Introduction to Sparse Modeling

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Introduction to Sparse Modeling

ee7PYHHQP AOPTQHP =YHQH 8TY BLJO 1 LH ] QYO V PT L 6 [V 1 JHSV M OL a8TT H P L @LYLH JO. 6 NLTLYPYd 9HT 0 2P]HQ

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http://sparse-modeling.jp

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) 9T] _ LRP L MWLNV S WP 0 NPY_P] ?.-

vyno wzu https://alma-telescope.jp/news/press/eht-201904

aAJPLTJLd 2 LHQ O [NO M OL ELH -

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M=(6.5 0.7)x 10 ⁹ Msun

eye sight

=3x10 ⁶

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Hisaaki Shinkai (OIT) 真貝寿明（大阪工大）

CTYR O bY :G PL]NS TYR 4 _ CPR]P TaP OPW

4C OPW

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ringdown search 60 mockdata

L_NSPO fW_P]TYR TWMP]_ LY E]LY ] L_T Y

4 _ CPR]P T Y ?P_S O P ]LW P_b ]V P_S O

NVOL_L NSLWWPYRP N [L]T Y

Nakano+, PRD99 (2019) 124032

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- 5P PTN H H ]P O RPTLH M[TJ

L N

Z ₁ = e ^{(r j!} ⁾ ^t Z ₂ = e ^(r+j ^! ⁾ ^t

x _n = Ae ^{rn t} cos(! n t)

5 10 15 20 25 30

-0.6 -0.4 -0.2 0.2 0.4 0.6 0.8

5 10 15

-0.5 0.5 1.0

NLY MP L[[WTPO LW _ Y T d OL_L Md LO _TYR

4 _ CPR]P TaP OPW TOPL

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. 5P PTN H H ]P O RPTLH M[TJ

c bT 2[ N SL O

c bT 5 4 bTHR V L PJ P T L SL O c L J TY [J ]H L YPNTHR M S b L M[TJ P T c HVVR 55B ]P O H IP H V LJPYP T

V ]L YVLJ [S

4 _ CPR]P TaP OPW ?P_S O RPYP]LW

JOH HJ L PY PJ LW

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/

4 _ CPR]P TaP OPWia DS ]_ 99E

PR PY_ 1 **,* PN 1 ,* [ TY_**

ST _ 1 ( PN 1 . [ TY_

]P B

4C

99E

8aPY ] S ]_ PR PY_

4C OPW S b []PNT P [ bP] [PN_]

FS JQ H H A @/( PTYVP HR VH G S _

BOL L JHT IL b L H f,

**L [WTYR ]L_P1* /,**

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10 @PTN ]T ALH JO I 1[ @LN LYYP L 1VV HJO = = V[IRPJ H H 6 B JH HR N[L

GW150914

> :A [L[P]

4C OPW

( - YHSVRPTN H L ]P

) () 7 bR L

YLNSLT / ( YLJ/ ( V PT Y YOPM / ) YLJ / , V PT Y

LY ]O ^{_T P}

]P I e

k P]RP]

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11 @PTN ]T ALH JO I 1[ @LN LYYP L 1VV HJO = = V[IRPJ H H 6 B JH HR N[L

Hanford (SNR=20.6)

GW150914

Livingston (SNR=14.2)

I e I e

4 D7I m e 4 D7I m e

_T P

I e I e

_T P

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@PTN ]T ALH JO I 1[ @LN LYYP L 1VV HJO = = V[IRPJ H H 6 B JH HR N[L

Hanford (SNR=20.6)

L100_SpectrogramAR H100_SpectrogramAR

GW150914

Livingston (SNR=14.2)

>F [L[P]

4C LY ]O 4C >TaTYR _ Y

f₂₂₀

= 271.8 Hz, f

₂₂₁

= 266.0 Hz, f

₂₂₂

= 254.7 Hz

f₂₁₀

= 380.7 Hz, f

₂₁₁

= 225.7 Hz, f

₂₀₀

= 252.8 Hz

f₃₃₀

= 430.9 Hz, f

₃₃₁

= 427.4 Hz, f

₃₃₂

= 421.1 Hz

f₃₂₀

= 387.9 Hz, f

₃₁₀

= 351.1 Hz, f

₃₀₀

= 320.3 Hz

M g :C VHVL g

>F [L[P]

4C LY ]O 4C >TaTYR _ Y

f̲real f̲imag

Mass Kerr

param

100 200 300 400 500 f_real(Hz)

20 40 60 80 100

f_imag(Hz) GW150914

20 40 60 80 100 Mass

0.2 0.4 0.6 0.8 1.0 Kerr parameter a

GW150914

source frame

detector frame

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D[L] P OPWTYR xrqstpj TY_] O N_T Y

5 10 15 20 25

-1.0 -0.5 0.5

Sin[i Pi/5 + Pi/3]

1.0

+ RandomReal[{-0.1, 0.1}]

5 10 15 20 25 30

-4 -2 2 4

5 10 15 20 25 30

-4 -2 2 4

5 10 15 20 25 30

-4 -2 2 4

25 pt data

fitting up to x^5 fitting up to x^10 fitting up to x^25

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5 10 15 20 25 30

-4 -2 2 4

5 10 15 20 25 30

-4 -2 2 4

5 10 15 20 25 30

-4 -2 2 4

fitting up to x^5 fitting up to x^10 fitting up to x^25 fitting up to x^7

5 10 15 20 25 30

-4 -2 2 4

D[L] P OPWTYR xrqstpj TY_] O N_T Y

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ﬁtting the data with noise

=> we do not need a ﬁtting function which passes all the data (overﬁtting, 過適合)

=> rather we should ﬁnd a ﬁtting as it has more zero-components D[L] P OPWTYR xrqstpj TY_] O N_T Y

data signal redundant dictionary

with many zero components

=> minimize

7PYHHQP AOPTQHP AOPT6HQ[9[ Y[ 2 HSV0ALT HP , - 15

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,

[L] P 1 L NS eP] N [ YPY_

>4DDA WPL _ LM W _P S]TYVLRP LYO PWPN_T Y [P]L_ ]

>4DDA

l P]] ]

k > Y ] k > Y ]

ETM ST]LYT //,

D[L] P ? OPWTYR

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-

-6 -4 -2 2 4 6

GSd > Y ] N Y _]LTY_ LVP [L] P W _T Y3

β ₁ β ₁

β ₂ β ₂

k > Y ] k >( Y ]

CTROP CPR]P T Y >4DDA

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.

>4DDA aL]TL_T Y

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/

]TRTYLW Y T PO

8cP]NT P T P CP aTYR

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( Y T PO

8cP]NT P T P CP aTYR

?PLY 5W ]

]P aP LWW NLWP Y T P > b BL fW_P]

8ORP T _SPO

S__[0 WLM PPN _ __ ]T LN [ O ?P MP] dL LOL A[PY6F S_ W [dK_ _ ]TLW [dKT R[] N [dKfW_P]TYR [dKfW_P]TYR S_ W

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((

Y T PO

8cP]NT P T P CP aTYR :L TLY 5W ]

6PY_P] NPWW T P [SL TePO

: O ] ]P aTYR bST_P Y T P

8ORP T _SPO

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() 8cP]NT P T P CP aTYR

>4DDA

S__[ 0 W[ _PNS YP_ L]_TNWP 6H(=Y

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(*

8cP]NT P T P CP aTYR

>4DDA

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( 8cP]NT P T P CP aTYR

E _LW FL]TL_T Y 5]PR LY _P]L_T Y

O VY. RV LJO TL H PJRLY Q 5 PL]

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(, 8cP]NT P T P CP aTYR

E _LW FL]TL_T Y 5]PR LY _P]L_T Y

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(-

]TRTYLW Y T PO

:L TLY >4DDA E _LW FL]TL_T Y

8cP]NT P T P CP aTYR

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(.

9T] _ LRP L MWLNV S WP 0 NPY_P] ?.-

vyno wzu https://alma-telescope.jp/news/press/eht-201904

aAJPLTJLd 2 LHQ O [NO M OL ELH -

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Introduction to Sparse Modeling