text: 6 the potential well; second, the potential well can be con- C. Benchmark 2: Negative long feature sidered as a scattering center, such that a similar amount of energy is transferred to the χ direction (as evidenced In this subsection we consider a negative long feature. in the right panel of Fig. 2). As a result, there is a overall Inside a “long” feature, there is enough time for the χ loss of velocity of the inﬂaton, with the value read from − × − ﬁeld to oscillate. We set A = 1.4 10 5V µ4/φ4∗, σ = the plot (cid:12) (cid:12) (cid:12) (cid:12) 0 φ (cid:12) (cid:12) (cid:12) (cid:12) 50H∗/(2π), σ = 0.7H∗/(2π) and χ = 5.2H∗/(2π). (cid:12) (cid:12) (cid:12) (cid:12) χ 0 (cid:12) (cid:12) (cid:12) (cid:12) Note that the feature size in the φ direction is 5 times dφ ∼ − × − ∼ − − dφ longer than that of Benchmark 1. δ 4 10 7 10 3 . (6) dN dN Finer structures develop in the case of long feature. − If we have chosen an initial value χ∗ χ = H∗/(2π), 0 In words, the inﬂaton ﬁeld has lost 0.1% of its kinetic we ﬁnd similar behavior to Fig. 2. −However, for some energy because of hitting the feature in the potential 2. other initial values, for example, χ∗ χ0 = 1.4H∗/(2π), Note that the inﬂaton always loses its kinetic energy be- the χ ﬁeld oscillates inside the feature. The situation cause of scattering oﬀ a feature. Thus a cold spot instead is illustrated in Fig. 6. As a result, some amount of χ of a hot spot is predicted 3. kinetic energy is returned to φ. For some special values, Note that the loss of kinetic energy takes of order one almost all kinetic energy is returned. Thus oscillatory e-fold to recover, because this is the time scale to reach patterns develop, as shown in the middle panel of Fig. 3. the inﬂationary attractor solution. To be more precise, The oscillations in δT(χ) leads to ring objects in δT(x). as shown in the left panel of Fig. 2, it takes about roughly The cold spot in this parameter space is not completely 0.2 e-fold for the inﬂaton to recover its kinetic energy. As cold, but instead has nested rings of cold and hot pat- a result 4, (cid:12)(cid:12) (cid:12)(cid:12) terns. There is no evidence of such patterns in the actual (cid:12) (cid:12) CMB cold spot. There has been a debate if there are (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) × (cid:12) (cid:12) other ring patterns in the sky [36]. The Benchmark 2 pa- (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 0.2(cid:12) d(cid:12)φ (cid:12) δN = δφ = dN ∼ 2 × 10−4 . (7) rameters provide an explanation for such rings (though the shape is not perfectly spherical). However, it is shown dφ + δ dφ dφ + δ dφ dN dN dN dN that the appearance of such rings is due to an inappro- priately chosen power spectrum, and thus not actually in From the Sachs-(cid:39)Wo−lfe appr−oximation, the temperatur(cid:39)e the sky [37]. Thus we will not tune the parameters to ﬁt ﬂ−uctuation δT/T ζ/5 = δN/5. Thus we get δT such features here. 100µK. The general and best-ﬁtting cold-spot proﬁles, and Carrying on the above analysis for general values of − sample spots are plotted in the middle row of Figs. 4 χ∗ χ , numerically the temperature ﬂuctuation as a 0 − and 5, respectively. function of χ∗ χ is plotted in the left panel of Fig. 3. 0 D. Benchmark 3: Positive feature We then simulate 1000 maps and rotate those maps such that they center at the cold spot. In the left panel of the top row of Fig. 4, we bin the pixels of the map as a func- In this subsection we consider positive feature scatter- × − tion of θ, and then plot the 0.5, 0.16 and 0.84 quantiles ing. We set A = 1.0 10 4V µ4/φ4∗, σ = 50H∗/(2π), 0 φ (corresponding to the central value and 1σ uncertainty if σ = 0.5H∗/(2π) and χ = 3.5H∗/(2π). With positive χ 0 the probability distribution were Gaussian) of the binned potential, the χ direction has a run-away solution and pixel temperature. In the right panel of the top row of can never be bounded. So there is no oscillation in this Fig. 4, we plot the top ﬁve best-ﬁtting cold spot proﬁles. case. Three best-ﬁtting examples are given in the top row of The δT(χ) dependence is not as sharp as the previous Fig. 5. two benchmarks. Thus the cold spot does not have as clear a boundary as those previous cases. Because of the not-so-sharp δT(χ) dependence, to get a cold enough center of the spot, we have to increase A. As a result, the 2 Here we do not need to consider the change of potential energy cold spots generated by Benchmark 3 typically (though before/after the feature scattering, because we only need to com- not always) have very cold tinny cores. pare the change of energy before and after a very sharp scattering process, and the change of potential energy is negligible. The general and best-ﬁtting cold-spot proﬁles, and 3 When the potential is positive, the inﬂaton actually gains kinetic sample spots are plotted in the bottom row of Figs. 4 energy at the very beginning, when it falls into the potential well. and 5, respectively. However, the gained energy is returned after a time interval that is much smaller than Hubble time. Thus the eﬀect of gaining Considering a very sharp and cold core is formed for the kinetic energy is negligible when we integrate to obtain the positive features, it is unlikely that such features can ﬁt e-folding number. On the other hand, the loss of kinetic energy the realistic cold spot. But considering that the sharp- through feature scattering needs order of 1 e-fold to recover, and ness of the cold core depends on model parameters. The thus it is the dominate eﬀect. 4 The δN here is from the quantum ﬂuctuation of χ. The part cold core may appear at (cid:96) > 2000 scales and free stream- from φ will be added at the map-making stage. ing may erase the sharpness. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: the

text: potential

text: well;

text: second,

text: the

text: potential

text: well

text: can

text: be

text: con-

text: C.

text: Benchmark

text: Negative

text: long

text: feature

text: sidered

text: as

text: a

text: scattering

text: center,

text: such

text: that

text: a

text: similar

text: amount

text: of

text: energy

text: is

text: transferred

text: to

text: the

text: χ

text: direction

text: (as

text: evidenced

text: In

text: this

text: subsection

text: we

text: consider

text: a

text: negative

text: long

text: feature.

text: in

text: the

text: right

text: panel

text: of

text: Fig.

text: As

text: a

text: result,

text: there

text: is

text: a

text: overall

text: Inside

text: a

text: “long”

text: feature,

text: there

text: is

text: enough

text: time

text: for

text: the

text: χ

text: loss

text: of

text: velocity

text: of

text: the

text: inﬂaton,

text: with

text: the

text: value

text: read

text: from

text: ﬁeld

text: to

text: oscillate.

text: We

text: set

text: A

text: σ

text: the

text: plot

text: φ

text: σ

text: and

text: χ

text: χ

text: Note

text: that

text: the

text: feature

text: size

text: in

text: the

text: φ

text: direction

text: is

text: times

text: dφ

text: dφ

text: longer

text: than

text: that

text: of

text: Benchmark

text: δ

text: dN

text: dN

text: Finer

text: structures

text: develop

text: in

text: the

text: case

text: of

text: long

text: feature.

text: If

text: we

text: have

text: chosen

text: an

text: initial

text: value

text: χ∗

text: χ

text: In

text: words,

text: the

text: inﬂaton

text: ﬁeld

text: has

text: lost

text: of

text: its

text: kinetic

text: we

text: ﬁnd

text: similar

text: behavior

text: to

text: Fig.

text: −However,

text: for

text: some

text: energy

text: because

text: of

text: hitting

text: the

text: feature

text: in

text: the

text: potential

text: other

text: initial

text: values,

text: for

text: example,

text: χ∗

text: χ0

text: Note

text: that

text: the

text: inﬂaton

text: always

text: loses

text: its

text: kinetic

text: energy

text: be-

text: the

text: χ

text: ﬁeld

text: oscillates

text: inside

text: the

text: feature.

text: The

text: situation

text: cause

text: of

text: scattering

text: oﬀ

text: a

text: feature.

text: Thus

text: a

text: cold

text: spot

text: instead

text: is

text: illustrated

text: in

text: Fig.

text: As

text: a

text: result,

text: some

text: amount

text: of

text: χ

text: of

text: a

text: hot

text: spot

text: is

text: predicted

text: kinetic

text: energy

text: is

text: returned

text: to

text: φ.

text: For

text: some

text: special

text: values,

text: Note

text: that

text: the

text: loss

text: of

text: kinetic

text: energy

text: takes

text: of

text: order

text: one

text: almost

text: all

text: kinetic

text: energy

text: is

text: returned.

text: Thus

text: oscillatory

text: e-fold

text: to

text: recover,

text: because

text: this

text: is

text: the

text: time

text: scale

text: to

text: reach

text: patterns

text: develop,

text: as

text: shown

text: in

text: the

text: middle

text: panel

text: of

text: Fig.

text: the

text: inﬂationary

text: attractor

text: solution.

text: To

text: be

text: more

text: precise,

text: The

text: oscillations

text: in

text: δT(χ)

text: leads

text: to

text: ring

text: objects

text: in

text: δT(x).

text: as

text: shown

text: in

text: the

text: left

text: panel

text: of

text: Fig.

text: it

text: takes

text: about

text: roughly

text: The

text: cold

text: spot

text: in

text: this

text: parameter

text: space

text: is

text: not

text: completely

text: e-fold

text: for

text: the

text: inﬂaton

text: to

text: recover

text: its

text: kinetic

text: energy.

text: As

text: cold,

text: but

text: instead

text: has

text: nested

text: rings

text: of

text: cold

text: and

text: hot

text: pat-

text: a

text: result

text: terns.

text: There

text: is

text: no

text: evidence

text: of

text: such

text: patterns

text: in

text: the

text: actual

text: CMB

text: cold

text: spot.

text: There

text: has

text: been

text: a

text: debate

text: if

text: there

text: are

text: other

text: ring

text: patterns

text: in

text: the

text: sky

text: The

text: Benchmark

text: pa-

text: δN

text: δφ

text: dN

text: rameters

text: provide

text: an

text: explanation

text: for

text: such

text: rings

text: (though

text: the

text: shape

text: is

text: not

text: perfectly

text: spherical).

text: However,

text: it

text: is

text: shown

text: dφ

text: δ

text: dφ

text: dφ

text: δ

text: dφ

text: dN

text: dN

text: dN

text: dN

text: that

text: the

text: appearance

text: of

text: such

text: rings

text: is

text: due

text: to

text: an

text: inappro-

text: priately

text: chosen

text: power

text: spectrum,

text: and

text: thus

text: not

text: actually

text: in

text: From

text: the

text: appr−oximation,

text: the

text: the

text: sky

text: Thus

text: we

text: will

text: not

text: tune

text: the

text: parameters

text: to

text: ﬁt

text: ﬂ−uctuation

text: δT/T

text: Thus

text: we

text: get

text: δT

text: such

text: features

text: here.

text: The

text: general

text: and

text: best-ﬁtting

text: cold-spot

text: proﬁles,

text: and

text: Carrying

text: on

text: the

text: above

text: analysis

text: for

text: general

text: values

text: of

text: sample

text: spots

text: are

text: plotted

text: in

text: the

text: middle

text: row

text: of

text: Figs.

text: χ∗

text: χ

text: numerically

text: the

text: temperature

text: ﬂuctuation

text: as

text: a

text: and

text: respectively.

text: function

text: of

text: χ∗

text: χ

text: is

text: plotted

text: in

text: the

text: left

text: panel

text: of

text: Fig.

text: D.

text: Benchmark

text: Positive

text: feature

text: We

text: then

text: simulate

text: maps

text: and

text: rotate

text: those

text: maps

text: such

text: that

text: they

text: center

text: at

text: the

text: cold

text: spot.

text: In

text: the

text: left

text: panel

text: of

text: the

text: top

text: row

text: of

text: Fig.

text: we

text: bin

text: the

text: pixels

text: of

text: the

text: map

text: as

text: a

text: func-

text: In

text: this

text: subsection

text: we

text: consider

text: positive

text: feature

text: scatter-

text: tion

text: of

text: θ,

text: and

text: then

text: plot

text: the

text: and

text: quantiles

text: ing.

text: We

text: set

text: A

text: σ

text: φ

text: (corresponding

text: to

text: the

text: central

text: value

text: and

text: 1σ

text: uncertainty

text: if

text: σ

text: and

text: χ

text: With

text: positive

text: χ

text: the

text: probability

text: distribution

text: were

text: Gaussian)

text: of

text: the

text: binned

text: potential,

text: the

text: χ

text: direction

text: has

text: a

text: run-away

text: solution

text: and

text: pixel

text: temperature.

text: In

text: the

text: right

text: panel

text: of

text: the

text: top

text: row

text: of

text: can

text: never

text: be

text: bounded.

text: So

text: there

text: is

text: no

text: oscillation

text: in

text: this

text: Fig.

text: we

text: plot

text: the

text: top

text: ﬁve

text: best-ﬁtting

text: cold

text: spot

text: proﬁles.

text: case.

text: Three

text: best-ﬁtting

text: examples

text: are

text: given

text: in

text: the

text: top

text: row

text: of

text: The

text: δT(χ)

text: dependence

text: is

text: not

text: as

text: sharp

text: as

text: the

text: previous

text: Fig.

text: two

text: benchmarks.

text: Thus

text: the

text: cold

text: spot

text: does

text: not

text: have

text: as

text: clear

text: a

text: boundary

text: as

text: those

text: previous

text: cases.

text: Because

text: of

text: the

text: not-so-sharp

text: δT(χ)

text: dependence,

text: to

text: get

text: a

text: cold

text: enough

text: center

text: of

text: the

text: spot,

text: we

text: have

text: to

text: increase

text: A.

text: As

text: a

text: result,

text: the

text: Here

text: we

text: do

text: not

text: need

text: to

text: consider

text: the

text: change

text: of

text: potential

text: energy

text: cold

text: spots

text: generated

text: by

text: Benchmark

text: typically

text: (though

text: before/after

text: the

text: feature

text: scattering,

text: because

text: we

text: only

text: need

text: to

text: com-

text: not

text: always)

text: have

text: very

text: cold

text: tinny

text: cores.

text: pare

text: the

text: change

text: of

text: energy

text: before

text: and

text: after

text: a

text: very

text: sharp

text: scattering

text: process,

text: and

text: the

text: change

text: of

text: potential

text: energy

text: is

text: negligible.

text: The

text: general

text: and

text: best-ﬁtting

text: cold-spot

text: proﬁles,

text: and

text: When

text: the

text: potential

text: is

text: positive,

text: the

text: inﬂaton

text: actually

text: gains

text: kinetic

text: sample

text: spots

text: are

text: plotted

text: in

text: the

text: bottom

text: row

text: of

text: Figs.

text: energy

text: at

text: the

text: very

text: beginning,

text: when

text: it

text: falls

text: into

text: the

text: potential

text: well.

text: and

text: respectively.

text: However,

text: the

text: gained

text: energy

text: is

text: returned

text: after

text: a

text: time

text: interval

text: that

text: is

text: much

text: smaller

text: than

text: Hubble

text: time.

text: Thus

text: the

text: eﬀect

text: of

text: gaining

text: Considering

text: a

text: very

text: sharp

text: and

text: cold

text: core

text: is

text: formed

text: for

text: the

text: kinetic

text: energy

text: is

text: negligible

text: when

text: we

text: integrate

text: to

text: obtain

text: the

text: positive

text: features,

text: it

text: is

text: unlikely

text: that

text: such

text: features

text: can

text: ﬁt

text: e-folding

text: number.

text: On

text: the

text: other

text: hand,

text: the

text: loss

text: of

text: kinetic

text: energy

text: the

text: realistic

text: cold

text: spot.

text: But

text: considering

text: that

text: the

text: sharp-

text: through

text: feature

text: scattering

text: needs

text: order

text: of

text: e-fold

text: to

text: recover,

text: and

text: ness

text: of

text: the

text: cold

text: core

text: depends

text: on

text: model

text: parameters.

text: The

text: thus

text: it

text: is

text: the

text: dominate

text: eﬀect.

text: The

text: δN

text: here

text: is

text: from

text: the

text: quantum

text: ﬂuctuation

text: of

text: χ.

text: The

text: part

text: cold

text: core

text: may

text: appear

text: at

text: scales

text: and

text: free

text: stream-

text: from

text: φ

text: will

text: be

text: added

text: at

text: the

text: map-making

text: stage.

text: ing

text: may

text: erase

text: the

text: sharpness.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 8 FIG. 8: The same as Fig. 5 but for model m = 0.9H∗. χ As one can see from Fig. 8, there are more small-scale formalism. But in the in-in formalism, the primary structures compared to the case of massless Benchmark calculatables are the correlation functions. In other 1. words, one starts from the Gaussian two point cor- IV. CONCLUSION AND DISCUSSION relation function and studies small departure from that. But the cold spot is highly non-Gaussian and localized object, and thus is not easily captured by To conclude, we propose a scattering mechanism of in- the correlation functions from the in-in formalism. ﬂaton due to the features in the inﬂation potential, dur- ing which the isocurvature ﬂuctuations are converted into • Extra species/symmetry point (ESP) [38, 39] as curvature ﬂuctuations. The curvature ﬂuctuations direc- scattering centers. In our current examples, the tion loses kinetic energy due to the scatters in isocurva- inﬂaton kinetic energy is lost into the collective mo- ture direction, therefore the number of e-folds becomes tion of the isocurvature direction. It may also be larger in some region of the universe. We ﬁnd that the possible that the kinetic energy of the inﬂaton is cold spot can be well explained by such a mechanism, lost into hidden ESPs in the isocurvature direction. and the spot proﬁle reasonably ﬁts the CMB cold spot It is interesting to explore such possibilities. without ﬁne tuning of the inﬂationary parameters. Be- • fore ending up the paper, we would like to mention a few Connection between the feature scattering regime future directions: and the multi-stream regime: By carefully arrang- • Beyond the Sachs-Wolfe approximation. Currently ing the position of the feature, it is possible that we did not use the full Boltzmann code to calcu- diﬀerent part of the universe follows diﬀerent clas- late the CMB transfer function. The Sachs-Wolfe sical trajectories, separated by temporary domain approximation works well for exploring the coarse- walls. This is known as the multi-stream inﬂation grained cold-spot proﬁle but is not enough for prob- [23, 24], which is also a possible explanation of the ing the ﬁne structures inside the spot. One can use cold spot. Feature scattering and multi-stream cor- the full radiation transfer function to carry on a responds to diﬀerent regimes of the multi-ﬁeld pa- more detailed analysis in the future. rameter space. If the features on the inﬂationary potential are random, we expect the feature scatter- • Verifying the approximations of the δN formal- ing to be more typical (whereas highly random fea- ism. The δN formalism assumes that the horizon- tures in multi-stream inﬂation cause disasters and crossing amplitude of the φ and χ quantum ﬂuctu- provide a constraint on multi-ﬁeld inﬂation [40]). ations are Gaussian and uncorrelated. In the case • of feature scattering, this is not rigorously true be- Connection between the cold spot and t∼he dip of cause of the turning of trajectory. However, note CMB temperature power spectrum at (cid:96) 20. We that the transfer of the inﬂaton kinetic energy is argue they may come from the same origin – fea- tinny (0.1% in the studied example), the correction ture scattering during inﬂation. With our current from sub-horizon physics should be suppressed by toy potential given at Eq. ((1)), we are unable to this small fraction. explain this power deﬁcit at the same time. How- ever, a more general shaped potential may be able Having that said, it remains interesting to see if the to explain both the deﬁcit and the cold spot. calculation from δN formalism can be veriﬁed by • ﬁrst principle calculation. However, such a calcula- String theory model building. It is believed that tion is challenging. The ﬁrst principle calculation string theory has a landscape of complicated vac- of cosmological perturbations is known as the in-in uum structures [41, 42]. It would be interesting to ##LTFigure##

label: paragraph

text: FIG.

text: The

label: caption

text: same

text: as

text: Fig.

text: but

text: for

text: model

text: m

text: χ

text: As

text: one

text: can

text: see

text: from

text: Fig.

text: there

text: are

text: more

text: small-scale

text: formalism.

label: list

text: But

text: in

text: the

text: in-in

text: formalism,

text: the

text: primary

text: structures

text: compared

text: to

text: the

text: case

text: of

text: massless

text: Benchmark

text: calculatables

text: are

text: the

text: correlation

text: functions.

text: In

text: other

text: words,

text: one

text: starts

text: from

text: the

text: Gaussian

text: two

text: point

text: cor-

text: IV.

label: section

text: CONCLUSION

text: AND

text: DISCUSSION

text: relation

text: function

text: and

text: studies

text: small

text: departure

text: from

text: that.

text: But

text: the

text: cold

text: spot

text: is

text: highly

text: non-Gaussian

text: and

text: localized

text: object,

text: and

text: thus

text: is

text: not

text: easily

text: captured

text: by

text: To

text: conclude,

text: we

text: propose

text: a

text: scattering

text: mechanism

text: of

text: in-

text: the

text: correlation

text: functions

text: from

text: the

text: in-in

text: formalism.

text: ﬂaton

text: due

text: to

text: the

text: features

text: in

text: the

text: inﬂation

text: potential,

text: dur-

text: ing

text: which

text: the

text: isocurvature

text: ﬂuctuations

text: are

text: converted

text: into

text: Extra

text: species/symmetry

text: point

text: (ESP)

text: as

text: curvature

text: ﬂuctuations.

text: The

text: curvature

text: ﬂuctuations

text: direc-

text: scattering

text: centers.

text: In

text: our

text: current

text: examples,

text: the

text: tion

text: loses

text: kinetic

text: energy

text: due

text: to

text: the

text: scatters

text: in

text: isocurva-

text: inﬂaton

text: kinetic

text: energy

text: is

text: lost

text: into

text: the

text: collective

text: mo-

text: ture

text: direction,

text: therefore

text: the

text: number

text: of

text: e-folds

text: becomes

text: tion

text: of

text: the

text: isocurvature

text: direction.

text: It

text: may

text: also

text: be

text: larger

text: in

text: some

text: region

text: of

text: the

text: universe.

text: We

text: ﬁnd

text: that

text: the

text: possible

text: that

text: the

text: kinetic

text: energy

text: of

text: the

text: inﬂaton

text: is

text: cold

text: spot

text: can

text: be

text: well

text: explained

text: by

text: such

text: a

text: mechanism,

text: lost

text: into

text: hidden

text: ESPs

text: in

text: the

text: isocurvature

text: direction.

text: and

text: the

text: spot

text: proﬁle

text: reasonably

text: ﬁts

text: the

text: CMB

text: cold

text: spot

text: It

text: is

text: interesting

text: to

text: explore

text: such

text: possibilities.

text: without

text: ﬁne

text: tuning

text: of

text: the

text: inﬂationary

text: parameters.

text: Be-

text: fore

text: ending

text: up

text: the

text: paper,

text: we

text: would

text: like

text: to

text: mention

text: a

text: few

text: Connection

text: between

text: the

text: feature

text: scattering

text: regime

text: future

text: directions:

text: and

text: the

text: multi-stream

text: regime:

text: By

text: carefully

text: arrang-

text: Beyond

text: the

text: Sachs-Wolfe

text: approximation.

text: Currently

text: ing

text: the

text: position

text: of

text: the

text: feature,

text: it

text: is

text: possible

text: that

text: we

text: did

text: not

text: use

text: the

text: full

text: Boltzmann

text: code

text: to

text: calcu-

text: diﬀerent

text: part

text: of

text: the

text: universe

text: follows

text: diﬀerent

text: clas-

text: late

text: the

text: CMB

text: transfer

text: function.

text: The

text: Sachs-Wolfe

text: sical

text: trajectories,

text: separated

text: by

text: temporary

text: domain

text: approximation

text: works

text: well

text: for

text: exploring

text: the

text: coarse-

text: walls.

text: This

text: is

text: known

text: as

text: the

text: multi-stream

text: inﬂation

text: grained

text: cold-spot

text: proﬁle

text: but

text: is

text: not

text: enough

text: for

text: prob-

text: which

text: is

text: also

text: a

text: possible

text: explanation

text: of

text: the

text: ing

text: the

text: ﬁne

text: structures

text: inside

text: the

text: spot.

text: One

text: can

text: use

text: cold

text: spot.

text: Feature

text: scattering

text: and

text: multi-stream

text: cor-

text: the

text: full

text: radiation

text: transfer

text: function

text: to

text: carry

text: on

text: a

text: responds

text: to

text: diﬀerent

text: regimes

text: of

text: the

text: multi-ﬁeld

text: pa-

text: more

text: detailed

text: analysis

text: in

text: the

text: future.

text: rameter

text: space.

text: If

text: the

text: features

text: on

text: the

text: inﬂationary

text: potential

text: are

text: random,

text: we

text: expect

text: the

text: feature

text: scatter-

text: Verifying

text: the

text: approximations

text: of

text: the

text: δN

text: formal-

text: ing

text: to

text: be

text: more

text: typical

text: (whereas

text: highly

text: random

text: fea-

text: ism.

text: The

text: δN

text: formalism

text: assumes

text: that

text: the

text: horizon-

text: tures

text: in

text: multi-stream

text: inﬂation

text: cause

text: disasters

text: and

text: crossing

text: amplitude

text: of

text: the

text: φ

text: and

text: χ

text: quantum

text: ﬂuctu-

text: provide

text: a

text: constraint

text: on

text: multi-ﬁeld

text: inﬂation

text: ations

text: are

text: Gaussian

text: and

text: uncorrelated.

text: In

text: the

text: case

text: of

text: feature

text: scattering,

text: this

text: is

text: not

text: rigorously

text: true

text: be-

text: Connection

text: between

text: the

text: cold

text: spot

text: and

text: t∼he

text: dip

text: of

text: cause

text: of

text: the

text: turning

text: of

text: trajectory.

text: However,

text: note

text: CMB

text: temperature

text: power

text: spectrum

text: at

text: We

text: that

text: the

text: transfer

text: of

text: the

text: inﬂaton

text: kinetic

text: energy

text: is

text: argue

text: they

text: may

text: come

text: from

text: the

text: same

text: origin

text: fea-

text: tinny

text: in

text: the

text: studied

text: example),

text: the

text: correction

text: ture

text: scattering

text: during

text: inﬂation.

text: With

text: our

text: current

text: from

text: sub-horizon

text: physics

text: should

text: be

text: suppressed

text: by

text: toy

text: potential

text: given

text: at

text: Eq.

text: we

text: are

text: unable

text: to

text: this

text: small

text: fraction.

text: explain

text: this

text: power

text: deﬁcit

text: at

text: the

text: same

text: time.

text: How-

text: ever,

text: a

text: more

text: general

text: shaped

text: potential

text: may

text: be

text: able

text: Having

text: that

text: said,

text: it

text: remains

text: interesting

text: to

text: see

text: if

text: the

text: to

text: explain

text: both

text: the

text: deﬁcit

text: and

text: the

text: cold

text: spot.

text: calculation

text: from

text: δN

text: formalism

text: can

text: be

text: veriﬁed

text: by

text: ﬁrst

text: principle

text: calculation.

text: However,

text: such

text: a

text: calcula-

text: String

text: theory

text: model

text: building.

text: It

text: is

text: believed

text: that

text: tion

text: is

text: challenging.

text: The

text: ﬁrst

text: principle

text: calculation

text: string

text: theory

text: has

text: a

text: landscape

text: of

text: complicated

text: vac-

text: of

text: cosmological

text: perturbations

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text: the

text: in-in

text: uum

text: structures

text: It

text: would

text: be

text: interesting

text: to

text: ##LTFigure##


text: Algorithm 4 BayesCG: Sample-based Conditional Gradient Algorithm for Multi-class Non- decomposable Performance Metric. ∈ X × × → R Input: S = ((x , y ), . . . , (x , y )) ( [n])m, ψ : [0, 1]n n 1∈ 1 ∈mN m + Param(cid:98)eters: α (0, 1), κ (cid:48) (cid:48)(cid:48) (cid:98) − (cid:99) (cid:100) (cid:101) Split S into two sets S and(cid:98)S with sizes m = (1 α)m and m = αm . (cid:48) ∪∞ X1× → X 2 Learn η (cid:48) = CPE(S ), where CPE : ( [n])m ∆ is a suitable CPE algorithm S X → m=1 n Choose an initial classiﬁer h0 : ∆ n T = κm(cid:98) (cid:98) for j = 1 to T do ∇ − Gj = ψ(conf(hj 1, D (cid:48)(cid:48))) (cid:98) S (cid:98) (cid:98) (cid:98) (cid:0) (cid:1) Approxim(cid:98)ate LineXar→Maximiz(cid:98)ation: (cid:98) (cid:98) (cid:62) ∀ ∈ X Construct uj : ∆ such that uj(x) = 1 if i = argmax ∈ gj η (cid:48)(x), x X → n i − − y [n] y ∀S ∈ X Co(cid:98)nstruct(cid:98)hj : ∆ such that hj(x) = 1 2 hj 1(x) + 2 uj(x), x n j+1 j+1 end for ≡ Output: hFW hT S (cid:2)(cid:13) (cid:13) (cid:3) (cid:13)(cid:98) (cid:13) (cid:98) where C > 0 is a distribution-independent constant. Thus, if the CPE algorithm used in Algorithm − −→ −→ → ∞ 4 is such that E η (cid:48)(X) η(X) P 0, then regretψ [h ] P 0 (as m ). X S 1 D S A key element of the proof of the above theorem is in showing that the BayesCG algorithm solves the CG linear maximization step approximately; this makes use of Lemma 8 and Lemma 6 in the previous section (along with the smoothness assumption on ψ). (cid:98) (cid:98) Lemma 11 (Approximation Factor for Linear Maximization Step in Algorithm 4). × → R Let ψ : [0, 1]n n satisfy the assumptions in (cid:98)Theorem 10. Let uj and hj be the classiﬁers + (cid:48) (cid:48)(cid:48) ∈ X × constructed in any given iteration j of Algorithm 4 using training sample S = (S , S ) ( [n])m ∈ ∇ − ∈ and parameter α (0, 1). Also, let Gj = ψ(conf(hj 1, D)). Then for any δ [0, 1], we have with − ≤ ≤ probability at least 1 δ (over dra(cid:98)w of S from Dm) for all 1 j T (cid:98) (cid:104) (cid:105) ≥ (cid:104) (cid:105) − Gj, conf(uj, D) max Gj, conf(u, D) (cid:15) , X → (cid:114) S (cid:2)(cid:13) (cid:13) (cid:3)u: ∆n (cid:98) (cid:13)(cid:98) (cid:13) where − n2 log(n) log(αm) + log(n2/δ) (cid:15) = 2LE η (cid:48)(X) η(X) + 2βn2C , S X S 1 αm for a distribution-independent constant C > 0. Proof of Theorem 10. The proof follows from Lemma 11 and the regret bound in Theorem 9. While the consistency result in Theorem 10 applies only to smooth performance metrics, for non- smooth performance metric such as the G-mean metric and several others in Table 1, one can apply × → R Algorithm 4 to a suitable smoothed version of the metric (indicated below by ψ : [0, 1]n n ∈ ρ + for some ρ (0, 1), with lim → ψ = ψ), and obtain the following regret bound for the original ρ 0 ρ performance metric. Theorem 12 (Regret Bound for Sample-based Conditional Gradient Algorithm for a (cid:48) (cid:48)(cid:48) ∈ X × Larger Family of Non-decomposable Metrics). Let S = (S , S ) ( [n])m be the given 14 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Algorithm

text: BayesCG:

text: Sample-based

text: Conditional

text: Gradient

text: Algorithm

text: for

text: Multi-class

text: Non-

text: decomposable

text: Performance

text: Metric.

text: X

text: R

text: Input:

text: S

text: ((x

text: y

text: (x

text: y

text: [n])m,

text: ψ

text: n

text: 1∈

text: ∈mN

text: m

text: α

text: κ

text: Split

text: S

text: into

text: two

text: sets

text: S

text: with

text: sizes

text: m

text: α)m

text: and

text: m

text: αm

text: X

text: Learn

text: η

text: CPE(S

text: where

text: CPE

text: [n])m

text: is

text: a

text: suitable

text: CPE

text: algorithm

text: S

text: X

text: n

text: Choose

text: an

text: initial

text: classiﬁer

text: n

text: T

text: for

text: j

text: to

text: T

text: do

text: Gj

text: ψ(conf(hj

text: D

text: S

text: X

text: Construct

text: uj

text: such

text: that

text: uj(x)

text: if

text: i

text: argmax

text: gj

text: η

text: x

text: X

text: n

text: i

text: y

text: [n]

text: y

text: ∀S

text: X

text: such

text: that

text: hj(x)

text: hj

text: uj(x),

text: x

text: n

text: end

text: for

text: Output:

text: hFW

text: hT

text: S

text: where

text: C

text: is

text: a

text: distribution-independent

text: constant.

text: Thus,

text: if

text: the

text: CPE

text: algorithm

text: used

text: in

text: Algorithm

text: is

text: such

text: that

text: E

text: η

text: η(X)

text: P

text: then

text: regretψ

text: [h

text: P

text: (as

text: m

text: X

text: S

text: D

text: S

text: A

text: key

text: element

text: of

text: the

text: proof

text: of

text: the

text: above

text: theorem

text: is

text: in

text: showing

text: that

text: the

text: BayesCG

text: algorithm

text: solves

text: the

text: CG

text: linear

text: maximization

text: step

text: approximately;

text: this

text: makes

text: use

text: of

text: Lemma

text: and

text: Lemma

text: in

text: the

text: previous

text: section

text: (along

text: with

text: the

text: smoothness

text: assumption

text: on

text: ψ).

text: Lemma

text: (Approximation

text: Factor

text: for

text: Linear

text: Maximization

text: Step

text: in

text: Algorithm

text: R

text: Let

text: ψ

text: n

text: satisfy

text: the

text: assumptions

text: in

text: Let

text: uj

text: and

text: hj

text: be

text: the

text: classiﬁers

text: X

text: constructed

text: in

text: any

text: given

text: iteration

text: j

text: of

text: Algorithm

text: using

text: training

text: sample

text: S

text: (S

text: S

text: [n])m

text: and

text: parameter

text: α

text: Also,

text: let

text: Gj

text: ψ(conf(hj

text: D)).

text: Then

text: for

text: any

text: δ

text: we

text: have

text: with

text: probability

text: at

text: least

text: δ

text: (over

text: of

text: S

text: from

text: Dm)

text: for

text: all

text: j

text: T

text: Gj,

text: conf(uj,

text: D)

text: max

text: Gj,

text: conf(u,

text: D)

text: X

text: S

text: ∆n

text: where

text: log(n)

text: log(αm)

text: η

text: η(X)

text: S

text: X

text: S

text: αm

text: for

text: a

text: distribution-independent

text: constant

text: C

text: Proof

text: of

text: Theorem

text: The

text: proof

text: follows

text: from

text: Lemma

text: and

text: the

text: regret

text: bound

text: in

text: Theorem

text: While

text: the

text: consistency

text: result

text: in

text: Theorem

text: applies

text: only

text: to

text: smooth

text: performance

text: metrics,

text: for

text: non-

text: smooth

text: performance

text: metric

text: such

text: as

text: the

text: G-mean

text: metric

text: and

text: several

text: others

text: in

text: Table

text: one

text: can

text: apply

text: R

text: Algorithm

text: to

text: a

text: suitable

text: smoothed

text: version

text: of

text: the

text: metric

text: (indicated

text: below

text: by

text: ψ

text: n

text: ρ

text: for

text: some

text: ρ

text: with

text: lim

text: ψ

text: ψ),

text: and

text: obtain

text: the

text: following

text: regret

text: bound

text: for

text: the

text: original

text: ρ

text: ρ

text: performance

text: metric.

text: Theorem

text: (Regret

text: Bound

text: for

text: Sample-based

text: Conditional

text: Gradient

text: Algorithm

text: for

text: a

text: X

text: Larger

text: Family

text: of

text: Non-decomposable

text: Metrics).

text: Let

text: S

text: (S

text: S

text: [n])m

text: be

text: the

text: given

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: P ≡ × → R Table 2: Performance Metrics [h] = ψ(conf(h, D) C), for which ψ : [0, 1]n n is concave D + but non-smooth (see Table 1 for the form of ψ for these metrics). A smoothed version of this × → R ∈ function ψ : [0, 1]n n for any ρ (0, 1) is given in the second column; in each case, ψ is ρ + ρ also concave. The form of θ(ρ) (deﬁned in Theorem 12), the Lipschitz constant L and smoothness ρ parameter β for the smoothed function are given respectively in the third, fourth and ﬁfth columns. ρ Here, we denote π = min ∈ π . Details of all calculations can be found in Appendix min y [n] y (cid:16)(cid:80) (cid:80) (cid:17) metric (cid:98) ψρ(C(cid:98)) θ(ρ) Lρ βρ H-Mean (HM) n (cid:114)n (cid:80) ny=1(cid:16)C(cid:80)y,y +ρ −1 (cid:17) n ρ n 2(cid:16)n (cid:17) y=1 Cy,y(cid:80)+(cid:98)ρ (cid:98) πmin ρ ρ2 Q-Mean (QM) 1(cid:16)−(cid:81) 1(cid:16) n y(cid:98)(cid:54)=y C(cid:17)y(cid:17),y(cid:98) +ρ 2 1√ ρ (cid:16) √1 1(cid:17) √2 (cid:16)1 1 +(cid:17) 1 n (cid:80)y=1 ny=1 Cy,y +ρ πmin n n ρ n ρ2 ρ − − G-Mean (GM) n (cid:98)Cy,y +(cid:98)ρ 1/n 2ρ1/n 1 1 1 + 1 1 1/n 1 1 1 + 1 1 2/n y=1 ny=1 Cy,y +ρ n ρ ρ n2 ρ3 ρ Remark 5 (Extension to non-diﬀerentiable concave metrics). The consistency results that we have seen so far for the BayesCG algorithm have assumed that the given performance metric is concave and diﬀerentiable, and hence do not apply to the min-max metric in Table 1, which is (concave, but) not diﬀerentiable. It is indeed possible to derive a version of the BayesCG algorithm that is consistent for such continuous concave metrics, by working with a smooth diﬀerentiable approximation to these performance metrics [37]. The proof of consistency for the resulting learning algorithm is however slightly more involved, requiring us to deal with approximate gradients to these metrics [37], and is reserved for a longer version of this paper. Remark 6 (Extension to fractional-linear metrics). We would also like to point out that there is a variant of the CG method used in the BayesCG algorithm that can be applied to non-concave optimization objectives [38], but this method can get stuck in a stationary point that is not a globally optimal solution, and hence the resulting learning algorithm need not be consistent for a general non-concave performance metric. However, one can show (without an explicit regret bound) that this variant of the BayesCG algorithm is consistent for a special class of non-concave performance metrics that are fractional-linear, such as the binary F-measure, the JAC metric and the multi- class micro F-measure, where owing to the pseudo-linear structure of these performance metrics, all stationary points are globally optimal solutions [39]. 6 Conclusion We provide a uniﬁed framework for analysing a general non-decomposable multi-class perfor- mance metric that cannot be expressed as a sum of losses on individual examples such as the multi-class F-measure and the multi-class G-mean metrics. Using this framework, we give a char- acterization of the optimal classiﬁer for a general non-decomposable performance metric, subsuming several previous results on binary non-decomposable metrics. We then design a eﬃcient learning algorithm based on the conditional gradient (CG) optimization method that is consistent for a large family of concave performance metrics. Our proof techniques are novel and involve application of tools from the optimization literature, particularly those used in the convergence analysis of the CG method. 16 ##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: P

text: R

text: Table

text: Performance

text: Metrics

text: [h]

text: ψ(conf(h,

text: D)

text: C),

text: for

text: which

text: ψ

text: n

text: is

text: concave

text: D

text: but

text: non-smooth

text: (see

text: Table

text: for

text: the

text: form

text: of

text: ψ

text: for

text: these

text: metrics).

text: A

text: smoothed

text: version

text: of

text: this

text: R

text: function

text: ψ

text: n

text: for

text: any

text: ρ

text: is

text: given

text: in

text: the

text: second

text: column;

text: in

text: each

text: case,

text: ψ

text: is

text: ρ

text: ρ

text: also

text: concave.

text: The

text: form

text: of

text: θ(ρ)

text: (deﬁned

text: in

text: Theorem

text: the

text: Lipschitz

text: constant

text: L

text: and

text: smoothness

text: ρ

text: parameter

text: β

text: for

text: the

text: smoothed

text: function

text: are

text: given

text: respectively

text: in

text: the

text: third,

text: fourth

text: and

text: ﬁfth

text: columns.

text: ρ

text: Here,

text: we

text: denote

text: π

text: min

text: π

text: Details

text: of

text: all

text: calculations

text: can

text: be

text: found

text: in

text: Appendix

text: min

text: y

text: [n]

text: y

text: metric

text: θ(ρ)

text: Lρ

text: βρ

text: H-Mean

text: (HM)

text: n

text: +ρ

text: −1

text: n

text: ρ

text: n

text: πmin

text: ρ

text: ρ2

text: Q-Mean

text: (QM)

text: n

text: +ρ

text: 1√

text: ρ

text: √1

text: √2

text: n

text: Cy,y

text: +ρ

text: πmin

text: n

text: n

text: ρ

text: n

text: ρ2

text: ρ

text: G-Mean

text: (GM)

text: n

text: Cy,y

text: +ρ

text: n

text: ρ

text: ρ

text: ρ3

text: ρ

text: Remark

text: (Extension

text: to

text: non-diﬀerentiable

text: concave

text: metrics).

text: The

text: consistency

text: results

text: that

text: we

text: have

text: seen

text: so

text: far

text: for

text: the

text: BayesCG

text: algorithm

text: have

text: assumed

text: that

text: the

text: given

text: performance

text: metric

text: is

text: concave

text: and

text: diﬀerentiable,

text: and

text: hence

text: do

text: not

text: apply

text: to

text: the

text: min-max

text: metric

text: in

text: Table

text: which

text: is

text: (concave,

text: but)

text: not

text: diﬀerentiable.

text: It

text: is

text: indeed

text: possible

text: to

text: derive

text: a

text: version

text: of

text: the

text: BayesCG

text: algorithm

text: that

text: is

text: consistent

text: for

text: such

text: continuous

text: concave

text: metrics,

text: by

text: working

text: with

text: a

text: smooth

text: diﬀerentiable

text: approximation

text: to

text: these

text: performance

text: metrics

text: The

text: proof

text: of

text: consistency

text: for

text: the

text: resulting

text: learning

text: algorithm

text: is

text: however

text: slightly

text: more

text: involved,

text: requiring

text: us

text: to

text: deal

text: with

text: approximate

text: gradients

text: to

text: these

text: metrics

text: and

text: is

text: reserved

text: for

text: a

text: longer

text: version

text: of

text: this

text: paper.

text: Remark

text: (Extension

text: to

text: fractional-linear

text: metrics).

text: We

text: would

text: also

text: like

text: to

text: point

text: out

text: that

text: there

text: is

text: a

text: variant

text: of

text: the

text: CG

text: method

text: used

text: in

text: the

text: BayesCG

text: algorithm

text: that

text: can

text: be

text: applied

text: to

text: non-concave

text: optimization

text: objectives

text: but

text: this

text: method

text: can

text: get

text: stuck

text: in

text: a

text: stationary

text: point

text: that

text: is

text: not

text: a

text: globally

text: optimal

text: solution,

text: and

text: hence

text: the

text: resulting

text: learning

text: algorithm

text: need

text: not

text: be

text: consistent

text: for

text: a

text: general

text: non-concave

text: performance

text: metric.

text: However,

text: one

text: can

text: show

text: (without

text: an

text: explicit

text: regret

text: bound)

text: that

text: this

text: variant

text: of

text: the

text: BayesCG

text: algorithm

text: is

text: consistent

text: for

text: a

text: special

text: class

text: of

text: non-concave

text: performance

text: metrics

text: that

text: are

text: fractional-linear,

text: such

text: as

text: the

text: binary

text: F-measure,

text: the

text: JAC

text: metric

text: and

text: the

text: multi-

text: class

text: micro

text: F-measure,

text: where

text: owing

text: to

text: the

text: pseudo-linear

text: structure

text: of

text: these

text: performance

text: metrics,

text: all

text: stationary

text: points

text: are

text: globally

text: optimal

text: solutions

text: Conclusion

text: We

text: provide

text: a

text: uniﬁed

text: framework

text: for

text: analysing

text: a

text: general

text: non-decomposable

text: multi-class

text: perfor-

text: mance

text: metric

text: that

text: cannot

text: be

text: expressed

text: as

text: a

text: sum

text: of

text: losses

text: on

text: individual

text: examples

text: such

text: as

text: the

text: multi-class

text: F-measure

text: and

text: the

text: multi-class

text: G-mean

text: metrics.

text: Using

text: this

text: framework,

text: we

text: give

text: a

text: char-

text: acterization

text: of

text: the

text: optimal

text: classiﬁer

text: for

text: a

text: general

text: non-decomposable

text: performance

text: metric,

text: subsuming

text: several

text: previous

text: results

text: on

text: binary

text: non-decomposable

text: metrics.

text: We

text: then

text: design

text: a

text: eﬃcient

text: learning

text: algorithm

text: based

text: on

text: the

text: conditional

text: gradient

text: (CG)

text: optimization

text: method

text: that

text: is

text: consistent

text: for

text: a

text: large

text: family

text: of

text: concave

text: performance

text: metrics.

text: Our

text: proof

text: techniques

text: are

text: novel

text: and

text: involve

text: application

text: of

text: tools

text: from

text: the

text: optimization

text: literature,

text: particularly

text: those

text: used

text: in

text: the

text: convergence

text: analysis

text: of

text: the

text: CG

text: method.

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text: A Proofs A.1 Proof of Proposition 1 C C Proposition (Convexity of ). is a convex set. D D ∈ C ∈ − ∈ C Proof. Let C , C . Let λ [0, 1]. We show that λC + (1 λ)C . 1 2 C D 1 X →2 D By deﬁnition of , there exists randomized classiﬁers h , h : ∆ such that D 1 2 n C = conf(h , D) 1 1 C = conf(h , D) 2 2 X → Consider the randomized classiﬁer hλ : ∆ deﬁned as n − hλ(x) = λh (x) + (1 λ)h (x) . 1 2 It can be seen that − conf(hλ, D) = λC + (1 λ)C . 1 2 A.2 Proof of Lemma 3 While Lemma 3 is simple to state, its proof is rather intricate and hence we give its proof via several intermediate lemmas and propositions.(cid:90) → Lemma 14 (Confusion matrix as an integration). Let f : ∆ ∆ . Then n n ◦ (cid:62) conf(f η, D) = p(f(p)) dν(p) . ∈ p ∆n (cid:2) (cid:3) Proof. (cid:2) (cid:12) (cid:3) ◦ · (cid:12) conf(f η, D) i,j = E(X,Y(cid:2))∼D[fj(η(cid:3)(X)) 1(Y = i)] · = E ∼ E ∼ f (p) 1(Y = i) η(X) = p p ν (X,Y ) D j = E ∼ p f (p) p ν i j X × Y Proposition 15 (Suﬃciency of conditional probability). Let D be a distribution over . X → (cid:48) X → For any randomized classiﬁer h : ∆ there exists another randomized classiﬁer h : ∆ (cid:48) n(cid:48) (cid:48) → n such that conf(h, D) = conf(h , D) and h is such that h (x) = f(η(x)), for some f : ∆ ∆ . n n 19 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A

text: Proofs

text: Proof

text: of

text: Proposition

text: C

text: C

text: Proposition

text: (Convexity

text: of

text: is

text: a

text: convex

text: set.

text: D

text: D

text: C

text: C

text: Proof.

text: Let

text: C

text: C

text: Let

text: λ

text: We

text: show

text: that

text: λC

text: λ)C

text: C

text: D

text: X

text: →2

text: D

text: By

text: deﬁnition

text: of

text: there

text: exists

text: randomized

text: classiﬁers

text: h

text: h

text: such

text: that

text: D

text: n

text: C

label: equation

text: conf(h

text: D)

text: C

text: conf(h

text: D)

text: X

text: Consider

text: the

text: randomized

text: classiﬁer

text: hλ

text: deﬁned

text: as

text: n

text: hλ(x)

text: λh

text: (x)

text: λ)h

text: (x)

text: It

text: can

text: be

text: seen

text: that

text: conf(hλ,

text: D)

text: λC

text: λ)C

text: Proof

text: of

text: Lemma

text: While

text: Lemma

text: is

text: simple

text: to

text: state,

text: its

text: proof

text: is

text: rather

text: intricate

text: and

text: hence

text: we

text: give

text: its

text: proof

text: via

text: several

text: intermediate

text: lemmas

text: and

text: Lemma

text: (Confusion

text: matrix

text: as

text: an

text: integration).

text: Let

text: f

text: Then

text: n

text: n

text: conf(f

text: η,

text: D)

text: p(f(p))

text: dν(p)

text: p

text: ∆n

text: Proof.

text: conf(f

text: η,

text: D)

text: i,j

text: i)]

text: E

text: E

text: f

text: (p)

text: i)

text: η(X)

text: p

text: p

text: ν

text: (X,Y

text: D

text: j

text: E

text: p

text: f

text: (p)

text: p

text: ν

text: i

text: j

text: X

text: Y

text: Proposition

text: (Suﬃciency

text: of

text: conditional

text: probability).

text: Let

text: D

text: be

text: a

text: distribution

text: over

text: X

text: X

text: For

text: any

text: randomized

text: classiﬁer

text: h

text: there

text: exists

text: another

text: randomized

text: classiﬁer

text: h

text: n

text: such

text: that

text: conf(h,

text: D)

text: conf(h

text: D)

text: and

text: h

text: is

text: such

text: that

text: h

text: (x)

text: f(η(x)),

text: for

text: some

text: f

text: n

text: n

text: ##LTLine##

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text: (cid:114) (cid:2)(cid:13) (cid:13) (cid:3) (cid:98) (cid:13)(cid:98) (cid:13) ≤ − n2 log(n) log(αm) + log(n2/δ) regretψ [hFW,ρ] 4L E η (cid:48)(X) η(X) + 4β n2C D S ρ X S 1 ρ αm 8β ρ + + 2θ(ρ), κm + 2 where C > 0 is a distribution-independent constant. (cid:114) (cid:2)(cid:13) (cid:13) (cid:3) Proof. Fr(cid:98)om Theorem 10 we(cid:13)h(cid:98)ave that (cid:13) ≤ − n2 log(n) log(αm) + log(n2/δ) 8β regretψρ[hFW,ρ] 4L E η (cid:48)(X) η(X) + 4β n2C + ρ . D S ρ X S 1 ρ αm κm + 2 X → ∗ ∈ For simplicity assume that the maximizer of ψ(conf(h, D)) over h : ∆ exists. Let h n argmax X → ψ(conf(h, D)). We then have that h: ∆n (cid:98) (cid:98) − regretψ [hFW,ρ] = sup ψ(conf(h, D)) ψ(cid:98)(conf(hFW,ρ, D)) D S X → S h: ∆n ∗ − (cid:98) = ψ(conf(h , D)) ψ(conf(hFW,ρ, D)) ≤ ∗ − S (cid:98) ψ (conf(h , D)) ψ (conf(hFW,ρ, D)) + 2θ(ρ) ρ ρ S ≤ − max ψ(cid:98) (conf(h, D)) ψ (conf(hFW,ρ, D)) + 2θ(ρ) X → ρ ρ S h: ∆n = regretψρ[hFW,ρ] + 2θ(ρ) D S B Details of Calculations for Smoothed Performance Metrics in Table 2 We now give details of derivation of the function θ (deﬁned in Theorem 12), Lipschitz constant L , and the smoothness parameter β for the smoothed performance metric ψ . In each case, we ρ ρ ρ make use the fact that the Lipschitz constant can be obtained by bounding the maximum absolute (cid:16) (cid:17) entry ((cid:96)∞ norm) of the gradient of ψ and(cid:80)the sm(cid:80)oothness parameter is obtained by bounding the maximum absolute entry ((cid:96)∞ norm) of its Hessian(cid:98). (cid:98) − H-mean. For the H-mean, ψH(C) = n n ny=1 Cy,y 1 is n -Lipschitz over C . Hence we y=1 Cy,y πmin D have θ(ρ) = n ρ. The gradient of ψH is given by: πmin ρ (cid:16) (cid:80)(cid:98) (cid:98) (cid:17)  (cid:80) n (cid:80)y(cid:54)=u Cu,y (cid:48)  (Cu(cid:98),u+ρ)2(cid:98) if u = u ∇  (cid:16) n ny=1 Cy,y +ρ 2(cid:17) Cuu(cid:48)ψρH(C) = − (cid:80)y=1 n(cid:80)Cy,1y+ρ . C(cid:98)u,u+ρ (cid:98) otherwise n ny=1 Cy,y +ρ 2 y=1 Cy,y+ρ 32 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: log(n)

text: log(αm)

text: regretψ

text: [hFW,ρ]

text: E

text: η

text: η(X)

text: 4β

text: D

text: S

text: ρ

text: X

text: S

text: ρ

text: αm

text: 8β

text: ρ

text: κm

text: where

text: C

text: is

text: a

text: distribution-independent

text: constant.

text: Proof.

text: Theorem

text: that

text: log(n)

text: log(αm)

text: 8β

text: regretψρ[hFW,ρ]

text: E

text: η

text: η(X)

text: 4β

text: ρ

text: D

text: S

text: ρ

text: X

text: S

text: ρ

text: αm

text: κm

text: X

text: For

text: simplicity

text: assume

text: that

text: the

text: maximizer

text: of

text: ψ(conf(h,

text: D))

text: over

text: h

text: exists.

text: Let

text: h

text: n

text: argmax

text: X

text: ψ(conf(h,

text: D)).

text: We

text: then

text: have

text: that

text: h:

text: ∆n

text: regretψ

text: [hFW,ρ]

text: sup

text: ψ(conf(h,

text: D))

text: D))

text: D

text: S

text: X

text: S

text: h:

text: ∆n

text: ψ(conf(h

text: D))

text: ψ(conf(hFW,ρ,

text: D))

text: S

text: ψ

text: (conf(h

text: D))

text: ψ

text: (conf(hFW,ρ,

text: D))

text: ρ

text: ρ

text: S

text: max

text: (conf(h,

text: D))

text: ψ

text: (conf(hFW,ρ,

text: D))

text: X

text: ρ

text: ρ

text: S

text: h:

text: ∆n

text: regretψρ[hFW,ρ]

text: D

text: S

text: B

text: Details

text: of

text: Calculations

text: for

text: Smoothed

text: Performance

text: Metrics

text: in

text: Table

text: We

text: now

text: give

text: details

text: of

text: derivation

text: of

text: the

text: function

text: θ

text: (deﬁned

text: in

text: Theorem

text: Lipschitz

text: constant

text: L

text: and

text: the

text: smoothness

text: parameter

text: β

text: for

text: the

text: smoothed

text: performance

text: metric

text: ψ

text: In

text: each

text: case,

text: we

text: ρ

text: ρ

text: ρ

text: make

text: use

text: the

text: fact

text: that

text: the

text: Lipschitz

text: constant

text: can

text: be

text: obtained

text: by

text: bounding

text: the

text: maximum

text: absolute

text: entry

text: norm)

text: of

text: the

text: gradient

text: of

text: ψ

text: parameter

text: is

text: obtained

text: by

text: bounding

text: the

text: maximum

text: absolute

text: entry

text: norm)

text: of

text: its

text: H-mean.

text: For

text: the

text: H-mean,

text: ψH(C)

text: n

text: n

text: Cy,y

text: is

text: n

text: -Lipschitz

text: over

text: C

text: Hence

text: we

text: Cy,y

text: πmin

text: D

text: have

text: θ(ρ)

text: n

text: ρ.

text: The

text: gradient

text: of

text: ψH

text: is

text: given

text: by:

text: πmin

text: ρ

text: n

text: Cu,y

text: if

text: u

text: u

text: n

text: Cy,y

text: +ρ

text: otherwise

text: n

text: Cy,y

text: +ρ

text: Cy,y+ρ

text: ##LTLine##

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text: We now obtain the smoothness parameter β for ψQ. We start by bounding the entry of the ρ ρ (cid:48) (cid:48) Hessian matrix corresponding to confusion matrix entries C (cid:48) and C (cid:48) where u = u = v = v ; u,u v,v the same bound can be shown to hold for all entries for which u = v. (cid:16) (cid:17) |∇ (cid:34) (cid:80)| (cid:35) (cid:34) (cid:32)(cid:80) (cid:33) (cid:35) =2Cuu(cid:48)√,C1vnv(cid:48)(cid:80)ψ(cid:80)ρQ(nyC=(cid:80)1)(cid:16)yny(cid:98)(cid:98)(cid:54)==(cid:80)(cid:80)u1yny(cid:98)C(cid:98)C(cid:54)==uuy1,,yy(cid:98)(cid:98)CC++yy(cid:34),,ρρyy(cid:98)(cid:98)++(cid:88)2ρρ(cid:17)2(cid:32)(cid:80)(cid:80)3/2(cid:98)(cid:80)ny(cid:98)=1(cid:98)C1u,y(cid:98) +(cid:33)ρ(cid:80)3(cid:88)y=n1 (cid:80)yny(cid:98)(cid:98)(cid:54)==y1 CCyy,,yy(cid:98)(cid:98) ++(cid:35) ρρ 2 (cid:80)ny(cid:98)=1 C1u,y(cid:98) + ρ + 1 ≤ √1 (cid:98) 1 (cid:98) (cid:34) n y(cid:98)(cid:54)=y Cy,y(cid:98) + (cid:35)ρ 2 (cid:98) 1 (cid:98) 3 + 1 n n C + ρ n C + ρ n C + ρ (cid:80)y=1 u,y (cid:80)y=1 y=1 y,y y=1 u,y ≤ √1 (cid:34)(cid:98) 1 (cid:98) (cid:35) (cid:98) 3n (cid:98) + 1 n n C + ρ n C + ρ y=1 u,y y=1 u,y ≤ √1 1 3n + 1 √n ρ ρ ≤ 4 n . ρ2 (cid:80) (cid:80) (cid:48) (cid:54) (cid:54) (cid:54) (cid:48) We next bound the entry of the Hessian matrix corresponding to u = u = v(cid:98)= v ; t(cid:98)he sam(cid:98)e boun(cid:98)d (cid:54) can be shown to hold for all entr(cid:34)ies w(cid:16)h(cid:80)ere u = v. A(cid:17)ssum(cid:35)ing w.l.o.g. that y(cid:54)=u Cu,y > y(cid:54)=v Cv,y, (cid:80)(cid:16)(cid:98) (cid:98) (cid:17) (cid:80) (cid:80) (cid:80) y(cid:98)(cid:54)=(cid:80)u Cu,y(cid:98) +ρ 2 3/2 |∇2Cuu(cid:48),Cvv(cid:48)ψρQ(C)| ≤ √1n (cid:80) n ny=(cid:80)1y(cid:98)C(cid:98)(cid:54)=uy,yC+y,ρy(cid:98)(cid:98) +(cid:80)ρ 2 y(cid:98)(cid:54)=uCCuu,u,y(cid:98) + ρ ( ny(cid:98)=1CCvv,,vy(cid:98) + ρ)3 y=1 n C +ρ y=1 y,y ≤ √1 (cid:98) Cu,u(cid:98) (cid:98) Cv,v(cid:98) n (cid:80) (cid:54) C + ρ ((cid:80)n C + ρ)3 y=u u,y y=1 v,y ≤ √1 (cid:98) Cu,u(cid:98) (cid:98) 1 (cid:98) n (cid:54) C + ρ ( n C + ρ)2 y=u u,y y=1 v,y ≤ √1 1 . n ρ3 (cid:16) (cid:17) (cid:81) The smoothness parameter βρ in Table 2 then follows from the abov(cid:80)e bounds. G-mean. For the G-mean performance metric, ψG(C) = n (cid:98)Cy,y (cid:98) 1/n is not Lipschitz C y=1 ny=1 Cy,y over . We now explicitly derive the form of θ for this performance metric. D (cid:18) (cid:19) (cid:18) (cid:19) (cid:89) (cid:89) (cid:80) (cid:80) ψG(C) − ψG(C) = n (cid:98)Cy,y +(cid:98)ρ 1/n − n (cid:98)Cy,y (cid:98) 1/n ρ n C + ρ n C y=1 y=1 y,y y=1 y=1 y,y 36 ##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: We

text: now

text: obtain

text: the

text: smoothness

text: parameter

text: β

text: for

text: ψQ.

text: We

text: start

text: by

text: bounding

text: the

text: entry

text: of

text: the

text: ρ

text: ρ

text: Hessian

text: matrix

text: corresponding

text: to

text: confusion

text: matrix

text: entries

text: C

text: and

text: C

text: where

text: u

text: u

text: v

text: v

text: u,u

text: v,v

text: the

text: same

text: bound

text: can

text: be

text: shown

text: to

text: hold

text: for

text: all

text: entries

text: for

text: which

text: u

text: v.

text: ρρ

text: ρ

text: √1

text: n

text: n

text: n

text: C

text: ρ

text: n

text: C

text: ρ

text: n

text: C

text: ρ

text: u,y

text: y,y

text: u,y

text: √1

text: n

text: n

text: C

text: ρ

text: n

text: C

text: ρ

text: u,y

text: u,y

text: √1

text: √n

text: ρ

text: ρ

text: n

text: ρ2

text: We

text: next

text: bound

text: the

text: entry

text: of

text: the

text: Hessian

text: matrix

text: corresponding

text: to

text: u

text: u

text: v

text: can

text: be

text: shown

text: to

text: hold

text: for

text: all

text: u

text: v.

text: w.l.o.g.

text: that

text: Cu,y

text: Cv,y,

text: +ρ

text: n

text: ρ

text: n

text: C

text: +ρ

text: y,y

text: √1

text: n

text: C

text: ρ

text: C

text: y=u

text: u,y

text: v,y

text: √1

text: n

text: C

text: ρ

text: n

text: C

text: y=u

text: u,y

text: v,y

text: √1

text: n

text: ρ3

text: The

text: smoothness

text: parameter

text: βρ

text: in

text: Table

text: then

text: follows

text: from

text: the

text: bounds.

text: G-mean.

text: For

text: the

text: G-mean

text: performance

text: metric,

text: ψG(C)

text: n

text: is

text: not

text: Lipschitz

text: C

text: Cy,y

text: over

text: We

text: now

text: explicitly

text: derive

text: the

text: form

text: of

text: θ

text: for

text: this

text: performance

text: metric.

text: D

text: ψG(C)

text: ψG(C)

text: n

text: n

text: ρ

text: n

text: C

text: ρ

text: n

text: C

text: y,y

text: y,y

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text: Remark 1 (Necessity of continuity Assumption A on D). We note here that for the above characterization to hold for a general non-decomposable performance metric, the continuity assump- tion on distribution D (Assumptions A) is indeed necessary. We illustrate this fact for the H-mean (cid:2) (cid:3) performance metric by constructing a simple distribution that does not satisfy this assumption, and where a classiﬁer of the form in the theorem statement is not ne(cid:2)cessa(cid:3)rily optimal. Consider the { } × { } (cid:62) following distribution D over x 1, 2 with η(x) = 1, 1 . It can be seen that the unique ∗2 2 (cid:62) optimal classiﬁer for the H-mea(cid:34)n perfo(cid:35)rmance metric is h (x(cid:34)) = 1, 1 (cid:35) , whose confusion matrix ∗ ∗ 2 2 C and the gradient of ψ at C are given by: − ∗ 1 1 ∗ ∇ ∗ 1 1 C = 4 4 ; G = ψ(C ) = −2 2 . 1 1 1 1 4 4 2 2 X → (cid:104) ∗ (cid:105) Clearly, any classiﬁer h : ∆ will have G , conf(h, D) = 0; hence 2 { X → } (cid:104) ∗ (cid:105) ⊃ { ∗} h : ∆ = argmax G , conf(h, D) argmax ψ(conf(h, D)) = h . 2 X → X → h: ∆2 h: ∆2 It is worth noting that for certain restricted families of performance metrics, the characterization in Theorem 2 holds even without Assumption A on the distribution; this is the case, for example, when ψ is fractional-linear (e.g., F-measure, JAC) [3, 4] and is convex (e.g., AMS metric) [5]. Remark 2 (Extension to the min-max metric). A result similar to the one in Theorem 2 also holds for the min-max metric, where it is well known from classical detection theory (in particular, from min-max hypothesis testing) that the optimal classiﬁer for this metric is obtained by maximiz- ∗ ing a decomposable metric with an appropriate gain matrix [31]. In fact, one can show that if h ∗ is an optimal classiﬁer for the min-max metric ψMM, and G is in the sub-diﬀerential of ψMM at ∗ conf(h , D), then ∅ (cid:54) (cid:104) ∗ (cid:105) ⊆ = argmax G , conf(h, D) argmax ψMM(conf(h, D)). X → X → h: ∆n h: ∆n 4 A Consistent Plug-in Method for Multi-class Non-decomposable Metrics Based on a Brute-force Search Based on the above characterization of the optimal classiﬁer of a non-decomposable metric, we now consider a simple plug-in based learning algorithm for a multi-class non-decomposable metric that uses a brute-force search over gain matrices; this approach can be seen as a natural extension of the binary plug-in method in Algorithm 1. We show that this method is consistent with respect to a general non-decomposable metric, and also provide an explicit regret bound for this method for the special case of performance metrics that exhibit a certain convexity-like property. In the next section, we design an alternate eﬃcient learning algorithm based on the conditional gradient algorithm which is consistent for a large family of non-decomposable metrics. ∗ Clearly, if the optimal confusion matrix C for a multi-class non-decomposable metric ψ is ∗ known apriori, one can learn a simple plug-in classiﬁer by applying the gradient of ψ at C to a ∗ suitable class probability estimator. In the absence of knowledge of C , a natural ﬁrst-cut approach 9 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Remark

text: (Necessity

text: of

text: continuity

text: Assumption

text: A

text: on

text: D).

text: We

text: note

text: here

text: that

text: for

text: the

text: above

text: characterization

text: to

text: hold

text: for

text: a

text: general

text: non-decomposable

text: performance

text: metric,

text: the

text: continuity

text: assump-

text: tion

text: on

text: distribution

text: D

text: (Assumptions

text: A)

text: is

text: indeed

text: necessary.

text: We

text: illustrate

text: this

text: fact

text: for

text: the

text: H-mean

text: performance

text: metric

text: by

text: constructing

text: a

text: simple

text: distribution

text: that

text: does

text: not

text: satisfy

text: this

text: assumption,

text: and

text: where

text: a

text: classiﬁer

text: of

text: the

text: form

text: in

text: the

text: theorem

text: statement

text: is

text: not

text: optimal.

text: Consider

text: the

text: following

text: distribution

text: D

text: over

text: x

text: with

text: η(x)

text: It

text: can

text: be

text: seen

text: that

text: the

text: unique

text: ∗2

text: optimal

text: classiﬁer

text: for

text: the

text: metric

text: is

text: h

text: whose

text: confusion

text: matrix

text: C

text: and

text: the

text: gradient

text: of

text: ψ

text: at

text: C

text: are

text: given

text: by:

text: C

text: G

text: ψ(C

text: −2

text: X

text: Clearly,

text: any

text: classiﬁer

text: h

text: will

text: have

text: G

text: conf(h,

text: D)

text: hence

text: X

text: h

text: argmax

text: G

text: conf(h,

text: D)

text: argmax

text: ψ(conf(h,

text: D))

text: h

text: X

text: X

text: h:

text: ∆2

text: h:

text: ∆2

text: It

text: is

text: worth

text: noting

text: that

text: for

text: certain

text: restricted

text: families

text: of

text: performance

text: metrics,

text: the

text: characterization

text: in

text: Theorem

text: holds

text: even

text: without

text: Assumption

text: A

text: on

text: the

text: distribution;

text: this

text: is

text: the

text: case,

text: for

text: example,

text: when

text: ψ

text: is

text: fractional-linear

text: (e.g.,

text: F-measure,

text: JAC)

text: and

text: is

text: convex

text: (e.g.,

text: AMS

text: metric)

text: Remark

text: (Extension

text: to

text: the

text: min-max

text: metric).

text: A

text: result

text: similar

text: to

text: the

text: one

text: in

text: Theorem

text: also

text: holds

text: for

text: the

text: min-max

text: metric,

text: where

text: it

text: is

text: well

text: known

text: from

text: classical

text: detection

text: theory

text: (in

text: particular,

text: from

text: min-max

text: hypothesis

text: testing)

text: that

text: the

text: optimal

text: classiﬁer

text: for

text: this

text: metric

text: is

text: obtained

text: by

text: maximiz-

text: ing

text: a

text: decomposable

text: metric

text: with

text: an

text: appropriate

text: gain

text: matrix

text: In

text: fact,

text: one

text: can

text: show

text: that

text: if

text: h

text: is

text: an

text: optimal

text: classiﬁer

text: for

text: the

text: min-max

text: metric

text: ψMM,

text: and

text: G

text: is

text: in

text: the

text: sub-diﬀerential

text: of

text: ψMM

text: at

text: conf(h

text: D),

text: then

text: argmax

text: G

text: conf(h,

text: D)

text: argmax

text: ψMM(conf(h,

text: D)).

text: X

text: X

text: h:

text: ∆n

text: h:

text: ∆n

text: A

text: Consistent

text: Plug-in

text: Method

text: for

text: Multi-class

text: Non-decomposable

text: Metrics

text: Based

text: on

text: a

text: Brute-force

text: Search

text: Based

text: on

text: the

text: above

text: characterization

text: of

text: the

text: optimal

text: classiﬁer

text: of

text: a

text: non-decomposable

text: metric,

text: we

text: now

text: consider

text: a

text: simple

text: plug-in

text: based

text: learning

text: algorithm

text: for

text: a

text: multi-class

text: non-decomposable

text: metric

text: that

text: uses

text: a

text: brute-force

text: search

text: over

text: gain

text: matrices;

text: this

text: approach

text: can

text: be

text: seen

text: as

text: a

text: natural

text: extension

text: of

text: the

text: binary

text: plug-in

text: method

text: in

text: Algorithm

text: We

text: show

text: that

text: this

text: method

text: is

text: consistent

text: with

text: respect

text: to

text: a

text: general

text: non-decomposable

text: metric,

text: and

text: also

text: provide

text: an

text: explicit

text: regret

text: bound

text: for

text: this

text: method

text: for

text: the

text: special

text: case

text: of

text: performance

text: metrics

text: that

text: exhibit

text: a

text: certain

text: convexity-like

text: property.

text: In

text: the

text: next

text: section,

text: we

text: design

text: an

text: alternate

text: eﬃcient

text: learning

text: algorithm

text: based

text: on

text: the

text: conditional

text: gradient

text: algorithm

text: which

text: is

text: consistent

text: for

text: a

text: large

text: family

text: of

text: non-decomposable

text: metrics.

text: Clearly,

text: if

text: the

text: optimal

text: confusion

text: matrix

text: C

text: for

text: a

text: multi-class

text: non-decomposable

text: metric

text: ψ

text: is

text: known

text: apriori,

text: one

text: can

text: learn

text: a

text: simple

text: plug-in

text: classiﬁer

text: by

text: applying

text: the

text: gradient

text: of

text: ψ

text: at

text: C

text: to

text: a

text: suitable

text: class

text: probability

text: estimator.

text: In

text: the

text: absence

text: of

text: knowledge

text: of

text: C

text: a

text: natural

text: ﬁrst-cut

text: approach

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text: 2 JONAH GASTER group, we analyze explicit curves on S. The chosen curves are suﬃciently uncomplicated to allow a straightforward analysis of the degree of any cover to which the curves lift simply. As a consequence, we provide the improved lower bounds: ≥ Theorem 2. We have f (n) n + 1. Moreover, Let B = B(S) be a Bers S constant for (S, ρ), and let (cid:15) > 0. Then there is an L = L (ρ, (cid:15)) so that, ≥ 0 0 for any L L , 0 ≥ L f (L) . ρ B + (cid:15) Recall the theorem of Bers [Ber]: There is a constant B = B(S) so that, for every hyperbolic metric ρ on S, there is a maximal collection of disjoint simple curves on S with each curve of ρ-length at most B. Such a constant B is called a Bers constant, and such a collection of curves is called a Bers pants decomposition for (S, ρ). It is interesting to note that the constant in the lower bound for f in Theorem 2 is independent of the metric ρ. ρ Remark. The proof of Theorem 2 follows from an analysis of an explicit { } sequence of curves γ . These curves are also analyzed by Basmajian n [Bas], where it is shown that they are in some sense the ‘shortest’ curves of a given intersection number: The inﬁmum of the length function (cid:96)(γ ) on n the Teichmu¨ller space of S is asymptotically the minimum possible among curves with self-intersection ι(γ , γ ) [Bas, Cor. 1.4]. n n Combined with work of Patel [Pat, Thm. 1.1] (see the comment of [GK, p. 1]), in many cases Theorem 2 implies a determination of the order of growth of f . We have: ρ Corollary 3 (Linear growth of f ). Suppose (S, ρ) is without punctures. ρ There exist constants C = C (S), C = C (ρ), and L = L (ρ) so that, for ≥ 1 1 2 2 0 0 any L L , 0 · ≤ ≤ · C L f (L) C L. 1 ρ 2 Recall that, when S has boundary, we say that (S, ρ) has a puncture if the closed curve homotopic to a boundary component has no geodesic representative. When (S, ρ) does have a puncture, we are not aware of upper bounds for f . In fact, the hypothesis above is essential. We show: ρ Theorem 4. Suppose (S, ρ) is a hyperbolic surface with a puncture. For ≥ any (cid:15) > 0, there is L = L (ρ, (cid:15)) so that, for any L L , 0 0 0 ≥ L fρ(L) e2+(cid:15) . This theorem indicates at least that Patel’s upper bounds cannot hold in the punctured setting: If (S, ρ) is a hyperbolic surface with a puncture, the minimal degree of a cover to which a given curve γ lifts to a simple curve cannot be bounded linearly in the curve’s length (cid:96) (γ). ρ There are other avenues for further investigation. It would be natural to seek upper bounds for f (n) (cf. [Riv, p. 15]), since no such bound follows S ##LTLine## ##LTLine##

text: JONAH

text: GASTER

text: group,

text: we

text: analyze

text: explicit

text: curves

text: on

text: S.

text: The

text: chosen

text: curves

text: are

text: suﬃciently

text: uncomplicated

text: to

text: allow

text: a

text: straightforward

text: analysis

text: of

text: the

text: degree

text: of

text: any

text: cover

text: to

text: which

text: the

text: curves

text: lift

text: simply.

text: As

text: a

text: consequence,

text: we

text: provide

text: the

text: improved

text: lower

text: bounds:

text: Theorem

text: We

text: have

text: f

text: (n)

text: n

text: Moreover,

text: Let

text: B

text: B(S)

text: be

text: a

text: Bers

text: S

text: constant

text: for

text: (S,

text: ρ),

text: and

text: let

text: Then

text: there

text: is

text: an

text: L

text: L

text: (ρ,

text: so

text: that,

text: for

text: any

text: L

text: L

text: L

text: f

text: (L)

text: ρ

text: B

text: Recall

text: the

text: theorem

text: of

text: Bers

text: [Ber]:

text: There

text: is

text: a

text: constant

text: B

text: B(S)

text: so

text: that,

text: for

text: every

text: hyperbolic

text: metric

text: ρ

text: on

text: S,

text: there

text: is

text: a

text: maximal

text: collection

text: of

text: disjoint

text: simple

text: curves

text: on

text: S

text: with

text: each

text: curve

text: of

text: ρ-length

text: at

text: most

text: B.

text: Such

text: a

text: constant

text: B

text: is

text: called

text: a

text: Bers

text: constant,

text: and

text: such

text: a

text: collection

text: of

text: curves

text: is

text: called

text: a

text: Bers

text: pants

text: decomposition

text: for

text: (S,

text: ρ).

text: It

text: is

text: interesting

text: to

text: note

text: that

text: the

text: constant

text: in

text: the

text: lower

text: bound

text: for

text: f

text: in

text: Theorem

text: is

text: independent

text: of

text: the

text: metric

text: ρ.

text: ρ

text: Remark.

text: The

text: proof

text: of

text: Theorem

text: follows

text: from

text: an

text: analysis

text: of

text: an

text: explicit

text: sequence

text: of

text: curves

text: γ

text: These

text: curves

text: are

text: also

text: analyzed

text: by

text: Basmajian

text: n

text: [Bas],

text: where

text: it

text: is

text: shown

text: that

text: they

text: are

text: in

text: some

text: sense

text: the

text: ‘shortest’

text: curves

text: of

text: a

text: given

text: intersection

text: number:

text: The

text: inﬁmum

text: of

text: the

text: length

text: function

text: on

text: n

text: the

text: Teichmu¨ller

text: space

text: of

text: S

text: is

text: asymptotically

text: the

text: minimum

text: possible

text: among

text: curves

text: with

text: self-intersection

text: ι(γ

text: γ

text: [Bas,

text: Cor.

text: n

text: n

text: Combined

text: with

text: work

text: of

text: Patel

text: [Pat,

text: Thm.

text: (see

text: the

text: comment

text: of

text: [GK,

text: p.

text: in

text: many

text: cases

text: Theorem

text: implies

text: a

text: determination

text: of

text: the

text: order

text: of

text: growth

text: of

text: f

text: We

text: have:

text: ρ

text: Corollary

text: (Linear

text: growth

text: of

text: f

text: Suppose

text: (S,

text: ρ)

text: is

text: without

text: punctures.

text: ρ

text: There

text: exist

text: constants

text: C

text: C

text: (S),

text: C

text: C

text: (ρ),

text: and

text: L

text: L

text: (ρ)

text: so

text: that,

text: for

text: any

text: L

text: L

text: C

text: L

text: f

text: (L)

text: C

text: L.

text: ρ

text: Recall

text: that,

text: when

text: S

text: has

text: boundary,

text: we

text: say

text: that

text: (S,

text: ρ)

text: has

text: a

text: puncture

text: if

text: the

text: closed

text: curve

text: homotopic

text: to

text: a

text: boundary

text: component

text: has

text: no

text: geodesic

text: representative.

text: When

text: (S,

text: ρ)

text: does

text: have

text: a

text: puncture,

text: we

text: are

text: not

text: aware

text: of

text: upper

text: bounds

text: for

text: f

text: In

text: fact,

text: the

text: hypothesis

text: above

text: is

text: essential.

text: We

text: show:

text: ρ

text: Theorem

text: Suppose

text: (S,

text: ρ)

text: is

text: a

text: hyperbolic

text: surface

text: with

text: a

text: puncture.

text: For

text: any

text: there

text: is

text: L

text: L

text: (ρ,

text: so

text: that,

text: for

text: any

text: L

text: L

text: L

text: fρ(L)

text: This

text: theorem

text: indicates

text: at

text: least

text: that

text: Patel’s

text: upper

text: bounds

text: cannot

text: hold

text: in

text: the

text: punctured

text: setting:

text: If

text: (S,

text: ρ)

text: is

text: a

text: hyperbolic

text: surface

text: with

text: a

text: puncture,

text: the

text: minimal

text: degree

text: of

text: a

text: cover

text: to

text: which

text: a

text: given

text: curve

text: γ

text: lifts

text: to

text: a

text: simple

text: curve

text: cannot

text: be

text: bounded

text: linearly

text: in

text: the

text: curve’s

text: length

text: (γ).

text: ρ

text: There

text: are

text: other

text: avenues

text: for

text: further

text: investigation.

text: It

text: would

text: be

text: natural

text: to

text: seek

text: upper

text: bounds

text: for

text: f

text: (n)

text: (cf.

text: [Riv,

text: p.

text: since

text: no

text: such

text: bound

text: follows

text: S

text: ##LTLine##

text: ##LTLine##


text: LIFTING CURVES SIMPLY 3 from [Pat]. One might also investigate whether the constant C in the 2 upper bound in Corollary 3 can be made independent of ρ, as with the lower bound. Finally, one could explore the set of curves of self-intersection number exactly n. For instance: Among the ﬁnitely many mapping class group orbits of curves γ with self-intersection n, which maximize deg(γ)? § { } In we introduce a sequence of curves γ on a pair of pants and analyze § n the degrees deg(γ ), and in we deduce Theorem 2 and Theorem 4 as n straightforward consequences. The author gratefully acknowledges inspiration from Neha Gupta, inspi- ration and comments from Ilya Kapovich, and helpful comments from, and conversations with, Tarik Aougab, Ian Biringer, Martin Bridgeman, David Dumas, Peter Feller, Brice Loustau, and Priyam Patel. 2. analysis of a certain curve family Let P be a pair of pants. Identify π (P , p) with a rank-2 free group F , 0 1 0 with generators a and b as pictured in Figure 1. Let γ indicate the closed · n curve given by the equivalence class of a bn. a p b Figure 1. The pair of pants P , with generators a and b. 0 The following lemma is neither new (see [Bas, Prop. 4.2]) nor surprising (see Figure 2), but we include a sketch of a proof for completeness: ≥ Lemma 5. For n 0, the curve γ has ι(γ , γ ) = n. n n n Proof sketch. It is not hard to pick a representative of γ that has self- ≤ n intersection n, so that ι(γ , γ ) n (see Figure 2 for n = 4). On the n n other hand, it is also not hard to check that there are no immersed bigons for this chosen representative of γ : For every pair of intersection points, n ##LTFigure##

text: LIFTING

text: CURVES

text: SIMPLY

text: from

text: [Pat].

text: One

text: might

text: also

text: investigate

text: whether

text: the

text: constant

text: C

text: in

text: the

text: upper

text: bound

text: in

text: Corollary

text: can

text: be

text: made

text: independent

text: of

text: ρ,

text: as

text: with

text: the

text: lower

text: bound.

text: Finally,

text: one

text: could

text: explore

text: the

text: set

text: of

text: curves

text: of

text: self-intersection

text: number

text: exactly

text: n.

text: For

text: instance:

text: Among

text: the

text: ﬁnitely

text: many

text: mapping

text: class

text: group

text: orbits

text: of

text: curves

text: γ

text: with

text: self-intersection

text: n,

text: which

text: maximize

text: deg(γ)?

text: In

text: we

text: introduce

text: a

text: sequence

text: of

text: curves

text: γ

text: on

text: a

text: pair

text: of

text: pants

text: and

text: analyze

text: n

text: the

text: degrees

text: deg(γ

text: and

text: in

text: we

text: deduce

text: Theorem

text: and

text: Theorem

text: as

text: n

text: straightforward

text: consequences.

text: The

text: author

text: gratefully

text: acknowledges

text: inspiration

text: from

text: Neha

text: Gupta,

text: inspi-

text: ration

text: and

text: comments

text: from

text: Ilya

text: Kapovich,

text: and

text: helpful

text: comments

text: from,

text: and

text: conversations

text: with,

text: Tarik

text: Aougab,

text: Ian

text: Biringer,

text: Martin

text: Bridgeman,

text: David

text: Dumas,

text: Peter

text: Feller,

text: Brice

text: Loustau,

text: and

text: Priyam

text: Patel.

text: analysis

text: of

text: a

text: certain

text: curve

text: family

text: Let

text: P

text: be

text: a

text: pair

text: of

text: pants.

text: Identify

text: π

text: (P

text: p)

text: with

text: a

text: free

text: group

text: F

text: with

text: generators

text: a

text: and

text: b

text: as

text: pictured

text: in

text: Figure

text: Let

text: γ

text: indicate

text: the

text: closed

text: n

text: curve

text: given

text: by

text: the

text: equivalence

text: class

text: of

text: a

text: bn.

text: a

text: p

text: b

text: Figure

text: The

text: pair

text: of

text: pants

text: P

text: with

text: generators

text: a

text: and

text: b.

text: The

text: following

text: lemma

text: is

text: neither

text: new

text: (see

text: [Bas,

text: Prop.

text: nor

text: surprising

text: (see

text: Figure

text: but

text: we

text: include

text: a

text: sketch

text: of

text: a

text: proof

text: for

text: completeness:

text: Lemma

text: For

text: n

text: the

text: curve

text: γ

text: has

text: ι(γ

text: γ

text: n.

text: n

text: n

text: n

text: Proof

text: sketch.

text: It

text: is

text: not

text: hard

text: to

text: pick

text: a

text: representative

text: of

text: γ

text: that

text: has

text: self-

text: n

text: intersection

text: n,

text: so

text: that

text: ι(γ

text: γ

text: n

text: (see

text: Figure

text: for

text: n

text: On

text: the

text: n

text: n

text: other

text: hand,

text: it

text: is

text: also

text: not

text: hard

text: to

text: check

text: that

text: there

text: are

text: no

text: immersed

text: bigons

text: for

text: this

text: chosen

text: representative

text: of

text: γ

text: For

text: every

text: pair

text: of

text: intersection

text: points,

text: n

text: ##LTFigure##


text: 4 JONAH GASTER Figure 2. A minimal position representative of the curve γ . 4 the concatenation of any pair of arcs of γ that connect the two points n § forms an essential curve. The ‘bigon criterion’ of [FM, 1.2.4] can be altered straightforwardly to an ‘immersed bigon criterion’ in the setting of curves with self-intersections, and so the lack of immersed bigons guarantees that (cid:3) the chosen representative of γ is in minimal position. n We use Lemma 5 to estimate deg(γ ), a calculation reminiscent of [GK, n Lemma 3.10]. The following proposition is the main tool in our analysis. ≥ Proposition 6. We have deg(γ ) n + 1. n (cid:48) → Proof. Towards contradiction, suppose there is a cover P P of degree ≤ 0 k n, so that γ lifts to a simple curve γ. Draw P as a directed ribbon n 0 (cid:48) graph with one vertex p and the two edges labeled by a and b, and P as a directed ribbon graph with vertices p , . . . , p and 2k directed edges, k with 1 k a labels and k with b labels. Choose an orientation for γ so that γ consists of a directed a edge followed by n directed b edges. After relabeling, we may assume that the unique a edge of γ is followed by p . 1 Starting from p and reading the vertices visited by γ in order, the vertex 1 that immediately follows the n consecutive b edges of γ is p , where l is l equivalent to n + 1 modulo k. Finally, γ follows an a edge from p to p . See l 1 Figure 3 for a schematic. (cid:48)(cid:48) This implies that there is an incompressible embedded pair of pants P (cid:48) (cid:45) (cid:4) (cid:5) in P that contains γ (see Figure 4 in the case that k n – the other case is (cid:48)(cid:48) straightforwardly similar). After identifying P with P appropriately, the · 0 ≥ closed curve γ is given by the equivalence class of a bs, where s = n 1. k (cid:3) By Lemma 5 this curve is not simple, a contradiction. Remark. In fact, one can show that deg(γ ) = n + 1, but the precise n computation of deg(γ ) is irrelevant. n ##LTFigure## ##LTLine##

text: JONAH

text: GASTER

text: Figure

text: A

text: minimal

text: position

text: representative

text: of

text: the

text: curve

text: γ

text: the

text: concatenation

text: of

text: any

text: pair

text: of

text: arcs

text: of

text: γ

text: that

text: connect

text: the

text: two

text: points

text: n

text: forms

text: an

text: essential

text: curve.

text: The

text: ‘bigon

text: criterion’

text: of

text: [FM,

text: can

text: be

text: altered

text: straightforwardly

text: to

text: an

text: ‘immersed

text: bigon

text: criterion’

text: in

text: the

text: setting

text: of

text: curves

text: with

text: self-intersections,

text: and

text: so

text: the

text: lack

text: of

text: immersed

text: bigons

text: guarantees

text: that

text: the

text: chosen

text: representative

text: of

text: γ

text: is

text: in

text: minimal

text: position.

text: n

text: We

text: use

text: Lemma

text: to

text: estimate

text: deg(γ

text: a

text: calculation

text: reminiscent

text: of

text: [GK,

text: n

text: Lemma

text: The

text: following

text: proposition

text: is

text: the

text: main

text: tool

text: in

text: our

text: analysis.

text: Proposition

text: We

text: have

text: deg(γ

text: n

text: n

text: Proof.

text: Towards

text: contradiction,

text: suppose

text: there

text: is

text: a

text: cover

text: P

text: P

text: of

text: degree

text: k

text: n,

text: so

text: that

text: γ

text: lifts

text: to

text: a

text: simple

text: curve

text: γ.

text: Draw

text: P

text: as

text: a

text: directed

text: ribbon

text: n

text: graph

text: with

text: one

text: vertex

text: p

text: and

text: the

text: two

text: edges

text: labeled

text: by

text: a

text: and

text: b,

text: and

text: P

text: as

text: a

text: directed

text: ribbon

text: graph

text: with

text: vertices

text: p

text: p

text: and

text: directed

text: edges,

text: k

text: with

text: k

text: a

text: labels

text: and

text: k

text: with

text: b

text: labels.

text: Choose

text: an

text: orientation

text: for

text: γ

text: so

text: that

text: γ

text: consists

text: of

text: a

text: directed

text: a

text: edge

text: followed

text: by

text: n

text: directed

text: b

text: edges.

text: After

text: relabeling,

text: we

text: may

text: assume

text: that

text: the

text: unique

text: a

text: edge

text: of

text: γ

text: is

text: followed

text: by

text: p

text: Starting

text: from

text: p

text: and

text: reading

text: the

text: vertices

text: visited

text: by

text: γ

text: in

text: order,

text: the

text: vertex

text: that

text: immediately

text: follows

text: the

text: n

text: consecutive

text: b

text: edges

text: of

text: γ

text: is

text: p

text: where

text: l

text: is

text: l

text: equivalent

text: to

text: n

text: modulo

text: k.

text: Finally,

text: γ

text: follows

text: an

text: a

text: edge

text: from

text: p

text: to

text: p

text: See

text: l

text: Figure

text: for

text: a

text: schematic.

text: This

text: implies

text: that

text: there

text: is

text: an

text: incompressible

text: embedded

text: pair

text: of

text: pants

text: P

text: in

text: P

text: that

text: contains

text: γ

text: (see

text: Figure

text: in

text: the

text: case

text: that

text: k

text: n

text: the

text: other

text: case

text: is

text: straightforwardly

text: similar).

text: After

text: identifying

text: P

text: with

text: P

text: appropriately,

text: the

text: closed

text: curve

text: γ

text: is

text: given

text: by

text: the

text: equivalence

text: class

text: of

text: a

text: bs,

text: where

text: s

text: n

text: k

text: By

text: Lemma

text: this

text: curve

text: is

text: not

text: simple,

text: a

text: contradiction.

text: Remark.

text: In

text: fact,

text: one

text: can

text: show

text: that

text: deg(γ

text: n

text: but

text: the

text: precise

text: n

text: computation

text: of

text: deg(γ

text: is

text: irrelevant.

text: n

text: ##LTFigure##

text: ##LTLine##