# Reuse your features: unifying retrieval and feature-metric alignment

Javier Morlana and J.M.M. Montiel This work was supported by the EU-H2020
grant 863146: ENDOMAPPER, the Spanish government grants PGC2018-096367-B-I00,
and by Aragón government grant DGA_T45-17R.The authors are with the Instituto
de Investigación en Ingeniería de Aragón (I3A), Universidad de Zaragoza, María
de Luna 1, 50018 Zaragoza, Spain. E-mail: {jmorlana<EMAIL_ADDRESS>
###### Abstract

We propose a compact pipeline to unify all the steps of Visual Localization:
image retrieval, candidate re-ranking and initial pose estimation, and camera
pose refinement. Our key assumption is that the deep features used for these
individual tasks share common characteristics, so we should reuse them in all
the procedures of the pipeline. Our DRAN (Deep Retrieval and image Alignment
Network) is able to extract global descriptors for efficient image retrieval,
use intermediate hierarchical features to re-rank the retrieval list and
produce an intial pose guess, which is finally refined by means of a feature-
metric optimization based on learned deep multi-scale dense features.

DRAN is the first single network able to produce the features for the three
steps of visual localization. DRAN achieves a competitive performance in terms
of robustness and accuracy specially in extreme day-night changes.

##  I Introduction

Feature extraction is a relevant step in most of computer vision tasks.
Traditional approaches rely on image gradients to extract _sparse features_ ,
for example, edges or keypoints in a fixed hand-crafted manner. In the last
decade, deep learning has taken the spot in feature extraction, with
Convolutional Neural Networks (CNNs) as the most successful method. CNNs apply
a set of convolutional filters to the image, obtaining a _dense hierarchy of
features_. Inherently, CNNs go deeper as the resolution decreases while
obtaining higher semantics, obtaining a pyramidal representation of the image.

In this work, we focus on visual localization for visual SLAM (Simultaneous
Localization And Mapping from visual sensors), i.e, we are assuming that a 3D
map of the scene is available, which has been built using SfM (Structure-from-
Motion) or Visual SLAM. The map is composed of 3D points, $\mathbf{P}_{i}$,
and keyframes. Keyframes, also called reference frames, are a selected set of
images from which the map geometry is computed by bundle adjustment. Per each
keyframe, we have available its image $\mathbf{I}_{j}$ and its camera pose
$\left\\{\mathbf{R}_{j},\mathbf{t}_{j}\right\\}$

Given a query image $\mathbf{I}_{q}$ and the map, the goal of camera location
is to efficiently retrieve the closest map keyframes, $\mathbf{I}_{k}$, and
the 6-DoF pose of the query image with respect to the
map,$\left\\{\mathbf{R}_{q},\mathbf{t}_{q}\right\\}$.

First we apply a _keyframe retrieval step_ , in which the typical deep
learning retrieval algorithm usually takes three sub-steps: i) the query image
is forwarded through the network encoder, a dense hierarchy of features, ii)
the deepest feature map is pooled [1, 2] obtaining a compact global
descriptor, and iii) the descriptor is compared against the other global
descriptors of the keyframes in the database, ranking the keyframes by
descriptor similarity. Images are translated into compact vectors based on the
deepest features, which typically encodes the high-level features of the
image. The comparison of global descriptors gives an initial list of candidate
keyframes ranked by similarity.

For the _camera feature-metric pose estimation step_ , we apply deep image
alignment techniques [3, 4, 5, 6] minimizing a feature-metric error of the
dense hierarchical features that encode the image at different resolutions,
estimating the relative pose between the query and a keyframe. These methods
are robust against illumination and point of view changes, and achieve high
precision as they can get subpixel accuracy, but they need a good initial pose
guess to converge to the correct minimum.

Figure 1: In a single forward pass our DRAN network extracts a unique
hierarchy of features to perform: image retrieval, candidate re-ranking and
feature-metric pose estimation. The image retrieval gives the most similar
keyframes to the query in a given 3D map. This list of candidates is re-ranked
by the matching of hypercolumns from the encoder. The top keyframe after re-
ranking gives the initial 6 DoF pose guess which is refined through feature-
metric optimization.

Most of the candidate keyframes in the list depict the same place as the query
and might provide a coarse pose initialization to the feature-metric
optimization, but their sorting criteria does not take into account the
conditions that most favour the pose optimization, i.e. point of view
similarity. For this reason, the intermediate _re-ranking step_ re-sort the
keyframes in the initial list to prioritize the overlap and point of view
similarity with the query. This step exploits the dense hierarchy of features
to produce matches between the query and the keyframes that yield 2D-3D
matches between the query and the map which are refined by a PnP RANSAC. After
this step, we have an initial pose guess which is closer to the true one, and
the feature-metric optimization can successfully converge.

Other works in literature deal with camera location employing different
networks for each of the setps. In contrast to them, we use a unique dense
hierarchy of features what is efficient and elegant, providing competitive
results of accuracy.

Figure 2: DRAN pipeline. DRAN provides the features for the three steps of
Visual Localization with a single network. The architecture follows a UNet
style with two heads: localization and retrieval. The blocks shown in green
are trained specifally for feature-metric alignment, while the ones shown in
blue are pretrained in a generic retrieval dataset and frozen during training.

Our key insight is that it is beneficial to use a unique hierarchy of learned
features for all the steps of the camera visual localization. Following these
ideas, we propose DRAN (Deep Retrieval and image Alignment Network, Figure 1),
a unified architecture that combines the knowledge from retrieval and camera
pose stages. Our contributions are:

  * •

DRAN, a multi-task single network providing, in a single forward pass, both an
image global descriptor for retrieval, hypercolumns for re-ranking and initial
pose estimation, and a multi-level dense feature hierarchy for feature-metric
camera pose refinement.

  * •

We demonstrate that well-localized features are already present in image
retrieval networks, so there is no need to train two separate pipelines for
the tasks of retrieval and localization.

  * •

An evaluation under challenging conditions, showing that our system achieves
better performance than other unified systems, and it is competitive against
feature matching pipelines that combine different networks.

##  II Related Work

We review the relevant work in the related areas of this paper: direct and
indirect SLAM, image retrieval, deep camera pose estimation and unified
methods.

Direct and Indirect SLAM is the widely used classification for traditional
SLAM techniques. Indirect SLAM [7, 8, 9, 10], processes the image to detect,
describe and match an sparse set of keypoints that are robust to illumination
and viewpoint changes, these matches are fed in a geometric Bundle Adjustment
to recover the scene geometry. In contrast, direct SLAM operates with raw
image intensities, relying on brightness consistency [11, 12, 13] to also
feed, in this case, a photometric Bundle Adjustment. Direct alignment allows
sub-pixel accuracy but is vulnerable to illumination changes and suffers from
small convergence basin. We propose the use of learned features to overcome
the challenges of the direct methods when applied to camera pose estimation.

Image retrieval stands for the task of efficiently retrieve an image among a
database of visited places, in our case the keyframes. To perform a quick
search, a compact image representation is needed. Before the advent of deep
learning, the image embedding was obtained by the aggregation of hand-crafted
local descriptors such as SIFT [14], ORB [15] in a Bag-of-Words [16, 17]
representation. Nowadays, the field is dominated by CNN representations that
aggregate feature maps into a global descriptor [2, 1, 18, 19, 20]. Thanks to
this embedding, two images can be compared efficiently with a single distance
computation between descriptors. Our work adopts the state-of-the-art pooling
method GeM for the image retrieval step[1].

Deep camera pose estimation mainly encompassed three approaches: pose
regression, image matching and deep direct alignment. Pose regression [21, 22]
learns to directly map the input to pose parameters without any 3D constraint.
They require a lot of training data and don’t generalize well to novel
domains. Differently, image matching extracts local features [23, 24, 25] and
perform data association by descriptor distance or learned matchers [26, 27].
The pose can be obtained with a PnP [28] algorithm if those local features are
present in a 3D reference model. Deep direct alignment [4, 3, 6, 29, 5] takes
an approach similar to direct SLAM, trying to minimize a photometric error
based on the deep features of a CNN, i.e. a feature-metric optimization. In
this work, we build on top of the recent framework PixLoc [3], which extracts
multi-scale dense descriptors that are aligned iteratively in a pyramidal
approach. The feature-metric error is evaluated on the 2D projections of the
corresponding 3D model built by Structure-from-Motion. S2DNet [30] is another
approach that also takes advantage of the map projections to match sparse deep
descriptors from the reference frame against the dense descriptors of the
query. In contrast to us, S2DNet is trained to solve only the matching step
between two images.

Unified approaches trying to join the local and global descriptor extraction
into a single system can be found in the literature. DELG [31] obtains the GeM
global descriptor provided by [1] and applies an attention mechanism [32] to
obtain a descriptor that reranks the initial retrieval candidates. HF-Net [25]
proposed a distillation framework in a teacher-student approach to learn from
NetVLAD [2] and SuperPoint [23], being able to perform retrieval and camera
localization. UR2KiD [33] also performs retrieval and matching, with the
benefit of being trained only with image labels. S2DHM [34] uses an encoder
pretrained for retrieval to perform re-ranking and extract an initial pose
under challenging conditions. We use the ideas of S2DHM but applying them to a
networks trained in a generic retrieval datasets (SfM120k [1]). Besides,
differently from them, our work unifies the tasks of image retrieval, local
matching and feature-metric image alignment, introducing, to the best of our
knowledge, the first system to combine them in the same network.

##  III Feature Hierarchy for Retrieval and Alignment

In this section, we detail the steps of visual localization using a single
network, giving birth to our Deep Retrieval and Image Alignment Network.

###  III-A Deep Retrieval and Image Alignment Network

We propose DRAN (Deep Retrieval and image Alignment Network) architecture,
whose main steps are described in Fig. 2. We argue that most of the meaningful
features are already extracted by image retrieval networks, so we don’t need
to train another full pipeline end-to-end to detect the features for the image
alignment that estimates the camera pose. For this purpose, we adopt a shared
encoder architecture with two heads: retrieval and localization. The shared
encoder is a VGG16 encoder pretrained for image retrieval in the SfM120k
dataset, which is splitted after the conv4 stage. We use SfM120k to pretrain
our encoder. This kind of data is easier to obtain than accurate 3D models and
allows to acquire invariance against challenging conditions.

The retrieval head incorporates the last stage of the truncated encoder
(conv5) and a Generalized-Mean pooling layer that aggregates the last feature
map into a compact vector. As we don’t want to affect the retrieval
performance, both the shared encoder and the retrieval head remain frozen
during the training stage, retaining their original weights fine-tuned for
retrieval.

The localization head has identical structure as the retrieval head (conv5 of
VGG16), with the difference that it is optimized during training (Figure 4).
It connects to a decoder network with skip connections in a UNet style.
Following [3], we optimize all the weights involved in order to learn features
for accurate camera localization (III-D). The output of the decoder is the
hierarchy of features, along with its uncertainty. The representation of an
image for each scale level $l$ is a feature map
$\mathbf{F}^{l}=\mathbb{R}^{W_{l}\times H_{l}\times D_{l}}$ and its per pixel
uncertainty $\mathbf{U}^{l}=\mathbb{R}^{W_{l}\times H_{l}}$. The decoder also
feeds the estimator for the damping factor of the Levenberg-Marquardt feature-
metric optimization.

To provide a good initial guess for the feature-metric camera pose
optimization, we extract hypercolumns [34] using the already computed features
of the shared encoder and the localization head. This allows us to filter the
retrieval candidates and obtain an initial pose by means of a PnP-Ransac. This
pose is better than the coarse pose obtained by image retrieval, boosting the
localization performance of the subsequent optimization.

###  III-B Compact Global Image Descriptor

We adopt the Generalized-Mean (GeM) [1] pooling layer as the method to
aggregate the activations from the last layer of the retrieval head. The GeM
operation for each channel of the $C\times H\times W$ activation maps is
described as

$\mathbf{f}=\left[f_{1}\ldots f_{k}\ldots
f_{K}\right]^{\top}\in\mathbb{R}^{512},\;\;f_{k}=\left(\frac{1}{\left|\mathcal{X}_{k}\right|}\sum_{x\in\mathcal{X}_{k}}x^{p_{k}}\right)^{\frac{1}{p_{k}}}$
(1)

Where $\mathcal{X}_{k}$ is the set of $HW$ activations of each channel and
$p_{k}$ is the learnt parameter that controls the pooling operation. Both the
encoder and $p_{k}$ is trained in the SfM120k [1] dataset, which depicts
popular landmarks clustered by COLMAP. It is trained using a siamese
architecture and the contrastive loss. The contrastive loss takes as input a
tuple containing 1 query, 1 positive example and 5 negative examples. Here the
objective is to obtain similar descriptors for images depicting the same
place, where the distance is computed by the dot product of two L2-normalized
global descriptors.

###  III-C Re-ranking and initial pose estimation

The image retrieval module focuses on finding keyframes depicting the same
place than the query, but in some cases, the first candidate is not the best
initialization for featuremetric alignment. Images can suffer from big changes
in scale or in the point of view, or little overlap, resulting in a bad pose
initialization than can hinder convergence to the optimal pose.

For this reason, we employed a re-ranking method that minimize this issue by
selecting the best candidate to initialize with, based on the number of
inliers. We brought the method S2DHM (Sparse-to-Dense Hypercolumn Matching)
proposed by Germain et al. [34] to our features. The goal is to obtain local
matches between the query and the keyframes with the features extracted by the
encoder, in order to filter the image retrieval candidates and estimate an
initial 6DoF pose guess.

Given a query $I_{q}$ and a keyframe $I_{k}$, we extract features from layers
conv_3_3, conv_4_1, conv_4_3, head_1 and head_3, in the same setting as S2DHM,
but using our trained localization head instead of the retrieval one. These
features are upsampled to match with the resolution of the earliest layer
(conv_3_3 in this case, resolution is 1/4 of the input image) and
concatenated. The result is the so called hypercolumns $H_{q}$ and $H_{k}$, a
dense descriptor that encodes information from different levels of the
network.

As we have an available map, we will only try to match the 3D points $P_{k}$
that were already detected in 2D locations $\\{m_{j}^{k}\\}_{j=1\ldots P_{k}}$
in the keyframe. We interpolate the hypercolumn $H_{k}$ at each location
$\\{m_{j}^{k}\\}$, obtaining a sparse set of descriptors
$\\{d_{j}^{k}\\}_{j=1\ldots P_{k}}$. To find the matches between the dense
query hypercolumn and the sparse keyframe descriptors, a dot product is
computed between every sparse descriptor $d_{j}^{k}$ and the dense
hypercolumn, obtaining a cross-correlation map $C_{q,j}^{k}=H_{q}*d_{j}^{k}$.
The tentative matches corresponds to the global maximum of the cross-
correlation map for each of the points $p_{k}$ considered. To avoid outliers
due to repetitive patterns or occlusions, a ratio test is conducted.

The resulting $P_{k}$ 2D-3D matches are fed into a PnP+RANSAC scheme, which
outputs the final inliers and the initial pose estimation. The keyframe with
the most inliers is the one selected, as well as the PnP pose estimated for
the query. This estimation is the seed given to the final optimization.

Figure 3: Exhaustive matching for re-ranking and initial pose. We only attempt
to match points present in the map, obtaining a sparse set of descriptors from
the dense hypercolumn $H_{k}$. Each descriptor $d_{j}^{k}$ is searched densely
in the query hypercolumn $H_{q}$, identifying the match as the global maxima
of the cross-correlation map.

###  III-D Deep direct feature-metric image alignment

We employ the method proposed in [3] to align our hierarchy of deep features.
As other recent works [4, 6, 5, 29], they treat deep image alignment as a
direct feature-metric optimization problem, in a similar way as DSO [11]
minimizes the photometric error. We describe it briefly, for further details
please refer to [3].

As PixLoc proposes, we extract $L$=3 feature maps from the UNet decoder, with
strides 1, 4 and 16. The shallow levels encode low texture cues while deeper
levels capture high level features and semantic content. This pyramidal
representation is similar as what traditional photometric alignment uses.
Learning this representation instead of relying on the raw photometric values
allows to overcome with the known limitations of direct image alignment:
illumination changes and small convergence basin.

For a feature level $l$, the feature-metric residual is defined as weighted
addition of the feature-metric error between the query and the retrieved
images for all the 3D map points, detected in the query image:

$\displaystyle E_{l}(\mathbf{R}_{q},\mathbf{t}_{q})$ $\displaystyle=$
$\displaystyle\sum_{i,k}w_{k}^{i}\rho(\left\|\mathbf{r}_{k}^{i}\right\|_{2}^{2})$
(2) $\displaystyle\mathbf{r}^{i}_{k}$ $\displaystyle=$
$\displaystyle\mathbf{F}^{l}_{q}\left[\mathbf{p}_{q}^{i}\right]-\mathbf{F}^{l}_{k}\left[\mathbf{p}_{k}^{i}\right]\in\mathbb{R}^{D}$
(3) $\displaystyle w_{k}^{i}$ $\displaystyle=$
$\displaystyle\frac{1}{1+\mathbf{U}^{l}_{q}\left[\mathbf{p}_{q}^{i}\right]}\frac{1}{1+\mathbf{U}^{l}_{k}\left[\mathbf{p}_{k}^{i}\right]}\in[0,1].$
(4)

Where $\mathbf{F}^{l}_{q}$ and $\mathbf{F}^{l}_{k}$ are the feature maps for
the query and the keyframe at a certain level $l$,
$\left[\mathbf{p}_{q}^{i}\right]$ and $\left[\mathbf{p}_{k}^{i}\right]$ are
the projections of the point $\mathbf{P}_{i}$ with subpixel accuracy, and
$\mathbf{U}^{l}_{q}$ and $\mathbf{U}^{l}_{k}$ are the predicted uncertainty
maps. $w_{k}^{i}$ learns to determine if the location of a 3D point is good
for localization or not. If the point projection has low uncertainty in both
the query and the reference image, $w_{k}^{i}$ will tend to 1. Otherwise,
$w_{k}^{i}$ will tend to 0, weighting down the residual in the optimization.
For $\mathit{N}$ points, (2) defines the goal function to be minimized with
the Levenberg-Marquadt (LM) algorithm. The network learns to find good
features to localize and wheter a point is reliable or not.

The initial guess for the iterative optimization is the one with the most
inliers found by the Sparse-to-Dense matching. The algorithm starts optimizing
the coarsest level (i.e. stride 16), the feature maps with lower resolution
but higher depth, and successively optimizes the finer levels, going from an
initial coarse estimation to a finer one. As the finer layer has te same
resolution as the input image, feature-metric optimization can achieve
subpixel accuracy, just like classis photometric alignmnt. The LM optimization
for the camera pose comes down to solve the linear system:

$\displaystyle\left(\mathbf{H}+{\boldsymbol{\lambda}}\,\mbox{diag}\left(\mathbf{H}\right)\right){\boldsymbol{\delta}}$
$\displaystyle=$ $\displaystyle\mathbf{J}^{\top}\mathbf{W}\mathbf{r}$ (5)
$\displaystyle\mathbf{H}$ $\displaystyle=$
$\displaystyle\mathbf{J}^{\top}\mathbf{W}\mathbf{J},$ (6)

where $\mathbf{J}$ is the Jacobian, $\mathbf{W}$ is weighting matrix depending
on (4) and $\boldsymbol{\delta}\in\mathbf{SE}(3)$ is the pose update of
parameterized by its Lie algebra.

The damping parameter $\boldsymbol{\lambda}$ is formulated as a fixed $6\times
6$ diagonal matrix coding the damping independently in each of the 6 DoF of
the camera pose, these diagonal values are also learnt.

Figure 4: Reprojection error loss. The training loss is computed by the
reprojection loss between the projected 3D points with the ground truth pose
(green dots) and the projection of the same points with the estimated pose
after the optimization (red dots).

Loss function The only supervision for learning is the ground truth pose for
the query images,
$\left\\{\bar{\mathbf{R}}_{q},\bar{\mathbf{t}}_{q}\right\\}$. The loss
function penalizes the Huber cost of distance in pixels the between the
reprojection of the map points in the ground camera pose (Figure 4), and the
reprojections of the same points in the feature-metric optimized camera
pose,$\left\\{\mathbf{R}_{l,q},\mathbf{t}_{l,q}\right\\}$, averaging among the
different scales.

$\mathcal{L}=\frac{1}{L}\sum_{l}\sum_{i}\left\|\Pi(\mathbf{R}_{l,q}\mathbf{P}_{i}+\mathbf{t}_{l,q})-\Pi(\bar{\mathbf{R}}_{q}\mathbf{P}_{i}+\bar{\mathbf{t}}_{q})\right\|_{\gamma},$
(7)

This formulation is not affected by the geometric scale of the scene, as it
only works with the reprojection of the points and not directly with camera
poses.

##  IV Experiments

In this section we evaluate the advantages of our retrieval deep alignment,
comparing it against other learned approaches. In section IV-A, we explain the
datasets used for training and evaluation, and the baselines used to compare.
We show the accuracy of the camera pose estimated by our system in large-scale
visual localization in section IV-B, performing better than the other unified
approaches and competitively against feature matching approaches. Finally, in
section IV-C, we perform an ablation study, showing the benefits of each one
of the elements of the pipeline.

###  IV-A Datasets and baselines

Figure 5: Examples from Aachen Day-Night. Each two columns depicts a query-
reference pair. The first row shows the Sparse-to-Dense matching of our
approach, showing impressive results in night conditions under severe
viewpoint change or occlusions. The second row shows the features $\mathbf{F}$
from the finest resolution learned in the decoder. The last row shows the also
learned uncertainty maps $\mathbf{U}$, where red means that points are more
reliable, and blue that points are ignored.

Training Our retrieval encoder is pretrained on SfM120k [1], which depicts
common landmarks around the world, downloaded from Flickr and clustered with
Structure-from-motion, using the contrastive loss and hard-negative mining for
retrieval. We experimented with the original version given by the authors,
which initializes their training with weights from custom ImageNet pretraining
on Caffe [35], but we found better convergence and performance when
initializing with PyTorch ImageNet pretraining. We use MegaDepth [36] dataset
for training the feature-metric alignment procedure. It contains about 1M of
images depicting popular landmarks, grouped into 196 scenes and reconstructed
by COLMAP [37]. MegaDepth provides depth maps and pose information for every
camera. We use the same split as D2-Net [24] for training and validation. The
training is performed for 20k iterations with the Adam optimizer, using a
constant learning rate of $5\times 10^{-6}$ and a batch size of 6.

Evaluation We selected Aachen Day-Night for evaluation, as it depicts the old
inner city of Aachen, testing visual localization under day-night condition.
Aachen Day-Night include SfM reconstructions: 3D point clouds/dense maps and
camera poses. As proposed by [1], we use multiscale and learned whitening to
obtain the retrieval candidates, for scales $1,\frac{1}{\sqrt{2}}$ and
$\frac{1}{2}$.

Baselines We compare against two groups of methods: Feature Matching (FM) and
Unified methods.

For Feature Matching, we consider methods for matching between two images,
where the image retrieval is given as external prior. In this group we found
the state-of-the-art methods for local matching as D2-Net [24] or the toolbox
hloc [25, 26], which uses SuperPoint [23] and SuperGlue [26]. We also compare
against PixLoc [3], which performs feature-metric optimization, and S2DNet
[30], a matching system for sparse-to-dense matching.

In the Unified methods, we consider methods able to perform retrieval and
recover a 6DoF pose with a single network. Here we found S2DHM [34], which
performs sparse-to-dense matching with an encoder trained for retrieval in a
subset of RobotCar Seasons and HF-Net [25], a pipeline that distilled
knowledge from NetVLAD [2] and SuperPoint [23] in a compact network.
UR2KiD[33] is also considered, as it is able to perform retrieval and local
matching while only being trained with image to image matches but not in image
to map matches.

###  IV-B Large-scale localization

Method | Aachen Day-Night
---|---
Day | Night
Pixloc | 64.3 / 69.3 / 77.4 | 51.0 / 55.1 / 67.3
S2DNet | 84.3 / 90.9 / 95.9 | 46.9 / 69.4 / 86.7
D2-Net | 84.3 / 91.9 / 96.2 | 75.5 / 87.8 / 95.9
hloc |  89.6 / 95.4 / 98.8 |  86.7 / 93.9 / 100
S2DHM | 56.3 / 72.9 / 90.9 | 30.6 / 56.1 / 78.6
HF-Net | 79.9 / 88.0 / 93.4 | 40.8 / 56.1 / 74.5
UR2KiD |  79.9 / 88.6 / 93.6 | 45.9 / 64.3 / 83.7
DRAN (ours) | 77.9 / 87.0 / 90.9 |  67.3 / 75.5 / 85.7

TABLE I: Results for Aachen Day-Night.

Aachen Day-Night objective is to evaluate visual localization under
challenging conditions. It is composed of 4,328 references images depicting
the city of Aachen, taken during daytime with hand-held devices. A 3D model is
reconstructed with these images, and 922 queries (824 daytime, 98 nighttime)
along with its 6DoF are provided. The benchmark protocol from [38] report the
percentage of queries localized for different thresholds for the camera
position and orientation error. Localization recall is provided for three
threshold levels: (25cm, 2º), (50cm, 5º) and (5m, 10º), which can be seen in
Table I.

Among the unified approaches, DRAN performs the best by a great margin in the
night-queries, while performing comparably as UR2KiD in the day ones. Feature
matching pipelines as hloc, that uses NetVLAD, SuperPoint and SuperGlue, are
more robust and precise than ours, with the drawback that they use several
networks to perform localization, while we only need one. DRAN obtains the
camera pose estimation using the top-3 candidates in the initial retrieval
list.

Aachen Day-Night reference poses are really sparse, providing a extremely
coarse initialization that makes difficult to pure feature-metric methods to
converge. Here is where the sparse-to-dense method shines, finding matches in
difficult situations (Figure 5) and providing an initial pose much more
precise to the feature-metric optimization. In contrast to PixLoc, which only
performs the optimization with a given retrieval, we are able to perform our
own retrieval and estimate more accurate and robust camera poses.

###  IV-C Ablation study

Method | Aachen Day-Night
---|---
Day | Night
DRAN (R+A) | 62.5 / 69.2 / 76.9 | 58.2 / 62.2 / 70.4
DRAN (R+P) | 73.4 / 83.3 / 90.8 | 63.3 / 74.5 / 85.7
DRAN (R+P+A) | 77.9 / 87.0 / 90.9 | 67.3 / 75.5 / 85.7

TABLE II: Ablation study in Aachen Day-Night.

We perform an ablation study of the different modules of our algorithm (Table
II.) Using our top scored keyframe of our retrieval network (R) as the intial
guess for deep image alignment (A), gives similar results on the day to
Pixloc, the most comparable method, but improves by an average of 10% in the
night queries.

Performing retrieval and only PnP with hypercolumns (R+P) allows to produce an
accurate pose by itself giving a huge boost to the performance. Having an
encoder trained in a generic retrieval dataset is benefitial for
generalization, specially in the night case, where DRAN doubles the recall of
S2DHM, which was trained on RobotCar Seasons.

Finally, using all the above modules (R+P+A), where the feature-metric
optimzation uses R+P as initial guess conforms a unified pipeline that beats
the state-of-the-art of multitasks methods in Aachen night queries, while
being competitive on the day condition. In Figure 6, we show an example where
DRAN can not converge using (R+A), but successfully localizes with the full
method (R+P+A), showing the increase in robustness.

Figure 6: The Sparse-to-Dense module allows convergence when reference poses
are sparse. Left column: reference image along with the 3D points that it
observes (shown in green). Right column: query image with points reprojected
using the initial pose (red points) and the optimized (green). With retrieval
initialization (top row) the optimization does not converge, while with PnP
pose (bottom row) it does.

###  IV-D Implementation

For the feature-metric optimization and training, we built DRAN upon the
original implementation of PixLoc, bringing there the retrieval and sparse-to-
dense ideas.

Running on a PC with a Intel® Core™ i7-10700K (3.80GHz) CPU and an Nvidia
GeForce RTX 2080 Ti, the typical times for one image with 2000 points are
300ms for the extraction, 700ms for the sparse-to-dense matching and between
200 to 400ms for the feature-metric optimization. The ratio test method is the
current bottleneck, averaging 0.36ms per point. This is not optimized and
could be substituted by a non-maxima suppresion method on GPU.

##  V Conclusions

We have presented a unified pipeline that performs all the tasks concerning
Visual Localization under challenging conditions. Compared with other
approaches in Aachen, DRAN, not only is competitive in day conditions to
feature matching pipelines that are not able to perform retrieval by
themselves, but also improves state-of-the-art in night queries among the
other unified approaches. In our opinion, SLAM pipelines should have a unique
feature extractor, able to provide good features for tracking and
relocalization. We see DRAN as a first step towards that paradigm. As a future
work, we would consider extending DRAN to other datasets and to train jointly
the tasks of retrieval and camera pose estimation.

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# Putting all eggs in one basket: some insights from a correlation inequality

Pradeep Dubey1,4, Siddhartha Sahi2,1, and Guanyang Wang3

1\. Center for Game Theory, Department of Economics
Stony Brook University, Stony Brook, NY 11790
2\. Department of Mathematics, Rutgers University,
3\. Department of Statistics, Rutgers University,
New Brunswick, NJ 08903
4\. Cowles Foundation, Yale University, New Haven, CT 06520

##  Abstract

We give examples of situations – stochastic production, military tactics,
corporate merger – where it is beneficial to concentrate risk rather than to
diversify it, i.e., to put all eggs in one basket. Our examples admit a dual
interpretation: as optimal strategies of a single player (the “principal”) or,
alternatively, as dominant strategies in a non-cooperative game with multiple
players (the “agents”).

The key mathematical result can be formulated in terms of a convolution
structure on the set of increasing functions on a Boolean lattice (the lattice
of subsets of a finite set). This generalizes the well-known Harris inequality
from statistical physics and discrete mathematics; we give a simple self-
contained proof of this result, and prove a _further_ generalization based on
the game-theoretic approach.

Keywords: Harris inequality, correlation inequality, increasing functions,
diversifying risk, concentrating risk, stochastic production, military
tactics, corporate merger, dominant strategy.

Mathematics Classification – MSC2020: 60E15, 91A10, 91A18, 91B43.

Economics Classification – JEL: C60, C61, C72, G10.

## Introduction

> “It is the part of a wise man to keep himself today for tomorrow, and not to
> venture all his eggs in one basket.” — M. Cervantes (Don Quixote, 1605)
>
> “Behold, the fool saith, “Put not all thine eggs in the one basket” — which
> is but a matter of saying, “Scatter your money and your attention”; but the
> wise man saith, “Put all your eggs in the one basket and – WATCH THAT
> BASKET.” — M. Twain (Pudd’nhead Wilson, 1894)

Although modern economic theory and practice widely advocate for risk
diversification—epitomized by the maxim of distributing ’eggs’ across various
’baskets’ and embraced alike by academic scholars and by financial experts
such as bankers, investors, and portfolio managers—there are circumstances
where it may be beneficial to concentrate risk. This is especially true in
cases where the potential gains from a single successful venture can eclipse
the losses from several failures, and where the magnitude of a possible reward
justifies the concentration of all resources into one endeavor. Our analysis
aims to illuminate various scenarios where adopting a strategy focused on a
single ’basket’ is advantageous.

In the realm of portfolio theory, the typical payoff function is the _sum_ of
returns from various investments, and then it is beneficial to create
strategic diversification by bundling together asset classes that exhibit no
correlation or negative correlation. In contrast, our study explores scenarios
where the payoff is the _product_ of exogenously given functions, and is
influenced by strategic choices that affect their joint distribution. The
pivotal insight is that positive correlation among these functions could make
it strategically beneficial to adopt a coordinated approach, which effectively
concentrates risk and increases the probability that the peaks and troughs of
the individual functions occur together.

We present a series of examples demonstrating that such scenarios can emerge
quite naturally in a strategic setting. Our examples span a variety of
contexts, including stochastic production with unpredictable input supplies,
military tactics aimed at disrupting enemy communication networks, and the
decision-making processes in corporate mergers. Each scenario can be
interpreted in two ways: either as an optimal strategy for an individual actor
(the ’principal’) or as dominant strategies in a non-cooperative setting
involving several participants (the ’agents’).

In each scenario presented, the principal’s potential strategies are subsets
$S$ of a finite set $H$. The case where $S=\emptyset$ represents the strategy
of maximum risk diversification, while $S=H$ signifies the strategy of maximum
risk concentration. We demonstrate that, in every instance, the payoff
function $\Pi\left(S\right)$ is an increasing function, meaning that if $S$
contains $T$ then $\Pi(S)\geq\Pi(T)$. Consequently, the optimal strategy is to
choose $S=H$, effectively ’putting all eggs in one basket’.

The unifying mathematical principle across the three examples is delineated as
follows: Let $\mathcal{H}$ be the power set of a finite set $H$, and let $A$
be the space of real-valued functions on $\mathcal{H}$. In (4) below, we
define a convolution operation $f\star g$ on $A$, predicated on a ’coin
tossing’ mechanism with probabilities $0\leq p_{h}\leq 1$ for each element
$h\in H$. Our first main result, Theorem 11, establishes that the convolution
of two increasing functions results in another increasing function. Now it
turns out that in each example, the payoff function $\Pi$ can be expressed as
the convolution of two increasing functions, confirming that $\Pi$ itself is
an increasing function.

While the initial examples are binary in nature, Section 5 expands the
discussion to a more complex scenario of stochastic production with multiple
inputs. In this broader context, the principal’s strategies correspond to
partitions of the input set. We establish that the optimal strategy involves
opting for the coarsest partition. To substantiate this, we recast the problem
as a strategic game among the agents. Theorem 20 asserts that selecting the
coarsest partition constitutes a dominant strategy for each agent—a notion
reinforced by Remark 7.4.

Theorem 20 has a further implication. In the context of stochastic production,
as detailed in Section 5, it may well happen that the principal does not have
the capacity to directly execute her optimal strategy. Instead, she must rely
upon each of her autonomous agents to “fall in line”, i.e., to voluntarily
choose to implement his component of her optimal strategy. Remark 7.5
illustrates that by allocating a modest share of her payoff to each agent, the
principal can effectively incentivize them to act in accordance with her
strategy. This strategic distribution of incentives ensures that the
principal’s optimal strategy is self-enforcing!

It is intriguing that Theorem 20 not only generalizes Theorem 11 but also
admits a very simple proof. In fact, one might well question the need for a
separate proof of Theorem 11. Our rationale is threefold: first, _both_ proofs
are concise and standalone. Second, Theorem 20, while broader, is more
technical; it involves a special class of games with elaborate assumptions on
strategies and payoffs. Most importantly, our exposition is crafted to be
accessible to both economists and probabilists; hopefully serving as a bridge
between the two disciplines.

Indeed, Theorem 11 belongs to the class of correlation inequalities that are
widely studied and applied in combinatorics, graph theory, and statistical
physics. In particular it readily implies the well-known Harris inequality
[8], which a pivotal concept in percolation theory and the Erdős-Rényi model
of random graphs [2]. We hope that our paper will serve to introduce the
beautiful subject of correlation inequalities to those previously unacquainted
with the topic.

The Harris inequality and its extensions, such as the Fortuin-Kasteleyn-
Ginibre inequality [7] and the Ahlswede-Daykin four-function inequaliy [1],
have been applied in economic theory to several areas including comparative
statics [3, 12], bargaining networks [4], and optimal assignments [9]. However
their role in risk concentration strategies, which is the central theme of our
paper, remains unexplored. While the paper [11] does discuss risk
concentration, it is underpinned by a different set of principles, and does
not employ correlation inequalities.

## Acknowledgement

The authors thank Ioannis Karatzas, Larry Samuelson, and Eran Shmaya for their
helpful and insightful comments on an earlier version of this paper. The
research of S. Sahi was partially supported by NSF grant DMS-2001537.

##  1 Example 1: Stochastic production with two inputs

An entrepreneur has a Cobb-Douglas (log-linear) production function of the
form

$f(x,y)=x^{\alpha}y^{\beta},\quad\alpha,\beta>0,$ (1)

involving two inputs $x,y$, which she sources from a finite set $H$ of
suppliers.

Each supplier $h\in H$ has $x_{h}$ units of the $x$-input and $y_{h}$ units of
the $y$-input which he needs to ship to the entrepreneur, either separately by
way of two independent shipments, or together in one “pooled” shipment. The
shipments have zero dollar cost (see, however, Remark 7.2) but are fraught
with risk: the probability that a shipment by $h$ will reach its destination
is $p_{h}.$ These probabilities are independent across the suppliers and also
across different shipments by the same supplier.

The entrepreneur can make a _separating contract_ with supplier $h$ for him to
ship $x_{h}$ and $y_{h}$ independently; or a _pooling_ _contract_ for him to
ship them jointly. She has to give all her suppliers sufficient advance notice
and make these contracts ex ante, prior to the realisation of any of their
deliveries, i.e., the luxury of telling a supplier $h$ what to do, conditional
on the deliveries obtained from other suppliers, is not available to her. Thus
her possible strategies are indexed by subsets $S\subseteq H$, where the
$S$_-strategy_ consists of making a pooling contract with each supplier in
$S$, and a separating contract with all the other suppliers.

_Which strategy $S\subseteq H$ will maximize the entrepreneur’s expected
output? _

There are clear-cut advantages of diversifying risk by means of separating
contracts. For suppose that many shipments fail under pooling, so that the
total $x$ and $y$ received by the entrepreneur are both small. Had she opted
for separating contracts, there would have remained a chance of less failure
of the $y$-shipments despite the widespread failure of the $x$-shipments,
enabling the production of medium output. Under pooling, $x$ and $y$ are
inexorably linked: if $x$ is small, so is $y,$ and therefore so is the output.
Why should the entrepreneur not diversify risk, instead of putting all the
eggs (inputs of $h$) in the same basket (shipment of $h$)? However we show
that she should do just that.

Indeed, consider the special case of a single supplier, with $H=\\{1\\}$. Then
the expected output is $px_{1}^{\alpha}y_{1}^{\beta}$, where $p$ is the
probability that the entrepreneur receives _both_ $x$ and $y$ from the
supplier. This probability is $p_{1}$ under pooling, and $p_{1}^{2}\leq p_{1}$
under separate shipments. Thus pooling leads to higher expected output.

We will show that the advantage of pooling persists with many suppliers, and
with a more general class of production functions that we now describe.

As before, we consider an entrepreneur who sources two kinds of input, $x$ and
$y$, from a finite set $H$ of suppliers. Each supplier $h$ can ship both kinds
of inputs in a single pooled shipment or in two separate shipments, and each
shipment by $h$ arrives with probability $p_{h}$. The $S$-strategy of the
entrepreneur consists of making a pooling contract with all $h\in S$ and a
separating contract with all $h\notin S$.

However we suppose now that the entrepreneur has a more general production
function of the form

$F(S_{1},S_{2})=F_{1}(S_{1})F_{2}(S_{2}),$ (2)

where $S_{1}$ and $S_{2}$ are the sets of suppliers from whom she receives
shipments of $x$ and $y$, respectively, and $F_{1}$ and $F_{2}$ are increasing
functions.

The $S$-strategy induces a probability distribution on pairs $(S_{1},S_{2})$,
and thus on the output, and we let $\Pi_{1}\left(S\right)$ denote the expected
output.

###### Claim 1

The expected output $\Pi_{1}(S)$ is an increasing function of $S$.

Proof. See Section 4.

###### Corollary 2

The $H$-strategy maximizes expected output.

###### Remark 3

The Cobb-Douglas function (1) is a special case of (2) with
$F_{1}(S_{1})=(\sum_{h\in S_{1}}x_{h})^{\alpha},\;F_{2}(S_{2})=(\sum_{h\in
S_{2}}y_{h})^{\beta}$. Clearly $F_{1},F_{2}$ are increasing in $S_{1},S_{2}$.

##  2 Example 2: Military tactics

Country I has two communication networks, $R$ (red) and $B$ (blue). Each
network is characterized by a pair
$\left(H_{\alpha},\mathcal{C}_{\alpha}\right),$
$\alpha\in\left\\{R,B\right\\}$. Here $H_{\alpha}$ is a finite set of sites
across the country at which there exist hubs of network $\alpha;$ and
$\mathcal{C}_{\alpha}$ is a collection of _critical subsets_ of $H_{\alpha}$,
so called because the destruction of all the hubs of $\alpha$ in
$S\in\mathcal{C}_{\alpha}$ will disable network $\alpha$. It is natural to
assume, and we will, that the $\mathcal{C}_{\alpha}$ are _increasing_ in the
sense __

$S\in\mathcal{C}_{\alpha}\text{ \& }S\subset T\implies
T\in\mathcal{C}_{\alpha}.$ (3)

Country II needs to disable _both_ networks. It knows only the set
$H=H_{R}\cup H_{B}$ of all sites and has no information regarding the pairs
$\left(H_{\alpha},\mathcal{C}_{\alpha}\right)$ other than that they exist. For
each $h\in H$ it has weapons, $r_{h}$ and $b_{h}$, which can destroy a red hub
or a blue hub respectively at site $h$, provided the hub exists there.111e.g.,
the hubs of $R$ (or, $B$) are above (or, below) ground; and $r_{h}$ detonates
above, while $b_{h}$ burrows into the earth and detonates below. It also
possesses several “$h$-missiles” which are trained to fire $r_{h}$ and $b_{h}$
at $h\in H;$ and each $h$-missile can carry either $r_{h}$ or $b_{h}$ or both,
but the probability that it will hit $h$ is $p_{h}$, independent of the
outcome of other missiles. Moreover, the missiles must be fired rapidly before
it can be known which ones hit their targets. The military is not concerned
with the costs of the weapons or of the missiles222However, see Remark 7.2..
It is pondering over its $2^{\left|H\right|}$ strategies, one for every
$S\subset H,$ where the $S$-strategy consists of firing the weapons jointly at
the sites in $S$ and separately at those in $H\diagdown S$.

_Which strategy should the military employ? And would it be of benefit for
country II to conduct espionage to find out_
$\left(H_{\alpha},\mathcal{C}_{\alpha}\right)$_in order to fine-tune its
strategy based on that information?_

As in the previous example, there seem to be some advantages to firing
separately. For suppose the weapons are fired jointly at all $h$, and it turns
out that the set $S$ of targets hit is in $\mathcal{C}_{R}$ but not in
$\mathcal{C}_{B}$, with the upshot that Country II fails in its objective.
Perhaps its prospects might have been better had it fired separately.
Conditional on the realisation of $S$, there would still have remained a
positive probability of hitting a critical set $T$ in $\mathcal{C}_{B}$.

Nevertheless, if $\Pi_{2}\left(S\right)$ denotes the probability that both
networks get disabled under the $S$-strategy then we prove the following.

###### Claim 4

The probability $\Pi_{2}\left(S\right)$ is an increasing function of $S$.

Proof. See Section 4.

###### Corollary 5

The $H$-strategy maximizes the probability of disabling both networks.

###### Remark 6

Note that this result holds no matter what
$\left(H_{R},\mathcal{C}_{R}\right)$ and $\left(H_{B},\mathcal{C}_{B}\right)$
are. Thus espionage is of no benefit. If the military _were_ concerned about
the costs of $r_{h},b_{h}$ then of course it would be of benefit to know
$H_{R},H_{B}$ in order to avoid firing $r_{h},b_{h}$ at $H\diagdown H_{R},$
$H\diagdown H_{B}$ respectively.

It turns out that firing jointly increases not just the probability of
disabling both networks, but _also_ the probability of disabling neither. What
decreases is the probability of disabling exactly one network. Here is the
precise statement.

###### Claim 7

Let $F\left(S\right)$ and $G\left(S\right)$ be the probabilities of disabling
neither network, or exactly one network, under the $S$-strategy; then $F$ is
an increasing function and $G$ is a decreasing function of $S$.

Proof. See Section 4.

##  3 Example 3: Corporate merger

Each individual $h\in H$ owns shares in company
$\alpha\in\left\\{A,B\right\\}$, and there are no shareholders outside of $H$.
The _voting game_ (aka “simple game”, see [13]) in company $\alpha$ is
described by a function

$f_{\alpha}:\mathcal{H\longrightarrow}\left\\{0,1\right\\}$

where $\mathcal{H}$ denotes the collection of all coalitions (subsets) of $H$.
The interpretation is that coalition $S\subset H$ is _winning_ (resp.,
_losing_) _in company_ $\alpha$ if $f_{\alpha}\left(S\right)=1$ (resp.,
$f_{\alpha}\left(S\right)=0$). We naturally assume that $f_{\alpha}$ is
_increasing,_ i.e., adding voters to a winning coalition cannot make it
losing; and (to avoid trivialities) that the empty coalition $\emptyset$ is
losing while the grand coalition $H$ is winning333A canonical example is
provided by a _weighted voting game_ in which player $h$ has
$r_{\alpha}^{h}\geq 0$ votes in company $\alpha,$ equal to his shares in
$\alpha$ (adhering to the “one dollar, one vote” principle), and
$f_{\alpha}\left(S\right)=1$ if, and only if, $\sum_{\alpha\in
S}r_{\alpha}^{h}\geq q_{\alpha}$ for some “quota”
$0<q_{\alpha}\leq\sum_{\alpha\in H}r_{\alpha}^{h}.$.

The possibility has arisen of the merger of $A$ and $B$ but this requires the
approval of owners of both companies, i.e., merger must be voted for by a
winning coalition in both $A$ and $B.$

We postulate (in the spirit of [5] and [10], see also [6] for a detailed
survey) that each $h$ has an exogenously given probability $p_{h}$ of
returning his ballot when called upon to vote $Y$ (yes) in favor of merger;
and that these probabilities are independent across $h\in H$ and across the
different occasions on which any particular $h$ may be asked to return a
ballot.

The management of both companies are strongly in favor of the merger. They can
send two separate ballots to any $h\in H,$ one of which is a vote of $Y$ in
company $A,$ and the other a vote of $Y$ in company $B$ (and this does seem to
be the norm in practice, where the decision-making inside any company is kept
independent of other companies). However, an interesting alternative presents
itself in the current context: they can send a “joint ballot” to $h,$ with the
explicit understanding that if $h$ returns that ballot it will mean that $h$
voted $Y$ in _both_ $A$ and $B.$

Thus, yet again, there is an $S$-strategy for every $S\subset H$: send joint
ballots to $h\in S$ and separate ballots to $h\in H\diagdown S$. If
$\Pi_{3}\left(S\right)$ is the probability of merger under the $S$-strategy
then the management is looking to maximize $\Pi_{3}\left(S\right)$.

_Which strategy $S\subset H$ will maximize the probability of merger?_

The advantages of risk-diversification notwithstanding, we show

###### Claim 8

The merger probability $\Pi_{3}\left(S\right)$ is an increasing function of
$S$.

Proof. See Section 4.

###### Corollary 9

The $H$-strategy maximizes the probability of a merger.

##  4 Convolution of increasing functions

We now formulate and prove the main result that underlies all our examples.

Let $\mathcal{H}$ be the set of subsets of a finite set $H$. For $S$ in
$\mathcal{H}$ we define a probabilty measure $\mu_{S}$ on
$\mathcal{H}\times\mathcal{H}$ as follows: $\mu_{S}(S_{1},S_{2})=0$ if $S\cap
S_{1}\neq S\cap S_{2}$, otherwise

$\mu_{S}(S_{1},S_{2})=P(S,S_{1})P(H\setminus S,S_{1})P(H\setminus S,S_{2})$

where $P(X,Y):=\prod_{h\in X\cap Y}(p_{h})\prod_{h\in X\setminus Y}(1-p_{h})$.
Then $\mu_{S}(S_{1},S_{2})$ is precisely the probability that the following
random procedure leads to the pair $(S_{1},S_{2})$:

  * •

For each $h$ in $S$ we toss a coin with probability $p_{h}$ of landing heads;
if heads we include $h$ in both $S_{1}$ and $S_{2}$, and if tails then we
exclude $h$ from both sets.

  * •

For each $h$ in $H\setminus S$ we toss the coin once to decide whether to
include $h$ in $S_{1}$, and once again, independently, to decide whether to
include $h$ in $S_{2}$.

###### Definition 10

If $f$ and $g$ are real valued functions on $\mathcal{H}$ then we define

$f\star g(S):=\sum\nolimits_{S_{1},S_{2}\subseteq
H}f(S_{1})g(S_{2})\mu_{S}(S_{1},S_{2}).$ (4)

As before, we say that $f$ is _increasing_ if $S\supseteq T$ implies
$f\left(S\right)\geq f\left(T\right)$.

###### Theorem 11

If $f$ and $g$ are increasing functions then so is $f\star g.$

Proof. It suffices to show $f\star g\left(S\right)\geq f\star g\left(T\right)$
in the case where $S\setminus T$ consists of a single element, $h$, say. Then
the two coin tossing procedures agree on $H^{\prime}=H\setminus\\{h\\}$ and by
grouping terms we can rewrite

$\displaystyle f\star g\left(S\right)$
$\displaystyle=\sum\nolimits_{T_{1},T_{2}\subseteq
H^{\prime}}A\left(T_{1},T_{2}\right)\mu_{T}^{\prime}(T_{1},T_{2})$
$\displaystyle f\star g\left(T\right)$
$\displaystyle=\sum\nolimits_{T_{1},T_{2}\subseteq
H^{\prime}}B\left(T_{1},T_{2}\right)\mu_{T}^{\prime}(T_{1},T_{2})$

where $\mu_{T}^{\prime}$ is the measure on $H^{\prime}$ induced by $T$, and
$A=A\left(T_{1},T_{2}\right)$ and $B=B\left(T_{1},T_{2}\right)$ are given
explicitly by the formulas

$\displaystyle A$
$\displaystyle=\left(1-p\right)ab+pa^{\prime}b^{\prime},\quad
B=\left[(1-p)a+pa^{\prime}\right]\left[(1-p)b+pb^{\prime}\right]$
$\displaystyle\text{with }a$
$\displaystyle=f\left(T_{1}\right),a^{\prime}=f\left(T_{1}\cup\left\\{h\right\\}\right),b=g\left(T_{2}\right),b^{\prime}=g\left(T_{2}\cup\left\\{h\right\\}\right),p=p_{h}.$

We claim that in fact $A\left(T_{1},T_{2}\right)\geq
B\left(T_{1},T_{2}\right)$ for all $T_{1},T_{2}$. Indeed, an easy calculation
shows that

$A-B=p\left(1-p\right)\left(a^{\prime}-a\right)\left(b^{\prime}-b\right).$

Now since $f$ and $g$ are increasing we have $a^{\prime}\geq a,b^{\prime}\geq
b$, and hence $A-B\geq 0.$

For the two extreme cases $S=H$ and $S=\emptyset$ we have

$f\star g\left(H\right)=\operatorname*{Exp}(fg),\quad f\star
g(\emptyset)=\operatorname*{Exp}(f)\operatorname*{Exp}(g),$

where $\operatorname*{Exp}$ is the expectation with respect to the measure
$\mu$ on $\mathcal{H}$ defined by

$\mu(S)={\textstyle\prod\nolimits_{h\in
S}}p_{h}{\textstyle\prod\nolimits_{h\notin S}}(1-p_{h})=P(H,S).$ (5)

Thus we obtain the following well-known inequality due to Harris [8].

###### Corollary 12

If $f$ and $g$ are increasing then we have

$\operatorname*{Exp}(fg)\geq\operatorname*{Exp}(f)\operatorname*{Exp}(g).$ (6)

If $\mathcal{F}=\\{f_{i},i\in I\\}$ is an _set_ of functions and
$\pi:I_{1}\sqcup\cdots\sqcup I_{k}$ is a partition of the index set $I$ then
we define
$E_{\mathcal{F}}\left(\pi\right)=\operatorname*{Exp}\left(\prod\nolimits_{i\in
I_{1}}f_{i}\right)\cdots\operatorname*{Exp}\left({\prod\nolimits_{i\in
I_{k}}}f_{i}\right).$

###### Corollary 13

Let $\mathcal{F}$ be a finite set of non-negative increasing functions and let
$\pi$ and $\pi^{\prime}$ be partitions of $I$ such that $\pi^{\prime}$ refines
$\pi$ then we have $E_{\mathcal{F}}\left(\pi^{\prime}\right)\leq
E_{\mathcal{F}}\left(\pi\right).$

Proof. If $f_{1},\ldots,f_{k}$ are nonnegative and increasing, then so is
their product. Now the result follows by iterated application of (6).

We now use these results to prove the various claims in the previous examples.

###  4.1 Proof of Claim 1

Proof. It is easy to see that the output for the $S$-strategy is
$\Pi_{1}=F_{1}\star F_{2}$. Thus the result follows from Theorem 11.

###### Remark 14

The proof shows that Claim 1 is _equivalent_ to Theorem 11.

###  4.2 Proof of Claim 4

Proof. Let $f$ and $g$ be the characteristic functions of $\mathcal{C}_{R}$
and $\mathcal{C}_{B}$, then by assumption $f,g$ are increasing. We have
$f(S_{1})g(S_{2})=1$ or $0$ according as the pair $(S_{1},S_{2})$ does or does
not succeed in disrupting both networks. It follows that the success
probability for the $S$-strategy is $\Pi_{2}=f\star g$, which is increasing by
Theorem 11.

###  4.3 Proof of Claim 7

Proof. The probabilty $F(S)$ of disabling neither network is given by

$F=(1-f)\star(1-g)=(f-1)\star(g-1),$

where $f$ and $g$ are as in the previous proof. Since $(f-1)$ and $(g-1)$ are
increasing functions so is $F$. This implies that $G=1-\Pi_{2}-F$ is a
decreasing function.

###  4.4 Proof of Claim 8

Proof. It is easy to see that the success probability of the $S$-strategy is
$\Pi_{3}=f_{A}\star f_{B}$, where $f_{A}$ and $f_{B}$ are the voting functions
of companies $A$ and $B$. Now $f_{A}$ and $f_{B}$ are increasing by assumption
and thus by Theorem 11 so is $\Pi_{2}$.

##  5 Example 4: Stochastic production - many inputs

We now consider production with inputs from a finite commodity set $K$.

An entrepreneur sources her inputs from a set $H$ of suppliers. Supplier $h$
agrees to supply commodities from a subset $K^{h}$ of $K$ — the sets $K^{h}$
need not be disjoint. She further enters into a “shipping contract” $P^{h}$
with $h$, which is a partition of $K^{h}$ into a disjoint union of shipments
$K^{h}=K_{1}^{h}\sqcup\cdots\sqcup K_{l}^{h}$. However there is uncertainity
in shipping and shipment $K_{i}^{h}$ arrives with independent probability
$p_{h}$.

After receiving the shipments, she produces an output given by

$F\left(\mathbf{S}\right)=\prod\nolimits_{k\in K}F_{k}\left(S_{k}\right),$

where $S_{k}$ is the set of players whose shipment of $k$ arrives
successfully, $\mathbf{S}$ is the “success” tuple $\left(S_{k}\right)_{k\in
K}$, and the $F_{k}$ are non-negative increasing functions on subsets of $H$,
that is to say $F_{k}\left(S\right)\geq F_{k}\left(S^{\prime}\right)$ if
$S\supset S^{\prime}$. Her goal is to choose the contract tuple
$\mathbf{P}=\left(P^{h}\right)_{h\in H}$ which maximizes the expected output
$\Phi\left(\mathbf{P}\right)$.

We now formulate a generalization of Claim 1, which involves a partial order
on shipping contract tuples. If $\mathbf{P}=\left(P^{h}\right)_{h\in H}$ and
$\mathbf{Q}=\left(Q^{h}\right)_{h\in H}$ are two such tuples we write
$\mathbf{P}\succeq\mathbf{Q}$ if $P^{h}$ is _coarser_ than $Q^{h}$ for all
$h$, that is if each $P^{h}$-shipment is a union of $Q^{h}$-shipments.

###### Claim 15

The expected output is monotonic: $\mathbf{P}\succeq\mathbf{Q}$ implies
$\Phi\left(\mathbf{P}\right)\geq\Phi\left(\mathbf{Q}\right)$.

Proof. See section 6.1.

###### Remark 16

For two commodities Claim 15 reduces to Claim 1.

The maximal element for $\mathbf{\succeq}$ is the “coarse” tuple
$\mathbf{C}=\left(C^{h}\right)_{h\in H}$ where $C^{h}$ is a single shipment of
all commodities in $K_{h}.$

###### Corollary 17

The coarse tuple $\mathbf{C}$ maximizes expected output.

###### Remark 18

Suppose supplier $h$ agrees to send $x_{k}^{h}$ units of commodity $k$. Then
the total amount of $k$ received by the entrepreneur is $x_{k}=\sum_{h\in
S_{k}}x_{k}^{h}$. If we set
$F_{k}\left(S_{k}\right)=\left(x_{k}\right)^{\alpha_{k}}$ for $a_{k}>0$ then
the $F_{k}$ are increasing functions, and
$F\left(\mathbf{S}\right)=\prod_{k\in K}\left(x_{k}\right)^{\alpha_{k}}$ is
the Cobb-Douglas production function.

##  6 A game-theoretic generalization

We now recast the previous example as a strategic game among the suppliers. We
use the same setup, $K,H,K^{h},P^{h},p^{h},S_{k},\ldots$, with two
differences.

  1. 1.

We think of the partition $P^{h}$ of $K^{h}$ as a _strategic_ choice of
shipments by $h$.

  2. 2.

If the success tuple is $\mathbf{S}=\left(S_{k}\right)_{k\in K}$ then player
$h$ receives the payoff

$F^{h}\left(\mathbf{S}\right)=\prod\nolimits_{k\in
K}F_{k}^{h}\left(S_{k}\right),$

where the $F_{k}^{h}$ are non-negative increasing functions that may depend on
$h$.

We will show that in this game444See Remarks 7.3, 7.4, 7.5 for more details on
the game. the coarse strategy $\mathbf{C}$ is _dominant_ in a very strong
sense. For this, we consider the following scenario. Suppose the shipping
tuple $\mathbf{P}$ has been played, in which $P^{h}$ corresponds to the
partition $K^{h}=K_{1}^{h}\sqcup\cdots\sqcup K_{l}^{h}$ with $l\geq 2$. Player
$h$ is informed of the success/failure of all shipments, including his own,
except for $K_{1}^{h}$ and $K_{2}^{h}.$ Let $\pi$ be his expected payoff
conditional on this information, and let $\pi^{\prime}$ be his expected payoff
if he chooses to combine the commodities in $K_{1}^{h}$ and $K_{2}^{h}$ into a
single shipment before sending them off.

###### Proposition 19

In the above scenario we have $\pi^{\prime}\geq\pi$.

Proof. Let $K^{\prime}$ be the set of commodities not in $K_{1}^{h}\cup
K_{2}^{h}$, then the sets $S_{k},k\in K^{\prime}$ are constant under the
conditioning assumption, and so is the product $c=\prod_{k\in
K^{\prime}}F_{k}^{h}\left(S_{k}\right)$. The products $a=\prod_{k\in
K_{1}^{h}}F_{k}^{h}\left(S_{k}\right)$ and $b=\prod_{k\in
K_{2}^{h}}F_{k}^{h}\left(S_{k}\right)$ have two possible values $a_{1}\geq
a_{0}$ and $b_{1}\geq b_{0}$ corresponding to the arrival/non-arrival of
shipments $K_{1}^{h}$ and $K_{2}^{h}$, respectively. When sent separately
these arrive with (independent) probability $p=p_{h},$ while sent together
they arrive together with probability $p$. The payoff is the expectation of
the product $abc$, and so we get

$\pi^{\prime}=\left[\left(1-p\right)a_{0}b_{0}+pa_{1}b_{1}\right]c,\text{\quad}\pi=\left[(1-p)a_{0}+pa_{1}\right]\left[(1-p)b_{0}+pb_{1}\right]c.$

Now a straightforward algebraic calculation shows that

$\pi^{\prime}-\pi=p\left(1-p\right)\left(a_{1}-a_{0}\right)\left(b_{1}-b_{0}\right)c$

Every factor in this expression is non-negative, which implies
$\pi^{\prime}-\pi\geq 0$.

###### Theorem 20

Fix a choice of strategies by all players except $h$ and let $\pi(P)$ denote
the expected payoff to $h$ if he chooses the partition $P$. If $P$ is coarser
than $Q$ then we have $\pi(P)\geq\pi(Q)$ . Thus the coarse partition is a
dominant strategy for h.

Proof. We can go from any partition to a coarser partition in a sequence of
steps, where at each step we combine two parts of a partition into a single
part. Clearly it is enough to show that the desired inequality holds at each
step of this procedure. But this follows from the previous proposition.

###  6.1 Proof of Claim 15

Proof. We specialize the strategic model to the ”symmetric” case in which
$F_{k}^{h}=F_{k}$ for all $h$. Then each player’s payoff is the same as that
of the principal in stochastic production model of the previous section. Thus
the two optimization problems are the same, and hence Claim 15 follows from
Theorem 20.

###  6.2 Another proof of Theorem 11

Proof. Claim 15 implies Claim 1, which is equivalent to Theorem 11.

##  7 Concluding Remarks

###  7.1 Robust Optimality

Pooling is optimal _for all possible characteristics_ of the population under
consideration, e.g., $\left(x_{h},y_{h},p_{h}\right)_{h\in H}$ in Example 1,
$\left(H_{h},\mathcal{C}_{h}\right)_{h\in\left\\{A,B\right\\}}$ in Example 2,
and so on. In short, the $H$-strategy is not only optimal but _robustly
optimal_. And therein lies its full value: pooling can be invoked with
impunity, without having detailed information of the population
characteristics.

###  7.2 Costs

We have assumed various monetary costs (such as those of firing missiles in
Example 1, or of shipping inputs in Example 2, etc.) to be zero. Suppose these
costs existed, but with “economies of scale”, i.e., the cost of joint action
is less than the sum of the costs of the separate actions. Then the pooling
strategy is even more efficient, as it diminishes costs, _over and above_ the
favorable probabilistic implications of Theorem 11.

###  7.3 Nash Equilibrium and Dominant Strategies

For completeness’ sake, we review some standard game-theoretic definitions.
First recall that once each supplier $h$ chooses his strategy $P^{h}$ (i.e., a
partition of $K^{h}),$ a probability distribution is induced on success tuples
$\mathbf{S=}\left(S_{k}\right)_{k\in K}\in\mathcal{H}^{K}$ via the independent
arrivals of the elements of the partitions in
$\mathbf{P}=\left(P^{h}\right)_{h\in H}$; and that the payoff to $h$ is the
expectation $\Phi^{h}\left(\mathbf{P}\right)$ of $\prod\nolimits_{k\in
K}F_{k}^{h}\left(S_{k}\right)$ w.r.t. this probability distribution. We are
now ready for the definitions (and the subsequent remarks below). A strategy
$P^{h}$ of player $h$ is a _best reply_ to the strategy-selection
$\left(P^{i}\right)_{i\in H\diagdown\left\\{h\right\\}}$ of the other players,
if $P^{h}$ maximizes the payoff to $h$ over all his strategies, conditional on
the fixed $\left(P^{i}\right)_{i\in H\diagdown\left\\{h\right\\}}.$ The
$H$-tuple of strategies $\mathbf{P}=\left(P^{h}\right)_{h\in H}$ is a _Nash
Equilibrium_(NE) if $P^{h}$ is a best reply to $\left(P^{i}\right)_{i\in
H\diagdown\left\\{h\right\\}}$ for every $h\in H.$ Next, $P^{h}$ is a
_dominant strategy_ of $h$ if $P^{h}$ is a best reply to _every_ strategy-
selection of the players in $H\diagdown\left\\{h\right\\}$. Finally $P^{h}$ is
a _strictly dominant strategy_ of $h$ if $P^{h}$ is the _unique_ best reply to
every strategy-selection by the players in $H\diagdown\left\\{h\right\\}$. It
is obvious that if $P^{h}$ is a strictly dominant strategy for every $h\in H$,
then $\mathbf{P}=\left(P^{h}\right)_{h\in H}$ is the unique NE of the game. It
is also obvious that if each $F_{k}^{h}$ is assumed to be _strictly_
increasing, then the coarse partition is _strictly_ dominant for every player
(This follows from the fact that $\pi^{\prime}-\pi$ is strictly positive in
the proof of Proposition 19). In this case the coarse partitions constitute
the _unique_ Nash Equilibrium of the game.

###  7.4 Ex Post Optimality

Our proof of Theorem 20 shows that the coarse partition not only maximizes the
expectation of $\prod\nolimits_{k\in K}F_{k}^{h}\left(S_{k}\right)$ for player
$h$, but in fact maximizes his expected payoff conditional on _every
realization_ of the success tuples in $\widetilde{\mathcal{H}}^{K}$ due to the
other players (where $\widetilde{\mathcal{H}}$ denotes the collection of
subsets of $H\diagdown\left\\{h\right\\}$). In other words, the coarse
strategy is not just dominant in the standard sense (ex ante optimal), but in
a much stronger sense (ex post optimal).

###  7.5 Incentive Compatibility

Theorem 20 is not only useful as a generalization of Theorem 11 (with a
transparent proof), but it considerably bolsters the economic plausibility of
our examples. To this end, consider Example 4. It is not easy for the
entrepreneur to _implement_ the optimal contract with her agents. She would
need to monitor their behavior and institute costly punishment for anyone who
breaks the contract, i.e., takes it into his head to choose a partition other
than the coarsest. Moreover she would need to make it credible to her agents
that the punishment will be forthcoming so that it acts as a deterrent. All
this is not achieveable without considerable cost and effort, if it is
achievable at all. However, our game-theoretic approach provides a way out of
this impasse. The entrepreneur can simply announce that she will part with a
(tiny!) fraction $\kappa^{h}$ of her profit $\prod\nolimits_{k\in
K}F_{k}\left(S_{k}\right)$ to each supplier $h.$ This engenders a game among
the suppliers with payoffs
$F^{h}\left(\mathbf{S}\right)=\kappa^{h}\prod\nolimits_{k\in
K}F_{k}\left(S_{k}\right)$ to $h\in H$, i.e., a game with common payoffs (upto
scalar multiplication). In this game it is a dominant strategy for each $h$ to
choose his coarsest partition by Theorem 20. Thus the optimal contract of the
principal is implemented by her agents of their own accord, at virtually no
cost to her!

## References

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# Evidence for top quark production in nucleus-nucleus collisions

G.K. Krintiras111On behalf of the CMS Collaboration22footnotemark: 2 The
University of Kansas  [

###### Abstract

Droplets of quark-gluon plasma (QGP), an exotic state of strongly interacting
quantum chromodynamics (QCD) matter, are routinely produced in heavy nuclei
high-energy collisions. Although the experimental signatures marked a paradigm
shift away from expectations of a weakly coupled QGP, a challenge remains as
to how the locally deconfined state with a lifetime of a few fm can be
resolved. The only colored particle that decays mostly within the QGP is the
top quark. Here we demonstrate, for the first time, that top quark decay
products are identified, irrespective of whether interacting with the medium
(bottom quarks) or not (leptonically decaying $\mathup{{{W}}}$ bosons). Using
$1.7\pm 0.1\,\mbox{$\,\text{nb}^{-1}$}$ of lead-lead ($A=208$) collision data
recorded by the CMS experiment at a nucleon-nucleon center-of-mass energy of
5.02 , we report evidence of top quark pair
($\mathup{{{t}}}$$\mathup{{\overline{{\mathup{{{t}}}}}}}$) production.
Dilepton final states are selected, and the cross section
($\sigma_{{\mathup{{{t}}}{}\mathup{{\overline{{\mathup{{{t}}}}}}}}}$) is
measured from a likelihood fit to a multivariate discriminator using lepton
kinematic variables. The
$\sigma_{{\mathup{{{t}}}{}\mathup{{\overline{{\mathup{{{t}}}}}}}}}$
measurement is additionally performed considering the jets originating from
the hadronization of bottom quarks, which improve the sensitivity to the
$\mathup{{{t}}}$$\mathup{{\overline{{\mathup{{{t}}}}}}}$ signal process. After
background subtraction and analysis corrections, the measured
$\sigma_{{\mathup{{{t}}}{}\mathup{{\overline{{\mathup{{{t}}}}}}}}}$ is
$2.56\pm 0.82\,\text{(tot)}$ and $2.02\pm 0.69\,\text{(tot)}$$\,\mu\text{b}$
in the two cases, respectively, consistent with predictions from perturbative
QCD.

###### keywords:

CMS, physics, heavy ions, top quark

colabcolabfootnotetext: Supported by the Nuclear Physics program DE-SC0019389
of the U.S. Department of Energy.

url]cern.ch/gkrintir

##  1 Introduction

Complementary methods are used to study an exotic form of matter, produced in
heavy nuclei high-energy collisions and referred to as the quark-gluon plasma
(QGP). The characteristic feature of experimental signatures is their
sensitivity to initial- or final-state effects integrated over the QGP
lifetime, the latter increasing as a function of nucleon-nucleon
($\mathrm{N}\mathrm{N}$) center-of-mass energy
$\sqrt{\smash[b]{s_{{}_{\mathrm{NN}}}}}$ and the atomic mass $A$ of the ions
being collided. At variance with measurements considered so far in the
literature, e.g., Ref. [1], top quark ($\mathup{{{t}}}$) offers the unique
opportunity to resolve the QGP thanks to its mean lifetime
$\tau_{\mathup{{{t}}}}\approx 0.15\,\,\text{fm}$. Although the feasibility of
top quark studies with nuclear collisions was demonstrated recently [2], they
remained inaccessible in nucleus-nucleus collisions because of the small
amount of data accumulated so far at the CERN LHC and the lower
$\sqrt{\smash[b]{s_{{}_{\mathrm{NN}}}}}$ available, e.g., at the BNL RHIC. At
the end of 2018, LHC was reconfigured to provide its four major experiments
with lead-lead (PbPb) collisions at
$\sqrt{\smash[b]{s_{{}_{\mathrm{NN}}}}}=5.02\,$ at an unprecedented collision
rate. The amount of data recorded by the CMS experiment [3] corresponds to an
integrated luminosity of about $1.7\pm 0.1\mbox{$\,\text{nb}^{-1}$}$ [4].

At the LHC, top quark cross sections are dominated by pair production
($\mathup{{{t}}}$$\mathup{{\overline{{\mathup{{{t}}}}}}}$) via gluon-gluon
($\mathup{{{g}}}$$\mathup{{{g}}}$) fusion, i.e., the partonic reaction
$\mathup{{{g}}}\mathup{{{g}}}\to{\mathup{{{t}}}{}\mathup{{\overline{{\mathup{{{t}}}}}}}}+X$,
and are computable with high accuracy in perturbative quantum chromodynamics
(QCD) [5, 6, 7]. The top quark is thus a theoretically precise probe of the
initial state and, in particular, the gluon parton distribution function (PDF)
[8]. Top quarks then decay mostly within the strongly interacting medium and
promptly without hadronizing into bottom ($\mathup{{{b}}}$) quarks and
$\mathup{{{W}}}$ bosons; the latter further decays to leptons or quarks and
defines the final state [9]. The detailed study of top quark decay products
provides novel insights into the mechanisms of medium-induced parton energy
loss, a final-state effect. On the one hand, $\mathup{{{b}}}$ quarks are
ideally suited to serve as a standard candle of the amount of energy
suppression for $\mathup{{{b}}}$ quark jets [10] emerged almost simultaneously
with the heavy ion collisions [11]. On the other hand, hadronically decaying
$\mathup{{{W}}}$ bosons are not immediately resolved by the medium, and hence
probe the QGP density evolution at different space-time scales [12, 13].

##  2 Methods

In these proceedings, we describe the evidence for
$\mathup{{{t}}}$$\mathup{{\overline{{\mathup{{{t}}}}}}}$ production in
nucleus-nucleus collisions [14]. We first identify
$\mathup{{{t}}}$$\mathup{{\overline{{\mathup{{{t}}}}}}}$ signal events based
on the leptonic $\mathup{{{W}}}^{\pm}\to\ell^{\pm}\mathup{}$ decays, with
$\ell$ being either an electron ($\mathup{{{e}}}$) or muon (). The resulting
dilepton final states ($\mathup{{{e}}}^{\pm}\mathup{}^{\mp}$,
$\mathup{}^{+}\mathup{}^{-}$, and $\mathup{{{e}}}^{+}\mathup{{{e}}}^{-}$)
present a typical signature of two energetic, oppositely charged (OS), and
isolated from nearby hadronic activity leptons (“$2\ell_{\text{OS}}$”),
momentum imbalance from the undetected neutrinos (), and two $\mathup{{{b}}}$
jets. Dilepton final states are of particular interest since leptons propagate
through the produced medium regardless of its nature, thereby providing
favorable conditions for the detection of
$\mathup{{{t}}}$$\mathup{{\overline{{\mathup{{{t}}}}}}}$ production. Since the
feasibility to reconstruct $\mathup{{{b}}}$ quarks out of the QGP may be
impacted by the sizeable suppression evidenced in data, the
“$2\ell_{\text{OS}}{}+{}\text{b-tags}$” method further relies on a data-based
estimate for the $\mathup{{{b}}}$ jet identification (“tagging”) performance,
measured in terms of the $\mathup{{{b}}}$ jet identification efficiency
$\varepsilon_{\mathup{{{b}}}}$. Finally, we extract the
$\mathup{{{t}}}$$\mathup{{\overline{{\mathup{{{t}}}}}}}$ cross section
($\sigma_{{\mathup{{{t}}}{}\mathup{{\overline{{\mathup{{{t}}}}}}}}}$) from a
combined maximum-likelihood fit to a multivariate discriminator using lepton
kinematic variables, independently with the $2\ell_{\text{OS}}$ and
$2\ell_{\text{OS}}{}+{}\text{b-tags}$ methods.

The
$\mathup{{{N}}}\mathup{{{N}}}\to{\mathup{{{t}}}{}\mathup{{\overline{{\mathup{{{t}}}}}}}}+X$
process ($\mathup{{{N}}}=\mathup{{{p}}},\mathup{{{n}}}$) is simulated at next-
to-leading order (NLO) using the EPPS16 NLO nuclear PDF [15]. Sources of the
dominant background, i.e., Drell–Yan production of quark-antiquark
annihilation into lepton-antilepton pairs through $\mathup{{{Z}}}$ boson or
virtual-photon exchange (referred to as “$\mathup{{{Z}}}{}/\gamma^{*}$”) and
nonprompt leptons, e.g., $\mathup{{{W}}}$ boson production with additional
jets ($\mathup{{{W}}}$+jets) are generated at NLO; both are further corrected
with scaling factors derived from data. Subdominant contributions from single
top quark plus $\mathup{{{W}}}$ boson events ($\mathup{{{t}}}\mathup{{{W}}}$)
and $\mathup{{{W}}}$$\mathup{{{W}}}$, $\mathup{{{W}}}$$\mathup{{{Z}}}$, and
$\mathup{{{Z}}}$$\mathup{{{Z}}}$ production (collectively referred to as
“$\mathrm{V}\mathrm{V}$”) are simulated at NLO too.

Gradient boosted decision trees (BDTs) are set up to maximally discriminate
genuine leptons with high transverse momentum ($p_{\mathrm{T}}$) between the
$\mathup{{{t}}}$$\mathup{{\overline{{\mathup{{{t}}}}}}}$ signal and background
processes. The BDTs exploit kinematic properties of the leading- and
subleading-$p_{\mathrm{T}}$ leptons (referred to as “$\ell_{1}$” and
“$\ell_{2}$”, respectively). We perform a maximum-likelihood fit to binned BDT
distributions in the $\mathup{{{e}}}^{\pm}\mathup{}^{\mp}$,
$\mathup{}^{+}\mathup{}^{-}$, and $\mathup{{{e}}}^{+}\mathup{{{e}}}^{-}$ final
states, accounting for all sources of uncertainty, i.e., statistical and
systematic, and their correlations. The simultaneous analysis of these events
allows us measuring the
$\mathup{{{t}}}$$\mathup{{\overline{{\mathup{{{t}}}}}}}$ signal strength
$\mu$, i.e., the ratio of the observed
$\sigma_{{\mathup{{{t}}}{}\mathup{{\overline{{\mathup{{{t}}}}}}}}}$ to the
expectation from theory. The best fit value of $\mu$ and its uncertainty
$\Delta\mu$ (corresponding to a 68% confidence level) are extracted
independently with the $2\ell_{\text{OS}}$ and
$2\ell_{\text{OS}}{}+{}\text{b-tags}$ methods. For the latter, we correlate
the number of $\mathup{{{t}}}$$\mathup{{\overline{{\mathup{{{t}}}}}}}$ signal
events in the $\mathup{{{b}}}$-tagged jet categories based on multinomial
probabilities, using $\varepsilon_{\mathup{{{b}}}}$ and a parameter
($\delta_{\text{QGP}}$) accounting for medium-induced suppression of
$\varepsilon_{\mathup{{{b}}}}$. We allow $\varepsilon_{\mathup{{{b}}}}$ to
receive different values for the two $\mathup{{{b}}}$ jets, i.e.,
$\varepsilon_{\mathup{{{b}}}}\to\varepsilon_{\mathup{{{b}}}}^{*}=(1-\delta_{\text{QGP}})\times\varepsilon_{\mathup{{{b}}}}$,
motivated by a path-length dependence of the parton energy loss.

##  3 Results

The postfit expected and observed BDT distributions are shown in Fig. 1 in the
$\mathup{{{t}}}$$\mathup{{\overline{{\mathup{{{t}}}}}}}$-enriched
$\mathup{{{e}}}^{\pm}\mathup{}^{\mp}$ final state with the $2\ell_{\text{OS}}$
($\mu=0.81^{+0.26}_{-0.23}$, left) and $2\ell_{\text{OS}}{}+{}\text{b-tags}$
($\mu=0.64^{+0.22}_{-0.20}$, middle) methods. We found the classifier to
separate well the $\mathup{{{t}}}$$\mathup{{\overline{{\mathup{{{t}}}}}}}$
signal from the $\mathup{{{Z}}}$/$\gamma^{*}$ background in the
$\mathup{}^{+}\mathup{}^{-}$ and $\mathup{{{e}}}^{+}\mathup{{{e}}}^{-}$ final
states, enhancing our confidence about its applicability to the
$\mathup{{{e}}}^{\pm}\mathup{}^{\mp}$ final state. The inclusive
$\sigma_{{\mathup{{{t}}}{}\mathup{{\overline{{\mathup{{{t}}}}}}}}}$ is then
obtained multiplying the best fit $\mu$ value by the theoretical expectation.
Accounting for the acceptance corrections, we measure
$\sigma_{{\mathup{{{t}}}{}\mathup{{\overline{{\mathup{{{t}}}}}}}}}$ to be
$2.56\pm 0.82\,\text{(tot)}$ and $2.02\pm 0.69\,\text{(tot)}$$\,\mu\text{b}$
in the combined $\mathup{{{e}}}^{\pm}\mathup{}^{\mp}$,
$\mathup{}^{+}\mathup{}^{-}$, and $\mathup{{{e}}}^{+}\mathup{{{e}}}^{-}$ final
states, with a relative total uncertainty of 32 and 34% with the
$2\ell_{\text{OS}}$ and $2\ell_{\text{OS}}{}+{}\text{b-tags}$ methods,
respectively.

Fig. 1:  Postfit expected (histograms) and observed (points) binned BDT (left,
middle) and sphericity (right) distributions in the
$\mathup{{{e}}}^{\pm}\mathup{}^{\mp}$ final state with the $2\ell_{\text{OS}}$
(left, right) and $2\ell_{\text{OS}}{}+{}\text{b-tags}$ (middle) methods. The
comparison between the
$\mathup{{{t}}}$$\mathup{{\overline{{\mathup{{{t}}}}}}}$ signal and the
background-subtracted data is shown for the postfit distributions as inset
panel (right). The vertical bars on the points represent the statistical
uncertainty in data (68% Clopper-Pearson intervals). The hatched regions show
the postfit uncertainty in the sum of
$\mathup{{{t}}}$$\mathup{{\overline{{\mathup{{{t}}}}}}}$ signal and
background. The lower panels display the ratio of the observed data to the
predictions, including the
$\mathup{{{t}}}$$\mathup{{\overline{{\mathup{{{t}}}}}}}$ signal, with bars and
bands, respectively, representing the statistical and total uncertainties in
the prediction [14].

The compatibility of the data with the background-only hypothesis is evaluated
using a profile-likelihood ratio as a test statistic, including all sources of
systematic uncertainty. The probability for the background to mimic an excess
of events as large or larger than that observed in data is measured using $p$
values. The latter values are then expressed in terms of Gaussian tail
probabilities, given in units of standard deviation ($\sigma$). The
background-only hypothesis is excluded with statistical significance of $3.8$
($4.8$) and $4.0$ ($6.0$) $\sigma$ with the $2\ell_{\text{OS}}$ and
$2\ell_{\text{OS}}{}+{}\text{b-tags}$ methods, respectively. The $p$ value of
the difference between observed and expected significance, given in
parentheses, is 0.18 and 0.05, respectively. To further examine the hypothesis
that the selected data are consistent with the production of top quarks, we
study the sphericity
$s=p_{\mathrm{T}}(\ell^{\pm}\ell^{\mp})/\left(p_{\mathrm{T}}(\ell_{1})+p_{\mathrm{T}}(\ell_{2})\right)$,
a global-event property. We found $s$ (Fig. 1, right) to perform well in terms
of separating the $\mathup{{{t}}}$$\mathup{{\overline{{\mathup{{{t}}}}}}}$
signal ($s\sim 1$) from the background ($s\sim 0$) and to exclude the
background-only hypothesis at a similar level.

Figure 2 presents the extracted
$\sigma_{{\mathup{{{t}}}{}\mathup{{\overline{{\mathup{{{t}}}}}}}}}$, including
the measurement at $\sqrt{\smash[b]{s}}=5.02$ [8] in proton-proton
($\mathup{{{p}}}\mathup{{{p}}}$) collisions, and compared to state-of-the-art
theoretical QCD predictions [6].

##  4 Summary

In summary, the top pair production cross section is measured for the first
time in nucleus-nucleus collisions, using lead-lead data at
$\sqrt{\smash[b]{s_{{}_{\mathrm{NN}}}}}=5.02\,$ with an integrated luminosity
of $1.7\pm 0.1\mbox{$\,\text{nb}^{-1}$}$. The measurement is performed
analyzing separately events containing at least one pair of oppositely charged
leptons (electrons or muons) and jets originating from the hadronization of
bottom quarks. The observed statistical significance of the
$\mathup{{{t}}}$$\mathup{{\overline{{\mathup{{{t}}}}}}}$ signal against the
background-only hypothesis is $3.8$ and $4.0$ standard deviations in the two
cases, respectively. The measured cross section is $2.56\pm
0.82\,\text{(tot)}$ and $2.02\pm 0.69\,\text{(tot)}$$\,\mu\text{b}$,
respectively, consistent with the expectations from scaled proton-proton data
as well as perturbative quantum chromodynamics calculations. The measurement
paves the way for detailed investigations of top quark production in nuclear
interactions with increased LHC heavy ion luminosities or future higher-energy
colliders. In particular, it provides a novel tool for probing nuclear parton
distribution functions, the mechanisms of parton energy loss, and the medium
opacity at different space-time scales.

Fig. 2: The inclusive $\mathup{{{t}}}$$\mathup{{\overline{{\mathup{{{t}}}}}}}$
cross section measured in the combined $\mathup{{{e}}}^{\pm}\mathup{}^{\mp}$,
$\mathup{}^{+}\mathup{}^{-}$, and $\mathup{{{e}}}^{+}\mathup{{{e}}}^{-}$final
states in $\mathrm{Pb}\mathrm{Pb}$ collisions (divided by the mass number
squared, $A^{2}$), compared to predictions [6], and
$\mathup{{{p}}}\mathup{{{p}}}$ results at
$\sqrt{\smash[b]{s_{{}_{\mathrm{NN}}}}}=5.02$ [8]. The total experimental
error bars (theoretical error bands) include statistical and systematic (PDF
and QCD scale) uncertainties added in quadrature [14].

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