text: 7 TABLE II. Sources observed by Sluse et al. [22], including the coordinates and redshifts listed in the reference. All observations were made using the EFOSC2 instrument. Source RA Dec. Redshift Polarization ζm(5a)x γmax ◦ ◦ − − − J2000 [ ] J2000 [ ] z [%] [1018GeV 2] [10 24GeV 1] SDSS J0242+0049 40.591 +0.820 2.071 1.47(24) 4.3 2.83 FIRST03133+0036 48.328 +0.606 1.250 1.48(29) 4.4 4.62 FIRST J0809+2753 122.256 +27.895 1.511 1.75(20) 3.9 3.42 PG 0946+301 147.421 +29.922 1.220 1.65(19) 4.0 4.30 − PKS 1124-186 171.768 17.045 1.048 11.68(36) 1.2 1.50 − He 1127-1304 172.583 12.653 0.634 1.32(13) 4.2 8.70 2QZ J114954+0012 177.479 +0.215 1.596 1.57(22) 4.1 3.42 SDSS J1206+0023 181.615 +0.393 2.331 0.94(15) 4.9 2.91 SDSS J1214-0001 183.673 +0.027 1.041 2.40(32) 3.5 4.39 PKS 1219+04 185.594 +4.221 0.965 5.56(15) 2.2 2.98 PKS 1222+037 186.218 +3.514 0.960 2.51(22) 3.3 4.49 TON 1530 186.364 +22.587 2.058 0.92(14) 4.9 3.25 SDSS J1234+0057 188.616 +0.966 1.532 1.35(23) 4.4 3.81 PG 1254+047 194.250 +4.459 1.018 0.84(15) 5.1 6.55 − PKS 1256-229 194.785 22.823 1.365 22.32(15) 1.0 0.97 SDSS J1302-0037 195.534 +0.626 1.672 1.37(20) 4.3 3.43 − PKS 1303-250 196.564 24.711 0.738 0.91(17) 5.0 8.87 FIRST J1312+2319 198.056 +23.333 1.508 1.10(16) 4.6 4.04 SDSS J1323-0038 200.769 +0.649 1.827 1.13(21) 4.7 3.46 − CTS J13.07 205.518 17.700 2.210 0.83(15) 5.1 3.17 SDSS J1409+0048 212.328 +0.807 1.999 3.91(28) 2.6 1.77 HS 1417+2547 215.055 +25.568 2.200 1.03(18) 4.8 3.00 FIRST J1427+2709 216.765 +27.161 1.170 1.35(25) 4.5 5.04 − FIRST J21079-0620 316.990 5.664 0.644 1.12(22) 4.8 9.79 − SDSS J2131-0700 322.912 6.996 2.048 1.78(32) 4.1 2.73 − PKS 2204-54 331.932 52.224 1.206 1.81(26) 3.9 4.24 − PKS 2227-445 337.735 43.725 1.326 5.26(48) 2.4 2.38 − PKS 2240-260 340.860 24.258 0.774 14.78(21) 1.1 1.86 PKS 2301+06 346.118 +6.336 1.268 3.69(26) 2.7 2.80 SDSS J2319-0024 349.995 +0.414 1.889 1.85(30) 4.0 2.86 − PKS 2320-035 350.883 2.715 1.411 9.56(20) 1.2 1.12 − PKS 2332-017 353.835 0.481 1.184 4.86(19) 2.3 2.55 − PKS 2335-027 354.489 1.484 1.072 3.55(30) 2.9 3.54 SDSS J2352+0105 358.159 +1.098 2.156 1.59(26) 4.2 2.67 SDSS J2356-0036 359.120 +0.601 2.936 1.81(34) 4.1 2.01 − QSO J2359-12 359.973 11.303 0.868 4.12(20) 2.5 3.76 TABLE III. Optical GRB measurements. Source Instrument RA Dec. Redshift Polarization ζm(5a)x γmax Refs. ◦ ◦ − − − J2000 [ ] J2000 [ ] z [%] [1018GeV 2] [10 24GeV 1] − ± GRB 990510 FORS1 R-band 204.532 80.497 1.619 1.6 0.2 7.00 5.76 [24] − ± GRB 990712 FORS1 R-band 337.971 73.408 0.430 2.9 0.4 5.50 17.0 [25] − ± GRB 020813 FORS1 V-band 296.674 19.601 1.25 1.42 0.25 4.60 4.83 [28] ± GRB 021004 NOT/ALFOSC 6.728 +18.928 2.330 2.1 0.6 7.50 4.46 [26] ± GRB 030329 NOT/ALFOSC 161.208 +21.522 0.169 2.4 0.4 6.30 51.6 [27] ± GRB 090102 RINGO 128.248 +33.107 1.547 10.2 1.3 1.30 1.12 [29] − ± GRB 091018 FORS2 32.192 57.55 0.97 3.25 0.35 7.55 1.02 [30] ± GRB 091208B HOWPol 29.410 16.881 1.06 10.4 2.5 7.30 9.00 [31] − ± GRB 121024A FORS2 70.467 12.268 2.298 4.83 0.20 5.95 3.58 [32] ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: TABLE

text: II.

text: Sources

text: observed

text: by

text: Sluse

text: et

text: al.

text: including

text: the

text: coordinates

text: and

text: redshifts

text: listed

text: in

text: the

text: reference.

text: All

text: observations

text: were

text: made

text: using

text: the

text: instrument.

text: Source

text: RA

text: Dec.

text: Redshift

text: Polarization

text: γmax

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text: PG

text: PKS

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text: SDSS

text: PKS

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text: SDSS

text: PG

text: PKS

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text: PKS

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text: CTS

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text: FIRST

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text: PKS

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text: SDSS

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text: SDSS

text: QSO

text: TABLE

text: III.

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text: GRB

text: measurements.

text: Source

text: Instrument

text: RA

text: Dec.

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text: Polarization

text: γmax

text: Refs.

text: z

text: GRB

text: R-band

text: GRB

text: R-band

text: GRB

text: V-band

text: GRB

text: NOT/ALFOSC

text: GRB

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text: 2 IEEE SIGNAL PROCESSING LETTERS (PREPRINT) denoising can be calculated exactly in ﬁnite-time [17], [18], IV. NON-CONVEX PENALTY [22], [29]. To strongly induce sparsity of Dx, we deﬁne a non-convex generalization of the standard TV penalty. The new penalty is III. MOREAU ENVELOPE deﬁned by subtracting a differentiable convex function from Before we deﬁne the non-differentiable non-convex penalty the standard penalty. in Sec. , we ﬁrst deﬁne a differentiable convex function. We Deﬁnition 3. Let α (cid:62) 0. We deﬁne the penalty ψ : RN → R α use the Moreau envelope from convex analysis [2]. as Deﬁnition 2. Let α (cid:62) 0. W(cid:8)e deﬁne S : RN → R(cid:9) as α (cid:107) (cid:107) − ψ (x) = Dx S (x) (11) (cid:107) (cid:107) (cid:107) − (cid:107) α 1 α S (x) = min Dv + α x v 2 (4) α ∈R 1 2 2 v N where D is the matrix (3) and S is deﬁned by (4). α where D is the ﬁrst-order difference matrix (3). The proposed penalty is upper bounded by the standard TV − If α > 0, then S is the Moreau envelope of index α 1 of penalty, which is recovered as a special case. (cid:55)→ (cid:107)α (cid:107) the function x Dx 1. Proposition 5. Let α (cid:62) 0. The penalty ψ satisﬁes α Proposition 1. The function S can be calculated by α (cid:107) (cid:107) ∀ ∈ R ψ (x) = Dx , x N (12) 0 1 S (x) = 0 (5) 0 (cid:107) (cid:107) and S (x) = D tvd(x; 1/α) α 1 (cid:107) − (cid:107) + α x tvd(x; 1/α) 2, α > 0. (6) (cid:54) (cid:54) (cid:107) (cid:107) ∀ ∈ R 2 2 0 ψ (x) Dx , x N. (13) α 1 Proof. For α = 0: Setting α = 0 and v = 0 in (4) gives (5). ∈ R Proof. It follows from (5) and (7). For α > 0: By the deﬁnition of TV denoising, the v N minimizing the function in (4) is the TV denoising of x, i.e., When a convex function is subtracted from another convex vopt = tvd(x, 1/α). function [as in (11)], the resulting function may well be (cid:62) negative on part of its domain. Inequality (13) states that the Proposition 2. Let α 0. The function S satisﬁes α proposed penalty ψ avoids this fate. This is relevant because α (cid:54) (cid:54) (cid:107) (cid:107) ∀ ∈ R the penalty function should be non-negative. 0 S (x) Dx , x N. (7) α 1 Figures in the supplemental material show examples of the proposed penalty ψ and the function S . α α (cid:54) (cid:107) (cid:107) (cid:107) − (cid:107) Proof. From (4), we have S (x) Dv + (α/2) x v 2 for all v ∈ RN. In particular,αv = x leads t1o S (x) (cid:54) (cid:107)Dx(cid:107) 2. V. ENHANCED TV DENOISING (cid:62) α 1 Also, S (x) 0 since S (x) is deﬁned as the minimum of a We deﬁne ‘Moreau-enhanced’ TV denoising. If α > 0, then α α non-negative function. the proposed penalty penalizes large amplitude values of Dx (cid:54) (cid:107) (cid:107) Proposition 3. Let α (cid:62) 0. The function Sα is convex and lleessss tlhikaenlythteo (cid:96)u1ndneorremstidmoeaste(ij.uem.,pψdαi(sxc)ontinuDitxies1.), hence it is differentiable. Deﬁnition 4. Given y ∈ RN, λ > 0, and α (cid:62) 0, we deﬁne Proof. It follows from Proposition 12.15 in Ref. [2]. Moreau-enhanced total variat(cid:8)ion denoising as (cid:9) (cid:62) Proposition 4. Let α 0. The gradient of S is given by α (cid:107) − (cid:107) mtvd(y; λ, α) = arg min 1 y x 2 + λψ (x) (14) (cid:0) (cid:1) ∈R 2 2 α ∇ x N S (x) = 0 (8) 0 ∇S (x) = α x − tvd(x; 1/α) , α > 0 (9) where ψα is given by (11). α The parameter α controls the non-convexity of the penalty. where tvd denotes total variation denoising (1). If α = 0, then the penalty is convex and Moreau-enhanced TV denoising reduces to TV denoising. Greater values of α make − Proof. Since S is the Moreau envelope of index α 1 of the the penalty more non-convex. What is the greatest value of (cid:55)→ (cid:107)α (cid:107) function x Dx 1 wh(cid:0)en α > 0, it follows by(cid:1) Proposition α that maintains convexity of the cost function? The critical 12.29 in Ref. [2] that value is given by Theorem 1. ∇S (x) = α x − prox (cid:107) ·(cid:107) (x) . (10) Theorem 1. Let λ > 0 and α (cid:62) 0. Deﬁne Fα : RN → R as α (1/α) D 1 (cid:107) − (cid:107) F (x) = 1 y x 2 + λψ (x) (15) This proximity operator is TV denoising, giving (9). α 2 2 α ##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: paragraph

text: IEEE

text: SIGNAL

text: PROCESSING

text: LETTERS

text: (PREPRINT)

text: denoising

text: can

text: be

text: calculated

text: exactly

text: in

text: ﬁnite-time

text: IV.

label: section

text: NON-CONVEX

text: PENALTY

text: To

text: strongly

text: induce

text: sparsity

text: of

text: Dx,

text: we

text: deﬁne

text: a

text: non-convex

text: generalization

text: of

text: the

text: standard

text: TV

text: penalty.

text: The

text: new

text: penalty

text: is

text: III.

text: MOREAU

text: ENVELOPE

text: deﬁned

text: by

text: subtracting

text: a

text: differentiable

text: convex

text: function

text: from

text: Before

text: we

text: deﬁne

text: the

text: non-differentiable

text: non-convex

text: penalty

text: the

text: standard

text: penalty.

text: in

text: Sec.

text: we

text: ﬁrst

text: deﬁne

text: a

text: differentiable

text: convex

text: function.

text: We

text: Deﬁnition

text: Let

text: α

text: We

text: deﬁne

text: the

text: penalty

text: ψ

text: RN

text: R

text: α

text: use

text: the

text: Moreau

text: envelope

text: from

text: convex

text: analysis

text: as

text: Deﬁnition

text: Let

text: α

text: deﬁne

text: S

text: RN

text: as

text: α

text: ψ

text: (x)

text: Dx

text: S

text: (x)

text: α

text: α

text: S

text: (x)

text: min

text: Dv

text: α

text: x

text: v

text: α

text: ∈R

text: v

text: N

text: where

text: D

text: is

text: the

text: matrix

text: and

text: S

text: is

text: deﬁned

text: by

text: α

text: where

text: D

text: is

text: the

text: ﬁrst-order

text: difference

text: matrix

text: The

text: proposed

text: penalty

text: is

text: upper

text: bounded

text: by

text: the

text: standard

text: TV

text: If

text: α

text: then

text: S

text: is

text: the

text: Moreau

text: envelope

text: of

text: index

text: α

text: of

text: penalty,

text: which

text: is

text: recovered

text: as

text: a

text: special

text: case.

text: the

text: function

text: x

text: Dx

text: Proposition

text: Let

text: α

text: The

text: penalty

text: ψ

text: satisﬁes

text: α

text: Proposition

text: The

text: function

text: S

text: can

text: be

text: calculated

text: by

text: α

text: R

text: ψ

text: (x)

text: Dx

text: x

text: N

text: S

text: (x)

text: and

text: S

text: (x)

text: D

text: tvd(x;

text: α

text: α

text: x

text: tvd(x;

text: α

text: R

text: ψ

text: (x)

text: Dx

text: x

text: N.

text: α

text: Proof.

text: For

text: α

text: Setting

text: α

text: and

text: v

text: in

text: gives

text: R

text: Proof.

text: It

text: follows

text: from

text: and

text: For

text: α

text: By

text: the

text: deﬁnition

text: of

text: TV

text: denoising,

text: the

text: v

text: N

text: minimizing

text: the

text: function

text: in

text: is

text: the

text: TV

text: denoising

text: of

text: x,

text: i.e.,

text: When

text: a

text: convex

text: function

text: is

text: subtracted

text: from

text: another

text: convex

text: vopt

text: tvd(x,

text: function

text: [as

text: in

text: the

text: resulting

text: function

text: may

text: well

text: be

text: negative

text: on

text: part

text: of

text: its

text: domain.

text: Inequality

text: states

text: that

text: the

text: Proposition

text: Let

text: α

text: The

text: function

text: S

text: satisﬁes

text: α

text: proposed

text: penalty

text: ψ

text: avoids

text: this

text: fate.

text: This

text: is

text: relevant

text: because

text: α

text: R

text: the

text: penalty

text: function

text: should

text: be

text: non-negative.

text: S

text: (x)

text: Dx

text: x

text: N.

text: α

text: Figures

text: in

text: the

text: supplemental

text: material

text: show

text: examples

text: of

text: the

text: proposed

text: penalty

text: ψ

text: and

text: the

text: function

text: S

text: α

text: α

text: Proof.

text: From

text: we

text: have

text: S

text: (x)

text: Dv

text: x

text: v

text: for

text: all

text: v

text: RN.

text: In

text: particular,αv

text: x

text: leads

text: S

text: (x)

text: V.

text: ENHANCED

text: TV

text: DENOISING

text: α

text: Also,

text: S

text: (x)

text: since

text: S

text: (x)

text: is

text: deﬁned

text: as

text: the

text: minimum

text: of

text: a

text: We

text: deﬁne

text: ‘Moreau-enhanced’

text: TV

text: denoising.

text: If

text: α

text: then

text: α

text: α

text: non-negative

text: function.

text: the

text: proposed

text: penalty

text: penalizes

text: large

text: amplitude

text: values

text: of

text: Dx

text: Proposition

text: Let

text: α

text: The

text: function

text: Sα

text: is

text: convex

text: and

text: lleessss

text: tlhikaenlythteo

text: hence

text: it

text: is

text: differentiable.

text: Deﬁnition

text: Given

text: y

text: RN,

text: λ

text: and

text: α

text: we

text: deﬁne

text: Proof.

text: It

text: follows

text: from

text: Proposition

text: in

text: Ref.

text: Moreau-enhanced

text: total

text: denoising

text: as

text: Proposition

text: Let

text: α

text: The

text: gradient

text: of

text: S

text: is

text: given

text: by

text: α

text: mtvd(y;

text: λ,

text: α)

text: arg

text: min

text: y

text: x

text: λψ

text: (x)

text: ∈R

text: α

text: x

text: N

text: S

text: (x)

text: ∇S

text: (x)

text: α

text: x

text: tvd(x;

text: α

text: where

text: ψα

text: is

text: given

text: by

text: α

text: The

text: parameter

text: α

text: controls

text: the

text: non-convexity

text: of

text: the

text: penalty.

text: where

text: tvd

text: denotes

text: total

text: variation

text: denoising

text: If

text: α

text: then

text: the

text: penalty

text: is

text: convex

text: and

text: Moreau-enhanced

text: TV

text: denoising

text: reduces

text: to

text: TV

text: denoising.

text: Greater

text: values

text: of

text: α

text: make

text: Proof.

text: Since

text: S

text: is

text: the

text: Moreau

text: envelope

text: of

text: index

text: α

text: of

text: the

text: the

text: penalty

text: more

text: non-convex.

text: What

text: is

text: the

text: greatest

text: value

text: of

text: function

text: x

text: Dx

text: α

text: it

text: follows

text: Proposition

text: α

text: that

text: maintains

text: convexity

text: of

text: the

text: cost

text: function?

text: The

text: critical

text: in

text: Ref.

text: that

text: value

text: is

text: given

text: by

text: Theorem

text: ∇S

text: (x)

text: α

text: x

text: prox

text: (x)

text: Theorem

text: Let

text: λ

text: and

text: α

text: Deﬁne

text: Fα

text: RN

text: R

text: as

text: α

text: D

text: F

text: (x)

text: y

text: x

text: λψ

text: (x)

text: This

text: proximity

text: operator

text: is

text: TV

text: denoising,

text: giving

text: α

text: α

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text: ##LTLine##

text: ##LTLine##


text: 3 where ψ is given by (11). If α (cid:107) − (cid:107) (cid:54) (cid:54) F (x) = 1 y x 2 + λψ (x), (25) 0 α 1/λ (16) α 2(cid:107) − (cid:107)2 (cid:107) α (cid:107) − = 1 y x 2 + λ Dx λS (x) (26) 2 2 1 α then F is convex. If 0 (cid:54) α < 1/λ then F is strongly = f1(x) + f2(x) (27) α α convex. where Proof. We write the cost function as (cid:107) − (cid:107) − F (x) = 1(cid:107)y − x(cid:107)2 + λ(cid:8)(cid:107)Dx(cid:107) − λS (x) (cid:9) (17) f1(x) = 12(cid:107)y (cid:107)x 22 λSα(x) (28a) α 2(cid:107) − (cid:107)2 (cid:107) (cid:107)1 α f2(x) = λ Dx 1. (28b) = 1 y x 2 + λ Dx 2 −(cid:8) 2 (cid:107) (cid:107)1 (cid:107) − (cid:107) λ min Dv + α x v 2 (18) ∈R 1 2 2 The gradient of f is given by v N (cid:9) 1 (cid:107) − (cid:107) (cid:107) (cid:107) = max 1 y x 2 + λ Dx (cid:0) (cid:1) v∈RN(cid:8)2 2 1 (cid:9) ∇ − − ∇ − λ(cid:107)Dv(cid:107) − λα(cid:107)x − v(cid:107)2 (19) f1(x) = x y λ Sα(x) (29) = max 1(1 − λα)(cid:107)x(cid:107)21+ λ(cid:107)2Dx(cid:107) + g2(x, v) (20) = x − y − λα x − tvd(x; 1/α) (30) ∈R 2 2 1 v N − (cid:107) (cid:107) (cid:107) (cid:107) = 1(1 λα) x 2 + λ Dx + max g(x, v) (21) 2 2 1 ∈R using Proposition 4. Subtracting S from f does not v N ∇ α 1 increase the Lipschitz constant of f , the value of which 1 where g(x, v) is afﬁne in x. The last term is convex as it is is 1. Hence, we may set 0 < µ < 2. the point-wise maximum of a set of convex functions. Hence, Using (28), the FBS(cid:2)algorithm (24) becomes − (cid:62) − F is a convex function if 1 λα 0. If 1 λα > 0, then (cid:0) (cid:1)(cid:3) α Fα is strongly convex (and strictly convex). z(k) = x(k) − µ(cid:8)x(k) − y (cid:9) − − λα x(k) tvd(x(k); 1/α) (31a) (cid:107) − (cid:107) (cid:107) (cid:107) VI. ALGORITHM x(k+1) = arg min 1 z(k) x 2 + µλ Dx . (31b) Proposition 6. Let y ∈ RN, λ > 0, and 0 < α < 1/λ. Then x 2 2 1 x(k) produced by the iter(cid:0)ation (cid:1) Note that (31b) is TV denoising (1). Using the value µ = 1 gives iteration (22). (Experimentally, we found this value − yields fast convergence.) z(k) = y + λα x(k) tvd(x(k); 1/α) (22a) x(k+1) = tvd(z(k); λ). (22b) Each iteration of (22) entails solving two standard TV denoising problems. In this work, we calculate TV denoising using the fast exact C language program by Condat [17]. Like converges to the solution of the Moreau-enhanced TV de- the iterative shrinkage/thresholding algorithm (ISTA) [19], noising problem (14). [25], algorithm (22) can be accelerated in various ways. Proof. If the cost function (15) is strongly convex, then We suggest not setting α too close to the critical value 1/λ because the FBS algorithm generally converges faster when the minimizer can be calculated using the forward-backward splitting (FBS) algorithm [2], [16]. This algorithm minimizes the cost function is more strongly convex (α < 1). In summary, the proposed Moreau-enhanced TV denoising a function of the form method comprises the steps: F(x) = f1(x) + f2(x) (23) 1) Set the regularization parameter λ (λ >(cid:54)0). 2) Set the non-convexity parameter α (0 α < 1/λ). ∇ 3) Initialize x(0) = 0. where both f and f are convex and f is additionally 4) Run iteration (22) until convergence. 1 2 1 Lipschitz continuous. Th(cid:2)e FBS algo(cid:3)rithm is given by (cid:8) (cid:9) VII. OPTIMALITY CONDITION − ∇ z(k) = x(k) µ f (x(k)) (24a) To avoid terminating the iterative algorithm too early, it is 1 (cid:107) − (cid:107) useful to verify convergence using an optimality condition. x(k+1) = arg mxin 12 z(k) x 22 + µf2(x) (24b) Proposition 7. Let y ∈ RN, λ > 0, and 0 < α < 1/λ. If x is a(cid:2)so(cid:0)lution to (14), then (cid:1)(cid:3) ∇ where 0 < µ < 2/ρ and ρ is the Lipschitz constant of f . 1 The iterates x(k) converge to a minimizer of F. − − ∈ C (x y)/λ + α(tvd(x; 1/α) x) sign([Dx] ) (32) To apply the FBS algorithm to the proposed cost function n n (15), we write it as ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: ψ

text: is

text: given

text: by

text: If

text: α

label: equation

text: F

text: (x)

text: y

text: x

text: λψ

text: (x),

text: α

text: α

text: α

text: y

text: x

text: λ

text: Dx

text: λS

text: (x)

text: α

text: then

text: F

text: is

text: convex.

text: If

text: α

text: then

text: F

text: is

text: strongly

text: α

text: α

text: convex.

text: where

text: Proof.

text: We

text: write

text: the

text: cost

text: function

text: as

text: F

text: (x)

text: λS

text: (x)

text: λSα(x)

text: α

text: α

text: λ

text: Dx

text: y

text: x

text: λ

text: Dx

text: λ

text: min

text: Dv

text: α

text: x

text: v

text: ∈R

text: The

text: gradient

text: of

text: f

text: is

text: given

text: by

text: v

text: N

text: max

text: y

text: x

text: λ

text: Dx

text: x

text: y

text: λ

text: Sα(x)

text: max

text: v)

text: x

text: y

text: λα

text: x

text: tvd(x;

text: ∈R

text: v

text: N

text: λα)

text: x

text: λ

text: Dx

text: max

text: g(x,

text: v)

text: ∈R

text: using

text: Proposition

text: Subtracting

text: S

text: from

text: f

text: does

text: not

text: v

text: N

text: α

text: increase

text: the

text: Lipschitz

text: constant

text: of

text: f

text: the

text: value

text: of

text: which

text: where

text: g(x,

text: v)

text: is

text: afﬁne

text: in

text: x.

text: The

text: last

text: term

text: is

text: convex

text: as

text: it

text: is

text: is

text: Hence,

text: we

text: may

text: set

text: µ

text: the

text: point-wise

text: maximum

text: of

text: a

text: set

text: of

text: convex

text: functions.

text: Hence,

text: Using

text: the

text: becomes

text: F

text: is

text: a

text: convex

text: function

text: if

text: λα

text: If

text: λα

text: then

text: α

text: Fα

text: is

text: strongly

text: convex

text: (and

text: strictly

text: convex).

text: z(k)

text: x(k)

text: y

text: λα

text: x(k)

text: tvd(x(k);

text: VI.

text: ALGORITHM

text: arg

text: min

text: z(k)

text: x

text: µλ

text: Dx

text: Proposition

text: Let

text: y

text: RN,

text: λ

text: and

text: α

text: Then

text: x

text: x(k)

text: produced

text: by

text: the

text: Note

text: that

text: is

text: TV

text: denoising

text: Using

text: the

text: value

text: µ

text: gives

text: iteration

text: (Experimentally,

text: we

text: found

text: this

text: value

text: yields

text: fast

text: convergence.)

text: z(k)

text: y

text: λα

text: x(k)

text: tvd(x(k);

text: tvd(z(k);

text: λ).

text: Each

text: iteration

text: of

text: entails

text: solving

text: two

text: standard

text: TV

text: denoising

text: problems.

text: In

text: this

text: work,

text: we

text: calculate

text: TV

text: denoising

text: using

text: the

text: fast

text: exact

text: C

text: language

text: program

text: by

text: Condat

text: Like

text: converges

text: to

text: the

text: solution

text: of

text: the

text: Moreau-enhanced

text: TV

text: de-

text: the

text: iterative

text: shrinkage/thresholding

text: algorithm

text: (ISTA)

text: noising

text: problem

text: algorithm

text: can

text: be

text: accelerated

text: in

text: various

text: ways.

text: Proof.

text: If

text: the

text: cost

text: function

text: is

text: strongly

text: convex,

text: then

text: We

text: suggest

text: not

text: setting

text: α

text: too

text: close

text: to

text: the

text: critical

text: value

text: because

text: the

text: FBS

text: algorithm

text: generally

text: converges

text: faster

text: when

text: the

text: minimizer

text: can

text: be

text: calculated

text: using

text: the

text: forward-backward

text: splitting

text: (FBS)

text: algorithm

text: This

text: algorithm

text: minimizes

text: the

text: cost

text: function

text: is

text: more

text: strongly

text: convex

text: (α

text: In

text: summary,

text: the

text: proposed

text: Moreau-enhanced

text: TV

text: denoising

text: a

text: function

text: of

text: the

text: form

text: method

text: comprises

text: the

text: steps:

text: F(x)

text: Set

text: the

text: regularization

text: parameter

text: λ

text: (λ

text: Set

text: the

text: non-convexity

text: parameter

text: α

text: α

text: Initialize

text: where

text: both

text: f

text: and

text: f

text: are

text: convex

text: and

text: f

text: is

text: additionally

text: Run

text: iteration

text: until

text: convergence.

text: Lipschitz

text: continuous.

text: FBS

text: is

text: given

text: by

text: VII.

text: OPTIMALITY

text: CONDITION

text: z(k)

text: x(k)

text: µ

text: f

text: (x(k))

text: To

text: avoid

text: terminating

text: the

text: iterative

text: algorithm

text: too

text: early,

text: it

text: is

text: useful

text: to

text: verify

text: convergence

text: using

text: an

text: optimality

text: condition.

text: arg

text: mxin

text: z(k)

text: x

text: Proposition

text: Let

text: y

text: RN,

text: λ

text: and

text: α

text: If

text: x

text: is

text: to

text: then

text: where

text: µ

text: and

text: ρ

text: is

text: the

text: Lipschitz

text: constant

text: of

text: f

text: The

text: iterates

text: x(k)

text: converge

text: to

text: a

text: minimizer

text: of

text: F.

text: C

text: (x

text: y)/λ

text: α(tvd(x;

text: x)

text: sign([Dx]

text: To

text: apply

text: the

text: FBS

text: algorithm

text: to

text: the

text: proposed

text: cost

text: function

text: n

text: n

text: we

text: write

text: it

text: as

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: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 6 IEEE SIGNAL PROCESSING LETTERS (PREPRINT) [34] H. Mohimani, M. Babaie-Zadeh, and C. Jutten. A fast approach for overcomplete sparse decomposition based on smoothed l0 norm. IEEE Trans. Signal Process., 57(1):289–301, January 2009. [35] T. Mo¨llenhoff, E. Strekalovskiy, M. Moeller, and D. Cremers. The primal-dual hybrid gradient method for semiconvex splittings. SIAM J. Imag. Sci., 8(2):827–857, 2015. [36] M. Nikolova. Estimation of binary images by minimizing convex criteria. In Proc. IEEE Int. Conf. Image Processing (ICIP), pages 108– 112 vol. 2, 1998. [37] M. Nikolova. Local strong homogeneity of a regularized estimator. SIAM J. Appl. Math., 61(2):633–658, 2000. [38] M. Nikolova. Analysis of the recovery of edges in images and signals by minimizing nonconvex regularized least-squares. Multiscale Model. Simul., 4(3):960–991, 2005. [39] M. Nikolova. Energy minimization methods. In O. Scherzer, editor, Handbook of Mathematical Methods in Imaging, chapter 5, pages 138– 186. Springer, 2011. [40] M. Nikolova, M. K. Ng, and C.-P. Tam. Fast nonconvex nonsmooth minimization methods for image restoration and reconstruction. IEEE Trans. Image Process., 19(12):3073–3088, December 2010. [41] A. Parekh and I. W. Selesnick. Convex denoising using non-convex tight frame regularization. IEEE Signal Processing Letters, 22(10):1786– 1790, October 2015. [42] A. Parekh and I. W. Selesnick. Enhanced low-rank matrix approxima- tion. IEEE Signal Processing Letters, 23(4):493–497, April 2016. [43] J. Portilla and L. Mancera. L0-based sparse approximation: two alternative methods and some applications. In Proceedings of SPIE, volume 6701 (Wavelets XII), San Diego, CA, USA, 2007. [44] P. Rodriguez and B. Wohlberg. Efﬁcient minimization method for a generalized total variation functional. IEEE Trans. Image Process., 18(2):322–332, February 2009. [45] L. Rudin, S. Osher, and E. Fatemi. Nonlinear total variation based noise removal algorithms. Physica D, 60:259–268, 1992. [46] I. W. Selesnick and I. Bayram. Sparse signal estimation by maximally sparse convex optimization. IEEE Trans. Signal Process., 62(5):1078– 1092, March 2014. [47] I. W. Selesnick and I. Bayram. Enhanced sparsity by non-separable regularization. IEEE Trans. Signal Process., 64(9):2298–2313, May 2016. [48] I. W. Selesnick, A. Parekh, and I. Bayram. Convex 1-D total variation denoising with non-convex regularization. IEEE Signal Processing Letters, 22(2):141–144, February 2015. [49] S. Setzer, G. Steidl, and T. Teuber. Inﬁmal convolution regularizations with discrete l1-type functionals. Commun. Math. Sci., 9(3):797–827, 2011. [50] M. Storath, A. Weinmann, and L. Demaret. Jump-sparse and sparse recovery using Potts functionals. IEEE Trans. Signal Process., 62(14):3654–3666, July 2014. [51] D. P. Wipf, B. D. Rao, and S. Nagarajan. Latent variable Bayesian models for promoting sparsity. IEEE Trans. Inform. Theory, 57(9):6236– 6255, September 2011. [52] C.-H. Zhang. Nearly unbiased variable selection under minimax concave penalty. The Annals of Statistics, pages 894–942, 2010. [53] H. Zou and R. Li. One-step sparse estimates in nonconcave penalized likelihood models. Ann. Statist., 36(4):1509–1533, 2008.

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text: SIGNAL

text: PROCESSING

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text: smoothed

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text: M.

text: Moeller,

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text: primal-dual

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text: Parekh,

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text: Bayram.

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text: Teuber.

text: Inﬁmal

text: convolution

text: regularizations

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text: discrete

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text: Math.

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text: M.

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text: A.

text: Weinmann,

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text: variable

text: Bayesian

text: models

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text: promoting

text: sparsity.

text: IEEE

text: Trans.

text: Inform.

text: Theory,

text: September

text: C.-H.

text: Zhang.

text: Nearly

text: unbiased

text: variable

text: selection

text: under

text: minimax

text: concave

text: penalty.

text: The

text: Annals

text: of

text: Statistics,

text: pages

text: H.

text: Zou

text: and

text: R.

text: Li.

text: One-step

text: sparse

text: estimates

text: in

text: nonconcave

text: penalized

text: likelihood

text: models.

text: Ann.

text: Statist.,


text: (cid:101) (cid:101) PRESENTATIONS OF NET MAPS 13 postcritical set of f, and let P be the inverse image of P in(cid:101)this union. So P consists of 2 (cid:48) 2 2 four points which are either in the interior of F (cid:101)or in the left half of its boundary. (cid:48) 1(cid:48) (cid:48) (cid:48) We begin to choose four line segments α , α , α , α . If P contains an element in th(cid:101)e (cid:48) 1 2 3 4 2 (cid:48) bottom of F , then we choose the element of P in the bottom of F which is nearest 0, 1 2 1 and we take one of our line segments to be the line segment joining 0 and this p(cid:101)oint. If P (cid:48) (cid:48) 2 contains another element in the bottom of F , then we choose the one nearest λ , and we 1 (cid:48) 1 take one of our line segments to be the line segment joining λ and this point. If P contains (cid:48) 1 (cid:48) 2 (cid:48) (cid:48) an element in the top of F , then we choose line segments analogously using(cid:101) λ and λ + λ (cid:48) 1 2 1 2 instead of 0 and λ . 1 Suppose that 0 is not yet in a line segment. Then we choose an element of P not yet chosen (cid:48) R (cid:48) 2(cid:48) with minimal λ -coord(cid:101)inate relative to the basis of 2 consisting of λ and λ . We take the 2 (cid:48) 1 2 line segment joining this point and 0. Similarly, if λ is not yet in a line segment, then we 1 (cid:48) choose an element of P not yet chosen with minimal λ -coordinate, and we take the line 2 (cid:48) 2 segment joining this point and λ . The two line segments chosen thus far in this paragraph 1 might meet. If so, then these two line segments are the diagonals of a (possibly degenerate) quadrilateral, and we simply exchange them for two opposite sides of that quadrilateral. We (cid:48) (cid:48) (cid:48) (cid:48) choose line segments for λ and λ + λ in the same way as for 0 and λ , using maximal (cid:48) 2 1 (cid:48) 2 1 λ -coordinates instead of minimal λ -coordinates. 2 R (cid:48) R 2 We identify 2/Γ and 2/Γ with S2 as usual. Although the instructions for choosing (cid:48) (cid:48) (cid:48) (cid:48) 1 2 (cid:48) (cid:48) (cid:48) (cid:48) α , α , α , α are somewhat verbose, it is easy to see that the images β , β , β , β in S2 of 1(cid:48) 2(cid:48) 3(cid:48) 4(cid:48) (cid:48) 1 2 3 4 α , α , α , α are disjoint arcs. The arcs β are like those in NET map data item (3). Let (cid:48)1 2 →3 4 (cid:48) i (cid:48) (cid:48) (cid:48) h : S2 S2 be a push map relative to β , β , β , β as in Section . ◦ 1 2 3 4 We have that f = h g, where h is a push map and g is the Euclidean NET map (cid:48) R → (cid:48) determined by Φ and the usual branched covering map π : 2 S2 corresponding to Γ . ◦ (cid:48) ◦ ◦ (cid:48)− 1 1 So f = ψ (h g), where ψ = h h 1 is an orientation-preserving homeomorphism which (cid:48) Z stabilizes the postcritical set P of f. Statement 2 of Lemma 2.2 with Λ = Λ = Λ = 2 2 R → R 2 easily implies that there exists an aﬃne isomorphism Ψ: 2 2 such that Ψ(Λ ) = Λ 2 2 and Ψ induces by means of the usual branched covering map π a homeomorphism from S2 2 to S2 which is isotopic to ψ rel P . After modifying f by an isotopy, we may assume that Ψ 2 induces ψ. Suppose that Ψ(x) = Cx + d. Now we apply Theorems 1.2 and 4.4. We conclude that f is Thurston equivalent to a NET map arising from NET map data with the following form. The lattice Λ is preserved. The (cid:48) 2 orientation-preserving aﬃne isomorphism has linear term A = CA . The associated green · (cid:48) line segments have the form α = C α . We take λ and λ to be the columns of A. Because (cid:48) (cid:48) i i 1 2 λ is divisible by m and λ is divisible by n, the corresponding statements are true for λ 1 2 1 and λ . 2 It easily follows that we now have a NET map presentation for our original NET map and even that data item (4) is satisﬁed. We obtain our diagram from this as discussed in Section . This proves Theorem 1.1. (cid:3)

text: PRESENTATIONS

text: OF

text: NET

text: MAPS

text: postcritical

text: set

text: of

text: f,

text: and

text: let

text: P

text: be

text: the

text: inverse

text: image

text: of

text: P

text: union.

text: So

text: P

text: consists

text: of

text: four

text: points

text: which

text: are

text: either

text: in

text: the

text: interior

text: of

text: F

text: in

text: the

text: left

text: half

text: of

text: its

text: boundary.

text: We

text: begin

text: to

text: choose

text: four

text: line

text: segments

text: α

text: α

text: α

text: α

text: If

text: P

text: contains

text: an

text: element

text: in

text: bottom

text: of

text: F

text: then

text: we

text: choose

text: the

text: element

text: of

text: P

text: in

text: the

text: bottom

text: of

text: F

text: which

text: is

text: nearest

text: and

text: we

text: take

text: one

text: of

text: our

text: line

text: segments

text: to

text: be

text: the

text: line

text: segment

text: joining

text: and

text: this

text: If

text: P

text: contains

text: another

text: element

text: in

text: the

text: bottom

text: of

text: F

text: then

text: we

text: choose

text: the

text: one

text: nearest

text: λ

text: and

text: we

text: take

text: one

text: of

text: our

text: line

text: segments

text: to

text: be

text: the

text: line

text: segment

text: joining

text: λ

text: and

text: this

text: point.

text: If

text: P

text: contains

text: an

text: element

text: in

text: the

text: top

text: of

text: F

text: then

text: we

text: choose

text: line

text: segments

text: analogously

text: λ

text: and

text: λ

text: λ

text: instead

text: of

text: and

text: λ

text: Suppose

text: that

text: is

text: not

text: yet

text: in

text: a

text: line

text: segment.

text: Then

text: we

text: choose

text: an

text: element

text: of

text: P

text: not

text: yet

text: chosen

text: R

text: with

text: minimal

text: λ

text: relative

text: to

text: the

text: basis

text: of

text: consisting

text: of

text: λ

text: and

text: λ

text: We

text: take

text: the

text: line

text: segment

text: joining

text: this

text: point

text: and

text: Similarly,

text: if

text: λ

text: is

text: not

text: yet

text: in

text: a

text: line

text: segment,

text: then

text: we

text: choose

text: an

text: element

text: of

text: P

text: not

text: yet

text: chosen

text: with

text: minimal

text: λ

text: -coordinate,

text: and

text: we

text: take

text: the

text: line

text: segment

text: joining

text: this

text: point

text: and

text: λ

text: The

text: two

text: line

text: segments

text: chosen

text: thus

text: far

text: in

text: this

text: paragraph

text: might

text: meet.

text: If

text: so,

text: then

text: these

text: two

text: line

text: segments

text: are

text: the

text: diagonals

text: of

text: a

text: (possibly

text: degenerate)

text: quadrilateral,

text: and

text: we

text: simply

text: exchange

text: them

text: for

text: two

text: opposite

text: sides

text: of

text: that

text: quadrilateral.

text: We

text: choose

text: line

text: segments

text: for

text: λ

text: and

text: λ

text: λ

text: in

text: the

text: same

text: way

text: as

text: for

text: and

text: λ

text: using

text: maximal

text: λ

text: -coordinates

text: instead

text: of

text: minimal

text: λ

text: -coordinates.

text: R

text: R

text: We

text: identify

text: and

text: with

text: as

text: usual.

text: Although

text: the

text: instructions

text: for

text: choosing

text: α

text: α

text: α

text: α

text: are

text: somewhat

text: verbose,

text: it

text: is

text: easy

text: to

text: see

text: that

text: the

text: images

text: β

text: β

text: β

text: β

text: in

text: of

text: α

text: α

text: α

text: α

text: are

text: disjoint

text: arcs.

text: The

text: arcs

text: β

text: are

text: like

text: those

text: in

text: NET

text: map

text: data

text: item

text: Let

text: →3

text: i

text: h

text: be

text: a

text: push

text: map

text: relative

text: to

text: β

text: β

text: β

text: β

text: as

text: in

text: Section

text: We

text: have

text: that

text: f

text: h

text: g,

text: where

text: h

text: is

text: a

text: push

text: map

text: and

text: g

text: is

text: the

text: Euclidean

text: NET

text: map

text: R

text: determined

text: by

text: Φ

text: and

text: the

text: usual

text: branched

text: covering

text: map

text: π

text: corresponding

text: to

text: Γ

text: So

text: f

text: ψ

text: (h

text: g),

text: where

text: ψ

text: h

text: h

text: is

text: an

text: orientation-preserving

text: homeomorphism

text: which

text: Z

text: stabilizes

text: the

text: postcritical

text: set

text: P

text: of

text: f.

text: Statement

text: of

text: Lemma

text: with

text: Λ

text: Λ

text: Λ

text: R

text: R

text: easily

text: implies

text: that

text: there

text: exists

text: an

text: aﬃne

text: isomorphism

text: Ψ:

text: such

text: that

text: Ψ(Λ

text: Λ

text: and

text: Ψ

text: induces

text: by

text: means

text: of

text: the

text: usual

text: branched

text: covering

text: map

text: π

text: a

text: homeomorphism

text: from

text: to

text: which

text: is

text: isotopic

text: to

text: ψ

text: rel

text: P

text: After

text: modifying

text: f

text: by

text: an

text: isotopy,

text: we

text: may

text: assume

text: that

text: Ψ

text: induces

text: ψ.

text: Suppose

text: that

text: Ψ(x)

text: Cx

text: d.

text: Now

text: we

text: apply

text: Theorems

text: and

text: We

text: conclude

text: that

text: f

text: is

text: Thurston

text: equivalent

text: to

text: a

text: NET

text: map

text: arising

text: from

text: NET

text: map

text: data

text: with

text: the

text: following

text: form.

text: The

text: lattice

text: Λ

text: is

text: preserved.

text: The

text: orientation-preserving

text: aﬃne

text: isomorphism

text: has

text: linear

text: term

text: A

text: CA

text: The

text: associated

text: green

text: line

text: segments

text: have

text: the

text: form

text: α

text: C

text: α

text: We

text: take

text: λ

text: and

text: λ

text: to

text: be

text: the

text: columns

text: of

text: A.

text: Because

text: i

text: i

text: λ

text: is

text: divisible

text: by

text: m

text: and

text: λ

text: is

text: divisible

text: by

text: n,

text: the

text: corresponding

text: statements

text: are

text: true

text: for

text: λ

text: and

text: λ

text: It

text: easily

text: follows

text: that

text: we

text: now

text: have

text: a

text: NET

text: map

text: presentation

text: for

text: our

text: original

text: NET

text: map

text: and

text: even

text: that

text: data

text: item

text: is

text: satisﬁed.

text: We

text: obtain

text: our

text: diagram

text: from

text: this

text: as

text: discussed

text: in

text: Section

text: This

text: proves

text: Theorem


text: PRESENTATIONS OF NET MAPS 19 Figure 14. A ﬁnite subdivision rule for f Figure 15. The boundary of the quadrilateral Q and its pullback under g 1 Step 3. We construct green arcs β , β , β , β in S2 as in Figure 15. Arrows indicate push 1 2 3 4 directions. In the right part of Figure 15, elements of P are marked with dots and elements × 1 of P are marked with ’s. 2 Step 4. We construct the quadrilateral Q so that its boundary is the union of the black 1 line segments and circular arc in the right side of Figure 15. We label the vertices of Q as ∞ 1 shown in Figure 15. The label λ is at . 1 − Step 5. The complex g 1(∂Q ) appears in the left side of Figure 15, correct up to isotopy ∪ 1 rel P P . Its edges are labeled with letters. The label of an edge is the label of its image 1 2 in ∂Q . 1 − ∞ Step 6. The interval [ 1/2, ] has label a in the right side of Figure 15. It is a core arc \ − for a simple closed curve γ in S2 P with slope 0. Figure 15 shows that g 1(γ) consists of 1 one connected component with slope 0. So p = 0 and d = 3. The line segment in the right q \ side of Figure 15 with label b is a core arc for a simple closed curve δ in S2 P with slope ∞ − 1 . Using Figure 15, we ﬁnd that g 1(δ) consists of one connected component with slope − − 3. So r = 3 and e = 3. (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) s Step 7. To get a matrix with positive determinant, we take − − − q s 1 1 1 1 3 1 A = d e = 3 3 = . p r 0 1 0 1 d e ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: PRESENTATIONS

text: OF

text: NET

text: MAPS

text: Figure

text: A

label: caption

text: ﬁnite

text: subdivision

text: rule

text: for

text: f

text: Figure

text: The

text: boundary

text: of

text: the

text: quadrilateral

text: Q

text: and

text: its

text: pullback

text: under

text: g

text: Step

text: We

text: construct

text: green

text: arcs

text: β

text: β

text: β

text: β

text: in

text: as

text: in

text: Figure

text: Arrows

text: indicate

text: push

text: directions.

text: In

text: the

text: right

text: part

text: of

text: Figure

text: elements

text: of

text: P

text: are

text: marked

text: with

text: dots

text: and

text: elements

text: of

text: P

text: are

text: marked

text: with

text: ’s.

text: Step

text: We

text: construct

text: the

text: quadrilateral

text: Q

text: so

text: that

text: its

text: boundary

text: is

text: the

text: union

text: of

text: the

text: black

text: line

text: segments

text: and

text: circular

text: arc

text: in

text: the

text: right

text: side

text: of

text: Figure

text: We

text: label

text: the

text: vertices

text: of

text: Q

text: as

text: shown

text: in

text: Figure

text: The

text: label

text: λ

text: is

text: at

text: Step

text: The

text: complex

text: g

text: appears

text: in

text: the

text: left

text: side

text: of

text: Figure

text: correct

text: up

text: to

text: isotopy

text: rel

text: P

text: P

text: Its

text: edges

text: are

text: labeled

text: with

text: letters.

text: The

text: label

text: of

text: an

text: edge

text: is

text: the

text: label

text: of

text: its

text: image

text: in

text: ∂Q

text: Step

text: The

text: interval

text: has

text: label

text: a

text: in

text: the

text: right

text: side

text: of

text: Figure

text: It

text: is

text: a

text: core

text: arc

text: for

text: a

text: simple

text: closed

text: curve

text: γ

text: in

text: P

text: with

text: slope

text: Figure

text: shows

text: that

text: g

text: consists

text: of

text: one

text: connected

text: component

text: with

text: slope

text: So

text: p

text: and

text: d

text: The

text: line

text: segment

text: in

text: the

text: right

text: q

text: side

text: of

text: Figure

text: with

text: label

text: b

text: is

text: a

text: core

text: arc

text: for

text: a

text: simple

text: closed

text: curve

text: δ

text: in

text: P

text: with

text: slope

text: Using

text: Figure

text: we

text: ﬁnd

text: that

text: g

text: consists

text: of

text: one

text: connected

text: component

text: with

text: slope

text: So

text: r

text: and

text: e

text: s

text: Step

text: To

text: get

text: a

text: matrix

text: with

text: positive

text: determinant,

text: we

text: take

text: q

text: s

text: A

text: d

text: e

text: p

text: r

text: d

text: e

text: ##LTFigure##

text: ##LTFigure##

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: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: PRESENTATIONS OF NET MAPS 3 R2 Φ (cid:47)(cid:47) R2 (cid:48) π π (cid:15)(cid:15) (cid:15)(cid:15) R φ (cid:47)(cid:47) R (cid:48) 2/Γ 2/Γ Figure 2. The maps of Lemmas 2.1 and 2.2 The remainder of the paper is devoted to showing that general NET maps admit NET map presentations. The basic idea is to show ﬁrst that an arbitrary NET map f has the (cid:48) ◦ (cid:48) form f = h g where g is a Euclidean NET map as above and where now h is a push-point homeomorphism along a set of possibly very complicated pairwise disjoint arcs. The proof (cid:48) of Theorem 1.1 then exploits the fact that h can be simpliﬁed at the expense of modifying the aﬃne map g. § discusses Euclidean NET maps in detail and shows that they arise as compositional § § factors of general NET maps. and attach preliminary data to NET maps and show how § this changes under twisting. proves our main result. The proof we give of Theorem 1.1, while constructive in principle, is nevertheless somewhat implicit. In Section we give an algorithm for ﬁnding NET map presentations, and illustrate it for two examples: the Douady rabbit quadratic polynomial, and the cubic rational map studied by Lodge [9]. Following this paper, [8] investigates modular groups, Hurwitz classes and dynamic por- traits of NET maps. Next, [7] is partly a survey of what we know, but mostly a compilation of special NET map results, both theoretical and computational. The NET map web site [11] contains papers, a census of many examples, and executable ﬁles for the computer program NETmap that generated the census data. The authors gratefully acknowledge support from the American Institute for Mathematics. Kevin Pilgrim was also supported by Simons grant #245269. 2. Lattices, tori, spheres, and affine maps R { (cid:55)→ ± ∈ } A lattice Λ < 2 determines a group of isometries Γ := x 2λ x : λ Λ generated Λ R → R by 180-degree rotations about elements of Λ and a quotient map π : 2 2/Γ . Given R Λ Λ an ordered basis (λ, µ) for Λ, the parallelogram in 2 spanned by 2λ and 2µ is a funda- R mental domain for the quotient torus 2/2Λ; the parallelogram spanned by 2λ and µ is a R fundamental domain F for the quotient sphere 2/Γ . Λ Λ The following two lemmas will be used repeatedly in the sequel; the proofs are straight- forward and are omitted. (cid:48) R (cid:48) Lemma 2.1. Let Λ and Λ be lattices in 2. Let Γ, respectively Γ , be the groups of Euclidean (cid:55)→ ± ∈ ∈ (cid:48) R → isometries of the form x 2λ x for some λ Λ, respectively λ Λ . Also let π : 2 R (cid:48) R → R (cid:48) R → R 2/Γ and π : 2 2/Γ be the canonical quotient maps. Let Φ: 2 2 be an aﬃne ⊆ (cid:48) R → isomorphism such that Φ(Λ) Λ . Then Φ induces a branched covering map φ: 2/Γ R (cid:48) ◦ (cid:48) ◦ 2/Γ such that φ π = π Φ as in Figure 2. The map Φ preserves orientation if and only (cid:48)(cid:48) (cid:48) if φ preserves orientation. The set Φ(Λ) is a coset of a sublattice Λ of Λ , and the degree of (cid:48) (cid:48)(cid:48) φ equals the index [Λ : Λ ]. ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: PRESENTATIONS

text: OF

text: NET

text: MAPS

text: Φ

font: default

text: π

text: π

text: R

text: φ

text: R

text: Figure

text: The

text: maps

text: of

text: Lemmas

text: and

text: The

text: remainder

text: of

text: the

text: paper

text: is

text: devoted

text: to

text: showing

text: that

text: general

text: NET

text: maps

text: admit

text: NET

text: map

text: presentations.

text: The

text: basic

text: idea

text: is

text: to

text: show

text: ﬁrst

text: that

text: an

text: arbitrary

text: NET

text: map

text: f

text: has

text: the

text: form

text: f

text: h

text: g

text: where

text: g

text: is

text: a

text: Euclidean

text: NET

text: map

text: as

text: above

text: and

text: where

text: now

text: h

text: is

text: a

text: push-point

text: homeomorphism

text: along

text: a

text: set

text: of

text: possibly

text: very

text: complicated

text: pairwise

text: disjoint

text: arcs.

text: The

text: proof

text: of

text: Theorem

text: then

text: exploits

text: the

text: fact

text: that

text: h

text: can

text: be

text: simpliﬁed

text: at

text: the

text: expense

text: of

text: modifying

text: the

text: aﬃne

text: map

text: g.

text: discusses

text: Euclidean

text: NET

text: maps

text: in

text: detail

text: and

text: shows

text: that

text: they

text: arise

text: as

text: compositional

text: factors

text: of

text: general

text: NET

text: maps.

text: and

text: attach

text: preliminary

text: data

text: to

text: NET

text: maps

text: and

text: show

text: how

text: this

text: changes

text: under

text: twisting.

text: proves

text: our

text: main

text: result.

text: The

text: proof

text: we

text: give

text: of

text: Theorem

text: while

text: constructive

text: in

text: principle,

text: is

text: nevertheless

text: somewhat

text: implicit.

text: In

text: Section

text: we

text: give

text: an

text: algorithm

text: for

text: ﬁnding

text: NET

text: map

text: presentations,

text: and

text: illustrate

text: it

text: for

text: two

text: examples:

text: the

text: Douady

text: rabbit

text: quadratic

text: polynomial,

text: and

text: the

text: cubic

text: rational

text: map

text: studied

text: by

text: Lodge

text: Following

text: this

text: paper,

text: investigates

text: modular

text: groups,

text: Hurwitz

text: classes

text: and

text: dynamic

text: por-

text: traits

text: of

text: NET

text: maps.

text: Next,

text: is

text: partly

text: a

text: survey

text: of

text: what

text: we

text: know,

text: but

text: mostly

text: a

text: compilation

text: of

text: special

text: NET

text: map

text: results,

text: both

text: theoretical

text: and

text: computational.

text: The

text: NET

text: map

text: web

text: site

text: contains

text: papers,

text: a

text: census

text: of

text: many

text: examples,

text: and

text: executable

text: ﬁles

text: for

text: the

text: computer

text: program

text: NETmap

text: that

text: generated

text: the

text: census

text: data.

text: The

text: authors

text: gratefully

text: acknowledge

text: support

text: from

text: the

text: American

text: Institute

text: for

text: Mathematics.

text: Kevin

text: Pilgrim

text: was

text: also

text: supported

text: by

text: Simons

text: grant

text: Lattices,

text: tori,

text: spheres,

text: and

text: affine

text: maps

text: R

text: A

text: lattice

text: Λ

text: determines

text: a

text: group

text: of

text: isometries

text: Γ

text: x

text: 2λ

text: x

text: λ

text: Λ

text: generated

text: Λ

text: R

text: R

text: by

text: rotations

text: about

text: elements

text: of

text: Λ

text: and

text: a

text: quotient

text: map

text: π

text: Given

text: R

text: Λ

text: Λ

text: an

text: ordered

text: basis

text: (λ,

text: µ)

text: for

text: Λ,

text: the

text: parallelogram

text: in

text: spanned

text: by

text: 2λ

text: and

text: 2µ

text: is

text: a

text: funda-

text: R

text: mental

text: domain

text: for

text: the

text: quotient

text: torus

text: the

text: parallelogram

text: spanned

text: by

text: 2λ

text: and

text: µ

text: is

text: a

text: R

text: fundamental

text: domain

text: F

text: for

text: the

text: quotient

text: sphere

text: Λ

text: Λ

text: The

text: following

text: two

text: lemmas

text: will

text: be

text: used

text: repeatedly

text: in

text: the

text: sequel;

text: the

text: proofs

text: are

text: straight-

text: forward

text: and

text: are

text: omitted.

text: R

text: Lemma

text: Let

text: Λ

text: and

text: Λ

text: be

text: lattices

text: in

text: Let

text: Γ,

text: respectively

text: Γ

text: be

text: the

text: groups

text: of

text: Euclidean

text: R

text: isometries

text: of

text: the

text: form

text: x

text: 2λ

text: x

text: for

text: some

text: λ

text: Λ,

text: respectively

text: λ

text: Λ

text: Also

text: let

text: π

text: R

text: R

text: R

text: R

text: R

text: and

text: π

text: be

text: the

text: canonical

text: quotient

text: maps.

text: Let

text: Φ:

text: be

text: an

text: aﬃne

text: R

text: isomorphism

text: such

text: that

text: Φ(Λ)

text: Λ

text: Then

text: Φ

text: induces

text: a

text: branched

text: covering

text: map

text: φ:

text: R

text: such

text: that

text: φ

text: π

text: π

text: Φ

text: as

text: in

text: Figure

text: The

text: map

text: Φ

text: preserves

text: orientation

text: if

text: and

text: only

text: if

text: φ

text: preserves

text: orientation.

text: The

text: set

text: Φ(Λ)

text: is

text: a

text: coset

text: of

text: a

text: sublattice

text: Λ

text: of

text: Λ

text: and

text: the

text: degree

text: of

text: φ

text: equals

text: the

text: index

text: [Λ

text: Λ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 4 WILLIAM FLOYD, WALTER PARRY, AND KEVIN M. PILGRIM (cid:48) R (cid:48) Lemma 2.2. Let Λ and Λ be lattices in 2. Let Γ, respectively Γ , be the group of Euclidean (cid:55)→ ± ∈ (cid:48) R → R isometries of the form x 2λ x for some λ Λ, respectively Λ . Also let π : 2 2/Γ (cid:48) R → R (cid:48) R → R (cid:48) and π : 2 2/Γ be the canonical quotient maps. Let φ: 2/Γ 2/Γ be a branched ⊆ (cid:48) (cid:48) covering map such that φ(π(Λ)) π (Λ ). Then we have the following three statements. R → R (1) There exists a homeomorphism Φ: 2 2 such that the restriction of Φ to Λ is ◦ (cid:48) ◦ (cid:48) ∈ (cid:48) aﬃne and φ π = π Φ. If π (0) φ(π(Λ)), then Φ(Λ) is a sublattice of Λ . R → R (2) There exists an aﬃne isomorphism Ψ: 2 2 such that the branched map of R R (cid:48) Lemma 2.1 which Ψ induces from 2/Γ to 2/Γ is φ up to isotopy rel π(Λ). If (cid:48) ∈ (cid:48) π (0) φ(π(Λ)), then Ψ(Λ) is a sublattice of Λ . (3) The maps Φ and Ψ are unique up to precomposing with an element of Γ. They are (cid:48) also unique up to postcomposing with an element of Γ . 3. From presentations and diagrams to maps Z R R → R In all that follows, Λ will denote the standard integer lattice 2 < 2, and Φ : 2 2 2 (cid:55)→ will denote an aﬃne map x Ax + b with the property that Λ := Φ(Λ ) is a proper 1 2 § sublattice of Λ . For i = 1, 2 we denote by π the corresponding quotient map as in and 2 i by S2 the corresponding quotient sphere; we put P := π (Λ ). The sphere S2 has an aﬃne i i i i i structure, making it a half-translation sphere with set of corner points P . i Precisely described, the data of a NET map presentation with associated diagram D consists of three things. Here are the ﬁrst two: (1) An ordered basis (λ , λ ) for Λ . This is indicated in D by drawing a parallelogram 1 2 1 F with sides joining the origin to lattice points 2λ , λ . The choice between the 1 1 2 two possible orderings of the basis elements is uniquely determined by the condition that the determinant of the column matrix A = [λ , λ ] is positive. The intersection { } 1 2 0, λ , 2λ , λ , λ + λ , 2λ + λ of Λ with F is indicated in D by large solid dots; 1 1 2 1 2∩ 1 2 1 1 note that P = π (Λ F ). 1 ∈ {1 1 1 } (2) An element b 0, λ , λ , λ + λ . It is indicated in D by circling the corresponding 1 2 1 2 point in the boundary of the parallelogram F . 1 The data of (1) and (2) above determines uniquely an orientation-preserving aﬃne iso- R → R morphism Φ : 2 2 sending Λ isomorphically to Λ and sending the origin to b; thus it 2 1 is linear iﬀ b = 0. § R R R From , we have the following. Here, 2 = 2 = 2; the subscripts are for clarity. We R 1 2 R → R set S2 = 2/Γ for i = 1, 2. The identity map id : 2 2 induces a degree d := [Λ : Λ ] i i i 1→ 2 2 1 orientation-preserving branched covering id : (S2, P ) (S2, P ). (Not only do we have that ⊆ 1 1 2 2 R → R id(P ) P but even that id(π (Λ )) = P .) The aﬃne isomorphism Φ : 2 2 induces 1 2 →1 2 2 ◦ →2 1 an isomorphism φ : (S2, P ) (S2, P ). The composition φ id : (S2, P ) (S2, P ), is a 2 2 1 1 1 1 1 1 Euclidean Thurston map, g. See Figure 3. The lower vertical arrows are the identity map. § We call the data in (1) and (2) above Euclidean NET map data. In the following , we show that up to equivalence, every Euclidean NET map arises in this way. The third and ﬁnal ingredient in a NET map presentation is the following. Recall that ∩ the four corners of the half-translation sphere form the set P = π (Λ ), and that Λ { } 1 1 1 1 F = 0, λ , 2λ , λ , λ + λ , 2λ + λ . For convenience of exposition, we denote P = { 1 1 } 1 2 1 2 1 2 1 p , p , p , p ; the choice of such a labeling is not part of the data of a NET map presentation. 1 2 3 4 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: WILLIAM

text: FLOYD,

text: WALTER

text: PARRY,

text: AND

text: KEVIN

text: M.

text: PILGRIM

text: R

text: Lemma

text: Let

text: Λ

text: and

text: Λ

text: be

text: lattices

text: in

text: Let

text: Γ,

text: respectively

text: Γ

text: be

text: the

text: group

text: of

text: Euclidean

text: R

text: R

text: isometries

text: of

text: the

text: form

text: x

text: 2λ

text: x

text: for

text: some

text: λ

text: Λ,

text: respectively

text: Λ

text: Also

text: let

text: π

text: R

text: R

text: R

text: R

text: and

text: π

text: be

text: the

text: canonical

text: quotient

text: maps.

text: Let

text: φ:

text: be

text: a

text: branched

text: covering

text: map

text: such

text: that

text: φ(π(Λ))

text: π

text: (Λ

text: Then

text: we

text: have

text: the

text: following

text: three

text: statements.

text: R

text: R

text: There

text: exists

text: a

text: homeomorphism

text: Φ:

text: such

text: that

text: the

text: restriction

text: of

text: Φ

text: to

text: Λ

text: is

text: aﬃne

text: and

text: φ

text: π

text: π

text: Φ.

text: If

text: π

text: φ(π(Λ)),

text: then

text: Φ(Λ)

text: is

text: a

text: sublattice

text: of

text: Λ

text: R

text: R

text: There

text: exists

text: an

text: aﬃne

text: isomorphism

text: Ψ:

text: such

text: that

text: the

text: branched

text: map

text: of

text: R

text: R

text: Lemma

text: which

text: Ψ

text: induces

text: from

text: to

text: is

text: φ

text: up

text: to

text: isotopy

text: rel

text: π(Λ).

text: If

text: π

text: φ(π(Λ)),

text: then

text: Ψ(Λ)

text: is

text: a

text: sublattice

text: of

text: Λ

text: The

text: maps

text: Φ

text: and

text: Ψ

text: are

text: unique

text: up

text: to

text: precomposing

text: with

text: an

text: element

text: of

text: Γ.

text: They

text: are

text: also

text: unique

text: up

text: to

text: postcomposing

text: with

text: an

text: element

text: of

text: Γ

text: From

text: presentations

text: and

text: diagrams

text: to

text: maps

text: Z

text: R

text: R

text: R

text: In

text: all

text: that

text: follows,

text: Λ

text: will

text: denote

text: the

text: standard

text: integer

text: lattice

text: and

text: Φ

text: will

text: denote

text: an

text: aﬃne

text: map

text: x

text: Ax

text: b

text: with

text: the

text: property

text: that

text: Λ

text: Φ(Λ

text: is

text: a

text: proper

text: sublattice

text: of

text: Λ

text: For

text: i

text: we

text: denote

text: by

text: π

text: the

text: corresponding

text: quotient

text: map

text: as

text: in

text: and

text: i

text: by

text: the

text: corresponding

text: quotient

text: sphere;

text: we

text: put

text: P

text: π

text: (Λ

text: The

text: sphere

text: has

text: an

text: aﬃne

text: i

text: i

text: i

text: i

text: i

text: structure,

text: making

text: it

text: a

text: half-translation

text: sphere

text: with

text: set

text: of

text: corner

text: points

text: P

text: i

text: Precisely

text: described,

text: the

text: data

text: of

text: a

text: NET

text: map

text: presentation

text: with

text: associated

text: diagram

text: D

text: consists

text: of

text: three

text: things.

text: Here

text: are

text: the

text: ﬁrst

text: two:

text: An

text: ordered

text: basis

text: (λ

text: λ

text: for

text: Λ

text: This

text: is

text: indicated

text: in

text: D

text: by

text: drawing

text: a

text: parallelogram

text: F

text: with

text: sides

text: joining

text: the

text: origin

text: to

text: lattice

text: points

text: 2λ

text: λ

text: The

text: choice

text: between

text: the

text: two

text: possible

text: orderings

text: of

text: the

text: basis

text: elements

text: is

text: uniquely

text: determined

text: by

text: the

text: condition

text: that

text: the

text: determinant

text: of

text: the

text: column

text: matrix

text: A

text: [λ

text: λ

text: is

text: positive.

text: The

text: intersection

text: λ

text: 2λ

text: λ

text: λ

text: λ

text: 2λ

text: λ

text: of

text: Λ

text: with

text: F

text: is

text: indicated

text: in

text: D

text: by

text: large

text: solid

text: dots;

text: 2∩

text: note

text: that

text: P

text: π

text: (Λ

text: F

text: An

text: element

text: b

text: λ

text: λ

text: λ

text: λ

text: It

text: is

text: indicated

text: in

text: D

text: by

text: circling

text: the

text: corresponding

text: point

text: in

text: the

text: boundary

text: of

text: the

text: parallelogram

text: F

text: The

text: data

text: of

text: and

text: above

text: determines

text: uniquely

text: an

text: orientation-preserving

text: aﬃne

text: iso-

text: R

text: R

text: morphism

text: Φ

text: sending

text: Λ

text: isomorphically

text: to

text: Λ

text: and

text: sending

text: the

text: origin

text: to

text: b;

text: thus

text: it

text: is

text: linear

text: iﬀ

text: b

text: R

text: R

text: R

text: From

text: we

text: have

text: the

text: following.

text: Here,

text: the

text: subscripts

text: are

text: for

text: clarity.

text: We

text: R

text: R

text: R

text: set

text: for

text: i

text: The

text: identity

text: map

text: id

text: induces

text: a

text: degree

text: d

text: [Λ

text: Λ

text: i

text: i

text: i

text: 1→

text: orientation-preserving

text: branched

text: covering

text: id

text: P

text: P

text: (Not

text: only

text: do

text: we

text: have

text: that

text: R

text: R

text: id(P

text: P

text: but

text: even

text: that

text: id(π

text: (Λ

text: P

text: The

text: aﬃne

text: isomorphism

text: Φ

text: induces

text: →1

text: →2

text: an

text: isomorphism

text: φ

text: P

text: P

text: The

text: composition

text: φ

text: id

text: P

text: P

text: is

text: a

text: Euclidean

text: Thurston

text: map,

text: g.

text: See

text: Figure

text: The

text: lower

text: vertical

text: arrows

text: are

text: the

text: identity

text: map.

text: We

text: call

text: the

text: data

text: in

text: and

text: above

text: Euclidean

text: NET

text: map

text: data.

text: In

text: the

text: following

text: we

text: show

text: that

text: up

text: to

text: equivalence,

text: every

text: Euclidean

text: NET

text: map

text: arises

text: in

text: this

text: way.

text: The

text: third

text: and

text: ﬁnal

text: ingredient

text: in

text: a

text: NET

text: map

text: presentation

text: is

text: the

text: following.

text: Recall

text: that

text: the

text: four

text: corners

text: of

text: the

text: half-translation

text: sphere

text: form

text: the

text: set

text: P

text: π

text: (Λ

text: and

text: that

text: Λ

text: F

text: λ

text: 2λ

text: λ

text: λ

text: λ

text: 2λ

text: λ

text: For

text: convenience

text: of

text: exposition,

text: we

text: denote

text: P

text: p

text: p

text: p

text: p

text: the

text: choice

text: of

text: such

text: a

text: labeling

text: is

text: not

text: part

text: of

text: the

text: data

text: of

text: a

text: NET

text: map

text: presentation.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: PRESENTATIONS OF NET MAPS 5 R2 id (cid:47)(cid:47) R2 Φ (cid:47)(cid:47) R2 1 2 1 π1 π2 π1 (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) R2/Γ id (cid:47)(cid:47) R2/Γ φ (cid:47)(cid:47) R2/Γ 1 1 2 2 1 1 id id (cid:15)(cid:15) (cid:15)(cid:15) g (cid:47)(cid:47) S2 S2 1 1 Figure 3. Deﬁning g ∈ (3) For each p P , a “green” (local) geodesic arc β joining p to an element of π (Λ ), j 1 j j 1 2 such that the geodesics β are pairwise disjoint. We allow the case of degenerate (zero j length) geodesics. We require that every β lifts to a line segment in F . The choice j 1 of the β ’s is represented in D by drawing all of their lifts in the closed fundamental j domain F as green line segments. Thus there may be up to eight such segments. 1 Let P be the set of four points in S2 consisting of the terminal endpoints of the β ’s. The 1 → j data in (3) determines a push-point homeomorphism h : (S2, P ) (S2, P ), unique up to 1 1 1 isotopy rel P , by pushing the four points of P along the (possibly trivial) oriented green 1 1 geodesics β . j The result below follows immediately from straightforward arguments. ◦ ◦ ◦ → Lemma 3.3. The map f = h g = h φ id: (S2, P ) (S2, P ) is a Thurston map; it is a 1 1 NET map if #P = 4. Once the data (1), (2), and (3) are ﬁxed, the following are uniquely f determined: ⊂ ⊂ (1) the subset P P S2; f 1 ⊂ (2) the isotopy class of f relative to P S2; 1 ◦ ◦ − ◦ ◦ → 1 (3) the isotopy class of the conjugate F := id f id = id h φ : (S2, P ) (S2, P ); ⊂ 2 2 2 2 P P and equality holds iﬀ F (equivalently, f) is a NET map; F 2 (4) the equivalence class of f. If #P = 4 (equivalently, if P = P ) then we call the data in (1), (2), (3) above NET map f f data and we say that (1), (2), (3) give a NET map presentation for f. This presentation is encoded in the diagram D. Remarks: (1) The dynamical plane—the sphere S2—and postcritical set of f depend on the data 1 in (1), (2), (3), while those of its conjugate F do not. The former representation is more convenient for computations of dynamical invariants like slope functions, while the latter is more convenient in other situations since the set of isotopy classes of Thurston maps with a common postcritical set P is naturally a biset over the pure 2 mapping class group P Γ(2). (2) Suppose a diagram D is ﬁxed; let P Γ(2).[D] be the orbit of its projective class under the natural free action of the pure mapping class group P Γ(2). Theorem 1.2 implies that the image of the orbit is a subset of isotopy classes of NET maps that naturally ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: PRESENTATIONS

text: OF

text: NET

text: MAPS

text: id

text: Φ

text: π1

text: π2

text: π1

text: id

text: φ

text: id

text: id

text: g

text: Figure

text: Deﬁning

text: g

label: list

text: For

text: each

text: p

text: P

text: a

text: “green”

text: (local)

text: geodesic

text: arc

text: β

text: joining

text: p

text: to

text: an

text: element

text: of

text: π

text: (Λ

text: j

text: j

text: j

text: such

text: that

text: the

text: geodesics

text: β

text: are

text: pairwise

text: disjoint.

text: We

text: allow

text: the

text: case

text: of

text: degenerate

text: (zero

text: j

text: length)

text: geodesics.

text: We

text: require

text: that

text: every

text: β

text: lifts

text: to

text: a

text: line

text: segment

text: in

text: F

text: The

text: choice

text: j

text: of

text: the

text: β

text: ’s

text: is

text: represented

text: in

text: D

text: by

text: drawing

text: all

text: of

text: their

text: lifts

text: in

text: the

text: closed

text: fundamental

text: j

text: domain

text: F

text: as

text: green

text: line

text: segments.

text: Thus

text: there

text: may

text: be

text: up

text: to

text: eight

text: such

text: segments.

text: Let

text: P

text: be

text: the

text: set

text: of

text: four

text: points

text: in

text: consisting

text: of

text: the

text: terminal

text: endpoints

text: of

text: the

text: β

text: ’s.

text: The

text: j

text: data

text: in

text: determines

text: a

text: push-point

text: homeomorphism

text: h

text: P

text: P

text: unique

text: up

text: to

text: isotopy

text: rel

text: P

text: by

text: pushing

text: the

text: four

text: points

text: of

text: P

text: along

text: the

text: (possibly

text: trivial)

text: oriented

text: green

text: geodesics

text: β

text: j

text: The

text: result

text: below

text: follows

text: immediately

text: from

text: straightforward

text: arguments.

text: Lemma

text: The

text: map

text: f

text: h

text: g

text: h

text: φ

text: id:

text: P

text: P

text: is

text: a

text: Thurston

text: map;

text: it

text: is

text: a

text: NET

text: map

text: if

text: #P

text: Once

text: the

text: data

text: and

text: are

text: ﬁxed,

text: the

text: following

text: are

text: uniquely

text: f

text: determined:

text: the

text: subset

text: P

text: P

text: f

text: the

text: isotopy

text: class

text: of

text: f

text: relative

text: to

text: P

text: the

text: isotopy

text: class

text: of

text: the

text: conjugate

text: F

text: id

text: f

text: id

text: id

text: h

text: φ

text: P

text: P

text: P

text: P

text: and

text: equality

text: holds

text: iﬀ

text: F

text: (equivalently,

text: f)

text: is

text: a

text: NET

text: map;

text: F

text: the

text: equivalence

text: class

text: of

text: f.

text: If

text: #P

text: (equivalently,

text: if

text: P

text: P

text: then

text: we

text: call

text: the

text: data

text: in

text: above

text: NET

text: map

text: f

text: f

text: data

text: and

text: we

text: say

text: that

text: give

text: a

text: NET

text: map

text: presentation

text: for

text: f.

text: This

text: presentation

text: is

text: encoded

text: in

text: the

text: diagram

text: D.

text: Remarks:

text: The

text: dynamical

text: plane—the

text: sphere

text: postcritical

text: set

text: of

text: f

text: depend

text: on

text: the

text: data

text: in

text: while

text: those

text: of

text: its

text: conjugate

text: F

text: do

text: not.

text: The

text: former

text: representation

text: is

text: more

text: convenient

text: for

text: computations

text: of

text: dynamical

text: invariants

text: like

text: slope

text: functions,

text: while

text: the

text: latter

text: is

text: more

text: convenient

text: in

text: other

text: situations

text: since

text: the

text: set

text: of

text: isotopy

text: classes

text: of

text: Thurston

text: maps

text: with

text: a

text: common

text: postcritical

text: set

text: P

text: is

text: naturally

text: a

text: biset

text: over

text: the

text: pure

text: mapping

text: class

text: group

text: P

text: Suppose

text: a

text: diagram

text: D

text: is

text: ﬁxed;

text: let

text: P

text: be

text: the

text: orbit

text: of

text: its

text: projective

text: class

text: under

text: the

text: natural

text: free

text: action

text: of

text: the

text: pure

text: mapping

text: class

text: group

text: P

text: Theorem

text: implies

text: that

text: the

text: image

text: of

text: the

text: orbit

text: is

text: a

text: subset

text: of

text: isotopy

text: classes

text: of

text: NET

text: maps

text: that

text: naturally

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: ##LTLine##

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text: ##LTLine##


text: PRESENTATIONS OF NET MAPS 7 F (cid:37)(cid:37) (cid:56)(cid:56) S2 h1 (cid:47)(cid:47) S2 id (cid:47)(cid:47) S2 1 1 2 S22 φφ(cid:48) A1 A1 A2 (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:38)(cid:38) S2(cid:48) h1(cid:48) (cid:47)(cid:47) S2(cid:48) id(cid:48) (cid:57)(cid:57)(cid:47)(cid:47) S2 1 1 2 (cid:48) F Figure 5. Proving Theorem 1.2 (cid:55)→ R → The translation x x + b on 2 induces a homeomorphism b : (S2, P ) (S2, P ), an 2 2 2 2 2 element of the modular group of (S2, P ). Because Γ contains all translations by elements 2 2 2 in 2Λ , this homeomorphism depends only on the coset b + 2Λ in Λ /2Λ . We identify 2 Z 2 2 Z 2 Z Λ /2Λ with 2 by means of the canonical group homomorphism from 2 to 2. In this 2 2 2 ∈ Z Z 2 way b determines an element b 2. We obtain another element of 2 from a NET map 2 2 presentation diagram D as follows. Let Λ be the lattice associated to D with ordered basis { 1 } (λ , λ ). One of the lattice points 0, λ , λ , λ + λ in the boundary of D is circled. Call it 1 2 1 →2Z 1 2 a. There is a group homomorphism Λ 2 which sends λ to (1, 0) and λ to (0, 1). Let Z 1 2 1 2 a denote the image in 2 of a under this group homomorphism. Now we have an element b Z 2 Z of 2 associated to b and an element a of 2 associated to D. We obtain a new NET map 2 (cid:48) (cid:48) 2 presentation diagram D such that D is the same as D except that now the translation term (cid:48) (cid:55)→ (cid:48) (cid:55)→ (cid:48) corresponding to D is b + a. The map D D extends to projective classes [D] [D ]. We have: ∈ Z (cid:48) (cid:48) → Theorem 4.4. Suppose b 2 sends [D] to [D ]. Let F, F : (S2, P ) (S2, P ) be the (cid:48) ◦ 2 →2 2 2 corresponding Thurston maps. Then F = b F , where b : (S2, P ) (S2, P ) is the map 2 2 2 2 2 2 induced by b. Z Proof. Let e = (1, 0) and e = (0, 1), the standard basis vectors in 2. We have a 1 { 2 } → { } straightforward bijection τ : 0, e , e , e + e 0, λ , λ , λ + λ . The homeomorphism 1 2 1 2 1 2 1 2 (cid:55)→ φ conjugates the element in the modular group of (S2, P ) induced by x x + e, where ∈ { } 2 2 (cid:55)→ e 0, e , e , e +e , to the element of the modular group of (S2, P ) induced by x x+τ(e). 1 2 1 2 1 1 Theorem 4.4 follows easily from this. (cid:3) 5. Presentation and equivalence of Euclidean NET maps Completeness of construction. In this paragraph, we show that the conditions in data § items (1) and (2) of obtains every Thurston equivalence class of Euclidean NET maps. So, → let g : S2 S2 be a Euclidean NET map. As in Section 1 of [3], we construct lattices Λ , R → R ∈ { }i groups Γ of Euclidean isometries and quotient maps π : 2 2/Γ = S2 for i 1, 2 . i i i i There exist identiﬁcations of S2 with the dynamical plane S2 of g so that π (Λ ) is the i i i ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: PRESENTATIONS

text: OF

text: NET

text: MAPS

text: F

text: id

text: F

text: Figure

text: Proving

text: Theorem

text: R

text: The

text: translation

text: x

text: x

text: b

text: on

text: induces

text: a

text: homeomorphism

text: b

text: P

text: P

text: an

text: element

text: of

text: the

text: modular

text: group

text: of

text: P

text: Because

text: Γ

text: contains

text: all

text: translations

text: by

text: elements

text: in

text: 2Λ

text: this

text: homeomorphism

text: depends

text: only

text: on

text: the

text: coset

text: b

text: 2Λ

text: in

text: Λ

text: We

text: identify

text: Z

text: Z

text: Z

text: Λ

text: with

text: by

text: means

text: of

text: the

text: canonical

text: group

text: homomorphism

text: from

text: to

text: In

text: this

text: Z

text: Z

text: way

text: b

text: determines

text: an

text: element

text: b

text: We

text: obtain

text: another

text: element

text: of

text: from

text: a

text: NET

text: map

text: presentation

text: diagram

text: D

text: as

text: follows.

text: Let

text: Λ

text: be

text: the

text: lattice

text: associated

text: to

text: D

text: with

text: ordered

text: basis

text: (λ

text: λ

text: One

text: of

text: the

text: lattice

text: points

text: λ

text: λ

text: λ

text: λ

text: in

text: the

text: boundary

text: of

text: D

text: is

text: circled.

text: Call

text: it

text: a.

text: There

text: is

text: a

text: group

text: homomorphism

text: Λ

text: which

text: sends

text: λ

text: to

text: and

text: λ

text: to

text: Let

text: Z

text: a

text: denote

text: the

text: image

text: in

text: of

text: a

text: under

text: this

text: group

text: homomorphism.

text: Now

text: we

text: have

text: an

text: element

text: b

text: Z

text: Z

text: of

text: associated

text: to

text: b

text: and

text: an

text: element

text: a

text: of

text: associated

text: to

text: D.

text: We

text: obtain

text: a

text: new

text: NET

text: map

text: presentation

text: diagram

text: D

text: such

text: that

text: D

text: is

text: the

text: same

text: as

text: D

text: except

text: that

text: now

text: the

text: translation

text: term

text: corresponding

text: to

text: D

text: is

text: b

text: a.

text: The

text: map

text: D

text: D

text: extends

text: to

text: projective

text: classes

text: [D]

text: [D

text: We

text: have:

text: Z

text: Theorem

text: Suppose

text: b

text: sends

text: [D]

text: to

text: [D

text: Let

text: F,

text: F

text: P

text: P

text: be

text: the

text: →2

text: corresponding

text: Thurston

text: maps.

text: Then

text: F

text: b

text: F

text: where

text: b

text: P

text: P

text: is

text: the

text: map

text: induced

text: by

text: b.

text: Z

text: Proof.

text: Let

text: e

text: and

text: e

text: the

text: standard

text: basis

text: vectors

text: in

text: We

text: have

text: a

text: straightforward

text: bijection

text: τ

text: e

text: e

text: e

text: e

text: λ

text: λ

text: λ

text: λ

text: The

text: homeomorphism

text: φ

text: conjugates

text: the

text: element

text: in

text: the

text: modular

text: group

text: of

text: P

text: induced

text: by

text: x

text: x

text: e,

text: where

text: e

text: e

text: e

text: e

text: +e

text: to

text: the

text: element

text: of

text: the

text: modular

text: group

text: of

text: P

text: induced

text: by

text: x

text: x+τ(e).

text: Theorem

text: follows

text: easily

text: from

text: this.

text: Presentation

text: and

text: equivalence

text: of

text: Euclidean

text: NET

text: maps

text: Completeness

text: of

text: construction.

text: In

text: this

text: paragraph,

text: we

text: show

text: that

text: the

text: conditions

text: in

text: data

text: items

text: and

text: of

text: obtains

text: every

text: Thurston

text: equivalence

text: class

text: of

text: Euclidean

text: NET

text: maps.

text: So,

text: let

text: g

text: be

text: a

text: Euclidean

text: NET

text: map.

text: As

text: in

text: Section

text: of

text: we

text: construct

text: lattices

text: Λ

text: R

text: R

text: }i

text: groups

text: Γ

text: of

text: Euclidean

text: isometries

text: and

text: quotient

text: maps

text: π

text: for

text: i

text: i

text: i

text: i

text: i

text: There

text: exist

text: identiﬁcations

text: of

text: with

text: the

text: dynamical

text: plane

text: of

text: g

text: so

text: that

text: π

text: (Λ

text: is

text: the

text: i

text: i

text: i

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: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: HSEARCH: protein motif search and clustering 19 X | | | | t H(C) = ci log ci (28) N N i=1 ##LTLine## ##LTLine##

text: HSEARCH:

text: protein

text: motif

text: search

text: and

text: clustering

text: X

text: t

text: H(C)

text: ci

text: log

text: ci

text: N

text: N

text: ##LTLine##

text: ##LTLine##


text: 6 H. Chen and T. Chen ( ) is the probability density function for half-normal distribution. [ ] is f y Pr c Y monotonically decreasing in when the bucket width is ﬁxed, which satisﬁes c w the propriety shown in Equation 3 In order to reduce the chance that two points far from each other hash to the same bucket, normally, random lines are selected. If two data points are K projected to the same bucket in all random lines, the two data points are K stored in the same bucket in a hash table. Additionally, to increase the chance that two points near to each other hash to the same bucket, generally, hash L tables are built on the data set. Thus, for any data point , the hash value ( ) p h p is shown as follow: ( ) = ( ( ) ( ) ( )) (6) h p h p , h p . . . , h p a1,b1 a2,b2 aK,bK To search neighbors for a data point , all data points in the same hash q bucket with in one of the hash tables will be validated. The probability that q two points with distance hash to the same bucket in of the of hash table is c L − − Pr [c] = 1 (1 Pr[c]K)L (7) K,L The increasing of could decrease the probability, and the increasing of could K L increase the probability. 3 Results 3.1 Protein Motif Finding − For a given protein database, each protein sequence is converted to + 1 l k -mers where is the length of protein sequence. A -mer is converted to a - k l k d dimensional data point. LSH tables are built on all data points from the protein database. A motif is also converted to a data point. The coordinate of a motif is obtained by the concatenation of coordinate in each position. The coordinate of a motif is also refer as the center point of the motif. The coordinate for each ≤ ≤ position (1 ) is shown in Equation 8. is the element in PWM j j l M ij which represents the probability of ith amino acid at jth position, and x is the i coordinate for ith amino acid. X 20 × = ( ) (8) C x M j i ij i=1 The query represents the motif and applies LSH technique to search all q k − k ≤ data points in the database where . p p q T Figure 3 shows the pairwise distance among sequences in a motif to the cen- ter point from data points in motifs and random data points. The data points in

text: H.

text: Chen

text: and

text: T.

text: Chen

text: is

text: the

text: probability

text: density

text: function

text: for

text: half-normal

text: distribution.

text: is

text: f

text: y

text: Pr

text: c

text: Y

text: monotonically

text: decreasing

text: in

text: when

text: the

text: bucket

text: width

text: is

text: ﬁxed,

text: which

text: satisﬁes

text: c

text: w

text: the

text: propriety

text: shown

text: in

text: Equation

text: In

text: order

text: to

text: reduce

text: the

text: chance

text: that

text: two

text: points

text: far

text: from

text: each

text: other

text: hash

text: to

text: the

text: same

text: bucket,

text: normally,

text: random

text: lines

text: are

text: selected.

text: If

text: two

text: data

text: points

text: are

text: K

text: projected

text: to

text: the

text: same

text: bucket

text: in

text: all

text: random

text: lines,

text: the

text: two

text: data

text: points

text: are

text: K

text: stored

text: in

text: the

text: same

text: bucket

text: in

text: a

text: hash

text: table.

text: Additionally,

text: to

text: increase

text: the

text: chance

text: that

text: two

text: points

text: near

text: to

text: each

text: other

text: hash

text: to

text: the

text: same

text: bucket,

text: generally,

text: hash

text: L

text: tables

text: are

text: built

text: on

text: the

text: data

text: set.

text: Thus,

text: for

text: any

text: data

text: point

text: the

text: hash

text: value

text: p

text: h

text: p

text: is

text: shown

text: as

text: follow:

text: h

text: p

text: h

text: p

text: h

text: p

text: h

text: p

text: aK,bK

text: To

text: search

text: neighbors

text: for

text: a

text: data

text: point

text: all

text: data

text: points

text: in

text: the

text: same

text: hash

text: q

text: bucket

text: with

text: in

text: one

text: of

text: the

text: hash

text: tables

text: will

text: be

text: validated.

text: The

text: probability

text: that

text: q

text: two

text: points

text: with

text: distance

text: hash

text: to

text: the

text: same

text: bucket

text: in

text: of

text: the

text: of

text: hash

text: table

text: is

text: c

text: L

text: Pr

text: [c]

text: Pr[c]K)L

text: K,L

text: The

text: increasing

text: of

text: could

text: decrease

text: the

text: probability,

text: and

text: the

text: increasing

text: of

text: could

text: K

text: L

text: increase

text: the

text: probability.

text: Results

text: Protein

text: Motif

text: Finding

text: For

text: a

text: given

text: protein

text: database,

text: each

text: protein

text: sequence

text: is

text: converted

text: to

text: l

text: k

text: -mers

text: where

text: is

text: the

text: length

text: of

text: protein

text: sequence.

text: A

text: -mer

text: is

text: converted

text: to

text: a

text: k

text: l

text: k

text: d

text: dimensional

text: data

text: point.

text: LSH

text: tables

text: are

text: built

text: on

text: all

text: data

text: points

text: from

text: the

text: protein

text: database.

text: A

text: motif

text: is

text: also

text: converted

text: to

text: a

text: data

text: point.

text: The

text: coordinate

text: of

text: a

text: motif

text: is

text: obtained

text: by

text: the

text: concatenation

text: of

text: coordinate

text: in

text: each

text: position.

text: The

text: coordinate

text: of

text: a

text: motif

text: is

text: also

text: refer

text: as

text: the

text: center

text: point

text: of

text: the

text: motif.

text: The

text: coordinate

text: for

text: each

text: position

text: is

text: shown

text: in

text: Equation

text: is

text: the

text: element

text: in

text: PWM

text: j

text: j

text: l

text: M

text: ij

text: which

text: represents

text: the

text: probability

text: of

text: ith

text: amino

text: acid

text: at

text: jth

text: position,

text: and

text: x

text: is

text: the

text: i

text: coordinate

text: for

text: ith

text: amino

text: acid.

text: X

text: C

text: x

text: M

text: j

text: i

text: ij

text: The

text: query

text: represents

text: the

text: motif

text: and

text: applies

text: LSH

text: technique

text: to

text: search

text: all

text: q

text: k

text: k

text: data

text: points

text: in

text: the

text: database

text: where

text: p

text: p

text: q

text: T

text: Figure

text: shows

text: the

text: pairwise

text: distance

text: among

text: sequences

text: in

text: a

text: motif

text: to

text: the

text: cen-

text: ter

text: point

text: from

text: data

text: points

text: in

text: motifs

text: and

text: random

text: data

text: points.

text: The

text: data

text: points

text: in