text: 7 TABLE I. Critical temperatures and excitonic gaps in pumped DMs. Dimensionless coupling constant α, average chemical potential µ− and lifetime of the transient excitonic state τ for graphene and 3DTIs were estimated based on existing literature (see Appendix for details). Critical temperature T and maximum value of the excitonic gap ∆ at T = 0 for given material c max parameters were calculated using the numerical model with screened Coulomb interaction. For the case of graphene we consider both the one valley (g = 2) and two valley (g = 4) cases. The last row corresponds to a hypothetical DM with parameters similar to graphene and g = 1. Material α µ− (meV) τ (ps) T (K) ∆ (meV) c max − 1.0 for g = 4 0.1 for g = 4 graphene (substrate) 0.4 1.0 500 0.1 − − 6 35 for g = 2 1 5 for g = 2 2 for g = 4 0.3 for g = 4 graphene (suspended) 2.2 500 0.1 70 for g = 2 10 for g = 2 − − − − 3DTI 0.1 1.0 100 1 106 0 30 0 3 − − − DM with g=1 0.4 2.2 500 - 50 500 10 100 a 20 b way to increase the lifetime of the inverted population is to use continuous pumping where electrons are con- 10 stantly injected into the empty states above the Dirac 0 node. However, such a scheme might result in high local -100 -50 0 50 100 µ+ [meV] electronic temperatures which will inhibit the formation c of the excitonic condensate. In 3DTIs the reported life- 20 times of the population inversion are much longer, e.g. of the order of a few ps17 and under certain conditions 10 0 200 400 even exceeding 4µs18. µ- [meV] In general, in order for the transient excitonic gaps to be observable in experiments, the timescale of formation FIG. 4. Stability diagram of the transient excitonic conden- (cid:126) of these collective states, τ = /∆ , should be small sate as a function of chemical potential mismatch. The max- ex max compared to the timescale τ on which the inverted pop- imum value of the order parameter ∆k evaluated at T = 0 el ≈ plotted as a function of (a) electron and hole chemical po- ulat≈ion is observed, τex << τel. For ∆max 10 meV, tentials, (b) the mismatch µ+ for a ﬁxed average chemical τex 60 fs, allowing for the observation of−these gaps in potential µ− = 0.1 meV, and (c) the average chemical po- graphene where τ is on the order of 100 200 fs. For el tential µ− for zero mismatch µ+ = 0. We used g = 1 and smaller gaps, larger lifetimes of inverted population are α = 0.7. necessary which could be realized in future experiments on graphene or 3DTIs. While our analysis focused on 2D DMs, like graphene carrier distribution decays toward equilibrium and, as a or 3DTI surface states, our theory also has implications result, the excitonic states in a pumped DM acquire a for 3D DMs such as Dirac46 and Weyl47,48 semimetals. N ∝ ﬁnite lifetime. We have analyzed the relaxation of the In a 3D DM, the DOS is quadratic in energy, E2, in order parameter using a dynamical approach based on contrast to the linear dependence in 2D, hence the eﬀec- semiconductor Bloch equations44,45 applied to a pumped tive coupling can be made even larger in these materials. DM, where both intraband relaxation and interband scat- However, since screening could also be much stronger in tering (recombination) is taken into account (see Ap- 3D a more detailed analysis would be necessary to make pendix for details). The result is that the characteristic quantitative predictions for these materials. It should timescale over which the population inversion is sustained be noted that the large valley degeneracy found in some provides an estimate of the lifetime of the transient exci- 3D Dirac systems (g = 24 in TaAs Weyl semimetal47) tonic gapped state. could be detrimental for the eﬀects discussed in this pa- per. Therefore, as in the case of 2D DMs, materials So far the lifetime of the inverted carrier distribution with smaller degeneracy will be the most promising can- observed in graphene has been limited to hundreds of fs15 didates. after which a single equilibrium Fermi-Dirac distribution V. CONSLUSIONS is reestablished via inverse Auger scattering (recombina- tion) and electron-phonon scattering16. However, these relatively short lifetimes were obtained for hole-doped In conclusion, we have shown that the energy-dependence graphene with the equilibrium chemical potential lying of the DOS in 2D Dirac materials allows for a tunable a few hundred meV below the Dirac node, as is typical enhancement of the strength of the Coulomb interaction for graphene on a substrate. One might expect more fa- relative to the values accessible in equilibrium. We have vorable conditions in undoped graphene, in which recom- demonstrated that this tunability allows for the genera- bination of carriers is suppressed due to reduced phase tion of transient excitonic states in optically-pumped 2D space near the Dirac node. Additionally, one possible Dirac materials leading to the formation of gaps in the ]Vem[ xam∆ ]Vem[ xam∆ ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: TABLE

text: I.

text: Critical

text: temperatures

text: and

text: excitonic

text: gaps

text: in

text: pumped

text: DMs.

text: Dimensionless

text: coupling

text: constant

text: α,

text: average

text: chemical

text: potential

text: µ−

text: and

text: lifetime

text: of

text: the

text: transient

text: excitonic

text: state

text: τ

text: for

text: graphene

text: and

text: were

text: estimated

text: based

text: on

text: existing

text: literature

text: (see

text: Appendix

text: for

text: details).

text: Critical

text: temperature

text: T

text: and

text: maximum

text: value

text: of

text: the

text: excitonic

text: gap

text: at

text: T

text: for

text: given

text: material

text: c

text: max

text: parameters

text: were

text: calculated

text: using

text: the

text: numerical

text: model

text: with

text: screened

text: Coulomb

text: interaction.

text: For

text: the

text: case

text: of

text: graphene

text: we

text: consider

text: both

text: the

text: one

text: valley

text: (g

text: and

text: two

text: valley

text: (g

text: cases.

text: The

text: last

text: row

text: corresponds

text: to

text: a

text: hypothetical

text: DM

text: with

text: parameters

text: similar

text: to

text: graphene

text: and

text: g

text: Material

text: α

text: µ−

text: (meV)

text: τ

text: (ps)

text: T

text: (K)

text: (meV)

text: c

text: max

text: for

text: g

text: for

text: g

text: graphene

text: (substrate)

text: for

text: g

text: for

text: g

text: for

text: g

text: for

text: g

text: graphene

text: (suspended)

text: for

text: g

text: for

text: g

text: DM

text: with

text: a

text: b

text: way

text: to

text: increase

text: the

text: lifetime

text: of

text: the

text: inverted

text: population

text: is

text: to

text: use

text: continuous

text: pumping

text: where

text: electrons

text: are

text: con-

text: stantly

text: injected

text: into

text: the

text: empty

text: states

text: above

text: the

text: Dirac

text: node.

text: However,

text: such

text: a

text: scheme

text: might

text: result

text: in

text: high

text: local

text: µ+

text: [meV]

text: electronic

text: temperatures

text: which

text: will

text: inhibit

text: the

text: formation

text: c

text: of

text: the

text: excitonic

text: condensate.

text: In

text: the

text: reported

text: life-

text: times

text: of

text: the

text: population

text: inversion

text: are

text: much

text: longer,

text: e.g.

text: of

text: the

text: order

text: of

text: a

text: few

text: and

text: under

text: certain

text: conditions

text: even

text: exceeding

text: µ-

text: [meV]

text: In

text: general,

text: in

text: order

text: for

text: the

text: transient

text: excitonic

text: gaps

text: to

text: be

text: observable

text: in

text: experiments,

text: the

text: timescale

text: of

text: formation

text: FIG.

text: Stability

text: diagram

text: of

text: the

text: transient

text: excitonic

text: conden-

text: of

text: these

text: collective

text: states,

text: τ

text: should

text: be

text: small

text: sate

text: as

text: a

text: function

text: of

text: chemical

text: potential

text: mismatch.

text: The

text: max-

text: ex

text: max

text: compared

text: to

text: the

text: timescale

text: τ

text: on

text: which

text: the

text: inverted

text: pop-

text: imum

text: value

text: of

text: the

text: order

text: parameter

text: ∆k

text: evaluated

text: at

text: T

text: el

text: plotted

text: as

text: a

text: function

text: of

text: (a)

text: electron

text: and

text: hole

text: chemical

text: po-

text: ulat≈ion

text: is

text: observed,

text: τex

text: τel.

text: For

text: ∆max

text: meV,

text: tentials,

text: (b)

text: the

text: mismatch

text: µ+

text: for

text: a

text: ﬁxed

text: average

text: chemical

text: τex

text: fs,

text: allowing

text: for

text: the

text: observation

text: of−these

text: gaps

text: in

text: potential

text: µ−

text: meV,

text: and

text: (c)

text: the

text: average

text: chemical

text: po-

text: graphene

text: where

text: τ

text: is

text: on

text: the

text: order

text: of

text: fs.

text: For

text: el

text: tential

text: µ−

text: for

text: zero

text: mismatch

text: µ+

text: We

text: used

text: g

text: and

text: smaller

text: gaps,

text: larger

text: lifetimes

text: of

text: inverted

text: population

text: are

text: α

text: necessary

text: which

text: could

text: be

text: realized

text: in

text: future

text: experiments

text: on

text: graphene

text: or

text: While

text: our

text: analysis

text: focused

text: on

text: DMs,

text: like

text: graphene

text: carrier

text: distribution

text: decays

text: toward

text: equilibrium

text: and,

text: as

text: a

text: or

text: surface

text: states,

text: our

text: theory

text: also

text: has

text: implications

text: result,

text: the

text: excitonic

text: states

text: in

text: a

text: pumped

text: DM

text: acquire

text: a

text: for

text: DMs

text: such

text: as

text: and

text: semimetals.

text: N

text: ﬁnite

text: lifetime.

text: We

text: have

text: analyzed

text: the

text: relaxation

text: of

text: the

text: In

text: a

text: DM,

text: the

text: DOS

text: is

text: quadratic

text: in

text: energy,

text: in

text: order

text: parameter

text: using

text: a

text: dynamical

text: approach

text: based

text: on

text: contrast

text: to

text: the

text: linear

text: dependence

text: in

text: hence

text: the

text: eﬀec-

text: semiconductor

text: Bloch

text: applied

text: to

text: a

text: pumped

text: tive

text: coupling

text: can

text: be

text: made

text: even

text: larger

text: in

text: these

text: materials.

text: DM,

text: where

text: both

text: intraband

text: relaxation

text: and

text: interband

text: scat-

text: However,

text: since

text: screening

text: could

text: also

text: be

text: much

text: stronger

text: in

text: tering

text: (recombination)

text: is

text: taken

text: into

text: account

text: (see

text: Ap-

text: a

text: more

text: detailed

text: analysis

text: would

text: be

text: necessary

text: to

text: make

text: pendix

text: for

text: details).

text: The

text: result

text: is

text: that

text: the

text: characteristic

text: quantitative

text: predictions

text: for

text: these

text: materials.

text: It

text: should

text: timescale

text: over

text: which

text: the

text: population

text: inversion

text: is

text: sustained

text: be

text: noted

text: that

text: the

text: large

text: valley

text: degeneracy

text: found

text: in

text: some

text: provides

text: an

text: estimate

text: of

text: the

text: lifetime

text: of

text: the

text: transient

text: exci-

text: Dirac

text: systems

text: (g

text: in

text: TaAs

text: Weyl

text: tonic

text: gapped

text: state.

text: could

text: be

text: detrimental

text: for

text: the

text: eﬀects

text: discussed

text: in

text: this

text: pa-

text: per.

text: Therefore,

text: as

text: in

text: the

text: case

text: of

text: DMs,

text: materials

text: So

text: far

text: the

text: lifetime

text: of

text: the

text: inverted

text: carrier

text: distribution

text: with

text: smaller

text: degeneracy

text: will

text: be

text: the

text: most

text: promising

text: can-

text: observed

text: in

text: graphene

text: has

text: been

text: limited

text: to

text: hundreds

text: of

text: didates.

text: after

text: which

text: a

text: single

text: equilibrium

text: Fermi-Dirac

text: distribution

text: V.

text: CONSLUSIONS

text: is

text: reestablished

text: via

text: inverse

text: Auger

text: scattering

text: (recombina-

text: tion)

text: and

text: electron-phonon

text: However,

text: these

text: relatively

text: short

text: lifetimes

text: were

text: obtained

text: for

text: hole-doped

text: In

text: conclusion,

text: we

text: have

text: shown

text: that

text: the

text: energy-dependence

text: graphene

text: with

text: the

text: equilibrium

text: chemical

text: potential

text: lying

text: of

text: the

text: DOS

text: in

text: Dirac

text: materials

text: allows

text: for

text: a

text: tunable

text: a

text: few

text: hundred

text: meV

text: below

text: the

text: Dirac

text: node,

text: as

text: is

text: typical

text: enhancement

text: of

text: the

text: strength

text: of

text: the

text: Coulomb

text: interaction

text: for

text: graphene

text: on

text: a

text: substrate.

text: One

text: might

text: expect

text: more

text: fa-

text: relative

text: to

text: the

text: values

text: accessible

text: in

text: equilibrium.

text: We

text: have

text: vorable

text: conditions

text: in

text: undoped

text: graphene,

text: in

text: which

text: recom-

text: demonstrated

text: that

text: this

text: tunability

text: allows

text: for

text: the

text: genera-

text: bination

text: of

text: carriers

text: is

text: suppressed

text: due

text: to

text: reduced

text: phase

text: tion

text: of

text: transient

text: excitonic

text: states

text: in

text: optically-pumped

text: space

text: near

text: the

text: Dirac

text: node.

text: Additionally,

text: one

text: possible

text: Dirac

text: materials

text: leading

text: to

text: the

text: formation

text: of

text: gaps

text: in

text: the

text: ]Vem[

text: xam∆

text: ]Vem[

text: xam∆

text: ##LTFigure##

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text: Sparse Kernel Canonical Correlation Analysis via ∗ (cid:96) -regularization 1 † Xiaowei Zhang 1, Delin Chu1, Li-Zhi Liao2 and Michael K. Ng2 1Department of Mathematics, National University of Singapore. 2Department of Mathematics, Hong Kong Baptist University. Abstract Canonical correlation analysis (CCA) is a multivariate statistical technique for ﬁnding the linear relationship between two sets of variables. The kernel generalization of CCA named kernel CCA has been proposed to ﬁnd nonlinear relations between datasets. Despite their wide usage, they have one common limitation that is the lack of sparsity in their solution. In this paper, we consider sparse kernel CCA and propose a novel sparse kernel CCA algorithm (SKCCA). Our algorithm is based on a relationship between kernel CCA and least squares. Sparsity of the dual transformations is introduced by penalizing the (cid:96) -norm of dual vectors. Experiments demonstrate that our algorithm 1 not only performs well in computing sparse dual transformations but also can alleviate the over-ﬁtting problem of kernel CCA. 1 Introduction The description of relationship between two sets of variables has long been an interesting topic to many researchers. Canonical correlation analysis (CCA), which was originally introduced in [26], is a multivariate statistical technique for ﬁnding the linear relationship between two sets of variables. Those two sets of variables can be considered as diﬀerent views of the same object or views of diﬀerent objects, and are assumed to contain some joint information in the correlations between them. CCA seeks a linear transformation for each of the two sets of variables in a way that the projected variables in the transfor{me}d spa∈ce are max{i(cid:2)ma}lly c∈orrela(cid:3)ted. (cid:2) (cid:3) Let xi ni=1 Rd1 and yi ni=1 Rd2 be n samples for variables x and y, respectively. Denote · · · ∈ × · · · ∈ × X = x1 xn Rd1 n, Y = y(cid:80)1 yn R(cid:80)d2 n, { } { } n n and assume both x n and y n have zero mean, i.e., x = 0 and y = 0. Then CCA solves i i=1 i i=1 i i i=1 i=1 the following optimization problem max wT XY T w x y wx,wy s.t. wT XXT w = 1, (1.1) x x wT Y Y T w = 1, y y to get the ﬁrst pair of weight vectors w and w , which are further utilized to obtain the ﬁrst pair x y of canonical variables wT X and wT Y, respectively. For the rest pairs of weight vectors and canonical x y variables, CCA solves sequentially the same problem as (1.1) with additional constraints of orthogonality ∗ Part of the material in this paper was presented in [10] and [56]. † Corresponding author: zxwtroy87@gmail.com 1 ##LTLine##

text: Sparse

label: title

text: Kernel

text: Canonical

text: Correlation

text: Analysis

text: via

text: -regularization

label: author

text: Xiaowei

text: Zhang

label: paragraph

text: Delin

text: Li-Zhi

text: and

text: Michael

text: K.

text: of

text: Mathematics,

text: National

text: University

text: of

text: Singapore.

text: of

text: Mathematics,

text: Hong

text: Kong

text: Baptist

text: University.

text: Abstract

text: Canonical

label: abstract

text: correlation

text: analysis

text: (CCA)

text: is

text: a

text: multivariate

text: statistical

text: technique

text: for

text: ﬁnding

text: the

text: linear

text: relationship

text: between

text: two

text: sets

text: of

text: variables.

text: The

text: kernel

text: generalization

text: of

text: CCA

text: named

text: kernel

text: CCA

text: has

text: been

text: proposed

text: to

text: ﬁnd

text: nonlinear

text: relations

text: between

text: datasets.

text: Despite

text: their

text: wide

text: usage,

text: they

text: have

text: one

text: common

text: limitation

text: that

text: is

text: the

text: lack

text: of

text: sparsity

text: in

text: their

text: solution.

text: In

text: this

text: paper,

text: we

text: consider

text: sparse

text: kernel

text: CCA

text: and

text: propose

text: a

text: novel

text: sparse

text: kernel

text: CCA

text: algorithm

text: (SKCCA).

text: Our

text: algorithm

text: is

text: based

text: on

text: a

text: relationship

text: between

text: kernel

text: CCA

text: and

text: least

text: squares.

text: Sparsity

text: of

text: the

text: dual

text: transformations

text: is

text: introduced

text: by

text: penalizing

text: the

text: -norm

text: of

text: dual

text: vectors.

text: Experiments

text: demonstrate

text: that

text: our

text: algorithm

text: not

text: only

text: performs

text: well

text: in

text: computing

text: sparse

text: dual

text: transformations

text: but

text: also

text: can

text: alleviate

text: the

text: over-ﬁtting

text: problem

text: of

text: kernel

text: CCA.

text: Introduction

text: The

text: description

text: of

text: relationship

text: between

text: two

text: sets

text: of

text: variables

text: has

text: long

text: been

text: an

text: interesting

text: topic

text: to

text: many

text: researchers.

text: Canonical

text: correlation

text: analysis

text: (CCA),

text: which

text: was

text: originally

text: introduced

text: in

text: is

text: a

text: multivariate

text: statistical

text: technique

text: for

text: ﬁnding

text: the

text: linear

text: relationship

text: between

text: two

text: sets

text: of

text: variables.

text: Those

text: two

text: sets

text: of

text: variables

text: can

text: be

text: considered

text: as

text: diﬀerent

text: views

text: of

text: the

text: same

text: object

text: or

text: views

text: of

text: diﬀerent

text: objects,

text: and

text: are

text: assumed

text: to

text: contain

text: some

text: joint

text: information

text: in

text: the

text: correlations

text: between

text: them.

text: CCA

text: seeks

text: a

text: linear

text: transformation

text: for

text: each

text: of

text: the

text: two

text: sets

text: of

text: variables

text: in

text: a

text: way

text: that

text: the

text: projected

text: variables

text: in

text: the

text: transfor{me}d

text: spa∈ce

text: are

text: Let

text: xi

text: and

text: yi

text: be

text: n

text: samples

text: for

text: variables

text: x

text: and

text: y,

text: respectively.

text: Denote

text: X

text: xn

text: n,

text: Y

text: yn

text: n,

text: n

text: n

text: and

text: assume

text: both

text: x

text: n

text: and

text: y

text: n

text: have

text: zero

text: mean,

text: i.e.,

text: x

text: and

text: y

text: Then

text: CCA

text: solves

text: i

text: i

text: i

text: i

text: the

text: following

text: optimization

text: problem

text: max

text: wT

text: XY

text: T

text: w

text: x

text: y

text: wx,wy

text: s.t.

text: wT

text: XXT

text: w

text: x

text: x

text: wT

text: Y

text: Y

text: T

text: w

text: y

text: y

text: to

text: get

text: the

text: ﬁrst

text: pair

text: of

text: weight

text: vectors

text: w

text: and

text: w

text: which

text: are

text: further

text: utilized

text: to

text: obtain

text: the

text: ﬁrst

text: pair

text: x

text: y

text: of

text: canonical

text: variables

text: wT

text: X

text: and

text: wT

text: Y,

text: respectively.

text: For

text: the

text: rest

text: pairs

text: of

text: weight

text: vectors

text: and

text: canonical

text: x

text: y

text: variables,

text: CCA

text: solves

text: sequentially

text: the

text: same

text: problem

text: as

text: with

text: additional

text: constraints

text: of

text: orthogonality

text: Part

text: of

text: the

text: material

text: in

text: this

text: paper

text: was

text: presented

text: in

text: and

text: Corresponding

text: author:

text: ##LTLine##


text: 1. The English-French corpus from the Europarl parallel corpus dataset [28]3, where we obtained 202 × × samples and generated a 23308 202 term-document matrix for English corpus and a 33986 202 term-document matrix for French corpus. 2. The Aligned Hansards of the 36th Parliament of Canada [18]4, which is a collection of text chunks (sentences or smaller fragments) in English and French from the 36th Parliament proceedings of Canada. In our experiments, we used only a part of text chunks to generate term-documents × × matrices and obtained a 5383 818 term-document matrix for English documents and a 8015 818 term-document matrix for French documents. For Europarl data 100 pairs of documents were used for training data and the rest for testing data while for Hansard data 200 pairs of documents were used for training data and the rest for testing data. In both experiments, the linear kernel (4.11) was employed to compute kernel matrices. We measure the precision of document retrieval by using average area under the ROC curve (AROC), and for a collection of queries we use the average of each query¡s retrieval precision as the average retrieval precision of this collection. More details about the acquisition of term-document matrix, data preprocessing and evaluation of retrieval performance can be found in [9]. Figure 5.2 presents the retrieval accuracy of KCCA, RKCCA and SKCCA on both datasets. 1 1 0.95 0.95 0.9 0.85 0.9 KCCA KCCA RKCCA RKCCA 0.8 SKCCA SKCCA CCA 0.85 CCA SCCA_l1 SCCA_l1 0.75 SCCA_LS SCCA_LS 0 20 40 60 80 100 0.80 50 100 150 200 Number of columns of (Wx,Wy) used Number of columns of (Wx,Wy) used (a) (b) Figure 5.2: Cross-language document retrieval using KCCA, RKCCA and SKCCA: (a) Europarl data with 100 training data, (b) Hansard data with 200training data. Figure 5.2 shows that all three algorithms achieve high precision for cross-language document retrieval, even though only a small number of training data were used. From the ﬁgures we also see that increasing l, the number of columns of W and W used in retrieval task, will usually assist in improving the x x precision. One possible explanation may be that when we increase l, more projections corresponding to nonzero canonical correlations are used for document retrieval and these added projections may carry information contained in the training data. Both ﬁgures in Figure 5.2 show that RKCCA and SKCCA outperform KCCA in terms of retrieval accuracy, though their diﬀerence is small. This indicates that both RKCCA and SKCCA have ability of avoiding data overﬁtting problem in ordinary kernel CCA, as stated in Section . Additional results are presented in the following table, where we recorded the retrieval precision (AROC) using projections corresponding to all nonzero canonical correlations, i.e, l = mˆ , summation W W of canonical correlations between testing data (Corr), sparsity of and , and violation of the x y orthogonality constraints measured by Err(W ) := (cid:107)WxTKx2√Wx−Il(cid:107)F and Err(W ) := (cid:107)WyTKy2√Wy−Il(cid:107)F . x y l l 3http://www.statmt.org/europarl/ 4http://www.isi.edu/natural-language/download/hansard/ 18 CORA egarevA CORA egarevA ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: list

text: The

text: English-French

text: corpus

text: from

text: the

text: Europarl

text: parallel

text: corpus

text: dataset

text: where

text: we

text: obtained

text: samples

text: and

text: generated

text: a

text: term-document

text: matrix

text: for

text: English

text: corpus

text: and

text: a

text: term-document

text: matrix

text: for

text: French

text: corpus.

text: The

text: Aligned

text: Hansards

text: of

text: the

text: Parliament

text: of

text: Canada

text: which

text: is

text: a

text: collection

text: of

text: text

text: chunks

text: (sentences

text: or

text: smaller

text: fragments)

text: in

text: English

text: and

text: French

text: from

text: the

text: Parliament

text: proceedings

text: of

text: Canada.

text: In

text: our

text: experiments,

text: we

text: used

text: only

text: a

text: part

text: of

text: text

text: chunks

text: to

text: generate

text: term-documents

text: matrices

text: and

text: obtained

text: a

text: term-document

text: matrix

text: for

text: English

text: documents

text: and

text: a

text: term-document

text: matrix

text: for

text: French

text: documents.

text: For

text: Europarl

text: data

text: pairs

text: of

text: documents

text: were

text: used

text: for

text: training

text: data

text: and

text: the

text: rest

text: for

text: testing

text: data

text: while

text: for

text: Hansard

text: data

text: pairs

text: of

text: documents

text: were

text: used

text: for

text: training

text: data

text: and

text: the

text: rest

text: for

text: testing

text: data.

text: In

text: both

text: experiments,

text: the

text: linear

text: kernel

text: was

text: employed

text: to

text: compute

text: kernel

text: matrices.

text: We

text: measure

text: the

text: precision

text: of

text: document

text: retrieval

text: by

text: using

text: average

text: area

text: under

text: the

text: ROC

text: curve

text: (AROC),

text: and

text: for

text: a

text: collection

text: of

text: queries

text: we

text: use

text: the

text: average

text: of

text: each

text: query¡s

text: retrieval

text: precision

text: as

text: the

text: average

text: retrieval

text: precision

text: of

text: this

text: collection.

text: More

text: details

text: about

text: the

text: acquisition

text: of

text: term-document

text: matrix,

text: data

text: preprocessing

text: and

text: evaluation

text: of

text: retrieval

text: performance

text: can

text: be

text: found

text: in

text: Figure

text: presents

text: the

text: retrieval

text: accuracy

text: of

text: KCCA,

text: RKCCA

text: and

text: SKCCA

text: on

text: both

text: datasets.

text: KCCA

text: KCCA

text: RKCCA

text: RKCCA

text: SKCCA

text: SKCCA

text: CCA

text: CCA

text: Number

text: of

text: columns

text: of

text: (Wx,Wy)

text: used

text: Number

text: of

text: columns

text: of

text: (Wx,Wy)

text: used

text: (a)

text: (b)

text: Figure

text: Cross-language

text: document

text: retrieval

text: using

text: KCCA,

text: RKCCA

text: and

text: SKCCA:

text: (a)

text: Europarl

text: data

text: with

text: training

text: data,

text: (b)

text: Hansard

text: data

text: with

text: data.

text: Figure

text: shows

text: that

text: all

text: three

text: algorithms

text: achieve

text: high

text: precision

text: for

text: cross-language

text: document

text: retrieval,

text: even

text: though

text: only

text: a

text: small

text: number

text: of

text: training

text: data

text: were

text: used.

text: From

text: the

text: ﬁgures

text: we

text: also

text: see

text: that

text: increasing

text: l,

text: the

text: number

text: of

text: columns

text: of

text: W

text: and

text: W

text: used

text: in

text: retrieval

text: task,

text: will

text: usually

text: assist

text: in

text: improving

text: the

text: x

text: x

text: precision.

text: One

text: possible

text: explanation

text: may

text: be

text: that

text: when

text: we

text: increase

text: l,

text: more

text: projections

text: corresponding

text: to

text: nonzero

text: canonical

text: correlations

text: are

text: used

text: for

text: document

text: retrieval

text: and

text: these

text: added

text: projections

text: may

text: carry

text: information

text: contained

text: in

text: the

text: training

text: data.

text: Both

text: ﬁgures

text: in

text: Figure

text: show

text: that

text: RKCCA

text: and

text: SKCCA

text: outperform

text: KCCA

text: in

text: terms

text: of

text: retrieval

text: accuracy,

text: though

text: their

text: diﬀerence

text: is

text: small.

text: This

text: indicates

text: that

text: both

text: RKCCA

text: and

text: SKCCA

text: have

text: ability

text: of

text: avoiding

text: data

text: overﬁtting

text: problem

text: in

text: ordinary

text: kernel

text: CCA,

text: as

text: stated

text: in

text: Section

text: Additional

text: results

text: are

text: presented

text: in

text: the

text: following

text: table,

text: where

text: we

text: recorded

text: the

text: retrieval

text: precision

text: (AROC)

text: using

text: projections

text: corresponding

text: to

text: all

text: nonzero

text: canonical

text: correlations,

text: i.e,

text: l

text: mˆ

text: summation

text: W

text: W

text: of

text: canonical

text: correlations

text: between

text: testing

text: data

text: (Corr),

text: sparsity

text: of

text: and

text: and

text: violation

text: of

text: the

text: x

text: y

text: orthogonality

text: constraints

text: measured

text: by

text: Err(W

text: and

text: Err(W

text: x

text: y

text: l

text: l

text: CORA

text: egarevA

text: CORA

text: egarevA

text: ##LTFigure##

text: ##LTFigure##

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text: max αT K K β x y α,β s.t. αT K2α = 1, (1.5) x βT K2β = 1. y The rest pairs of dual vectors are obtained via sequentially solving the same problem as (1.5) with extra constraints of orthogonality. More details on the derivation of kernel CCA are presented in Section . W W ∈ × Suppose we have obtained dual transformations , Rn l and corresponding CCA transfor- ∈ N × ∈ N × x y mations Wx R x l and Wy R y l in feature spaces, then projection of data paiHr (x, y) Honto the kernel CCA directions can be computed by ﬁrst mapping x and y into the feature space and , then x y evaluate their inner products with W and W . More speciﬁcally, projections can be carried out as x y (cid:104) (cid:105) (cid:104) W (cid:105) W (cid:2) W , φ (x) = Φ (cid:3) , φ (x) = T K (X, x), (1.6) x x x x x x x · · · T with K (X, x) = κ (x , x) κ (x , x) , and x x 1 x n (cid:104) (cid:105) (cid:104) W (cid:105) W (cid:2) W , φ (y) = Φ(cid:3) , φ (y) = T K (Y, y), (1.7) y y y y y y y · · · T with K (Y, y) = κ (y , y) κ (y , y) . y y 1 y n Both optimization problems (1.1) and (1.5) can be solved by considering generalized eigenvalue prob- lems [4] of the form Ax = λBx, (1.8) where A, B are symmetric positive semi-deﬁnite. This generalized eigenvalue problem can be solved eﬃciently using approaches from numerical linear algebra [19]. CCA and kernel CCA have been suc- − − cessfully applied in many ﬁelds, including cross language documents retrieval [47], content based image retrieval [25], bioinformatics [46, 53], independent component analysis [2, 17], computation of principal angles between linear subspaces [6, 20]. Despite the wide usage of CCA and kernel CCA, they have one common limitation that is lack W W of sparseness in transformation matrices W and W and dual transformation matrices and . x y x y Equation (1.2) shows that projections of the data pair x and y are linear combinations of themselves which make interpretation of the extracted features diﬃcult if the transformation matrices W and W x y are dense. Similarly, from (1.6) and (1.7) we can see that the kernel functions κ (x , x) and κ (y , y) must { } { } W x i W y i be evaluated for all x n and y n when dual transformation matrices and are dense, which i i=1 i i=1 x y can lead to excessive computational time to compute projections of new data. To handle the limitation of CCA, researchers suggested to incorporate sparsity into weight vectors and many papers have studied sparse CCA [9, 23, 35, 40, 41, 48, 50, 51, 52]. Similarly, we shall ﬁnd sparse solutions for kernel CCA so that projections of new data can be computed by evaluating the kernel function at a subset of the training data. Although there are many sparse kernel approaches [5], such as support vector machines [37], relevance vector machine [45] and sparse kernel partial least squares [14, 34], seldom can be found in the area of sparse kernel CCA [13, 43]. In this paper we ﬁrst consider a new sparse CCA approach and then generalize it to incorporate sparsity into kernel CCA. A relationship between CCA and least squares is established so that CCA solutions can be obtained by solving a least squares problem. We attempt to introduce sparsity by penal- izing (cid:96) -norm of the solutions, which eventually leads to a (cid:96) -norm penalized least squares optimization 1 1 problem of the form 1(cid:107) − (cid:107) (cid:107) (cid:107) min Ax b 2 + λ x , x∈Rd 2 2 1 where λ > 0 is a regularizer controlling the sparsity of x. We adopt a ﬁxed-point continuation (FPC) method [21, 22] to solve the (cid:96) -norm regularized least squares above, which results in a new sparse CCA 1 3 ##LTLine##

text: max

label: equation

text: αT

text: K

text: K

text: β

text: x

text: y

text: α,β

text: s.t.

text: αT

text: x

text: βT

text: y

text: The

text: rest

text: pairs

text: of

text: dual

text: vectors

text: are

text: obtained

text: via

text: sequentially

text: solving

text: the

text: same

text: problem

text: as

text: with

text: extra

text: constraints

text: of

text: orthogonality.

text: More

text: details

text: on

text: the

text: derivation

text: of

text: kernel

text: CCA

text: are

text: presented

text: in

text: Section

text: W

text: W

text: Suppose

text: we

text: have

text: obtained

text: dual

text: transformations

text: Rn

text: l

text: and

text: corresponding

text: CCA

text: transfor-

text: N

text: N

text: x

text: y

text: mations

text: Wx

text: R

text: x

text: l

text: and

text: Wy

text: R

text: y

text: l

text: in

text: feature

text: spaces,

text: then

text: projection

text: of

text: data

text: paiHr

text: (x,

text: y)

text: Honto

text: the

text: kernel

text: CCA

text: directions

text: can

text: be

text: computed

text: by

text: ﬁrst

text: mapping

text: x

text: and

text: y

text: into

text: the

text: feature

text: space

text: and

text: then

text: x

text: y

text: evaluate

text: their

text: inner

text: products

text: with

text: W

text: and

text: W

text: More

text: speciﬁcally,

text: projections

text: can

text: be

text: carried

text: out

text: as

text: x

text: y

text: W

text: W

text: W

text: φ

text: (x)

text: Φ

text: φ

text: (x)

text: T

text: K

text: (X,

text: x),

text: x

text: x

text: x

text: x

text: x

text: x

text: x

text: T

text: with

text: K

text: (X,

text: x)

text: κ

text: (x

text: x)

text: κ

text: (x

text: x)

text: and

text: x

text: x

text: x

text: n

text: W

text: W

text: W

text: φ

text: (y)

text: φ

text: (y)

text: T

text: K

text: (Y,

text: y),

text: y

text: y

text: y

text: y

text: y

text: y

text: y

text: T

text: with

text: K

text: (Y,

text: y)

text: κ

text: (y

text: y)

text: κ

text: (y

text: y)

text: y

text: y

text: y

text: n

text: Both

text: optimization

text: problems

text: and

text: can

text: be

text: solved

text: by

text: considering

text: generalized

text: eigenvalue

text: prob-

text: lems

text: of

text: the

text: form

text: Ax

text: λBx,

text: where

text: A,

text: B

text: are

text: symmetric

text: positive

text: semi-deﬁnite.

text: This

text: generalized

text: eigenvalue

text: problem

text: can

text: be

text: solved

text: eﬃciently

text: using

text: approaches

text: from

text: numerical

text: linear

text: algebra

text: CCA

text: and

text: kernel

text: CCA

text: have

text: been

text: suc-

text: cessfully

text: applied

text: in

text: many

text: ﬁelds,

text: including

text: cross

text: language

text: documents

text: retrieval

text: content

text: based

text: image

text: retrieval

text: bioinformatics

text: independent

text: component

text: analysis

text: computation

text: of

text: principal

text: angles

text: between

text: linear

text: subspaces

text: Despite

text: the

text: wide

text: usage

text: of

text: CCA

text: and

text: kernel

text: CCA,

text: they

text: have

text: one

text: common

text: limitation

text: that

text: is

text: lack

text: W

text: W

text: of

text: sparseness

text: in

text: transformation

text: matrices

text: W

text: and

text: W

text: and

text: dual

text: transformation

text: matrices

text: and

text: x

text: y

text: x

text: y

text: Equation

text: shows

text: that

text: projections

text: of

text: the

text: data

text: pair

text: x

text: and

text: y

text: are

text: linear

text: combinations

text: of

text: themselves

text: which

text: make

text: interpretation

text: of

text: the

text: extracted

text: features

text: diﬃcult

text: if

text: the

text: transformation

text: matrices

text: W

text: and

text: W

text: x

text: y

text: are

text: dense.

text: Similarly,

text: from

text: and

text: we

text: can

text: see

text: that

text: the

text: kernel

text: functions

text: κ

text: (x

text: x)

text: and

text: κ

text: (y

text: y)

text: must

text: W

text: x

text: i

text: W

text: y

text: i

text: be

text: evaluated

text: for

text: all

text: x

text: n

text: and

text: y

text: n

text: when

text: dual

text: transformation

text: matrices

text: and

text: are

text: dense,

text: which

text: i

text: i

text: x

text: y

text: can

text: lead

text: to

text: excessive

text: computational

text: time

text: to

text: compute

text: projections

text: of

text: new

text: data.

text: To

text: handle

text: the

text: limitation

text: of

text: CCA,

text: researchers

text: suggested

text: to

text: incorporate

text: sparsity

text: into

text: weight

text: vectors

text: and

text: many

text: papers

text: have

text: studied

text: sparse

text: CCA

text: Similarly,

text: we

text: shall

text: ﬁnd

text: sparse

text: solutions

text: for

text: kernel

text: CCA

text: so

text: that

text: projections

text: of

text: new

text: data

text: can

text: be

text: computed

text: by

text: evaluating

text: the

text: kernel

text: function

text: at

text: a

text: subset

text: of

text: the

text: training

text: data.

text: Although

text: there

text: are

text: many

text: sparse

text: kernel

text: approaches

text: such

text: as

text: support

text: vector

text: machines

text: relevance

text: vector

text: machine

text: and

text: sparse

text: kernel

text: partial

text: least

text: squares

text: seldom

text: can

text: be

text: found

text: in

text: the

text: area

text: of

text: sparse

text: kernel

text: CCA

text: In

text: this

text: paper

text: we

text: ﬁrst

text: consider

text: a

text: new

text: sparse

text: CCA

text: approach

text: and

text: then

text: generalize

text: it

text: to

text: incorporate

text: sparsity

text: into

text: kernel

text: CCA.

text: A

text: relationship

text: between

text: CCA

text: and

text: least

text: squares

text: is

text: established

text: so

text: that

text: CCA

text: solutions

text: can

text: be

text: obtained

text: by

text: solving

text: a

text: least

text: squares

text: problem.

text: We

text: attempt

text: to

text: introduce

text: sparsity

text: by

text: penal-

text: izing

text: -norm

text: of

text: the

text: solutions,

text: which

text: eventually

text: leads

text: to

text: a

text: -norm

text: penalized

text: least

text: squares

text: optimization

text: problem

text: of

text: the

text: form

text: min

text: Ax

text: b

text: λ

text: x

text: x∈Rd

text: where

text: λ

text: is

text: a

text: regularizer

text: controlling

text: the

text: sparsity

text: of

text: x.

text: We

text: adopt

text: a

text: ﬁxed-point

text: continuation

text: (FPC)

text: method

text: to

text: solve

text: the

text: -norm

text: regularized

text: least

text: squares

text: above,

text: which

text: results

text: in

text: a

text: new

text: sparse

text: CCA

text: ##LTLine##


text: [35] E. Parkhomenko, D. Tritchler, and J. Beyene. Sparse canonical correlation analysis with application to genomic data integration. Statistical Applications in Genetics and Molecular Biology, 8, 2009. Issue 1, Article 1. 3 [36] R. Rosipal and L. J. Trejo. Kernel partial least squares regression in reproducing kernel Hilbert space. Journal of Machine Learning Research, 2:97–123, 2001. 2 [37] B. Sch¨olkopf and A. J. Smola. Learning with kernels : support vector machines, regularization, optimization, and beyond. MIT Press, 2002. 2, 3, 6, 7 [38] B. Sch¨olkopf, A. J. Smola, and K.-R. Mu¨ller. Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation, 10:1299–1319, 1998. 2, 6, 7 [39] J. Shawe-Taylor and N. Cristianini. Kernel Methods for Pattern Analysis. Cambridge University Press, 2004. 2 [40] B. K. Sriperumbudur, D. A. Torres, and G. R. G. Lanckriet. Sparse eigen methods by d.c. program- ming. In The 24th International Conference on Machine Learning, pages 831–838, 2007. 3 [41] B. K. Sriperumbudur, D. A. Torres, and G. R. G. Lanckriet. A majorization-minimization approach to the sparse generalized eigenvalue problem. Machine Learning, 85(1-2):3–39, 2011. 3 [42] L. Sun, S. Ji, and J. Ye. Canonical correlation analysis for multilabel classiﬁcation: A least-squares formulation, extensions, and analysis. IEEE Transactions on Pattern Analysis and Machine Intelli- gence, 33(1):194–200, 2011. 7 [43] L. Tan and C. Fyfe. Sparse kernel canonical correlation analysis. In Proceedings of 9th European Symposium on Artiﬁcial Neural Networks, pages 335–340, 2001. 3 [44] R. Tibshirani. Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society (Series B), 58:267–288, 1996. 8, 14 [45] M. E. Tipping. Sparse bayesian learning and the relevance vector machine. Journal of Machine Learning Research, 1, 2001. 3 [46] J.-P. Vert and M. Kanehisa. Graph-driven features extraction from microarray data using diﬀusion kernels and kernel cca. In S. Becker, S. Thrun, and K. Obermayer, editors, Advances in neural information processing systems, volume 15. MIT Press, 2003. 3 [47] A. Vinokourov, J. Shawe-taylor, and N. Cristianini. Inferring a semantic representation of text via cross-language correlation analysis. In S. Becker, S. Thrun, and K. Obermayer, editors, Advances in neural information processing systems, volume 15, pages 1473–1480. MIT Press, 2003. 3, 17 [48] S. Waaijenborg, P. C. V. de Witt Hamer, and A. H. Zwinderman. Quantifying the association between gene expressions and dna-markers by penalized canonical correlation analysis. Statistical Applications in Genetics and Molecular Biology, 7, 2008. Issue 1, Article 3. 3 [49] G. Wahba. Support vector machines, reproducing kernel Hilbert spaces, and randomized GACV, − pages 69–88. Advances in kernel methods Support Vector Learning. MIT Press, 1999. 2 [50] A. Wiesel, M. Kliger, and A. O. Hero. A greedy approach to sparse canonical correlation analysis. 2008. Available at http://arxiv.org/abs/0801.2748. 3 [51] D. M. Witten and R. Tibshirani. Extensions of sparse canonical correlation analysis with applications to genomic data. Statistical Applications in Genetics and Molecular Biology, 8, 2009. Issue 1, Article 28. 3 [52] D. M. Witten, R. Tibshirani, and T. Hastie. A penalized matrix decomposition, with applications to sparse principal components and canonical correlation analysis. Biostatistics, 10(3):515–534, 2009. 3 24

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text: Neural

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text: Research,

text: J.-P.

text: Vert

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text: M.

text: Kanehisa.

text: Graph-driven

text: features

text: extraction

text: from

text: microarray

text: data

text: using

text: diﬀusion

text: kernels

text: and

text: kernel

text: cca.

text: In

text: S.

text: Becker,

text: S.

text: Thrun,

text: and

text: K.

text: Obermayer,

text: editors,

text: Advances

text: in

text: neural

text: information

text: processing

text: systems,

text: volume

text: MIT

text: Press,

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text: Vinokourov,

text: J.

text: Shawe-taylor,

text: and

text: N.

text: Cristianini.

text: Inferring

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text: semantic

text: representation

text: of

text: text

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text: cross-language

text: correlation

text: analysis.

text: In

text: S.

text: Becker,

text: S.

text: Thrun,

text: and

text: K.

text: Obermayer,

text: editors,

text: Advances

text: in

text: neural

text: information

text: processing

text: systems,

text: volume

text: pages

text: MIT

text: Press,

text: S.

text: Waaijenborg,

text: P.

text: C.

text: V.

text: de

text: Witt

text: Hamer,

text: and

text: A.

text: H.

text: Zwinderman.

text: Quantifying

text: the

text: association

text: between

text: gene

text: expressions

text: and

text: dna-markers

text: by

text: penalized

text: canonical

text: correlation

text: analysis.

text: Statistical

text: Applications

text: in

text: Genetics

text: and

text: Molecular

text: Biology,

text: Issue

text: Article

text: G.

text: Wahba.

text: Support

text: vector

text: machines,

text: reproducing

text: kernel

text: Hilbert

text: spaces,

text: and

text: randomized

text: GACV,

text: pages

text: Advances

text: in

text: kernel

text: methods

text: Support

text: Vector

text: Learning.

text: MIT

text: Press,

text: A.

text: Wiesel,

text: M.

text: Kliger,

text: and

text: A.

text: O.

text: Hero.

text: A

text: greedy

text: approach

text: to

text: sparse

text: canonical

text: correlation

text: analysis.

text: Available

text: at

text: D.

text: M.

text: Witten

text: and

text: R.

text: Tibshirani.

text: Extensions

text: of

text: sparse

text: canonical

text: correlation

text: analysis

text: with

text: applications

text: to

text: genomic

text: data.

text: Statistical

text: Applications

text: in

text: Genetics

text: and

text: Molecular

text: Biology,

text: Issue

text: Article

text: D.

text: M.

text: Witten,

text: R.

text: Tibshirani,

text: and

text: T.

text: Hastie.

text: A

text: penalized

text: matrix

text: decomposition,

text: with

text: applications

text: to

text: sparse

text: principal

text: components

text: and

text: canonical

text: correlation

text: analysis.

text: Biostatistics,


text: 2.2 Kernel canonical correlation analysis Now, we look at some details on the derivation of kernel CCA. Note from Theorem 2.1 that each solution (W , W ) of CCA can be expressed as x y W ⊥ W ⊥ W = X + W , W = Y + W , x x x y y y ⊥ ⊥ where W and W are orthogonal to the range space of X and Y , respectively. Since, intrinsically, kernel x y CCA is performing ordinary CCA on Φ and Φ , it follows that the solutions of kernel CCA should be x y obtained by virtually solving max Trace(WT Φ Φ W ) x x y y Wx,Wy ∈ N × s.t. WxT ΦxΦTx Wx = I, Wx ∈ RNx×l, (2.11) WyT ΦyΦTy Wy = I, Wy R y l, Similar to ordinary CCA, each solution (W , W ) of (2.11) shall be represented as x y W ⊥ W ⊥ W = Φ + W , W = Φ + W , (2.12) x x x x y y y y W W ∈ × ⊥ ⊥ where , Rn l are usually called dual transformation matrices, W and W are orthogonal x y x y to the range space of Φ and Φ , respectively. x y Substituting (2.12) into (2.11), we have W W W W W W WT Φ Φ W = T K K , WT Φ ΦT W = T K2 , WT Φ ΦT W = T K2 . x x y y x x y y x x x x x x x y y y y y y y Thus, the computation of transformations of kernel CCA can be converted to the computation of dual W W transformation matrices and by solving the following optimization problem x y W W max Trace( T K K ) W W x x y y x, y W W W ∈ × s.t. T K2 = I, Rn l, (2.13) Wx xWx Wx ∈ × T K2 = I, Rn l, y y y y which is used as the criterion of kernel CCA in this paper. (cid:2) (cid:3) ⊥ ⊥ As can be seen from the analysis above, terms W and W in (2.12) do not contribute to the x y canonical correlations between Φ and Φ , thus, are usually neglected in practice. Therefore, when we x y · · · (cid:2) (cid:3) are given a set of testing data X = x1 xN consisting of N points, the projection of X onto t t t H t kernel CCA direction W can be performed by ﬁrst mapping X into feature space , then compute its x t · · · x inner product with W . More speciﬁcally, suppose Φ = φ (x1) φ (xN) is the projection of X H x x,t x t x t t in feature space , then the projection of X onto kernel CCA direction W is given by x t x W WT Φ = T K , x x,t x x,t (cid:104) (cid:105) ∈ × where K = Φ , Φ = [κ (x , xj)]j=1:N Rn N is the matrix consisting of the kernel evaluations of x,t x x,t x i t i=1:n X with all training data X. Similar process can be adopted to compute projections of new data drawn t from variable y. In the process of deriving (2.13), we assumed data Φ and Φ have been centered (that is, the column x y mean of both Φ and Φ are zero), otherwise, we need to perform data centering before applying kernel x y CCA. Unlike data centering of X and Y , we can not perform data centering directly on Φ and Φ since x y we do not know their explicit coordinates. However, as shown in [38, 37], data centering in RKHS can be accomplished via some operations on kernel matrices. To center Φ , a natural idea should be computing − x Φ = Φ (I eneTn ), where e denotes column vector in Rn with all entries being 1. However, since x,c x n n kernel CCA makes use of the data through kernel matrix K , the centering process can be performed on x K as x (cid:104) (cid:105) − e eT (cid:104) (cid:105) − e eT − e eT − e eT K = Φ , Φ = (I n n ) Φ , Φ (I n n ) = (I n n )K (I n n ). (2.14) x,c x,c x,c x x x n n n n 6 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: section

text: Kernel

text: canonical

text: correlation

text: analysis

text: Now,

text: we

text: look

text: at

text: some

text: details

text: on

text: the

text: derivation

text: of

text: kernel

text: CCA.

text: Note

text: from

text: Theorem

text: that

text: each

text: solution

text: (W

text: W

text: of

text: CCA

text: can

text: be

text: expressed

text: as

text: x

text: y

text: W

text: W

text: W

text: X

text: W

text: W

text: Y

text: W

text: x

text: x

text: x

text: y

text: y

text: y

text: where

text: W

text: and

text: W

text: are

text: orthogonal

text: to

text: the

text: range

text: space

text: of

text: X

text: and

text: Y

text: respectively.

text: Since,

text: intrinsically,

text: kernel

text: x

text: y

text: CCA

text: is

text: performing

text: ordinary

text: CCA

text: on

text: Φ

text: and

text: Φ

text: it

text: follows

text: that

text: the

text: solutions

text: of

text: kernel

text: CCA

text: should

text: be

text: x

text: y

text: obtained

text: by

text: virtually

text: solving

text: max

text: Trace(WT

text: Φ

text: Φ

text: W

text: x

text: x

text: y

text: y

text: Wx,Wy

text: N

text: s.t.

text: WxT

text: ΦxΦTx

text: Wx

text: I,

text: Wx

text: RNx×l,

text: WyT

text: ΦyΦTy

text: Wy

text: I,

text: Wy

text: R

text: y

text: l,

text: Similar

text: to

text: ordinary

text: CCA,

text: each

text: solution

text: (W

text: W

text: of

text: shall

text: be

text: represented

text: as

text: x

text: y

text: W

text: W

text: W

text: Φ

text: W

text: W

text: Φ

text: W

text: x

text: x

text: x

text: x

text: y

text: y

text: y

text: y

text: W

text: W

text: where

text: Rn

text: l

text: are

text: usually

text: called

text: dual

text: transformation

text: matrices,

text: W

text: and

text: W

text: are

text: orthogonal

text: x

text: y

text: x

text: y

text: to

text: the

text: range

text: space

text: of

text: Φ

text: and

text: Φ

text: respectively.

text: x

text: y

text: Substituting

text: into

text: we

text: have

text: W

text: W

text: W

text: W

text: W

text: W

text: WT

text: Φ

text: Φ

text: W

text: T

text: K

text: K

text: WT

text: Φ

text: ΦT

text: W

text: T

text: WT

text: Φ

text: ΦT

text: W

text: T

text: x

text: x

text: y

text: y

text: x

text: x

text: y

text: y

text: x

text: x

text: x

text: x

text: x

text: x

text: x

text: y

text: y

text: y

text: y

text: y

text: y

text: y

text: Thus,

text: the

text: computation

text: of

text: transformations

text: of

text: kernel

text: CCA

text: can

text: be

text: converted

text: to

text: the

text: computation

text: of

text: dual

text: W

text: W

text: transformation

text: matrices

text: and

text: by

text: solving

text: the

text: following

text: optimization

text: problem

text: x

text: y

text: W

text: W

text: max

text: Trace(

text: T

text: K

text: K

text: W

text: W

text: x

text: x

text: y

text: y

text: x,

text: y

text: W

text: W

text: W

text: s.t.

text: T

text: I,

text: Rn

text: l,

text: Wx

text: xWx

text: Wx

text: T

text: I,

text: Rn

text: l,

text: y

text: y

text: y

text: y

text: which

text: is

text: used

text: as

text: the

text: criterion

text: of

text: kernel

text: CCA

text: in

text: this

text: paper.

text: As

text: can

text: be

text: seen

text: from

text: the

text: analysis

text: above,

text: terms

text: W

text: and

text: W

text: in

text: do

text: not

text: contribute

text: to

text: the

text: x

text: y

text: canonical

text: correlations

text: between

text: Φ

text: and

text: Φ

text: thus,

text: are

text: usually

text: neglected

text: in

text: practice.

text: Therefore,

text: when

text: we

text: x

text: y

text: are

text: given

text: a

text: set

text: of

text: testing

text: data

text: X

text: xN

text: consisting

text: of

text: N

text: points,

text: the

text: projection

text: of

text: X

text: onto

text: t

text: t

text: t

text: H

text: t

text: kernel

text: CCA

text: direction

text: W

text: can

text: be

text: performed

text: by

text: ﬁrst

text: mapping

text: X

text: into

text: feature

text: space

text: then

text: compute

text: its

text: x

text: t

text: x

text: inner

text: product

text: with

text: W

text: More

text: speciﬁcally,

text: suppose

text: Φ

text: φ

text: φ

text: (xN)

text: is

text: the

text: projection

text: of

text: X

text: H

text: x

text: x,t

text: x

text: t

text: x

text: t

text: t

text: in

text: feature

text: space

text: then

text: the

text: projection

text: of

text: X

text: onto

text: kernel

text: CCA

text: direction

text: W

text: is

text: given

text: by

text: x

text: t

text: x

text: W

text: WT

text: Φ

text: T

text: K

text: x

text: x,t

text: x

text: x,t

text: where

text: K

text: Φ

text: Φ

text: [κ

text: (x

text: Rn

text: N

text: is

text: the

text: matrix

text: consisting

text: of

text: the

text: kernel

text: evaluations

text: of

text: x,t

text: x

text: x,t

text: x

text: i

text: t

text: X

text: with

text: all

text: training

text: data

text: X.

text: Similar

text: process

text: can

text: be

text: adopted

text: to

text: compute

text: projections

text: of

text: new

text: data

text: drawn

text: t

text: from

text: variable

text: y.

text: In

text: the

text: process

text: of

text: deriving

text: we

text: assumed

text: data

text: Φ

text: and

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text: Draft version May 29, 2020 Preprint typeset using LATEX style emulateapj v. 12/16/11 SOLAR ELLERMAN BOMBS IN 1-D RADIATIVE HYDRODYNAMICS A. Reid1,4, M. Mathioudakis1, A. Kowalski2,3, J. G. Doyle4, J. C. Allred5 0 1. Astrophysics Research Centre, School of Mathematics and Physics, Queen’s University Belfast, BT7 1NN, Northern Ireland, UK; e-mail: aaron.reid@qub.ac.uk 2. Department of Astrophysical and Planetary Sciences, University of Colorado Boulder, 2000 Colorado Ave, Boulder, CO 80305, USA. 3. National Solar Observatory, University of Colorado Boulder, 3665 Discovery Drive, Boulder, CO 80303, USA 4. Armagh Observatory and Planetarium, College Hill, Armagh, BT61 9DG, UK 5. NASA/Goddard Space Flight Center, Code 671, Greenbelt, MD 20771 Draft version May 29, 2020 ABSTRACT Recent observations from the Interface Region Imaging Spectrograph (IRIS) appear to show impulsive bright- enings in high temperature lines, which when combined with simultaneous ground based observations in Hα, appear co-spatial to Ellerman Bombs (EBs). We use the RADYN 1-dimensional radiative transfer code in an attempt to try and reproduce the observed line proﬁles and simulate the atmospheric conditions of these events. Combined with the MULTI/RH line synthesis codes, we compute the Hα, Ca II 8542 Å, and Mg II h & k lines for these simulated events and compare them to previous observations. Our ﬁndings hint that the presence of superheated regions in the photosphere (>10,000 K) is not a plausible explanation for the production of EB signatures. While we are able to recreate EB-like line proﬁles in Hα, Ca II 8542 Å, and Mg II h & k, we cannot achieve agreement with all of these simultaneously. Keywords: Sun: Activity — Hydrodynamics — Sun: Photosphere 1. INTRODUCTION Matsumoto et al. 2008b; Hashimoto et al. 2010; Nelson et al. 2013; Vissers et al. 2013). Recent spectropolarimetric Ellerman Bombs (EBs henceforth) were ﬁrst noticed in inversions of these events show evidence for ﬂux cancella- Ellerman (1917), who noted brightenings in the wings of the tion (Reid et al. 2016), with magnetic energies comparable to Hα, Hβ, and Hγ lines. These events have also been observed that of the radiative energy losses. EBs have been estimated in the wings of Ca II 8542 Å (Fang et al. 2006; Socas-Navarro to form in the temperature minimum region (Nelson et al. et al. 2006; Pariat et al. 2007; Vissers et al. 2015; Li et al. 2015), with foot-points reported to form as low as 300 km 2015), as well as in Ca II H images (Matsumoto et al. 2008a; (Watanabe et al. 2011). As such, they are thought to appear Hashimoto et al. 2010). They show no observable enhance- due to photospheric magnetic reconnection, occurring around ment in the core of these line proﬁles as these are formed in the temperature minimum region, where this process is most the overlying chromospheric canopy. EBs are considered a eﬃcient (Litvinenko 1999). solely photospheric/lower chromospheric phenomenon (Vis- Three-dimensional numerical modeling of photospheric sers et al. 2013). reconnection has shown local temperature increases in the EB brightenings are also observable in the Solar Dynamics photosphere by a factor of 1.1-1.5 relative to quiet Sun, along Observatory (SDO) 1700 Å and 1600 Å channels (Qiu et al. with a density increase by a factor of 4 at the magnetic in- 2000; Georgoulis et al. 2002; Pariat et al. 2007; Berlicki et version line (Archontis & Hood 2009). The Archontis & al. 2010; Vissers et al. 2013), though to a lesser degree than Hood (2009) model has also shown bi-directional ﬂows in the Hα line wings due to the broad passbands of these ﬁlters the region, with values of 2-4 km s−1. Semi-empirical models encompassing a wide range of atmospheric heights (Vissers for EBs show localized temperature enhancements of 600- et al. 2013). While the 1600 Å channel oﬀers better con- 3000 K around the temperature minimum region (Fang et trast than the 1700 Å channel (Rutten et al. 2013; Vissers et al. 2006; Berlicki & Heinzel 2014). These temperature en- al. 2013), EB signatures are more diﬃcult to observe in the hancements lead to intensity enhancements in the wings of 1600 Å channel due to contamination eﬀects from C IV emis- the Hα and Ca II 8542 Å lines. Other EB studies also ﬁnd sion with transition region temperatures. similar temperature enhancements (200-3000K) in the photo- EBs typically last for a few minutes, and appear rather sphere/temperature minimum region (Georgoulis et al. 2002; impulsively in comparison with other photospheric brighten- Isobe et al. 2007; Yang et al. 2013; Hong et al. 2014; Li et al. ings such as moving magnetic features (MMFs). Reid et al. 2015). (2016) have shown that by introducing an impulsivity crite- EBs have also been found within quiet Sun regions of the rion, it is possible to distinguish between pseudo-EBs and photosphere (Rouppe van der voort et al. 2016). The obser- EBs. EBs are generally observed with co-spatial blue-shifts, vational signatures of these QSEBs are not as pronounced as or bi-directional Doppler shifts (Matsumoto et al. 2008a; those found in the vicinity of active regions, though they are Watanabe et al. 2011). still located in regions of opposite polarity magnetic ﬂux. This EBs are generally found in regions of opposite polarity suggests that these events are identical to EBs occurring at lo- magnetic ﬂux (Georgoulis et al. 2002; Watanabe et al. 2008; cations of weaker magnetic ﬂux cancellation. More recently, observations using the IRIS instrument indi-

text: Draft

font: SJGRZE+rtxsc

text: version

text: May

text: Preprint

text: typeset

text: using

text: LATEX

text: style

text: emulateapj

text: v.

text: SOLAR

text: ELLERMAN

text: BOMBS

text: IN

text: RADIATIVE

text: HYDRODYNAMICS

text: A.

text: M.

text: A.

text: J.

text: G.

text: J.

text: C.

text: Astrophysics

text: Research

text: Centre,

text: School

text: of

text: Mathematics

text: and

text: Physics,

text: Queen’s

text: University

text: Belfast,

text: Northern

text: Ireland,

text: UK;

text: e-mail:

text: aaron.reid@qub.ac.uk

text: Department

text: of

text: Astrophysical

text: and

text: Planetary

text: Sciences,

text: University

text: of

text: Colorado

text: Boulder,

text: Colorado

text: Ave,

text: Boulder,

text: CO

text: USA.

text: National

text: Solar

text: Observatory,

text: University

text: of

text: Colorado

text: Boulder,

text: Discovery

text: Drive,

text: Boulder,

text: CO

text: USA

text: Armagh

text: Observatory

text: and

text: Planetarium,

text: College

text: Hill,

text: Armagh,

text: UK

text: NASA/Goddard

text: Space

text: Flight

text: Center,

text: Code

text: Greenbelt,

text: MD

text: Draft

text: version

text: May

text: ABSTRACT

text: Recent

text: observations

text: from

text: the

text: Interface

text: Region

text: Imaging

text: Spectrograph

text: (IRIS)

text: appear

text: to

text: show

text: impulsive

text: bright-

text: enings

text: in

text: high

text: temperature

text: lines,

text: which

text: when

text: combined

text: with

text: simultaneous

text: ground

text: based

text: observations

text: in

text: Hα,

text: appear

text: co-spatial

text: to

text: Ellerman

text: Bombs

text: (EBs).

text: We

text: use

text: the

text: RADYN

text: radiative

text: transfer

text: code

text: in

text: an

text: attempt

text: to

text: try

text: and

text: reproduce

text: the

text: observed

text: line

text: proﬁles

text: and

text: simulate

text: the

text: atmospheric

text: conditions

text: of

text: these

text: events.

text: Combined

text: with

text: the

text: MULTI/RH

text: line

text: synthesis

text: codes,

text: we

text: compute

text: the

text: Hα,

text: Ca

text: II

text: Å,

text: and

text: Mg

text: II

text: h

text: k

text: lines

text: for

text: these

text: simulated

text: events

text: and

text: compare

text: them

text: to

text: previous

text: observations.

text: Our

text: ﬁndings

text: hint

text: that

text: the

text: presence

text: of

text: superheated

text: regions

text: in

text: the

text: photosphere

text: K)

text: is

text: not

text: a

text: plausible

text: explanation

text: for

text: the

text: production

text: of

text: EB

text: signatures.

text: While

text: we

text: are

text: able

text: to

text: recreate

text: EB-like

text: line

text: proﬁles

text: in

text: Hα,

text: Ca

text: II

text: Å,

text: and

text: Mg

text: II

text: h

text: k,

text: we

text: cannot

text: achieve

text: agreement

text: with

text: all

text: of

text: these

text: simultaneously.

text: Keywords:

text: Sun:

text: Activity

text: Hydrodynamics

text: Sun:

text: Photosphere

text: INTRODUCTION

text: Matsumoto

text: et

text: al.

text: Hashimoto

text: et

text: al.

text: Nelson

text: et

text: al.

text: Vissers

text: et

text: al.

text: Recent

text: spectropolarimetric

text: Ellerman

text: Bombs

text: (EBs

text: henceforth)

text: were

text: ﬁrst

text: noticed

text: in

text: inversions

text: of

text: these

text: events

text: show

text: evidence

text: for

text: ﬂux

text: cancella-

text: Ellerman

text: who

text: noted

text: brightenings

text: in

text: the

text: wings

text: of

text: the

text: tion

text: (Reid

text: et

text: al.

text: with

text: magnetic

text: energies

text: comparable

text: to

text: Hα,

text: Hβ,

text: and

text: Hγ

text: lines.

text: These

text: events

text: have

text: also

text: been

text: observed

text: that

text: of

text: the

text: radiative

text: energy

text: losses.

text: EBs

text: have

text: been

text: estimated

text: in

text: the

text: wings

text: of

text: Ca

text: II

text: Å

text: (Fang

text: et

text: al.

text: Socas-Navarro

text: to

text: form

text: in

text: the

text: temperature

text: minimum

text: region

text: (Nelson

text: et

text: al.

text: et

text: al.

text: Pariat

text: et

text: al.

text: Vissers

text: et

text: al.

text: Li

text: et

text: al.

text: with

text: foot-points

text: reported

text: to

text: form

text: as

text: low

text: as

text: km

text: as

text: well

text: as

text: in

text: Ca

text: II

text: H

text: images

text: (Matsumoto

text: et

text: al.

text: (Watanabe

text: et

text: al.

text: As

text: such,

text: they

text: are

text: thought

text: to

text: appear

text: Hashimoto

text: et

text: al.

text: They

text: show

text: no

text: observable

text: enhance-

text: due

text: to

text: photospheric

text: magnetic

text: reconnection,

text: occurring

text: around

text: ment

text: in

text: the

text: core

text: of

text: these

text: line

text: proﬁles

text: as

text: these

text: are

text: formed

text: in

text: the

text: temperature

text: minimum

text: region,

text: where

text: this

text: process

text: is

text: most

text: the

text: overlying

text: chromospheric

text: canopy.

text: EBs

text: are

text: considered

text: a

text: eﬃcient

text: (Litvinenko

text: solely

text: photospheric/lower

text: chromospheric

text: phenomenon

text: (Vis-

text: Three-dimensional

text: numerical

text: modeling

text: of

text: photospheric

text: sers

text: et

text: al.

text: reconnection

text: has

text: shown

text: local

text: temperature

text: increases

text: in

text: the

text: EB

text: brightenings

text: are

text: also

text: observable

text: in

text: the

text: Solar

text: Dynamics

text: photosphere

text: by

text: a

text: factor

text: of

text: relative

text: to

text: quiet

text: Sun,

text: along

text: Observatory

text: (SDO)

text: Å

text: and

text: Å

text: channels

text: (Qiu

text: et

text: al.

text: with

text: a

text: density

text: increase

text: by

text: a

text: factor

text: of

text: at

text: the

text: magnetic

text: in-

text: Georgoulis

text: et

text: al.

text: Pariat

text: et

text: al.

text: Berlicki

text: et

text: version

text: line

text: (Archontis

text: Hood

text: The

text: Archontis

text: al.

text: Vissers

text: et

text: al.

text: though

text: to

text: a

text: lesser

text: degree

text: than

text: Hood

text: model

text: has

text: also

text: shown

text: bi-directional

text: ﬂows

text: in

text: the

text: Hα

text: line

text: wings

text: due

text: to

text: the

text: broad

text: passbands

text: of

text: these

text: ﬁlters

text: the

text: region,

text: with

text: values

text: of

text: km

text: Semi-empirical

text: models

text: encompassing

text: a

text: wide

text: range

text: of

text: atmospheric

text: heights

text: (Vissers

text: for

text: EBs

text: show

text: localized

text: temperature

text: enhancements

text: of

text: et

text: al.

text: While

text: the

text: Å

text: channel

text: oﬀers

text: better

text: con-

text: K

text: around

text: the

text: temperature

text: minimum

text: region

text: (Fang

text: et

text: trast

text: than

text: the

text: Å

text: channel

text: (Rutten

text: et

text: al.

text: Vissers

text: et

text: al.

text: Berlicki

text: Heinzel

text: These

text: temperature

text: en-

text: al.

text: EB

text: signatures

text: are

text: more

text: diﬃcult

text: to

text: observe

text: in

text: the

text: hancements

text: lead

text: to

text: intensity

text: enhancements

text: in

text: the

text: wings

text: of

text: Å

text: channel

text: due

text: to

text: contamination

text: eﬀects

text: from

text: C

text: IV

text: emis-

text: the

text: Hα

text: and

text: Ca

text: II

text: Å

text: lines.

text: Other

text: EB

text: studies

text: also

text: ﬁnd

text: sion

text: with

text: transition

text: region

text: temperatures.

text: similar

text: temperature

text: enhancements

text: in

text: the

text: photo-

text: EBs

text: typically

text: last

text: for

text: a

text: few

text: minutes,

text: and

text: appear

text: rather

text: sphere/temperature

text: minimum

text: region

text: (Georgoulis

text: et

text: al.

text: impulsively

text: in

text: comparison

text: with

text: other

text: photospheric

text: brighten-

text: Isobe

text: et

text: al.

text: Yang

text: et

text: al.

text: Hong

text: et

text: al.

text: Li

text: et

text: al.

text: ings

text: such

text: as

text: moving

text: magnetic

text: features

text: (MMFs).

text: Reid

text: et

text: al.

text: have

text: shown

text: that

text: by

text: introducing

text: an

text: impulsivity

text: crite-

text: EBs

text: have

text: also

text: been

text: found

text: within

text: quiet

text: Sun

text: regions

text: of

text: the

text: rion,

text: it

text: is

text: possible

text: to

text: distinguish

text: between

text: pseudo-EBs

text: and

text: photosphere

text: (Rouppe

text: van

text: der

text: voort

text: et

text: al.

text: The

text: obser-

text: EBs.

text: EBs

text: are

text: generally

text: observed

text: with

text: co-spatial

text: blue-shifts,

text: vational

text: signatures

text: of

text: these

text: QSEBs

text: are

text: not

text: as

text: pronounced

text: as

text: or

text: bi-directional

text: Doppler

text: shifts

text: (Matsumoto

text: et

text: al.

text: those

text: found

text: in

text: the

text: vicinity

text: of

text: active

text: regions,

text: though

text: they

text: are

text: Watanabe

text: et

text: al.

text: still

text: located

text: in

text: regions

text: of

text: opposite

text: polarity

text: magnetic

text: ﬂux.

text: This

text: EBs

text: are

text: generally

text: found

text: in

text: regions

text: of

text: opposite

text: polarity

text: suggests

text: that

text: these

text: events

text: are

text: identical

text: to

text: EBs

text: occurring

text: at

text: lo-

text: magnetic

text: ﬂux

text: (Georgoulis

text: et

text: al.

text: Watanabe

text: et

text: al.

text: cations

text: of

text: weaker

text: magnetic

text: ﬂux

text: cancellation.

text: More

text: recently,

text: observations

text: using

text: the

text: IRIS

text: instrument

text: indi-