text: (cid:80) (cid:80) (cid:80) 8 | − | | − | | − | and bending interactions. Overall, we compute: 1 r 21/6σ , 1 r 0.96σ and 1 θ θ . NLJ α α NFENE β β Nbend γ γ 0 The results are given in Figure 5a-c for LJ, FENE and bending interactions respectively. As bending stiﬀness κ /κ increases, the chains increasingly conform to θ independently of chain length bend LJ 0 (Figure 5c). This requires changes in structure that must be accommodated by small additional deviations of LJ and FENE bonds from their resting positions. For LJ interactions (Figure 5a) the deviations increase steadily with stiﬀness. This eﬀect is more marked for long chains, which have a larger number of bending constraints per bead. For FENE bonds (Figure 5b), meanwhile, the behavior is more subtle. The primary eﬀect is similar to that observed for LJ interactions: increasing bending stiﬀness increases the deviation of FENE bonds from their rest positions. For short and ﬂexible chains, however, there is an anomalous secondary eﬀect: for κ /κ < 1, the deviation of FENE bonds bend LJ increases with decreasing chain length. As discussed above, the ubiquity of LJ interactions means they may dictate the positions of FENE bonded beads regardless of their considerably weaker potential. As chain length decreases, the average number of FENE bonds per bead decreases steadily, while the number of LJ interactions per bead remains unchanged. This means that LJ interactions can have more inﬂuence on FENE positions as chain length decreases, thus leading to additional stretching. This eﬀect is stronger for ﬂexible chains, which don’t have the additional eﬀect of angular constraints. We further show that deviations from rest positions lead directly to potential energy being stored in the system in each case. The potential energy is calculated by summing U , U and U over every interaction in the LJ FENE bend minimized conﬁguration, and is given as a function of chain length and stiﬀness in Figure 5d. The main eﬀect is ∼ that increasing chain length introduces more FENE bonds into the system, whose individual deviations of 0.02σ (Figure 5b) contribute signiﬁcantly to increasing stored potential energy. There is an additional eﬀect whereby stored potential energy increases with increasing stiﬀness. This has contributions from LJ and FENE, in line with their deviations shown in Figures 5a-b, and also from bending potentials. Although the triplet conﬁgurations increasingly conform to their rest positions with increasing stiﬀness, remaining deviations become progressively more costly as κ /κ increases, leading to a contribution to the stored potenital energy. bend LJ D. Overview of structural changes The evolution of density and internal stresses (as parameterized by deviations from interaction energy minima) are strong functions of the bending stiﬀness and chain length. Upon increasing the stiﬀness, all angles between adjacent bonds tend to approach the minimum of the bending potential at θ . However, this eﬀect competes with the tendency 0 of neighboring beads to stay close to the minima of LJ and FENE interactions. This competition leads to an increase of potential energy due to pairs of beads drifting subtly away from LJ and FENE minima. The eﬀect of increasing ∼ the chain length is to insert more FENE bonds into the system. Since these are typically 0.02σ from their resting positions, this leads to a sharp increase in stored potential energy. We have checked that increasing chain length further above L = 50 does not bring any further evolution, and we can safely conclude that chain length has a non-negligible eﬀect only for L < 50. IV. VIBRATIONAL DENSITY OF STATES We next investigate the connection between chain length, angular potential and the VDOS. Importantly, changes in the spectrum due to increasing the angular potential are not only related to the associated increase in angular forces, but also to the structural changes that arise as discussed above. In what follows, we ﬁrst outline our formalism for obtaining the VDOS, including a description of how we decompose it into various contributions. We then give an overview of the generic features of the VDOS of polymeric glasses, before focussing speciﬁcally on the behavior with respect to chain length and bending stiﬀness. A. Formalism for obtaining the VDOS ∗ Since we prepared the glasses well below T , we can ignore any eﬀects of thermal noise and hence work in the g harmonic approximation, where the displacements of the system around energy minima are small. The equation of ##LTLine## ##LTLine## ##LTLine##

text: and

text: bending

text: interactions.

text: Overall,

text: we

text: compute:

text: r

text: r

text: and

text: θ

text: θ

text: NLJ

text: α

text: α

text: NFENE

text: β

text: β

text: Nbend

text: γ

text: γ

text: The

text: results

text: are

text: given

text: in

text: Figure

text: for

text: LJ,

text: FENE

text: and

text: bending

text: interactions

text: respectively.

text: As

text: bending

text: stiﬀness

text: κ

text: /κ

text: increases,

text: the

text: chains

text: increasingly

text: conform

text: to

text: θ

text: independently

text: of

text: chain

text: length

text: bend

text: LJ

text: (Figure

text: This

text: requires

text: changes

text: in

text: structure

text: that

text: must

text: be

text: accommodated

text: by

text: small

text: additional

text: deviations

text: of

text: LJ

text: and

text: FENE

text: bonds

text: from

text: their

text: resting

text: positions.

text: For

text: LJ

text: interactions

text: (Figure

text: the

text: deviations

text: increase

text: steadily

text: with

text: stiﬀness.

text: This

text: eﬀect

text: is

text: more

text: marked

text: for

text: long

text: chains,

text: which

text: have

text: a

text: larger

text: number

text: of

text: bending

text: constraints

text: per

text: bead.

text: For

text: FENE

text: bonds

text: (Figure

text: meanwhile,

text: the

text: behavior

text: is

text: more

text: subtle.

text: The

text: primary

text: eﬀect

text: is

text: similar

text: to

text: that

text: observed

text: for

text: LJ

text: interactions:

text: increasing

text: bending

text: stiﬀness

text: increases

text: the

text: deviation

text: of

text: FENE

text: bonds

text: from

text: their

text: rest

text: positions.

text: For

text: short

text: and

text: ﬂexible

text: chains,

text: however,

text: there

text: is

text: an

text: anomalous

text: secondary

text: eﬀect:

text: for

text: κ

text: /κ

text: the

text: deviation

text: of

text: FENE

text: bonds

text: bend

text: LJ

text: increases

text: with

text: decreasing

text: chain

text: length.

text: As

text: discussed

text: above,

text: the

text: ubiquity

text: of

text: LJ

text: interactions

text: means

text: they

text: may

text: dictate

text: the

text: positions

text: of

text: FENE

text: bonded

text: beads

text: regardless

text: of

text: their

text: considerably

text: weaker

text: potential.

text: As

text: chain

text: length

text: decreases,

text: the

text: average

text: number

text: of

text: FENE

text: bonds

text: per

text: bead

text: decreases

text: steadily,

text: while

text: the

text: number

text: of

text: LJ

text: interactions

text: per

text: bead

text: remains

text: unchanged.

text: This

text: means

text: that

text: LJ

text: interactions

text: can

text: have

text: more

text: inﬂuence

text: on

text: FENE

text: positions

text: as

text: chain

text: length

text: decreases,

text: thus

text: leading

text: to

text: additional

text: stretching.

text: This

text: eﬀect

text: is

text: stronger

text: for

text: ﬂexible

text: chains,

text: which

text: don’t

text: have

text: the

text: additional

text: eﬀect

text: of

text: angular

text: constraints.

text: We

text: further

text: show

text: that

text: deviations

text: from

text: rest

text: positions

text: lead

text: directly

text: to

text: potential

text: energy

text: being

text: stored

text: in

text: the

text: system

text: in

text: each

text: case.

text: The

text: potential

text: energy

text: is

text: calculated

text: by

text: summing

text: U

text: U

text: and

text: U

text: over

text: every

text: interaction

text: in

text: the

text: LJ

text: FENE

text: bend

text: minimized

text: conﬁguration,

text: and

text: is

text: given

text: as

text: a

text: function

text: of

text: chain

text: length

text: and

text: stiﬀness

text: in

text: Figure

text: The

text: main

text: eﬀect

text: is

text: that

text: increasing

text: chain

text: length

text: introduces

text: more

text: FENE

text: bonds

text: into

text: the

text: system,

text: whose

text: individual

text: deviations

text: of

text: (Figure

text: contribute

text: signiﬁcantly

text: to

text: increasing

text: stored

text: potential

text: energy.

text: There

text: is

text: an

text: additional

text: eﬀect

text: whereby

text: stored

text: potential

text: energy

text: increases

text: with

text: increasing

text: stiﬀness.

text: This

text: has

text: contributions

text: from

text: LJ

text: and

text: FENE,

text: in

text: line

text: with

text: their

text: deviations

text: shown

text: in

text: Figures

text: and

text: also

text: from

text: bending

text: potentials.

text: Although

text: the

text: triplet

text: conﬁgurations

text: increasingly

text: conform

text: to

text: their

text: rest

text: positions

text: with

text: increasing

text: stiﬀness,

text: remaining

text: deviations

text: become

text: progressively

text: more

text: costly

text: as

text: κ

text: /κ

text: increases,

text: leading

text: to

text: a

text: contribution

text: to

text: the

text: stored

text: potenital

text: energy.

text: bend

text: LJ

text: D.

text: Overview

text: of

text: structural

text: changes

text: The

text: evolution

text: of

text: density

text: and

text: internal

text: stresses

text: (as

text: parameterized

text: by

text: deviations

text: from

text: interaction

text: energy

text: minima)

text: are

text: strong

text: functions

text: of

text: the

text: bending

text: stiﬀness

text: and

text: chain

text: length.

text: Upon

text: increasing

text: the

text: stiﬀness,

text: all

text: angles

text: between

text: adjacent

text: bonds

text: tend

text: to

text: approach

text: the

text: minimum

text: of

text: the

text: bending

text: potential

text: at

text: θ

text: However,

text: this

text: eﬀect

text: competes

text: with

text: the

text: tendency

text: of

text: neighboring

text: beads

text: to

text: stay

text: close

text: to

text: the

text: minima

text: of

text: LJ

text: and

text: FENE

text: interactions.

text: This

text: competition

text: leads

text: to

text: an

text: increase

text: of

text: potential

text: energy

text: due

text: to

text: pairs

text: of

text: beads

text: drifting

text: subtly

text: away

text: from

text: LJ

text: and

text: FENE

text: minima.

text: The

text: eﬀect

text: of

text: increasing

text: the

text: chain

text: length

text: is

text: to

text: insert

text: more

text: FENE

text: bonds

text: into

text: the

text: system.

text: Since

text: these

text: are

text: typically

text: from

text: their

text: resting

text: positions,

text: this

text: leads

text: to

text: a

text: sharp

text: increase

text: in

text: stored

text: potential

text: energy.

text: We

text: have

text: checked

text: that

text: increasing

text: chain

text: length

text: further

text: above

text: L

text: does

text: not

text: bring

text: any

text: further

text: evolution,

text: and

text: we

text: can

text: safely

text: conclude

text: that

text: chain

text: length

text: has

text: a

text: non-negligible

text: eﬀect

text: only

text: for

text: L

text: IV.

text: VIBRATIONAL

text: DENSITY

text: OF

text: STATES

text: We

text: next

text: investigate

text: the

text: connection

text: between

text: chain

text: length,

text: angular

text: potential

text: and

text: the

text: VDOS.

text: Importantly,

text: changes

text: in

text: the

text: spectrum

text: due

text: to

text: increasing

text: the

text: angular

text: potential

text: are

text: not

text: only

text: related

text: to

text: the

text: associated

text: increase

text: in

text: angular

text: forces,

text: but

text: also

text: to

text: the

text: structural

text: changes

text: that

text: arise

text: as

text: discussed

text: above.

text: In

text: what

text: follows,

text: we

text: ﬁrst

text: outline

text: our

text: formalism

text: for

text: obtaining

text: the

text: VDOS,

text: including

text: a

text: description

text: of

text: how

text: we

text: decompose

text: it

text: into

text: various

text: contributions.

text: We

text: then

text: give

text: an

text: overview

text: of

text: the

text: generic

text: features

text: of

text: the

text: VDOS

text: of

text: polymeric

text: glasses,

text: before

text: focussing

text: speciﬁcally

text: on

text: the

text: behavior

text: with

text: respect

text: to

text: chain

text: length

text: and

text: bending

text: stiﬀness.

text: A.

text: Formalism

text: for

text: obtaining

text: the

text: VDOS

text: Since

text: we

text: prepared

text: the

text: glasses

text: well

text: below

text: T

text: we

text: can

text: ignore

text: any

text: eﬀects

text: of

text: thermal

text: noise

text: and

text: hence

text: work

text: in

text: the

text: g

text: harmonic

text: approximation,

text: where

text: the

text: displacements

text: of

text: the

text: system

text: around

text: energy

text: minima

text: are

text: small.

text: The

text: equation

text: of

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Multisensor Online Transfer Learning for 3D LiDAR-based Human Classiﬁcation with a Mobile Robot Zhi Yan1, Li Sun2, Tom Duckett2, and Nicola Bellotto2 Abstract— Human detection and tracking is an essential task for service robots, where the combined use of multiple sensors has potential advantages that are yet to be exploited. In this paper, we introduce a framework allowing a robot to learn a new 3D LiDAR-based human classiﬁer from other sensors over time, taking advantage of a multisensor tracking system. The main innovation is the use of different detectors for existing sensors (i.e. RGB-D camera, 2D LiDAR) to train, online, a new 3D LiDAR-based human classiﬁer, exploiting a so-called trajectory probability. Our framework uses this probability to check whether new detections belongs to a human trajectory, estimated by different sensors and/or detectors, and to learn a human classiﬁer in a semi-supervised fashion. The framework has been implemented and tested on a real-world dataset collected by a mobile robot. We present experiments illustrating that our system is able to effectively learn from different Fig. 1. Left: our robot platform including a Pioneer 3-AT mobile robot with four wheels for indoor use and a built-in odometry sensor, a Velodyne sensors and from the environment, and that the performance VLP-16 3D LiDAR (1), an ASUS Xtion PRO LIVE RGB-D camera (2), of the 3D LiDAR-based human classiﬁcation improves with the and a HOKUYO UTM-30LX 2D LiDAR (3). Right: visualized sensor number of sensors/detectors used. data (only depth information is shown for RGB-D camera) annotated with corresponding sensor numbers. I. INTRODUCTION Human detection and tracking is an important task in do not. We wish the latter to learn such a detection ability service robotics, where knowledge of human motion proper- from the existing sensors, and ultimately improve the overall ties such as position, velocity and direction can be used to performance of people tracking. In this paper we focus on the improve the behavior of the robot, for example to improve ﬁrst part only and learn a classiﬁer to distinguish between its collision avoidance and adapt its velocity to that of the human and non-human 3D LiDAR detections, leaving the surrounding people. second part for future work. Using multiple sensors to track people has advantages over a single one. The most obvious one is that multiple sensors Typically, data collection and training of the classiﬁer are can often do the task with a wider ﬁeld of view and thus track done ofﬂine, with obvious labour cost and potential human more people within a larger range [1], [2]. Another advantage errors. With our new framework, instead, a 3D LiDAR-based is that multiple sensors providing redundant information can human classiﬁer is learned online using tracking information increase tracking accuracy and reliability [3], [4]. Different and pre-trained human detectors for 2D LiDAR and RGB- sensors have different properties, including ﬁeld of view. D sensors. In general, the framework allows a new sensor The 3D LiDAR in our robot platform (Fig. 1) has 16 scan to learn from the trajectories of the tracking system, each channels, 360◦ horizontal and 30◦ vertical ﬁelds of view, one with an associated probability of being human-generated, and up to 100 m range. However, the measurement returned and fusing both model-based (labeled) and model-free (un- by this sensor are discrete points in space (i.e. point cloud), labeled) detections from different detectors according to a from which it is difﬁcult to identify people because some semi-supervised learning scheme [5]. In contrast to previous low-level features, such as texture and colour, are missing. approaches [6], [7], our solution does not need any hand- The 2D LiDAR has similar problems. The RGB-D camera, labeled data, performing online learning completely from instead, can detect humans more reliably but only within a scratch. This feature also makes our system easily adaptable short range and small ﬁeld of view (see Fig. 2). to the environment where the robot is deployed. We consider a particular problem in multisensor people The contributions of this paper are as follows: 1) we tracking: there are different types of sensors, some of which propose an online transfer learning framework for multisen- already do have good human detectors, while other sensors sor people tracking based on a trajectory probability, which takes into account both sensor independence in the detection 1Laboratoire Electronique Informatique Image (Le2i), University of Tech- and multisensor interaction in the trajectory estimation; 2) nology of Belfort-Montbe´liard (UTBM), France zhi.yan@utbm.fr we present an experimental evaluation of our system for 2Lincoln Centre for Autonomous { Systems (L-CAS), 3D LiDAR-based human classiﬁcation with a mobile robot University of Lincoln, UK lsun, tduckett, } nbellotto @lincoln.ac.uk on a real-world dataset using different sensor combinations. ##LTFigure## ##LTFigure##

text: Multisensor

label: title

text: Online

text: Transfer

text: Learning

text: for

text: LiDAR-based

text: Human

text: Classiﬁcation

text: with

text: a

text: Mobile

text: Robot

text: Zhi

label: author

text: Li

text: Tom

text: and

text: Nicola

text: Abstract—

label: paragraph

text: Human

label: abstract

text: detection

text: and

text: tracking

text: is

text: an

text: essential

text: task

text: for

text: service

text: robots,

text: where

text: the

text: combined

text: use

text: of

text: multiple

text: sensors

text: has

text: potential

text: advantages

text: that

text: are

text: yet

text: to

text: be

text: exploited.

text: In

text: this

text: paper,

text: we

text: introduce

text: a

text: framework

text: allowing

text: a

text: robot

text: to

text: learn

text: a

text: new

text: LiDAR-based

text: human

text: classiﬁer

text: from

text: other

text: sensors

text: over

text: time,

text: taking

text: advantage

text: of

text: a

text: multisensor

text: tracking

text: system.

text: The

text: main

text: innovation

text: is

text: the

text: use

text: of

text: different

text: detectors

text: for

text: existing

text: sensors

text: (i.e.

text: RGB-D

text: camera,

text: LiDAR)

text: to

text: train,

text: online,

text: a

text: new

text: LiDAR-based

text: human

text: classiﬁer,

text: exploiting

text: a

text: so-called

text: trajectory

text: probability.

text: Our

text: framework

text: uses

text: this

text: probability

text: to

text: check

text: whether

text: new

text: detections

text: belongs

text: to

text: a

text: human

text: trajectory,

text: estimated

text: by

text: different

text: sensors

text: and/or

text: detectors,

text: and

text: to

text: learn

text: a

text: human

text: classiﬁer

text: in

text: a

text: semi-supervised

text: fashion.

text: The

text: framework

text: has

text: been

text: implemented

text: and

text: tested

text: on

text: a

text: real-world

text: dataset

text: collected

text: by

text: a

text: mobile

text: robot.

text: We

text: present

text: experiments

text: illustrating

text: that

text: our

text: system

text: is

text: able

text: to

text: effectively

text: learn

text: from

text: different

text: Fig.

text: Left:

text: our

text: robot

text: platform

text: including

text: a

text: Pioneer

text: mobile

text: robot

text: with

text: four

text: wheels

text: for

text: indoor

text: use

text: and

text: a

text: built-in

text: odometry

text: sensor,

text: a

text: Velodyne

text: sensors

text: and

text: from

text: the

text: environment,

text: and

text: that

text: the

text: performance

text: LiDAR

text: an

text: ASUS

text: Xtion

text: PRO

text: LIVE

text: RGB-D

text: camera

text: of

text: the

text: LiDAR-based

text: human

text: classiﬁcation

text: improves

text: with

text: the

text: and

text: a

text: HOKUYO

text: LiDAR

text: Right:

text: visualized

text: sensor

text: number

text: of

text: sensors/detectors

text: used.

text: data

text: (only

text: depth

text: information

text: is

text: shown

text: for

text: RGB-D

text: camera)

text: annotated

text: with

text: corresponding

text: sensor

text: numbers.

text: I.

text: INTRODUCTION

text: Human

text: detection

text: and

text: tracking

text: is

text: an

text: important

text: task

text: in

text: do

text: not.

text: We

text: wish

text: the

text: latter

text: to

text: learn

text: such

text: a

text: detection

text: ability

text: service

text: robotics,

text: where

text: knowledge

text: of

text: human

text: motion

text: proper-

text: from

text: the

text: existing

text: sensors,

text: and

text: ultimately

text: improve

text: the

text: overall

text: ties

text: such

text: as

text: position,

text: velocity

text: and

text: direction

text: can

text: be

text: used

text: to

text: performance

text: of

text: people

text: tracking.

text: In

text: this

text: paper

text: we

text: focus

text: on

text: the

text: improve

text: the

text: behavior

text: of

text: the

text: robot,

text: for

text: example

text: to

text: improve

text: ﬁrst

text: part

text: only

text: and

text: learn

text: a

text: classiﬁer

text: to

text: distinguish

text: between

text: its

text: collision

text: avoidance

text: and

text: adapt

text: its

text: velocity

text: to

text: that

text: of

text: the

text: human

text: and

text: non-human

text: LiDAR

text: detections,

text: leaving

text: the

text: surrounding

text: people.

text: second

text: part

text: for

text: future

text: work.

text: Using

text: multiple

text: sensors

text: to

text: track

text: people

text: has

text: advantages

text: over

text: a

text: single

text: one.

text: The

text: most

text: obvious

text: one

text: is

text: that

text: multiple

text: sensors

text: Typically,

text: data

text: collection

text: and

text: training

text: of

text: the

text: classiﬁer

text: are

text: can

text: often

text: do

text: the

text: task

text: with

text: a

text: wider

text: ﬁeld

text: of

text: view

text: and

text: thus

text: track

text: done

text: ofﬂine,

text: with

text: obvious

text: labour

text: cost

text: and

text: potential

text: human

text: more

text: people

text: within

text: a

text: larger

text: range

text: Another

text: advantage

text: errors.

text: With

text: our

text: new

text: framework,

text: instead,

text: a

text: LiDAR-based

text: is

text: that

text: multiple

text: sensors

text: providing

text: redundant

text: information

text: can

text: human

text: classiﬁer

text: is

text: learned

text: online

text: using

text: tracking

text: information

text: increase

text: tracking

text: accuracy

text: and

text: reliability

text: Different

text: and

text: pre-trained

text: human

text: detectors

text: for

text: LiDAR

text: and

text: RGB-

text: sensors

text: have

text: different

text: properties,

text: including

text: ﬁeld

text: of

text: view.

text: D

text: sensors.

text: In

text: general,

text: the

text: framework

text: allows

text: a

text: new

text: sensor

text: The

text: LiDAR

text: in

text: our

text: robot

text: platform

text: (Fig.

text: has

text: scan

text: to

text: learn

text: from

text: the

text: trajectories

text: of

text: the

text: tracking

text: system,

text: each

text: channels,

text: horizontal

text: and

text: vertical

text: ﬁelds

text: of

text: view,

text: one

text: with

text: an

text: associated

text: probability

text: of

text: being

text: human-generated,

text: and

text: up

text: to

text: m

text: range.

text: However,

text: the

text: measurement

text: returned

text: and

text: fusing

text: both

text: model-based

text: (labeled)

text: and

text: model-free

text: (un-

text: by

text: this

text: sensor

text: are

text: discrete

text: points

text: in

text: space

text: (i.e.

text: point

text: cloud),

text: labeled)

text: detections

text: from

text: different

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text: to

text: a

text: from

text: which

text: it

text: is

text: difﬁcult

text: to

text: identify

text: people

text: because

text: some

text: semi-supervised

text: learning

text: scheme

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text: contrast

text: to

text: previous

text: low-level

text: features,

text: such

text: as

text: texture

text: and

text: colour,

text: are

text: missing.

text: approaches

text: our

text: solution

text: does

text: not

text: need

text: any

text: hand-

text: The

text: LiDAR

text: has

text: similar

text: problems.

text: The

text: RGB-D

text: camera,

text: labeled

text: data,

text: performing

text: online

text: learning

text: completely

text: from

text: instead,

text: can

text: detect

text: humans

text: more

text: reliably

text: but

text: only

text: within

text: a

text: scratch.

text: This

text: feature

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text: makes

text: our

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text: easily

text: adaptable

text: short

text: range

text: and

text: small

text: ﬁeld

text: of

text: view

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text: Fig.

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text: the

text: environment

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text: the

text: robot

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text: deployed.

text: We

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text: problem

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text: multisensor

text: people

text: The

text: contributions

text: of

text: this

text: paper

text: are

text: as

text: follows:

text: we

text: tracking:

text: there

text: are

text: different

text: types

text: of

text: sensors,

text: some

text: of

text: which

text: propose

text: an

text: online

text: transfer

text: learning

text: framework

text: for

text: multisen-

text: already

text: do

text: have

text: good

text: human

text: detectors,

text: while

text: other

text: sensors

text: sor

text: people

text: tracking

text: based

text: on

text: a

text: trajectory

text: probability,

text: which

text: takes

text: into

text: account

text: both

text: sensor

text: independence

text: in

text: the

text: detection

text: Electronique

text: Informatique

text: Image

text: University

text: of

text: Tech-

text: and

text: multisensor

text: interaction

text: in

text: the

text: trajectory

text: estimation;

text: nology

text: of

text: Belfort-Montbe´liard

text: (UTBM),

text: France

text: zhi.yan@utbm.fr

text: we

text: present

text: an

text: experimental

text: evaluation

text: of

text: our

text: system

text: for

text: Centre

text: for

text: Autonomous

text: Systems

text: (L-CAS),

text: LiDAR-based

text: human

text: classiﬁcation

text: with

text: a

text: mobile

text: robot

text: University

text: of

text: Lincoln,

text: UK

text: lsun,

text: tduckett,

text: nbellotto

text: @lincoln.ac.uk

text: on

text: a

text: real-world

text: dataset

text: using

text: different

text: sensor

text: combinations.

text: ##LTFigure##

text: ##LTFigure##


text: tracking in close vicinity to the robot. The proposed tracking system tracks people by detecting legs extracted from a 2D LiDAR and fusing this with the faces or the upper-bodies detected with a camera using a sequential implementation of the Unscented Kalman Filter (UKF). The combination with 3D LiDAR is increasing with the development of the 3D LiDAR technology. Taking advantage of its high accuracy, [4] developed an algorithm to align 3D LiDAR data with high-resolution camera images obtained from ﬁve cameras, in order to accurately track moving vehicles. Other reported results include [13] and [14], which mainly focused on pedestrian detection rather than tracking. In addition, earlier work presented multitarget tracking with a mobile robot equipped with two 2D LiDARs, respectively located at the front and back [1]. Thus the robot can have ◦ a 360 horizontal ﬁeld of view, where each scan of these two sensors covers the whole surrounding of the robot at an ◦ Fig. 2. Different horizontal ﬁelds of view of different sensors. Vertical angular resolution of 1 . ﬁelds of view, not shown in this ﬁgure, are also different. The use of machine learning algorithms for tracking has particular advantages. The closest work to ours is [7], where the authors proposed a semi-supervised learning approach to The remainder of this paper is organized as follows: the problem of track classiﬁcation in 3D LiDAR data, based Section gives an overview of the related literature. Sec- on Expectation-Maximization (EM) algorithm. In contrast to tion formulates the problem in multisensor background. our approach, their learning procedure needs a small set of Section presents our solution framework for online transfer seed tracks and a large set of background tracks, that need learning. Section describes the application of the proposed to be manually or semi-manually labeled at ﬁrst, whereas we framework to the problem of 3D LiDAR-based human clas- do not need any hand-labeled data. siﬁcation. Section illustrates the experimental results for To our knowledge, no existing work in the robotics ﬁeld different sensor conﬁgurations. Finally, Section concludes exploits information from multisensor-based tracking to im- the paper summarizing the contributions and suggesting plement transfer learning between different sensors as in future research work. this paper. Our work combines the advantages of multiple II. RELATED WORK sensors with the efﬁciency of semi-supervised learning, and The problem of multitarget and multisensor tracking has integrates them into an single online framework applied to been extensively studied during the past few decades. Most 3D LiDAR-based human classiﬁcation. of present systems are based on Bayesian methods [8], which III. PROBLEM FORMULATION compute an estimate of the correspondence between features detected in the sensor data and the different humans to be In this section, we model the problem of online learning tracked. Regarding robotic applications, multiple sensors can for human classiﬁcation in a multitarget-multisensor track- be deployed in single- or multi-robot systems [9], while the ing system. The formulation is generalized which can be former is the concern of this paper. extended to objects rather than humans. We consider online RGB/RGB-D camera plus 2D LiDAR is the most fre- learning as a detection-and-tracking-based iterative process. quently used combination in the literature. [3] presented two Whilst this idea has been presented in the computer vision different methods for mobile robot tracking and following literature [15], it has yet to be applied in robotics. In addition, of a fast-moving person in an outdoor environment. The we put the problem into a multisensor system context, which robot was equipped with an omnidirectional camera and a distinguishes this formulation from previous work. 2D LiDAR. [10] presented an integrated system to detect We assume ﬁrstly that, in terms of hardware, a robot and track people and cars in outdoor scenarios, based on the perceives the environment through a ﬁnite set S of sensors. ∈ information retrieved from a camera and a 2D LiDAR on an Sensors s , s , . . . , s S may have completely different 1 2 n autonomous car. [2] introduced a people tracking system for perception mechanisms and may produce a variety of types mobile robots in very crowded and dynamic environments. of sensory data such as point clouds, laser scans, images, Their system was evaluated with a robot equipped with two etc. We assume secondly that, in terms of software, the robot RGB-D cameras, a stereo camera and two 2D LiDARs. detects humans through a ﬁnite set D of detectors. Detectors ∈ Our previous work with the aforementioned sensor com- d , d , . . . , d D can be different. For example d can be 1 2 n 1 bination includes [11] and [12]. The former presented a for human face detection, d for legs, d for whole body, human tracking system for mobile service robots in popu- etc. Moreover, f : D −1→:1 S2is not necess3ary, meaning that lated environments, while the latter extended this system to a sensor can have zero or several detectors, and a detector a fully integrated perception pipeline for people detection and can also be shared by several sensors. This statement is the ##LTFigure##

text: tracking

text: in

text: close

text: vicinity

text: to

text: the

text: robot.

text: The

text: proposed

text: tracking

text: system

text: tracks

text: people

text: by

text: detecting

text: legs

text: extracted

text: from

text: a

text: LiDAR

text: and

text: fusing

text: this

text: with

text: the

text: faces

text: or

text: the

text: upper-bodies

text: detected

text: with

text: a

text: camera

text: using

text: a

text: sequential

text: implementation

text: of

text: the

text: Unscented

text: Kalman

text: Filter

text: (UKF).

text: The

text: combination

text: with

text: LiDAR

text: is

text: increasing

text: with

text: the

text: development

text: of

text: the

text: LiDAR

text: technology.

text: Taking

text: advantage

text: of

text: its

text: high

text: accuracy,

text: developed

text: an

text: algorithm

text: to

text: align

text: LiDAR

text: data

text: with

text: high-resolution

text: camera

text: images

text: obtained

text: from

text: ﬁve

text: cameras,

text: in

text: order

text: to

text: accurately

text: track

text: moving

text: vehicles.

text: Other

text: reported

text: results

text: include

text: and

text: which

text: mainly

text: focused

text: on

text: pedestrian

text: detection

text: rather

text: than

text: tracking.

text: In

text: addition,

text: earlier

text: work

text: presented

text: multitarget

text: tracking

text: with

text: a

text: mobile

text: robot

text: equipped

text: with

text: two

text: LiDARs,

text: respectively

text: located

text: at

text: the

text: front

text: and

text: back

text: Thus

text: the

text: robot

text: can

text: have

text: a

text: horizontal

text: ﬁeld

text: of

text: view,

text: where

text: each

text: scan

text: of

text: these

text: two

text: sensors

text: covers

text: the

text: whole

text: surrounding

text: of

text: the

text: robot

text: at

text: an

text: Fig.

text: Different

text: horizontal

text: ﬁelds

text: of

text: view

text: of

text: different

text: sensors.

text: Vertical

text: angular

text: resolution

text: of

text: ﬁelds

text: of

text: view,

text: not

text: shown

text: in

text: this

text: ﬁgure,

text: are

text: also

text: different.

text: The

text: use

text: of

text: machine

text: learning

text: algorithms

text: for

text: tracking

text: has

text: particular

text: advantages.

text: The

text: closest

text: work

text: to

text: ours

text: is

text: where

text: the

text: authors

text: proposed

text: a

text: semi-supervised

text: learning

text: approach

text: to

text: The

text: remainder

text: of

text: this

text: paper

text: is

text: organized

text: as

text: follows:

text: the

text: problem

text: of

text: track

text: classiﬁcation

text: in

text: LiDAR

text: data,

text: based

text: Section

text: gives

text: an

text: overview

text: of

text: the

text: related

text: literature.

text: Sec-

text: on

text: Expectation-Maximization

text: (EM)

text: algorithm.

text: In

text: contrast

text: to

text: tion

text: formulates

text: the

text: problem

text: in

text: multisensor

text: background.

text: our

text: approach,

text: their

text: learning

text: procedure

text: needs

text: a

text: small

text: set

text: of

text: Section

text: presents

text: our

text: solution

text: framework

text: for

text: online

text: transfer

text: seed

text: tracks

text: and

text: a

text: large

text: set

text: of

text: background

text: tracks,

text: that

text: need

text: learning.

text: Section

text: describes

text: the

text: application

text: of

text: the

text: proposed

text: to

text: be

text: manually

text: or

text: semi-manually

text: labeled

text: at

text: ﬁrst,

text: whereas

text: we

text: framework

text: to

text: the

text: problem

text: of

text: LiDAR-based

text: human

text: clas-

text: do

text: not

text: need

text: any

text: hand-labeled

text: data.

text: siﬁcation.

text: Section

text: illustrates

text: the

text: experimental

text: results

text: for

text: To

text: our

text: knowledge,

text: no

text: existing

text: work

text: in

text: the

text: robotics

text: ﬁeld

text: different

text: sensor

text: conﬁgurations.

text: Finally,

text: Section

text: concludes

text: exploits

text: information

text: from

text: multisensor-based

text: tracking

text: to

text: im-

text: the

text: paper

text: summarizing

text: the

text: contributions

text: and

text: suggesting

text: plement

text: transfer

text: learning

text: between

text: different

text: sensors

text: as

text: in

text: future

text: research

text: work.

text: this

text: paper.

text: Our

text: work

text: combines

text: the

text: advantages

text: of

text: multiple

text: II.

text: RELATED

text: WORK

text: sensors

text: with

text: the

text: efﬁciency

text: of

text: semi-supervised

text: learning,

text: and

text: The

text: problem

text: of

text: multitarget

text: and

text: multisensor

text: tracking

text: has

text: integrates

text: them

text: into

text: an

text: single

text: online

text: framework

text: applied

text: to

text: been

text: extensively

text: studied

text: during

text: the

text: past

text: few

text: decades.

text: Most

text: LiDAR-based

text: human

text: classiﬁcation.

text: of

text: present

text: systems

text: are

text: based

text: on

text: Bayesian

text: methods

text: which

text: III.

text: PROBLEM

text: FORMULATION

text: compute

text: an

text: estimate

text: of

text: the

text: correspondence

text: between

text: features

text: detected

text: in

text: the

text: sensor

text: data

text: and

text: the

text: different

text: humans

text: to

text: be

text: In

text: this

text: section,

text: we

text: model

text: the

text: problem

text: of

text: online

text: learning

text: tracked.

text: Regarding

text: robotic

text: applications,

text: multiple

text: sensors

text: can

text: for

text: human

text: classiﬁcation

text: in

text: a

text: multitarget-multisensor

text: track-

text: be

text: deployed

text: in

text: single-

text: or

text: multi-robot

text: systems

text: while

text: the

text: ing

text: system.

text: The

text: formulation

text: is

text: generalized

text: which

text: can

text: be

text: former

text: is

text: the

text: concern

text: of

text: this

text: paper.

text: extended

text: to

text: objects

text: rather

text: than

text: humans.

text: We

text: consider

text: online

text: RGB/RGB-D

text: camera

text: plus

text: LiDAR

text: is

text: the

text: most

text: fre-

text: learning

text: as

text: a

text: detection-and-tracking-based

text: iterative

text: process.

text: quently

text: used

text: combination

text: in

text: the

text: literature.

text: presented

text: two

text: Whilst

text: this

text: idea

text: has

text: been

text: presented

text: in

text: the

text: computer

text: vision

text: different

text: methods

text: for

text: mobile

text: robot

text: tracking

text: and

text: following

text: literature

text: it

text: has

text: yet

text: to

text: be

text: applied

text: in

text: robotics.

text: In

text: addition,

text: of

text: a

text: fast-moving

text: person

text: in

text: an

text: outdoor

text: environment.

text: The

text: we

text: put

text: the

text: problem

text: into

text: a

text: multisensor

text: system

text: context,

text: which

text: robot

text: was

text: equipped

text: with

text: an

text: omnidirectional

text: camera

text: and

text: a

text: distinguishes

text: this

text: formulation

text: from

text: previous

text: work.

text: LiDAR.

text: presented

text: an

text: integrated

text: system

text: to

text: detect

text: We

text: assume

text: ﬁrstly

text: that,

text: in

text: terms

text: of

text: hardware,

text: a

text: robot

text: and

text: track

text: people

text: and

text: cars

text: in

text: outdoor

text: scenarios,

text: based

text: on

text: the

text: perceives

text: the

text: environment

text: through

text: a

text: ﬁnite

text: set

text: S

text: of

text: sensors.

text: information

text: retrieved

text: from

text: a

text: camera

text: and

text: a

text: LiDAR

text: on

text: an

text: Sensors

text: s

text: s

text: s

text: S

text: may

text: have

text: completely

text: different

text: n

text: autonomous

text: car.

text: introduced

text: a

text: people

text: tracking

text: system

text: for

text: perception

text: mechanisms

text: and

text: may

text: produce

text: a

text: variety

text: of

text: types

text: mobile

text: robots

text: in

text: very

text: crowded

text: and

text: dynamic

text: environments.

text: of

text: sensory

text: data

text: such

text: as

text: point

text: clouds,

text: laser

text: scans,

text: images,

text: Their

text: system

text: was

text: evaluated

text: with

text: a

text: robot

text: equipped

text: with

text: two

text: etc.

text: We

text: assume

text: secondly

text: that,

text: in

text: terms

text: of

text: software,

text: the

text: robot

text: RGB-D

text: cameras,

text: a

text: stereo

text: camera

text: and

text: two

text: LiDARs.

text: detects

text: humans

text: through

text: a

text: ﬁnite

text: set

text: D

text: of

text: detectors.

text: Detectors

text: Our

text: previous

text: work

text: with

text: the

text: aforementioned

text: sensor

text: com-

text: d

text: d

text: d

text: D

text: can

text: be

text: different.

text: For

text: example

text: d

text: can

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text: n

text: bination

text: includes

text: and

text: The

text: former

text: presented

text: a

text: for

text: human

text: face

text: detection,

text: d

text: for

text: legs,

text: d

text: for

text: whole

text: body,

text: human

text: tracking

text: system

text: for

text: mobile

text: service

text: robots

text: in

text: popu-

text: etc.

text: Moreover,

text: f

text: D

text: not

text: meaning

text: that

text: lated

text: environments,

text: while

text: the

text: latter

text: extended

text: this

text: system

text: to

text: a

text: sensor

text: can

text: have

text: zero

text: or

text: several

text: detectors,

text: and

text: a

text: detector

text: a

text: fully

text: integrated

text: perception

text: pipeline

text: for

text: people

text: detection

text: and

text: can

text: also

text: be

text: shared

text: by

text: several

text: sensors.

text: This

text: statement

text: is

text: the

text: ##LTFigure##


text: 4 FIG. 2. Schematic representation of the experimental setup. Only commercial oﬀ-the-shelf devices were used. and the quantum min-entropy is lower bounded by phase shift. These optical signals, detected by a couple of high-bandwidth balanced detectors (1.6 GHz Thorlabs- π H (X |E) ≥ log . (8) PDB480C), are proportional to the quadratures of the min δ 2 δ δ q p signal, Re[α], Im[α]. We sampled both signals coming from the detectors us- Hence, in the real-life implementation, the min-entropy ing a fast oscilloscope with a sampling rate of 10 GSam- of the random numbers is bounded by a function that ples/s at 10 bits of resolution (Lecroy HDO 9404). Each depends on the measurement resolution only. The mea- samples contains 20 bits of raw data, 10-bit for Re[α] surement, in this scenario, is under control of the user: and 10-bit for Im[α]. The raw signals of the ADC are Alice can readily obtain the min-entropy (8) by measur- proportional to the quadratures and directly sample the ing δ and δ of her well characterized apparatus. The p q Q-function in the phase space, as shown in Fig. 3. The min-entropy is constant and Alice does not need to worry resolution of the ADC can be directly converted to the updating its value, as long as she trusts the apparatus. equivalent resolution in the phase space; in our case we IV. EXPERIMENTAL RESULTS − got δRe [α] = (14.05±0.02)·10 3 and δIm [α] = (14.14± − 0.02) · 10 3, respectively. The proposed new protocol has been implemented with an all-ﬁber setup at telecom wavelength with the scheme in Fig. 2; in this way is possible to exploit the availability of fast oﬀ-the-shelf components for classical telecommu- nication while keeping the setup compact. The heart of the experiment lies in the heterodyne detection of the vacuum state, that samples the Q function with the help of a coherent ﬁeld |α(cid:105) of a Local Oscillator (LO). We em- ployed a narrow linewidth ECL laser at 1550nm (Thor- labs SFL1550) followed by and electronically-controlled Variable Optical Attenuator (VOA) and a in-line Polar- ization Controller (PC). In this way we were able to ﬁnely control the intensity and the polarization of our LO, be- sides making the calibration procedure automatized. Before entering the heterodyne measurement, 10% of the LO is sent to a photodetector, for a continuous monitor of its intensity. By doing that, any anomaly to the normal functioning of the LO can be noticed in real-time, and deviations can be compensated during the post-processing. FIG. 3. The plot shows the Husimi function for the vacuum The optical heterodyne was realized with a commercial (meshed curve) and the measured state (colored histogram). ﬁber integrated “90 degree hybrid”: one port is coupled The measured variance is slightly larger that the one expected to the LO while from the other is entering the vacuum for the vacuum due to the electronic noise that widens the state. However, we work in the SDI scenario and, from distribution. the point of view of security, this port can be fully con- trolled by Eve, since we don’t assume anything about The raw data are then digitally ﬁltered, taking only the source. The 90 degree hybrid mixes the signal with a 1.25 GHz window in the central part of the spectrum the LO and returns two pairs of outputs, featuring a π/2 obtained by the detectors. This is necessary in order to ##LTFigure## ##LTFigure## ##LTLine##

text: FIG.

text: Schematic

label: caption

text: representation

text: of

text: the

text: experimental

text: setup.

text: Only

text: commercial

text: oﬀ-the-shelf

text: devices

text: were

text: used.

text: and

text: the

text: quantum

text: min-entropy

text: is

text: lower

text: bounded

text: by

text: phase

text: shift.

text: These

text: optical

text: signals,

text: detected

text: by

text: a

text: couple

text: of

text: high-bandwidth

text: balanced

text: detectors

text: GHz

text: Thorlabs-

text: π

text: H

text: (X

text: |E)

text: log

text: are

text: proportional

text: to

text: the

text: quadratures

text: of

text: the

text: min

text: δ

text: δ

text: δ

text: q

text: p

text: signal,

text: Re[α],

text: Im[α].

text: We

text: sampled

text: both

text: signals

text: coming

text: from

text: the

text: detectors

text: us-

text: Hence,

text: in

text: the

text: real-life

text: implementation,

text: the

text: min-entropy

text: ing

text: a

text: fast

text: oscilloscope

text: with

text: a

text: sampling

text: rate

text: of

text: GSam-

text: of

text: the

text: random

text: numbers

text: is

text: bounded

text: by

text: a

text: function

text: that

text: ples/s

text: at

text: bits

text: of

text: resolution

text: (Lecroy

text: HDO

text: Each

text: depends

text: on

text: the

text: measurement

text: resolution

text: only.

text: The

text: mea-

text: samples

text: contains

text: bits

text: of

text: raw

text: data,

text: for

text: Re[α]

text: surement,

text: in

text: this

text: scenario,

text: is

text: under

text: control

text: of

text: the

text: user:

text: and

text: for

text: Im[α].

text: The

text: raw

text: signals

text: of

text: the

text: ADC

text: are

text: Alice

text: can

text: readily

text: obtain

text: the

text: min-entropy

text: by

text: measur-

text: proportional

text: to

text: the

text: quadratures

text: and

text: directly

text: sample

text: the

text: ing

text: δ

text: and

text: δ

text: of

text: her

text: well

text: characterized

text: apparatus.

text: The

text: p

text: q

text: Q-function

text: in

text: the

text: phase

text: space,

text: as

text: shown

text: in

text: Fig.

text: The

text: min-entropy

text: is

text: constant

text: and

text: Alice

text: does

text: not

text: need

text: to

text: worry

text: resolution

text: of

text: the

text: ADC

text: can

text: be

text: directly

text: converted

text: to

text: the

text: updating

text: its

text: value,

text: as

text: long

text: as

text: she

text: trusts

text: the

text: apparatus.

text: equivalent

text: resolution

text: in

text: the

text: phase

text: space;

text: in

text: our

text: case

text: we

text: IV.

text: EXPERIMENTAL

text: RESULTS

text: got

text: δRe

text: [α]

text: and

text: δIm

text: [α]

text: respectively.

text: The

text: proposed

text: new

text: protocol

text: has

text: been

text: implemented

text: with

text: an

text: all-ﬁber

text: setup

text: at

text: telecom

text: wavelength

text: with

text: the

text: scheme

text: in

text: Fig.

text: in

text: this

text: way

text: is

text: possible

text: to

text: exploit

text: the

text: availability

text: of

text: fast

text: oﬀ-the-shelf

text: components

text: for

text: classical

text: telecommu-

text: nication

text: while

text: keeping

text: the

text: setup

text: compact.

text: The

text: heart

text: of

text: the

text: experiment

text: lies

text: in

text: the

text: heterodyne

text: detection

text: of

text: the

text: vacuum

text: state,

text: that

text: samples

text: the

text: Q

text: function

text: with

text: the

text: help

text: of

text: a

text: coherent

text: ﬁeld

text: of

text: a

text: Local

text: Oscillator

text: (LO).

text: We

text: em-

text: ployed

text: a

text: narrow

text: linewidth

text: ECL

text: laser

text: at

text: (Thor-

text: labs

text: followed

text: by

text: and

text: electronically-controlled

text: Variable

text: Optical

text: Attenuator

text: (VOA)

text: and

text: a

text: in-line

text: Polar-

text: ization

text: Controller

text: (PC).

text: In

text: this

text: way

text: we

text: were

text: able

text: to

text: ﬁnely

text: control

text: the

text: intensity

text: and

text: the

text: polarization

text: of

text: our

text: LO,

text: be-

text: sides

text: making

text: the

text: calibration

text: procedure

text: automatized.

text: Before

text: entering

text: the

text: heterodyne

text: measurement,

text: of

text: the

text: LO

text: is

text: sent

text: to

text: a

text: photodetector,

text: for

text: a

text: continuous

text: monitor

text: of

text: its

text: intensity.

text: By

text: doing

text: that,

text: any

text: anomaly

text: to

text: the

text: normal

text: functioning

text: of

text: the

text: LO

text: can

text: be

text: noticed

text: in

text: real-time,

text: and

text: deviations

text: can

text: be

text: compensated

text: during

text: the

text: post-processing.

text: FIG.

text: The

text: plot

text: shows

text: the

text: Husimi

text: function

text: for

text: the

text: vacuum

text: The

text: optical

text: heterodyne

text: was

text: realized

text: with

text: a

text: commercial

text: (meshed

text: curve)

text: and

text: the

text: measured

text: state

text: (colored

text: histogram).

text: ﬁber

text: integrated

text: degree

text: hybrid”:

text: one

text: port

text: is

text: coupled

text: The

text: measured

text: variance

text: is

text: slightly

text: larger

text: that

text: the

text: one

text: expected

text: to

text: the

text: LO

text: while

text: from

text: the

text: other

text: is

text: entering

text: the

text: vacuum

text: for

text: the

text: vacuum

text: due

text: to

text: the

text: electronic

text: noise

text: that

text: widens

text: the

text: state.

text: However,

text: we

text: work

text: in

text: the

text: SDI

text: scenario

text: and,

text: from

text: distribution.

text: the

text: point

text: of

text: view

text: of

text: security,

text: this

text: port

text: can

text: be

text: fully

text: con-

text: trolled

text: by

text: Eve,

text: since

text: we

text: don’t

text: assume

text: anything

text: about

text: The

text: raw

text: data

text: are

text: then

text: digitally

text: ﬁltered,

text: taking

text: only

text: the

text: source.

text: The

text: degree

text: hybrid

text: mixes

text: the

text: signal

text: with

text: a

text: GHz

text: window

text: in

text: the

text: central

text: part

text: of

text: the

text: spectrum

text: the

text: LO

text: and

text: returns

text: two

text: pairs

text: of

text: outputs,

text: featuring

text: a

text: obtained

text: by

text: the

text: detectors.

text: This

text: is

text: necessary

text: in

text: order

text: to

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##


text: 7 Appendix C: Statistical Tests 5.18·1010 random numbers from an initial set of 7.5·1010 raw numbers. We tested them with the NIST [30] and In order to check for problems in our implementa- dieharder suite [31]: in both cases all the tests were tion we performed some statistical test on the generated passed, as we can see in Table I. Passing these tests numbers. First, we implemented the fast computable doesn’t certify the randomness, but only shows that some two-universal hash function introduced in [29], then we patterns are not present in the analyzed data. However, used it to extract the ﬁnal numbers from the ﬁltered since our QRNG is supposed to pass all of them, is a way samples. We calibrated the extractor with the value to double-check that our setup is working as expected. of H (X|E) of eq. (10) and then we extracted ≈ min min Test’s name P-value Result diehard birthdays 0.398 PASSED diehard operm5 0.391 PASSED diehard rank 32x32 0.414 PASSED diehard rank 6x8 0.767 PASSED diehard bitstream 0.529 PASSED Test’s name P-value Result diehard opso 0.655 PASSED diehard oqso 0.758 PASSED Frequency 0.980 PASSED diehard dna 0.731 PASSED BlockFrequency 0.323 PASSED diehard count 1s str 0.482 PASSED CumulativeSums 0.819 PASSED diehard count 1s byt 0.361 PASSED CumulativeSums 0.265 PASSED diehard parking lot 0.515 PASSED Runs 0.187 PASSED diehard 2dsphere 0.484 PASSED LongestRun 0.864 PASSED diehard 3dsphere 0.739 PASSED Rank 0.372 PASSED diehard squeeze 0.580 PASSED DFT 0.341 PASSED diehard sums 0.140 PASSED NonOverlappingTemplate 0.016 PASSED diehard runs 0.478 PASSED OverlappingTemplate 0.748 PASSED diehard runs 0.316 PASSED Universal 0.381 PASSED diehard craps 0.348 PASSED ApproximateEntropy 0.509 PASSED diehard craps 0.937 PASSED RandomExcursions 0.315 PASSED marsaglia tsang gcd 0.504 PASSED RandomExcursionsVariant 0.047 PASSED marsaglia tsang gcd 0.444 PASSED Serial 0.318 PASSED sts monobit 0.204 PASSED LinearComplexity 0.373 PASSED sts runs 0.716 PASSED sts serial 0.151 PASSED rgb bitdist 0.056 PASSED rgb minimum distance 0.043 PASSED rgb permutations 0.068 PASSED rgb lagged sum 0.019 PASSED TABLE I. Result of Dieharder (left) and NIST (right) test suite on the extracted random numbers. In the case of multiple tests in a category, the smallest have been reported. ∗ Shalm, ArXiv e-prints (2017), arXiv:1702.05178 [quant-ph]. [1] R. Konig, R. Renner, and C. Schaﬀner, IEEE Transactions on[9]InYfo.rLmiua,tiXon. YTuhaeno,ryM5.-5H,.4L3i3,7W(2.0Z0h9a)n. g, Q. Zhao, J. Zhong, Y. Cao, Y.-H. Li, L.-K. Chen, H. Li, T. Peng, Y.-A. Chen, C.-Z. [2] J. Rarity, P. Owens, and P. Tapster, J. Mod. Opt. 41, 2435 (199P4)e.ng, S.-C. Shi, Z. Wang, L. You, X. Ma, J. Fan, Q. Zhang, and J.-W. Pan, Phys. Rev. Lett. 120, 010503 (2018). [3] A. Stefanov, N. Gisin, O. Guinnard, L. Guinnard, and H. Zb[1in0d] eSn.,GJoo´umrenza,l Aof. MMoadttearnr, OEp.tSic.sG4´o7m, 5ez9,5D(2.0C00a)v.alcanti, O. Jim´enez Far´ıas, A. Ac´ın, and G. Lima, ArXiv e-prints (2017), [4] T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and A. ZeialrinXgievr:,17R1e1v.i1e0w29o4f S[qcuieanntti-ﬁpchI].nstruments 71, 1675 (2000). [5] G. Vallone, D. G. Marangon, M. Tomasin, and P. Villoresi, [P1h1y] sX. .RMeva.,AX.90Y,u0a5n2,3Z2.7C(a2o0,14B).. Qi, and Z. Zhang, npj Quantum Information 2, 16021 (2016). [6] S. Pironio, A. Ac´ın, S. Massar, A. B. De La Giroday, D. N[.12M]aTts.uLkuevnigchhi,, PJ.. BM.aBunrazs,kS,.CO. lCm.scWhe.nCk., WD.. HLiamye,sQ, L. .LaLvuiog,ne, J. Bowles, A. Martin, H. Zbinden, and N. Brunner, Physical T. A. Manning, and C. Monroe, Nature 464, 1021 (2010). Review Letters 114, 150501 (2015). [7] B. G. Christensen, K. T. McCusker, J. B. Altepeter, B. Calki[n13s,] TG..GCearn˜riatss,, JA..CEa.rLin˜itea,,EA..SM. iGllo´erm, eLz., KJ.. SFh.aBlmar,rYa,. AZh. aCnagb,ello, G. B. Xavier, G. Lima, and M. Pawl(cid:32)owski, ArXiv e-prints S. W. Nam, N. Brunner, C. C. W. Lim, N. Gisin, and P. G. Kwi(a2t0,1P4h),ysa.rXRievv:.14L1e0t.t3.414131[,q1u3a0n4t0-p6h(]2. 013). [8] P. Bierhorst, E. Knill, S. Glancy, A. Mink, S. Jordan, A. R[o1m4]mDa.l,GY. .M-Ka.raLniugo, nB,.GC.hVraisltloennese,na, nSd. PW. .VNilalomre,si,aPndhyLsi.cKal.Review Letters 118, 060503 (2017). ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Appendix

label: section

text: C:

text: Statistical

text: Tests

text: random

text: numbers

text: from

text: an

text: initial

text: set

text: of

text: raw

text: numbers.

text: We

text: tested

text: them

text: with

text: the

text: NIST

text: and

text: In

text: order

text: to

text: check

text: for

text: problems

text: in

text: our

text: implementa-

text: dieharder

text: suite

text: in

text: both

text: cases

text: all

text: the

text: tests

text: were

text: tion

text: we

text: performed

text: some

text: statistical

text: test

text: on

text: the

text: generated

text: passed,

text: as

text: we

text: can

text: see

text: in

text: Table

text: I.

text: Passing

text: these

text: tests

text: numbers.

text: First,

text: we

text: implemented

text: the

text: fast

text: computable

text: doesn’t

text: certify

text: the

text: randomness,

text: but

text: only

text: shows

text: that

text: some

text: two-universal

text: hash

text: function

text: introduced

text: in

text: then

text: we

text: patterns

text: are

text: not

text: present

text: in

text: the

text: analyzed

text: data.

text: However,

text: used

text: it

text: to

text: extract

text: the

text: ﬁnal

text: numbers

text: from

text: the

text: ﬁltered

text: since

text: our

text: QRNG

text: is

text: supposed

text: to

text: pass

text: all

text: of

text: them,

text: is

text: a

text: way

text: samples.

text: We

text: calibrated

text: the

text: extractor

text: with

text: the

text: value

text: to

text: double-check

text: that

text: our

text: setup

text: is

text: working

text: as

text: expected.

text: of

text: H

text: (X|E)

text: of

text: eq.

text: and

text: then

text: we

text: extracted

text: min

text: min

text: Test’s

text: name

text: P-value

text: Result

text: diehard

text: birthdays

text: PASSED

text: diehard

text: PASSED

text: diehard

text: rank

text: PASSED

text: diehard

text: rank

text: PASSED

text: diehard

text: bitstream

text: PASSED

text: Test’s

text: name

text: P-value

text: Result

text: diehard

text: opso

text: PASSED

text: diehard

text: oqso

text: PASSED

text: Frequency

text: PASSED

text: diehard

text: dna

text: PASSED

text: BlockFrequency

text: PASSED

text: diehard

text: count

text: str

text: PASSED

text: CumulativeSums

text: PASSED

text: diehard

text: count

text: byt

text: PASSED

text: CumulativeSums

text: PASSED

text: diehard

text: parking

text: lot

text: PASSED

text: Runs

text: PASSED

text: diehard

text: PASSED

text: LongestRun

text: PASSED

text: diehard

text: PASSED

text: Rank

text: PASSED

text: diehard

text: squeeze

text: PASSED

text: DFT

text: PASSED

text: diehard

text: sums

text: PASSED

text: NonOverlappingTemplate

text: PASSED

text: diehard

text: runs

text: PASSED

text: OverlappingTemplate

text: PASSED

text: diehard

text: runs

text: PASSED

text: Universal

text: PASSED

text: diehard

text: craps

text: PASSED

text: ApproximateEntropy

text: PASSED

text: diehard

text: craps

text: PASSED

text: RandomExcursions

text: PASSED

text: marsaglia

text: tsang

text: gcd

text: PASSED

text: RandomExcursionsVariant

text: PASSED

text: marsaglia

text: tsang

text: gcd

text: PASSED

text: Serial

text: PASSED

text: sts

text: monobit

text: PASSED

text: LinearComplexity

text: PASSED

text: sts

text: runs

text: PASSED

text: sts

text: serial

text: PASSED

text: rgb

text: bitdist

text: PASSED

text: rgb

text: minimum

text: distance

text: PASSED

text: rgb

text: permutations

text: PASSED

text: rgb

text: lagged

text: sum

text: PASSED

text: TABLE

text: I.

text: Result

text: of

text: Dieharder

text: (left)

text: and

text: NIST

text: (right)

text: test

text: suite

text: on

text: the

text: extracted

text: random

text: numbers.

text: In

text: the

text: case

text: of

text: multiple

text: tests

text: in

text: a

text: category,

text: the

text: smallest

text: have

text: been

text: reported.

text: Shalm,

text: ArXiv

text: e-prints

text: [quant-ph].

text: R.

text: Konig,

text: R.

text: Renner,

text: and

text: C.

text: Schaﬀner,

text: IEEE

text: Transactions

text: g,

text: Q.

text: Zhao,

text: J.

text: Zhong,

text: Y.

text: Cao,

text: Y.-H.

text: Li,

text: L.-K.

text: Chen,

text: H.

text: Li,

text: T.

text: Peng,

text: Y.-A.

text: Chen,

text: C.-Z.

text: J.

text: Rarity,

text: P.

text: Owens,

text: and

text: P.

text: Tapster,

text: J.

text: Mod.

text: Opt.

text: S.-C.

text: Shi,

text: Z.

text: Wang,

text: L.

text: You,

text: X.

text: Ma,

text: J.

text: Fan,

text: Q.

text: Zhang,

text: and

text: J.-W.

text: Pan,

text: Phys.

text: Rev.

text: Lett.

text: A.

text: Stefanov,

text: N.

text: Gisin,

text: O.

text: Guinnard,

text: L.

text: Guinnard,

text: and

text: H.

text: eSn.,GJoo´umrenza,l

text: Aof.

text: MMoadttearnr,

text: O.

text: Jim´enez

text: Far´ıas,

text: A.

text: Ac´ın,

text: and

text: G.

text: Lima,

text: ArXiv

text: e-prints

text: T.

text: Jennewein,

text: U.

text: Achleitner,

text: G.

text: Weihs,

text: H.

text: Weinfurter,

text: and

text: A.

text: S[qcuieanntti-ﬁpchI].nstruments

text: G.

text: Vallone,

text: D.

text: G.

text: Marangon,

text: M.

text: Tomasin,

text: and

text: P.

text: Villoresi,

text: sX.

text: Qi,

text: and

text: Z.

text: Zhang,

text: npj

text: Quantum

text: Information

text: S.

text: Pironio,

text: A.

text: Ac´ın,

text: S.

text: Massar,

text: A.

text: B.

text: De

text: La

text: Giroday,

text: D.

text: PJ..

text: BM.aBunrazs,kS,.CO.

text: lCm.scWhe.nCk.,

text: WD..

text: HLiamye,sQ,

text: L.

text: .LaLvuiog,ne,

text: J.

text: Bowles,

text: A.

text: Martin,

text: H.

text: Zbinden,

text: and

text: N.

text: Brunner,

text: Physical

text: T.

text: A.

text: Manning,

text: and

text: C.

text: Monroe,

text: Nature

text: Review

text: Letters

text: B.

text: G.

text: Christensen,

text: K.

text: T.

text: McCusker,

text: J.

text: B.

text: Altepeter,

text: B.

text: TG..GCearn˜riatss,,

text: JA..CEa.rLin˜itea,,EA..SM.

text: iGllo´erm,

text: eLz.,

text: KJ..

text: SFh.aBlmar,rYa,.

text: AZh.

text: aCnagb,ello,

text: G.

text: B.

text: Xavier,

text: G.

text: Lima,

text: and

text: M.

text: ArXiv

text: e-prints

text: S.

text: W.

text: Nam,

text: N.

text: Brunner,

text: C.

text: C.

text: W.

text: Lim,

text: N.

text: Gisin,

text: and

text: P.

text: G.

text: P.

text: Bierhorst,

text: E.

text: Knill,

text: S.

text: Glancy,

text: A.

text: Mink,

text: S.

text: Jordan,

text: A.

text: .M-Ka.raLniugo,

text: nB,.GC.hVraisltloennese,na,

text: nSd.

text: PW.

text: .VNilalomre,si,aPndhyLsi.cKal.Review

text: Letters

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: MNRAS 000, 1–7 (0000) Preprint 30 May 2020 Compiled using MNRAS LATEX style ﬁle v3.0 Cosmic String Detection with Tree-Based Machine Learning A. Vafaei Sadr1,2,3, M. Farhang1, S. M. S. Movahed1,4 (cid:63), B. Bassett3,5,6,7, M. Kunz2 1 Department of Physics, Shahid Beheshti University, Velenjak, Tehran 19839, Iran 2 D´epartement de Physique Th´eorique and Center for Astroparticle Physics, Universit´e de Gen`eve, 24 Quai Ernest Ansermet, 1211 Gen`eve 4, Switzerland 3 African Institute for Mathematical Sciences, 6 Melrose Road, Muizenberg, 7945, South Africa 4 School of Physics, Institute for Research in Fundamental Sciences, (IPM), P. O. Box 19395-5531, Tehran, Iran 5 South African Astronomical Observatory, Observatory, Cape Town, 7925, South Africa 6 SKA South Africa, The Park, Park Road, Pinelands, Cape Town 7405, South Africa 7 Department of Maths and Applied Maths, University of Cape Town, Cape Town, South Africa 30 May 2020 ABSTRACT We explore the use of random forest and gradient boosting, two powerful tree-based machine learning algorithms, for the detection of cosmic strings in maps of the cosmic microwave background (CMB), through their unique Gott-Kaiser-Stebbins eﬀect on the temperature anisotropies.The information in the maps is compressed into feature vectors before being passed to the learning units. The feature vectors contain various statistical measures of processed CMB maps that boost the cosmic string detectability. Our proposed classiﬁers, after training, give results improved over or similar to the claimed detectability levels of the existing methods for string tension, Gµ. They can (cid:38) × − (cid:48) make 3σ detection of strings with Gµ 2.1 10 10 for noise-free, 0.9 -resolution CMB (cid:38) × − observations. The minimum detectable tension increases to Gµ 3.0 10 8 for a more realistic, CMB S4-like (II) strategy, still a signiﬁcant improvement over the previous results. Key words: Cosmic string, Machine learning, Tree based models, Curvelet, CMB, 1 INTRODUCTION eﬀort has been put into developing powerful statistical tools for cosmic string network detection and putting tight upper The inﬂationary paradigm is the most widely accepted bounds on the CS tension, parametrized by Gµ, where G scenario for seeding the structures in the Universe, so far and µ represent Newton’s constant and the string’s tension, passing observational tests with ﬂying colors. There is, how- respectively. The string tension is intimately related to the ever, both theoretical and observational room for contri- energy of the phase transition epoch, butions from alternative, well-motivated scenarios. Among (cid:18) (cid:19) these are perturbations sourced by cosmic topological de- fects formed at cosmological phase transitions. In particular, cosmic strings (CS) are theoretically expected to be pro- Gµ O (cid:36)2 = , (1) duced in the early Universe (Kibble 1976; Zeldovich 1980; c2 M2 Planck Vilenkin 1981; Vachaspati & Vilenkin 1984; Vilenkin 1985; Shellard 1987; Hindmarsh & Kibble 1995; Vilenkin & Shel- (cid:112) lard 2000; Sakellariadou 2007; Bevis et al. 2008; Depies 2009; where (cid:36) is the symmetry breaking energy scale, c is the Bevis et al. 2010; Copeland et al. 1994; Sakellariadou 1997; ≡ (cid:126) speed of light and M c/G represents the Planck Sarangi & Tye 2002; Copeland et al. 2004; Pogosian et al. Planck (cid:126) mass. In this paper we work in natural units with = c = 1. 2003; Majumdar & Christine-Davis 2002; Dvali & Vilenkin A CS network would leave various imprints on cos- 2004; Kibble 2004; Henry Tye 2008). The detection of CS mic microwave background (CMB) anisotropies. The Gott- would open a unique window to the physics of the early Uni- Kaiser-Stebbins (KS) eﬀect (Kaiser & Stebbins 1984; verse (Kibble 1976; Zeldovich 1980; Vilenkin 1981; Vilenkin Gott III 1985; Stebbins 1988; Bouchet et al. 1988; Allen & Shellard 2000; Firouzjahi & Tye 2005). Therefore a lot of et al. 1997; Pen et al. 1997; Ringeval & Bouchet 2012) corresponds to the integrated Sachs-Wolfe eﬀect caused by (cid:63) E-mail: m.s.movahed@ipm.ir moving strings. It produces line-like discontinuities on the (cid:13) c 0000 The Authors ##LTLine## ##LTLine## ##LTLine##

text: MNRAS

text: Preprint

text: May

text: Compiled

text: using

text: MNRAS

text: LATEX

text: style

text: ﬁle

text: Cosmic

text: String

text: Detection

text: with

text: Tree-Based

text: Machine

text: Learning

text: A.

text: Vafaei

text: M.

text: S.

text: M.

text: S.

text: B.

text: M.

text: Department

text: of

text: Physics,

text: Shahid

text: Beheshti

text: University,

text: Velenjak,

text: Tehran

text: Iran

text: D´epartement

text: de

text: Physique

text: Th´eorique

text: and

text: Center

text: for

text: Astroparticle

text: Physics,

text: Universit´e

text: de

text: Gen`eve,

text: Quai

text: Ernest

text: Ansermet,

text: Gen`eve

text: Switzerland

text: African

text: Institute

text: for

text: Mathematical

text: Sciences,

text: Melrose

text: Road,

text: Muizenberg,

text: South

text: Africa

text: School

text: of

text: Physics,

text: Institute

text: for

text: Research

text: in

text: Fundamental

text: Sciences,

text: (IPM),

text: P.

text: O.

text: Box

text: Tehran,

text: Iran

text: South

text: African

text: Astronomical

text: Observatory,

text: Observatory,

text: Cape

text: Town,

text: South

text: Africa

text: SKA

text: South

text: Africa,

text: The

text: Park,

text: Park

text: Road,

text: Pinelands,

text: Cape

text: Town

text: South

text: Africa

text: Department

text: of

text: Maths

text: and

text: Applied

text: Maths,

text: University

text: of

text: Cape

text: Town,

text: Cape

text: Town,

text: South

text: Africa

text: May

text: ABSTRACT

text: We

text: explore

text: the

text: use

text: of

text: random

text: forest

text: and

text: gradient

text: boosting,

text: two

text: powerful

text: tree-based

text: machine

text: learning

text: algorithms,

text: for

text: the

text: detection

text: of

text: cosmic

text: strings

text: in

text: maps

text: of

text: the

text: cosmic

text: microwave

text: background

text: (CMB),

text: through

text: their

text: unique

text: Gott-Kaiser-Stebbins

text: eﬀect

text: on

text: the

text: temperature

text: anisotropies.The

text: information

text: in

text: the

text: maps

text: is

text: compressed

text: into

text: feature

text: vectors

text: before

text: being

text: passed

text: to

text: the

text: learning

text: units.

text: The

text: feature

text: vectors

text: contain

text: various

text: statistical

text: measures

text: of

text: processed

text: CMB

text: maps

text: that

text: boost

text: the

text: cosmic

text: string

text: detectability.

text: Our

text: proposed

text: classiﬁers,

text: after

text: training,

text: give

text: results

text: improved

text: over

text: or

text: similar

text: to

text: the

text: claimed

text: detectability

text: levels

text: of

text: the

text: existing

text: methods

text: for

text: string

text: tension,

text: Gµ.

text: They

text: can

text: make

text: 3σ

text: detection

text: of

text: strings

text: with

text: Gµ

text: for

text: noise-free,

text: -resolution

text: CMB

text: observations.

text: The

text: minimum

text: detectable

text: tension

text: increases

text: to

text: Gµ

text: for

text: a

text: more

text: realistic,

text: CMB

text: (II)

text: strategy,

text: still

text: a

text: signiﬁcant

text: improvement

text: over

text: the

text: previous

text: results.

text: Key

text: words:

text: Cosmic

text: string,

text: Machine

text: learning,

text: Tree

text: based

text: models,

text: Curvelet,

text: CMB,

text: INTRODUCTION

text: eﬀort

text: has

text: been

text: put

text: into

text: developing

text: powerful

text: statistical

text: tools

text: for

text: cosmic

text: string

text: network

text: detection

text: and

text: putting

text: tight

text: upper

text: The

text: inﬂationary

text: paradigm

text: is

text: the

text: most

text: widely

text: accepted

text: bounds

text: on

text: the

text: CS

text: tension,

text: parametrized

text: by

text: Gµ,

text: where

text: G

text: scenario

text: for

text: seeding

text: the

text: structures

text: in

text: the

text: Universe,

text: so

text: far

text: and

text: µ

text: represent

text: Newton’s

text: constant

text: and

text: the

text: string’s

text: tension,

text: passing

text: observational

text: tests

text: with

text: ﬂying

text: colors.

text: There

text: is,

text: how-

text: respectively.

text: The

text: string

text: tension

text: is

text: intimately

text: related

text: to

text: the

text: ever,

text: both

text: theoretical

text: and

text: observational

text: room

text: for

text: contri-

text: energy

text: of

text: the

text: phase

text: transition

text: epoch,

text: butions

text: from

text: alternative,

text: well-motivated

text: scenarios.

text: Among

text: these

text: are

text: perturbations

text: sourced

text: by

text: cosmic

text: topological

text: de-

text: fects

text: formed

text: at

text: cosmological

text: phase

text: transitions.

text: In

text: particular,

text: cosmic

text: strings

text: (CS)

text: are

text: theoretically

text: expected

text: to

text: be

text: pro-

text: Gµ

text: O

text: duced

text: in

text: the

text: early

text: Universe

text: (Kibble

text: Zeldovich

text: Planck

text: Vilenkin

text: Vachaspati

text: Vilenkin

text: Vilenkin

text: Shellard

text: Hindmarsh

text: Kibble

text: Vilenkin

text: Shel-

text: lard

text: Sakellariadou

text: Bevis

text: et

text: al.

text: Depies

text: where

text: is

text: the

text: symmetry

text: breaking

text: energy

text: scale,

text: c

text: is

text: the

text: Bevis

text: et

text: al.

text: Copeland

text: et

text: al.

text: Sakellariadou

text: speed

text: of

text: light

text: and

text: M

text: c/G

text: represents

text: the

text: Planck

text: Sarangi

text: Tye

text: Copeland

text: et

text: al.

text: Pogosian

text: et

text: al.

text: Planck

text: mass.

text: In

text: this

text: paper

text: we

text: work

text: in

text: natural

text: units

text: with

text: c

text: Majumdar

text: Christine-Davis

text: Dvali

text: Vilenkin

text: A

text: CS

text: network

text: would

text: leave

text: various

text: imprints

text: on

text: cos-

text: Kibble

text: Henry

text: Tye

text: The

text: detection

text: of

text: CS

text: mic

text: microwave

text: background

text: (CMB)

text: anisotropies.

text: The

text: Gott-

text: would

text: open

text: a

text: unique

text: window

text: to

text: the

text: physics

text: of

text: the

text: early

text: Uni-

text: Kaiser-Stebbins

text: (KS)

text: eﬀect

text: (Kaiser

text: Stebbins

text: verse

text: (Kibble

text: Zeldovich

text: Vilenkin

text: Vilenkin

text: Gott

text: III

text: Stebbins

text: Bouchet

text: et

text: al.

text: Allen

text: Shellard

text: Firouzjahi

text: Tye

text: Therefore

text: a

text: lot

text: of

text: et

text: al.

text: Pen

text: et

text: al.

text: Ringeval

text: Bouchet

text: corresponds

text: to

text: the

text: integrated

text: Sachs-Wolfe

text: eﬀect

text: caused

text: by

text: E-mail:

text: m.s.movahed@ipm.ir

text: moving

text: strings.

text: It

text: produces

text: line-like

text: discontinuities

text: on

text: the

text: c

text: The

text: Authors

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Cosmic String Detection with Tree-Based Machine Learning 7 Opitz D., Maclin R., 1999, Journal of Artiﬁcial Intelligence Re- Sakellariadou M., 2007, Lect. Notes Phys., 718, 247 search, 11, 169 Sarangi S., Tye S. H. H., 2002, Phys. Lett., B536, 185 Pen U.-L., Seljak U., Turok N., 1997, Phys. Rev. Lett., 79, 1611 Shellard E. P. S., 1987, Nucl. Phys., B283, 624 Pogosian L., Tye S. H. H., Wasserman I., Wyman M., 2003, Phys. Stebbins A., 1988, Astrophys. J., 327, 584 Rev., D68, 023506 Stebbins A., Veeraraghavan S., 1995, Phys. Rev., D51, 1465 Pogosyan D., Pichon C., Gay C., Prunet S., Cardoso J. F., Sousbie Stewart A., Brandenberger R., 2009, JCAP, 0902, 009 T., Colombi S., 2009, Mon. Not. Roy. Astron. Soc., 396, 635 Vachaspati T., Vilenkin A., 1984, Phys. Rev., D30, 2036 Polikar R., 2006, IEEE Circuits and systems magazine, 6, 21 Vafaei Sadr A., Movahed S. M. S., Farhang M., Ringeval C., Quinlan J. R., 1986, Machine learning, 1, 81 Bouchet F. R., 2017 Regan D., Hindmarsh M., 2015, JCAP, 1510, 030 Vilenkin A., 1981, Phys. Rev. Lett., 46, 1169 Rice S. O., 1944, Bell Labs Technical Journal, 23, 282 Vilenkin A., 1985, Phys. Rept., 121, 263 Ringeval C., 2010, Adv. Astron., 2010, 380507 Vilenkin A., Shellard E. P. S., 2000, Cosmic Strings and Other Ringeval C., Bouchet F. R., 2012, Phys. Rev., D86, 023513 Topological Defects. Cambridge University Press, http: Ringeval C., Suyama T., 2017, arXiv:1709.03845 //www.cambridge.org/mw/academic/subjects/physics/ Ringeval C., Sakellariadou M., Bouchet F., 2007, JCAP, 0702, theoretical-physics-and-mathematical-physics/ 023 cosmic-strings-and-other-topological-defects?format= Ringeval C., Yamauchi D., Yokoyama J., Bouchet F. R., 2016, PB JCAP, 1602, 033 White M. J., Carlstrom J. E., Dragovan M., 1999, Astrophys. J., Rokach L., 2010, Artiﬁcial Intelligence Review, 33, 1 514, 12 Ryden B. S., 1988, The Astrophysical Journal, 333, 78 Zeldovich Ya. B., 1980, Mon. Not. Roy. Astron. Soc., 192, 663 Ryden B. S., Melott A. L., Craig D. A., Gott III J. R., Wein- berg D. H., Scherrer R. J., Bhavsar S. P., Miller J. M., 1989, Astrophys. J., 340, 647 This paper has been typeset from a TEX/LATEX ﬁle prepared by Sakellariadou M., 1997, Int. J. Theor. Phys., 36, 2503 the author. MNRAS 000, 1–7 (0000)

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text: Machine

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text: Holomorphic Hartree–Fock Theory: The Nature of Two-Electron Problems Hugh G. A. Burton,∗,† Mark Gross,‡ and Alex J. W. Thom† †Department of Chemistry, Lensﬁeld Road, Cambridge, CB2 1EW, UK ‡DPMMS, CMS, Wilberforce Road, Cambridge, CB3 0WB, UK E-mail: \markcomm{footercolor}{hb407@cam.ac.uk} Abstract lished self-consistent ﬁeld method (SCF), the Hartree–Fock energy is minimised with respect We explore the existence and behaviour of holo- to variations of a set of orbitals expressed in morphic restricted Hartree–Fock (h-RHF) solu- a given ﬁnite basis set. This optimal set of tions for two–electron problems. Through alge- orbitals therefore exists as a stationary point braic geometry, the exact number of solutions of the energy.2,3 However, it is less widely ap- with n basis functions is rigorously identiﬁed preciated that the nonlinear form of the SCF as 1(3n − 1), proving that states must exist for equations can lead to convergence onto a range 2 all molecular geometries. A detailed study on of diﬀerent solutions.4,5 These solutions repre- the h-RHF states of HZ (STO-3G) then demon- sent additional local minima, maxima or saddle strates both the conservation of holomorphic points of the energy. solutions as geometry or atomic charges are var- Through methods including SCF metady- ied and the emergence of complex h-RHF solu- namics5 and the maximum overlap method tions at coalescence points. Using catastrophe (MOM),6 locating higher energy stationary theory, the nature of these coalescence points is points has become relatively routine, and sev- described, highlighting the inﬂuence of molec- eral authors have sought to interpret these as ular symmetry. The h-RHF states of HHeH2+ physical excited states.6–10 However, for many and HHeH (STO-3G) are then compared, illus- systems there exist multiple low energy solu- trating the isomorphism between systems with tions that may cross as the geometry changes, two electrons and two electron holes. Finally, presenting a dilemma when correlation tech- we explore the h-RHF states of ethene (STO- niques require a single reference determinant 3G) by considering the π electrons as a two– to be chosen.11 Recently there has been in- electron problem, and employ NOCI to iden- creasing interest in using multiple Hartree–Fock tify a crossing of the lowest energy singlet and states as a basis for non-orthogonal conﬁgura- triplet states at the perpendicular geometry. tion interaction (NOCI) calculations, providing a more egalitarian treatment of individual low energy SCF solutions.11–16 1 Introduction Since each Hartree–Fock state is itself an mean-ﬁeld optimised solution, excited con- Hartree–Fock theory is ubiquitous in quan- ﬁgurations can be accurately represented in tum chemistry. Representing the many- the NOCI basis set.15 Consequently, NOCI electron wave function as a single Slater de- also provides an alternative to the multi- terminant, the Hartree–Fock approximation conﬁgurational Complete Active Space SCF provides a mean-ﬁeld description of molecular (CASSCF)17 approach using an “active space” electronic structure.1 Through the long estab- 1 ##LTLine##

text: Holomorphic

text: Hartree–Fock

text: Theory:

text: The

text: Nature

text: of

text: Two-Electron

text: Problems

text: Hugh

text: G.

text: A.

text: Burton,∗,†

text: Mark

text: Gross,‡

text: and

text: Alex

text: J.

text: W.

text: Thom†

text: †Department

text: of

text: Chemistry,

text: Lensﬁeld

text: Road,

text: Cambridge,

text: UK

text: ‡DPMMS,

text: CMS,

text: Wilberforce

text: Road,

text: Cambridge,

text: UK

text: E-mail:

text: Abstract

text: lished

text: self-consistent

text: ﬁeld

text: method

text: (SCF),

text: the

text: Hartree–Fock

text: energy

text: is

text: minimised

text: with

text: respect

text: We

text: explore

text: the

text: existence

text: and

text: behaviour

text: of

text: holo-

text: to

text: variations

text: of

text: a

text: set

text: of

text: orbitals

text: expressed

text: in

text: morphic

text: restricted

text: Hartree–Fock

text: (h-RHF)

text: solu-

text: a

text: given

text: ﬁnite

text: basis

text: set.

text: This

text: optimal

text: set

text: of

text: tions

text: for

text: two–electron

text: problems.

text: Through

text: alge-

text: orbitals

text: therefore

text: exists

text: as

text: a

text: stationary

text: point

text: braic

text: geometry,

text: the

text: exact

text: number

text: of

text: solutions

text: of

text: the

text: However,

text: it

text: is

text: less

text: widely

text: ap-

text: with

text: n

text: basis

text: functions

text: is

text: rigorously

text: identiﬁed

text: preciated

text: that

text: the

text: nonlinear

text: form

text: of

text: the

text: SCF

text: as

text: proving

text: that

text: states

text: must

text: exist

text: for

text: equations

text: can

text: lead

text: to

text: convergence

text: onto

text: a

text: range

text: all

text: molecular

text: geometries.

text: A

text: detailed

text: study

text: on

text: of

text: diﬀerent

text: These

text: solutions

text: repre-

text: the

text: h-RHF

text: states

text: of

text: HZ

text: then

text: demon-

text: sent

text: additional

text: local

text: minima,

text: maxima

text: or

text: saddle

text: strates

text: both

text: the

text: conservation

text: of

text: holomorphic

text: points

text: of

text: the

text: energy.

text: solutions

text: as

text: geometry

text: or

text: atomic

text: charges

text: are

text: var-

text: Through

text: methods

text: including

text: SCF

text: metady-

text: ied

text: and

text: the

text: emergence

text: of

text: complex

text: h-RHF

text: solu-

text: and

text: the

text: maximum

text: overlap

text: method

text: tions

text: at

text: coalescence

text: points.

text: Using

text: catastrophe

text: locating

text: higher

text: energy

text: stationary

text: theory,

text: the

text: nature

text: of

text: these

text: coalescence

text: points

text: is

text: points

text: has

text: become

text: relatively

text: routine,

text: and

text: sev-

text: described,

text: highlighting

text: the

text: inﬂuence

text: of

text: molec-

text: eral

text: authors

text: have

text: sought

text: to

text: interpret

text: these

text: as

text: ular

text: symmetry.

text: The

text: h-RHF

text: states

text: of

text: physical

text: excited

text: However,

text: for

text: many

text: and

text: HHeH

text: are

text: then

text: compared,

text: illus-

text: systems

text: there

text: exist

text: multiple

text: low

text: energy

text: solu-

text: trating

text: the

text: isomorphism

text: between

text: systems

text: with

text: tions

text: that

text: may

text: cross

text: as

text: the

text: geometry

text: changes,

text: two

text: electrons

text: and

text: two

text: electron

text: holes.

text: Finally,

text: presenting

text: a

text: dilemma

text: when

text: correlation

text: tech-

text: we

text: explore

text: the

text: h-RHF

text: states

text: of

text: ethene

text: (STO-

text: niques

text: require

text: a

text: single

text: reference

text: determinant

text: by

text: considering

text: the

text: π

text: electrons

text: as

text: a

text: two–

text: to

text: be

text: Recently

text: there

text: has

text: been

text: in-

text: electron

text: problem,

text: and

text: employ

text: NOCI

text: to

text: iden-

text: creasing

text: interest

text: in

text: using

text: multiple

text: Hartree–Fock

text: tify

text: a

text: crossing

text: of

text: the

text: lowest

text: energy

text: singlet

text: and

text: states

text: as

text: a

text: basis

text: for

text: non-orthogonal

text: conﬁgura-

text: triplet

text: states

text: at

text: the

text: perpendicular

text: geometry.

text: tion

text: interaction

text: (NOCI)

text: calculations,

text: providing

text: a

text: more

text: egalitarian

text: treatment

text: of

text: individual

text: low

text: energy

text: SCF

text: Introduction

text: Since

text: each

text: Hartree–Fock

text: state

text: is

text: itself

text: an

text: mean-ﬁeld

text: optimised

text: solution,

text: excited

text: con-

text: Hartree–Fock

text: theory

text: is

text: ubiquitous

text: in

text: quan-

text: ﬁgurations

text: can

text: be

text: accurately

text: represented

text: in

text: tum

text: chemistry.

text: Representing

text: the

text: many-

text: the

text: NOCI

text: basis

text: Consequently,

text: NOCI

text: electron

text: wave

text: function

text: as

text: a

text: single

text: Slater

text: de-

text: also

text: provides

text: an

text: alternative

text: to

text: the

text: multi-

text: terminant,

text: the

text: Hartree–Fock

text: approximation

text: conﬁgurational

text: Complete

text: Active

text: Space

text: SCF

text: provides

text: a

text: mean-ﬁeld

text: description

text: of

text: molecular

text: approach

text: using

text: an

text: “active

text: space”

text: electronic

text: Through

text: the

text: long

text: estab-

text: ##LTLine##