees of pneumonia
infectious organisms [25, 10], cell nuclei [17, 18], skin lesions [43, 9], and
diabetic retinopathy severity [28, 24]. The quantity of instances belonging to
novel classes is inherently much smaller compared with their base counterpart
and naturally formalize the long-tailed class distribution. More details can
be found in the supplemental material, where extended studies on general
concept discovery are further presented and analyzed.

Implementation Details and Evaluation Protocol. We adopt the ViT-B/16 [12]
with DINO pre-trained weights [5] as the backbone network for fair comparisons
with the previous methods [44, 36]. Its output of $\mathtt{[CLS]}$ token with
a dimension of 768 is used for latent embedding. Please refer to the attached
code for details.

For evaluation, we measure the clustering accuracy between the predicted class
assignment and the ground-truth label. Following the protocol in [44], we
match the two class sets by searching for the optimal permutation using the
Hungarian assignment algorithm [26]. We adopt the accuracy and F1 scores as
metrics and report the averaged score across the entire class set.
Particularly, we also present the scores specific to the base known
$\mathcal{S_{B}}$ and novel $\mathcal{S}_{N}$ subset of classes in a separated
manner.

Figure 3: Results of class number estimation in the unlabeled set. For
pneumonia and cell nuclei, we further estimate the number of fine-grained
subclasses.

###  4.2 Results and Discussions

Main Results. We compare our method with state-of-the-art approaches for novel
class discovery (NCD), including RankStats+ [21], UNO+ [16], GCD [44], and
DCCL [36] which are specifically devised for the challenging generalized NCD
setting, as well as SLF [50] and BYOP [48] which take account of the
non-i.i.d. data and non-uniform label distribution issues, respectively. The
overall quantitative results over four diverse biomedical concept discovery
tasks are presented in Table 1 and 2. All methods are implemented based upon
the same ViT-B/16 backbone with DINO pre-trained weights [44]. It can be
observed that our proposed method consistently and significantly outperforms
the state-of-the-art novel class discovery approaches over all four tasks.
Notably, as highlighted in the gray columns which correspond to the clustering
metrics specific to novel classes, our method surpasses the comparison
approaches by a substantial margin especially on the identification and
grouping of unseen phenotypes. The results demonstrate the advantages and
effectiveness of the proposed geometry-constrained probabilistic modeling for
discovering novel concepts despite the data and label distribution biases
intrinsically present in the biomedical domain.

We further assess the efficacy of the spectral-based class number estimation
method, as presented in Fig. 3. The proposed approach attains promising
results that the estimated number is identical or very close to the ground-
truth value. The versatile method also holds an appealing merit that it
supports taxonomy-adaptive estimation, which means it is capable of estimating
the number of potential fine-grained subclasses, a practical aspect not
previously considered in the literature at all. Additionally, it is noteworthy
that our method does not rely on recursive clustering as in [44] and is
therefore computationally efficient. On the pneumonia benchmark, our method
only takes 0.121s, which is more than $\mathbf{10\times}$ faster than the
comparison approach [44] consuming 2.854s.

Table 3: Key component analysis of our method. $\mathtt{Uni}$, $\mathtt{Bnd}$, $\mathtt{Disp}$, and $\mathtt{Str}$ correspond to uniform proxies, base space bounding, open space dispersion and structuring, respectively.  Components | Pneumonia | Cell Nuclei
---|---|---
$\mathtt{Uni}$ | $\mathtt{Bnd}$ | $\mathtt{Disp}$ | $\mathtt{Str}$ |  $\mathsf{F_{1}}$-all |  $\mathsf{F_{1}}$-known |  $\mathsf{F_{1}}$-novel |  $\mathsf{F_{1}}$-all |  $\mathsf{F_{1}}$-known |  $\mathsf{F_{1}}$-novel
|  |  |  | 22.82 | 27.37 | 18.28 | 16.21 | 20.76 | 8.64
$\checkmark$ | $\checkmark$ |  |  | 35.84 | 48.14 | 23.53 | 34.30 | 38.76 | 27.72
$\checkmark$ | $\checkmark$ | $\checkmark$ |  | 36.92 | 45.89 | 27.94 | 33.02 | 31.83 | 33.47
$\checkmark$ | $\checkmark$ |  | $\checkmark$ | 28.71 | 36.51 | 20.91 | 38.23 | 40.49 | 35.59
| $\checkmark$ | $\checkmark$ | $\checkmark$ | 40.23 | 37.69 | 31.42 | 31.97 | 34.85 | 29.74
$\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | 46.73 | 44.27 | 49.18 | 42.64 | 41.48 | 43.27

Ablation of Key Components. Table 3 presents the ablation study results to
demonstrate the effectiveness of key components in our proposed method. It can
be remarked that each component is indispensable and holds a positive impact
on novel class discovery. Notably, the combination of pre-defined uniform
proxies and base space bounding reaches the peak accuracy in terms of
$\mathsf{F_{1}}$-known on the pneumonia infectious organism benchmark.
However, it suffers from inferior efficacy in identifying and clustering
potential novel concepts. The observation is in line with our argument that
the consideration and explicit modeling of the dark information regarding
unknown classes are crucial to ensure the generalizability of learned
representations for discovering unseen concepts. Solely imposing constraints
on the base space is insufficient to calibrate the learning process biased
towards the dominant known classes. In contrast, our method proposes two
supplementary geometric constraints which cooperatively shape the layout of
embedding space and compensate for the open space risk, which in turn leads to
unbiased and discriminative modeling of novel class representations.

Figure 4: Hyperspherical and distributional embeddings of the skin lesion test
set for two variants of our proposed method. For distributional visualization,
we overlay data points over the super-level-set ellipses of the associated
probabilistic distributions.

Impacts on Geometric Layout of Embedding Space. To perspicuously illustrate
the effectiveness of our method towards shaping the geometric structure of
embedding space, we visualize the learned representations on the 3D spherical
manifold and its associated probabilistic characterizations, as shown in Fig.
4. The covariance structure of data points is modeled as a group of super-
level-set ellipses to indicate the uncertainty of constructed clusters. We
observe that the removal of open space dispersion and structuring inevitably
deteriorates the degree of inter-class angular separation and results in
ambiguous semantic representations. The comparative analysis between the two
variants of our proposed method substantiates the significance of
incorporating geometric inductive biases to regulate the underlying
organization of the embedding space.

##  5 Conclusion

In this work, we propose a geometry-constrained probabilistic modeling
framework to address challenging aspects of novel class discovery in the
biomedical domain. Specifically, we suggest to model the latent representation
of each instance as a directional distribution to exclude the task-irrelevant
data variations arising from inconsistent imaging protocols across cohorts. We
then identify several geometric properties critical for organic embedding
space structuring. By explicitly integrating those principles to regularize
representation layout, the open space risk associated with the discovery of
novel classes can be effectively bounded. Additionally, a spectral graph-
theoretic algorithm is developed to estimate the number of categories in
unlabeled data, which inherits taxonomy adaptivity and high efficiency. Our
method exhibits appealing performance on a diverse suite of novel concept
discovery applications in biomedical domains.

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# Charged-hadron suppression in Pb+Pb and Xe+Xe collisions measured with the
ATLAS detector

Petr Balek (for the ATLAS Collaboration)  Dept. of Particle Physics and
Astrophysics, Faculty of Physics, Weizmann Institute of Science, 234 Herzl
Street, Rehovot 76100, Israel

###### Abstract

The ATLAS detector at the LHC recorded 0.49 nb-1 of Pb+Pb collisions and 25
pb-1 of $pp$ collisions, both at the center-of-mass energy 5.02 TeV per
nucleon pair. Recently, ATLAS also recorded 3 $\mu$b-1 of Xe+Xe collisions at
the center-of-mass energy 5.44 TeV, which provides a new opportunity to study
the system-size dependence of the charged-hadron production in heavy-ion
collisions. The large acceptance of the ATLAS detector allows to measure the
spectra of charged hadrons in a wide range of pseudorapidity and transverse
momentum. The nuclear modification factors $R_{\mathrm{AA}}$ are constructed
as a ratio of the spectra measured in Pb+Pb or Xe+Xe collisions to that
measured in $pp$ collisions. The $R_{\mathrm{AA}}$ obtained in the two systems
are presented for different centrality intervals and the results are
discussed.

keywords: xenon–xenon collisions, lead–lead collisions, charged-hadron
production, nuclear modification factor

##  1 Introduction

The Xe+Xe collisions delivered by the LHC in 2017 offer a unique opportunity
to study properties of the quark-gluon plasma in systems with different
geometries [1, 2]. Previous measurements [3, 4] show that the yields of
charged hadrons are suppressed in the Pb+Pb collisions relative to the $pp$
collisions in a centrality-dependent way, when accounted for an increased
parton flux in the Pb+Pb collisions. The new Xe+Xe data allow to study the
centrality dependence of this suppression at a whole new angle.

The suppression of charged-hadron production is quantified using the nuclear
modification factor $R_{\mathrm{AA}}$:

$\mbox{$R_{\mathrm{AA}}$}=\frac{1}{\mbox{$\langle\mbox{$T_{\mathrm{AA}}$}\rangle$}}\frac{1/N_{\mathrm{evt}}\
\mathrm{d}^{2}N_{\mathrm{ch}}/\mathrm{d}\eta\mathrm{d}\mbox{$p_{\mathrm{T}}$}}{\mathrm{d}^{2}\sigma_{pp}/\mathrm{d}\eta\mathrm{d}\mbox{$p_{\mathrm{T}}$}},$
(1)

where $\langle\mbox{$T_{\mathrm{AA}}$}\rangle$ is the nuclear thickness
function which accounts for the fact that in a nucleus–nucleus collision, a
nucleon can interact with more than one nucleon from the other nucleus;
$1/N_{\mathrm{evt}}\
\mathrm{d}^{2}N_{\mathrm{ch}}/\mathrm{d}\eta\mathrm{d}\mbox{$p_{\mathrm{T}}$}$
is the per-event yield of charged hadrons in Xe+Xe or Pb+Pb collisions
measured differentially in pseudorapidity $\eta$ and transverse momentum
$p_{\mathrm{T}}$; and
$\mathrm{d}^{2}\sigma_{pp}/\mathrm{d}\eta\mathrm{d}\mbox{$p_{\mathrm{T}}$}$ is
the differential $pp$ cross-section.

##  2 Analysis

The measurement [5] described in this proceeding uses Xe+Xe data recorded by
the ATLAS detector [6] at $\mbox{$\sqrt{s_{\mathrm{NN}}}$}=5.44$ TeV with the
total integrated luminosity of 3 $\mu$b-1. The $pp$ cross-section is obtained
by extrapolation of $\mbox{$\sqrt{s}$}=5.02$ TeV data [4] to the same center-
of-mass energy.

The measurement is performed using the inner detector, calorimeters, muon
spectrometer, trigger system and data acquisition system. The tracking
information is provided by the inner detector covering $|\eta|<2.5$. It is
immersed in a 2 T axial magnetic field. The calorimeter system consists of an
electromagnetic calorimeter covering $|\eta|<3.2$, hadronic calorimeters
covering also $|\eta|<3.2$ and forward calorimeters covering $3.1<|\eta|<4.9$.
The muon spectrometer covers $|\eta|<2.7$. The Xe+Xe events were recorded with
two minimum-bias triggers. They required the total transverse energy deposited
in the calorimeters to be more than 4 GeV or to have at least one track
reconstructed in the inner detector.

The centrality of the collisions is characterized by the total transverse
energy in the forward calorimeters (FCal $E_{\mathrm{T}}$), whose distribution
is divided into percentiles of the inelastic cross-section. If the nuclei
overlap significantly, the collision is called “central”, while collisions
with a small overlap are called “peripheral”. A Monte Carlo Glauber model
simulation [7, 8] is used to estimate the mean number of nucleons
participating in the collision, $\langle\mbox{$N_{\mathrm{part}}$}\rangle$,
the mean number of binary nucleon–nucleon collisions,
$\langle\mbox{$N_{\mathrm{coll}}$}\rangle$, the nuclear thickness function,
$\langle\mbox{$T_{\mathrm{AA}}$}\rangle$, as well as their uncertainties.

A particle emerging from the interaction point and passing through the inner
detector typically crosses 4 layers of the pixel detector, 4 double-sided
modules of the semiconductor tracker (SCT) and around 36 straw tubes of the
transition radiation tracker. Reconstructed tracks are required to have at
least 9 (11) hits, if they are within $|\eta|\leq 1.65$ ($|\eta|>1.65$). At
least one hit is required to be in one of the two innermost layers of the
pixel detector, if the tracks passed through active sensors. Tracks shall not
have any missing hits in the pixel or SCT detectors if such hits are expected
from the track trajectory. Tracks are also required to emerge from the
collisions vertex. Tracks with $\mbox{$p_{\mathrm{T}}$}>40$ GeV are further
required to be matched to a jet within $\Delta
R=\sqrt{\Delta^{2}\eta+\Delta^{2}\phi}<0.4$. The jets are reconstructed in the
hadronic calorimeters using the anti-$k_{t}$ algorithm [9] with the radius
parameter of $R=0.4$. Tracks are required not to exceed the $p_{\mathrm{T}}$
of the matched jets by more than 30% in order to suppress mis-measured tracks.
Such tracks are suppressed by enforcing conservation of energy, however track
and jet momentum resolutions are taken into account as well.

Monte Carlo simulations are used to study the detector response effects. Hard-
scattering $pp$ collisions generated by Pythia 8 [10] are overlaid onto Xe+Xe
collisions produced by Hijing [11]. The resulting events are reconstructed in
the same way as data. A total of $3\cdot 10^{6}$ events are generated in
different exclusive kinematic intervals of leading charged-hadron
$p_{\mathrm{T}}$, allowing sufficient statistics over the whole
$p_{\mathrm{T}}$ range.

There are several corrections applied to the measured spectra. First, leptons
from the decays of electroweak bosons are subtracted as they do not follow the
same suppression pattern as hadrons [12]. Then, secondary and fake tracks are
subtracted. The former ones are tracks matched to secondary particles, and the
later ones are the tracks that are coming from the spurious combination of
hits not associated with a single particle. Their fraction is estimated from
the simulations. It does not exceed 1% at $\mbox{$p_{\mathrm{T}}$}\approx 1$
GeV and is even less at $\mbox{$p_{\mathrm{T}}$}\gtrsim 1$ GeV. The spectra
are also corrected for the $p_{\mathrm{T}}$ resolution and for the track
reconstruction efficiency by the bin-by-bin unfolding. The efficiency, which
is also estimated from the simulations, is about 75% at
$\mbox{$p_{\mathrm{T}}$}\approx 1$ GeV, $|\eta|\lesssim 1$ and in peripheral
collisions. At $|\eta|\approx 2.5$, the efficiency decreases down to about
60%. Another reduction, which is less than 15%, is observed in the most
central collisions. A small increase of the efficiency with increasing
$p_{\mathrm{T}}$ is also present, however it is no more than 5%.

The $pp$ cross-section measured at $\mbox{$\sqrt{s}$}=5.02$ TeV is
extrapolated to $\mbox{$\sqrt{s}$}=5.44$ TeV by the ratio of the samples
generated by Pythia 8 with $1.9\cdot 10^{7}$ events in each energy regime. The
ratio shows an increase of the cross-section section by about 4% at
$\mbox{$p_{\mathrm{T}}$}\approx 1$ GeV and up to 26% at the highest
$p_{\mathrm{T}}$ and $|\eta|$ measured.

There are several sources of the systematic uncertainties affecting the
results. The analysis parameters are varied independently and the resulting
outcomes are compared to that of the default setup. The correlated components
are varied consistently in numerator and denominator in order to estimate the
uncertainty on $R_{\mathrm{AA}}$. Variation of the track selection
requirements introduces an uncertainty not exceeding 5%. The analysis
corrections depend on a matching of the reconstructed tracks to the generated
particles. The uncertainty covering ambiguities in the matching procedure is
about 1%. The bin-by-bin correction uncertainty has three sources. Limited
statistics of the simulation samples yield an uncertainty of no more than 7%.
The difference of the shape of charged-hadron spectra in data and Pythia
results in an uncertainty of 2%. Due to the limited description of the
inactive material of the detector, an uncertainty of up to 6% has to be
assigned. An uncertainty of the geometric parameter
$\langle\mbox{$T_{\mathrm{AA}}$}\rangle$ is largest for the peripheral
collisions where it reaches about 8%. In the central collisions, it is less
than 1%. A half of the difference between the $pp$ cross-sections at
$\mbox{$\sqrt{s}$}=5.02$ TeV and 5.44 TeV is assigned as a systematic
uncertainty of the extrapolation.

##  3 Results

The left panel of Fig. 1 shows the nuclear modification factors,
$R_{\mathrm{AA}}$, for Xe+Xe and Pb+Pb collisions in the same centrality
intervals. They have a characteristic curvature which is more pronounced in
the central collisions. Curves reach a maximum at
$\mbox{$p_{\mathrm{T}}$}\approx 2$ GeV, then a minimum at around 7 GeV and
then increase up to around 60 GeV. The behavior of $R_{\mathrm{AA}}$ in Xe+Xe
collisions above this value is difficult to ascertain due to the low
statistics. In Pb+Pb collisions, the slope of $R_{\mathrm{AA}}$ above
$\mbox{$p_{\mathrm{T}}$}\approx 60$ GeV diminishes. The stronger suppression
in Pb+Pb than in Xe+Xe collisions for the same centrality intervals is
expected because size of Pb+Pb collision system is larger than that of Xe+Xe
collision.

Figure 1: Nuclear modification factors $R_{\mathrm{AA}}$ as a function of
$p_{\mathrm{T}}$ measured in Xe+Xe collisions (closed markers) and in Pb+Pb
collisions (open markers). The intervals of the same marker styles have the
same centrality (left) or comparable deposited energy in the forward
calorimeter (right). The statistical uncertainties are shown as the bars;
systematic uncertainties are shown by the brackets.

Figure 2: Nuclear modification factors $R_{\mathrm{AA}}$ as a function of
$p_{\mathrm{T}}$ measured in Xe+Xe collisions (closed markers) and in Pb+Pb
collisions (open markers). The centrality intervals of the same marker styles
have comparable $\langle\mbox{$N_{\mathrm{part}}$}\rangle$ (left) or
$\langle\mbox{$N_{\mathrm{coll}}$}\rangle$ (right). The statistical
uncertainties are shown as the bars; systematic uncertainties are shown by the
brackets.

The right panel of Fig. 1 shows $R_{\mathrm{AA}}$ for Xe+Xe and Pb+Pb
collisions in centrality intervals corresponding to approximately the same
FCal $E_{\mathrm{T}}$. Collisions with the same FCal $E_{\mathrm{T}}$ have
about the same size. The observed suppressions are consistent between the two
systems within the systematics uncertainties, suggesting scaling with the
system size.

Figure 2 shows comparison of nuclear modification factors for Xe+Xe and Pb+Pb
collisions for centrality intervals of similar
$\langle\mbox{$N_{\mathrm{part}}$}\rangle$ (left) and
$\langle\mbox{$N_{\mathrm{coll}}$}\rangle$ (right). The production rate of
low-$p_{\mathrm{T}}$ (high-$p_{\mathrm{T}}$) particles is rather proportional
to $N_{\mathrm{part}}$ ($N_{\mathrm{coll}}$), and therefore the size of the
two systems is expected to be comparable at similar $N_{\mathrm{part}}$.
However, the agreement between the systems is still worse than in the right
panel of Fig. 1. At $p_{\mathrm{T}}$ around 7 GeV, the Xe+Xe results on the
left panel of Fig. 2 show slightly stronger suppression for the central
events, but slightly milder suppression for peripheral events. This feature is
demonstrated in Fig. 3 where it is clearly visible. At higher $p_{\mathrm{T}}$
(26–30 GeV), the suppressions are comparable within uncertainties and
$R_{\mathrm{AA}}$ measured in both Xe+Xe and Pb+Pb collisions follow the same
dependency, which suggests the suppression scales with the system size.

Figure 3: Nuclear modification factors $R_{\mathrm{AA}}$ as a function of
$N_{\mathrm{part}}$ for two selected $p_{\mathrm{T}}$ ranges measured in Xe+Xe
collisions (closed markers) and in Pb+Pb collisions (open markers). The
statistical uncertainties are shown as the bars, and systematic uncertainties
are shown by the brackets. The width of the brackets represents the systematic
uncertainty of $N_{\mathrm{part}}$. The lines are only to help guide the eye.

##  4 Summary & Acknowledgements

Measurement of charged-hadron spectra and the nuclear modification factor in
Xe+Xe collisions has been presented. The $R_{\mathrm{AA}}$ is compared between
the Xe+Xe collisions at $\mbox{$\sqrt{s_{\mathrm{NN}}}$}=5.44$ TeV and Pb+Pb
collisions at $\mbox{$\sqrt{s_{\mathrm{NN}}}$}=5.02$ TeV measured by the ATLAS
detector at the LHC.

The data suggest that $R_{\mathrm{AA}}$ scales with the system size. Other
aspects of the collisions, such as center-of-mass energy or initial energy
density, may not play a significant role for the comparison presented in this
proceedings. However, they may become important when comparing collisions at
the LHC energies to those at e.g. RHIC energies.

This research is supported by the Israel Science Foundation (grant 1065/15)
and by the MINERVA Stiftung with the funds from the BMBF of the Federal
Republic of Germany.

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