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721 7711 7217712 - Idiap Research...

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ESEARCH R EP R ORT IDIAP +41-27-721 77 11 +41-27-721 77 12
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Page 1: 721 7711 7217712 - Idiap Research Institutepublications.idiap.ch/downloads/reports/2003/rr03-42.pdf · e s e a r c h r e p r o r t i d i a p ! "$# %%&' )(+*,-/. $& $ ! / 0 %*1&' %2

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LM(x, y) =1

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g (xi, yi)[Tu�`

2)(D5

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LV (x, y)

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����� ��� � ��. )$!��-# 12!$. ) � ��1�������# ������� �� ¥ ¨ =^4h2�*�+-P�V;(D=AO�8;+�6�&)*,=AKx2C*,+?5B0�dB4�*�2L*�=A4�*,=^PF2):c:A+-2)I,(;=A(;H�*�V;+F&.I,d�2C(D5BVD2.4r0�+F+-(J4�8DP-P$+?4,4�RT8;:A:>dB2CS;SD:>=A+-5�*�&

(�8D6�+-I,&.8D4qP$:^2)4�4�=>�3P-2L*,=A&)(�*�2.4�M@4�j lCuD]cl.v k _ybVD+�M)+Fd7=^5@+?2!&)R� ¥ ¨ =A4N*�&�=A6�S;:A=APF=<*�:>dQS;I,&)WX+-P$*N*�V;+�=>(DS;8@*�4,SD2)PF+=>(�*,&92�4,SD2)PF+g&CRyV;=AH)V;+-I'5@=A6�+F(D4,=A&)(J[TS�&.4�4,=>0;:Ad�=>(;�D(;=>*,+?`fegV;+FI�+g*�V;+UPF:A2.4,4,+-4'2CI�+g6�&.I,+h:>=A(;+-2)I,:Ad�4�+-SD2CI�2C0;:A+)_.bV;=A4S;I,&)WX+-P$*,=A&)(�=A4o=A6�S;:A=APF=<*"0�+-P-2C8D4,+i*,V;+h:>+?2CI�(;=>(DHq2)(D5�5@+-PF=A4,=>&.(�S;I,&@PF+-4�4�+?4�&.(;:Ad�=A(�K.&):AK)+i=>(D(;+FI"5;&C*fS;I�&@5@8DPY*f=A(�*,V;+RT+-2L*�8;I�+q4,SD2.P$+U*,VD2C*gP-2C(�03+ZPF&)6�S;8@*�+-5!8D4�=A(;H�29M)+FI�(;+-:�_@nr(B+FE�*�+F(D4,=AK)+q5@=A4�P$834,4,=>&.(z&)Rw� ¥ ¨J4gPF2)(!0�+NRT&)8;(D5!=A(j � ¤ k _ ¢ (�*�V;=A4wS32CS�+FI?]Le'+eg=A:>:@&.(;:>d9P$&)(34�=^5@+FIo2N0;=A(D2CI�d�P$:^2)4�4,=<�3P-2L*�=>&.(�*�2.4�MZeg=>*,V

nt

:^2C0�+F:A+-59*�I�2)=>(;=A(;HN+$E;2C6�S;:A+-4 (z1, y1) , (z2, y2) , . . . , (znt

, ynt)];egV;+-I,+

zi

5@+F(;&)*,+Z2C(�=A(;S;8@*rK.+-PY*�&)Ig2C(35yi = ±1

=A(D5@=^PF2C*,+?4=<*�4gP$:^2)4�4:A2)03+-:�_¢ R'*�V;+�5;2L*�2�4,+$*�=^4q:A=>(;+?2CI�:>dQ4�+-SD2CI�2C0;:A+)]�*�V;+FI�+�+$E@=A4�*�4Z2C*�:>+?2)4�*Z&)(;+�VGdGS3+-I,S;:^2C(D+

w · z + b = 0[\5@+?P$=^4�=A&)(

4�8;I,R\2)PF+?`"*,V32L*h4,+FSD2)I�2C*,+-42):>:�*�V;+q*,I�2C=A(;=A(;H�+$E;2C6�S;:A+-4�

yi (zi · w + b) ≥ 1 ∀i[ � l)`

egV;+FI�+w

2C(D5b2CI�+N(D&)I�6�2):>=A{F+?5t0Gdt*�V;+qRT&):A:A&Leg=>(;H�P$&.(D4�*,I�2C=A(.*�

mini=1,...,nt

|w · zi + b| = 1[ � m.`

bV;+�6�2)I,H.=>(Q&CR4�8DP�V�2�V�dGS�+FI�S;:^2C(;+w · z + b = 0

=^4q5@+$�3(;+-5�0GdB*,V;+�4,8;6�&)R"*,VD+�4,V;&.I�*�+-4�*q5@=^4�*�2C(3P$+�RTI,&.6*,V;+9V�dGS�+FI�S;:^2C(;+q*�&t*,VD+�P$:A&.4,+-4�*hS�&.4,=<*�=>K.+�+$E;2C6�S;:A+�2C(35B*,V;+94�V;&.I�*�+-4�*U5@=^4X*�2C(DPF+ZRTI�&)6 *,V;+9V�dGS�+FI�S;:^2C(;+q*�&�*,V;+P$:A&.4,+-5J(D+FH.2C*,=AK)+9+$E;2C6�S;:A+)_�bV;=^4U6�2)I,H.=>(Q=A4q4,=>6�S;:Ad 2

‖w‖

]�egV;+-I,+ ‖w‖ =A4N*�V;+ s 8DP$:A=A5;+-2C(�(;&.I,6�&CRw

_ybV�834F]*,V;+!6�2CE@=>698;6 6�2CI�H)=A(�PF2)(�03+t&.0@*�2)=>(D+-5�0GdQ6�=A(;=A6�=A{F=A(;H ‖w‖2 4,8;0@WX+?PY*�*�&7*,VD+zP$&.(D4X*�I�2)=>(�*�4 s O�_ � l@_�bVD+&)S@*�=>6�2C:

w∗PF2)(!0�+�5@+$�3(;+-5�0Gdt8D4,=>(DH�2�4,+$*r&)R�ei+F=AH)V�*

λj

&)(�*,VD+U*�I�2)=>(D=>(;H�+FE@2)6�S;:>+?4g2)4�

w∗ =

nt∑

j=1

λjyjzj

[ � uG`

bV;+F(y];*,V;+�PF:A2.4,4,=<��PF2L*�=>&.(z&)R�2C(B8;(DM�(D&Leg(!+FE@2)6�S;:>+z

=^40D2)4,+-5!&)(�*�V;+�4�=AH)(�&CR�*,V;+qRT8D(DPY*�=>&.(

G(z) =

nt∑

j=1

λjyjz · zj + b∗[ � v.`

egV;+FI�+b∗ = yj −w

∗ · zj , λj 6= 0_

bV;=A4�6�+$*�V;&@5�PF2)(�03+'H)+-(;+FI�2C:A=>{-+-5N*�&r*,VD+"(;&.(;:>=A(;+?2CI�P-2)4,+o*,VDI,&.8;H)VZ*�V;+"6�2)S;S;=A(;Hh&)R@*,V;+"*,I�2C=A(;=>(DHr+$E;2C6�S;:A+-4zi

=A(�*,&�2C(72C:>*,+-I,(32L*,=AK)+NV;=AH)V75@=A6�+F(D4,=>&.(D2C:�RT+?2L*�8;I,+Z4,SD2)PF+φ (zi)

=>6�S;:A=^P$=>*,:Ad!2)P�VD=>+-K)+-5!0GdzP�V;&G&.4,=A(;H�2�M.+FI�(;+F: K (zi, zj) = φ (zi) · φ (zj)

_DbV;+Z5@+?P$=^4�=A&)(!*�2.4�Mz=A4*�V;+F(74,=>6�S;:Adt*�I�2)(D4�RT&)I�6�+?5t=A(�*,&

G(z) =

nt∑

j=1

λjyjK (z, zj) + b∗[ � ��`

���� � ) � ������� ,bV;+N5D2L*�2)0D2)4,+Ne'+N8D4,+-5�P$&)(34�=^4X*�4i&CRc*�+$EG*r2C(D5!(;&.(@/�*�+$EG*+$E;2C6�S;:A+-4iI�+-4,8;:<*�=>(DH�RTI�&)6 *�V;+q:>&@P-2C:A=>{?2L*,=A&)(!4X*�+FSy_

� +g5@=AKG=A5@+?5�*�V;+5;2L*�2C0D2.4�+'+-O�8D2C:A:AdZ=>(�*,&q2h*,I�2C=A(;=>(DHN4,+$*o2C(D5�2h*,+-4�*o4�+F*-_Cb�I�2)=>(;=A(;HN2C(D5�*,+-4�*,=A(;Hqei+FI�+"S�+FI,RT&)I�6�+?50�dz8D4�=A(;Ht2�¨J© � &.Ir2�� ¥ ¨ P$:^2)4�4,=<�D+-I-_

¢ (9*�V;+hP-2)4,+i&)R�¨�© � ]C*�V;+r(;+F*Xe'&.I,M�PF&)(D4,=A4�*�4f&CRp2C(�=>(DS;8@*":^2xd)+-I-]L2qV;=^5;5@+-(�:A2xd.+FIo2C(D5�2)(�&.8@*,SD8@*"=A(D5@=^PF2L*�=>(DHP$:^2)4�4�+?4q&CR*,+FE�*�2C(D5�(;&.(@/1*,+$EG*?_cbV;+z¨J© � =A4�*,I�2C=A(;+-5�0GdQ*,V;+z|2.P�M./}S;I�&)SD2)H.2C*,=A&)(�2):>H.&)I�=<*�V;6B_ybV;+t(G8;6�0�+FI&CRV;=A5D5@+F(�(;+-8;I,&.(D4-]pegV;=AP�V�=A(D5@=^PF2C*,+�*,VD+tPF2CS32)P$=>*XdJ&)R'*�V;+t¨J© � ]y=^4�P�V;&�4�+-(Q0GdQS3+-I�RT&.I,6�=A(;H�27¡N/1RT&):^5�P$I�&.4�4Kx2):>=^5;2C*,=A&)(�&)(!*,V;+q*�I�2)=>(;=A(;H�4,+$*-_ ¢ *r=A4g5@+?4,PFI,=A0�+-5!03+-:>&LeZ_

¢ (z*�V;+�PF2.4�+N&CRf� ¥ ¨�]@*,V;+N*�I�2)=>(;=A(;H�=^4S3+-I�RT&.I,6�+-5!0Gdt8D4,=>(DH�2�O�8D2.5@I�2C*,=^PhSDI,&.H)I�2C6�6�=>(;H�2C:AH)&.I,=>*,V;6B_;nU4g2P�V;&)=^P$+NRT&)I*�V;+ZM)+-I,(D+F:1]�ei+Z8D4,+N*Xd�SD=AP-2C:y�r2.5@=A2):p0D2)4,=^4iRT8;(DPY*�=>&.(�[\�r| 5 `Y]D5;+$�D(;+?5�0�d

K(z1, z2) = e−||z1−z2||2

2σ2

[ � �)`

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������������ z|}� ���y ¤

egV;+FI�+�*,V;+�M.+FI�(;+F:"0D2C(D5;eg=A5G*�Vσ=^4�2BSD2)I�2)6�+$*,+-Iq*,VD2C*�=A4�5@+-PF=A5@+?5�0�dQ8D4,=>(;HJ¡N/1RT&):^5�P$I�&.4�4X/}KL2C:A=A5D2L*,=A&)(c_�bVD+

¡N/�RT&.:A5�P$I�&.4�4X/}KL2C:A=A5D2L*,=A&)(z8D4�=A(;H�2�� ¥ ¨ 2.42C(B+$E;2C6�S;:A+ZPF2)(!0�+�5@+-4�P$I�=A03+?5z:A=>M.+N*,VD2C* � _ � 2CI,*,=>*,=A&)(t*�V;+q*,I�2C=A(;=A(;H�5;2L*�2�4,+$*g=A(�*,&�¡ SD2)I�*�4i&CR�+?O�8D2C:y4,=>{-+)]GI�+$RT+FI*�&�2)4 ��RT&.:A5;4��L_@bV;+-(y]D2.4,4,=>H.(t+-2.P�VRT&.:A5�2�S3&�4,4,=A0;:>+qKL2C:A8;+q&CR

σ_

l@_ 5 &.Ii �

� *�&�¡�]G*�I�2)=>(z*,V;+�� ¥ ¨ 834�=A(;H�2C:A:�*,V;+NRT&):^5;4i+$E;PF+FS@**,VD+i*�V!2)4'*�V;+q*,I�2C=A(;=A(;H94,+$*r2C(35t*�V;+

i*�V

σ2)4SD2)I�2)6�+$*,+-I-_ s KL2):>8D2C*,+N*,VD+Z+FI�I,&.Ii&)Rc*�V;+Z&)8@*�S;8@*rVGdGS3&)*,V;+?4�=^4g8D4,=A(;H�*,V;+Z=>*,V�RT&):^5�2)4i*,VD+q*,+-4�*h4,+$*-_

m;_gb�2)M)+�*,VD+7Kx2):>8D+�&)RσP$&.I,I�+-4,S�&)(D5@=A(;HQ*,&�*�V;+7:>&Lei+-4�*�+FI�I,&.I�I�2C*,+7P$&.6�SD8@*,+?5�=A( [\m�`�2)4�*,VD+7&)S@*�=>6�2C:

SD2)I�2)6�+$*,+-Ig2C(D5z*,I�2C=A(z*�V;+�� ¥ ¨ 834�=A(;H�2C:A:p*,VD+q*,I�2C=A(;=>(DH�5;2C*�2�*�&�H.+$*r2�H)&G&@5!4,8;S;S�&)I,*rK)+?PY*�&)Ig4�+F*-_

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P$:^2)4�4�=>�D+FI�2CI�+J+$EG*�I�2.PY*,+?5�RTI�&)6 *,V;+Q(;&)I�6�2C:A=>{-+-5�P-2C(D5@=^5;2C*,+J*�+$EG*B:>=A(;+�&.(16 × 16

4�:A=A5;=>(;H�eg=>(35@&Ler4teg=>*,V2Q4,:>=^5@+74�*,+FS�&CRUuQS;=>E@+F:^4F_fbVG8D4-]�RT&)I�+-2.P�V�P-2C(D5@=^5;2C*,+!*,+$EG*�:A=>(;+

r

]oei+!&.0@*�2)=>(;+?5�2�4,+$*�&CRhRT+?2L*�8;I,+�K)+?PY*�&)I�4Zr = (zr

1, . . . , zrl )_�©c+$*

G(zri )5@+-(;&C*�+-4q*�V;+t&)8;*,S;8@*9&CR*,V;+!¨J© � &.Iq*,V;+t6�2)H)(;=>*,8D5;+t&CR*,VD+z� ¥ ¨�]�egV;=^P�V

=>(D5;=AP-2L*,+?4y*�V;+P$&)(;�35@+F(3P$+f*�VD2L*�*,V;+iK)+?PY*�&)Iz

ri

0�+F:A&)(;H�4y*�&U2r*,+$EG*w:>=A(;+._?bV;+iP$&.(@�35@+-(DP$+'&CR;*�V;+"egV;&.:>+iPF2)(D5@=^5;2L*�+*,+$EG*r:A=>(D+

r

=^4i*,V;+-(B5;+$�D(;+?5B2)4�

Conf(r) =∑

zri∈Zr

G(zri ).

1√2πσ0

e−

d2

i

2σ20

[ � ��`

egV;+FI�+di

=^4g*,VD+95@=^4X*�2C(DPF+ZRTI�&)6 *,V;+9H)+F&.6�+$*,I�=^PZP$+F(�*�+FIh&)Rw*,V;+ith

4�:A=A5;=>(;Hzeg=>(D5;&Le�*�&�*,VD+�H)+-&)6�+$*�I,=^PZP$+-(�*,+FI&CR"*�V;+�*,+FEG*q:A=A(;+ � ]�2C(D5

σ0=^4Z2!4�PF2):>+�R\2.PY*,&.IZ5@+FS�+F(D5;=>(;HB&.(�*�V;+�*,+FE�*Z:A=>(D+�:A+F(DHC*,Vc_ 5 =>(D2):>:Ad)]�*�V;+�PF2)(D5@=^5;2L*�+

*,+$EG*r:A=>(D+ � =^4gP$:^2)4�4�=>�D+-5�2)4g2�I�+-2):�*,+$EG*rI�+FH.=>&.(z=>Rf%'&.(@R(r) ≥ 0

_

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