Outlinekernel Methods

SVM

feature mapKd RP

a Ollie

fitting hood Oton using gradientdescentwith 01 uh as features

0 0Loop0 Oth IECgc 80cm loln

Issue 0 Oln CIRP can be very highdimensionalruntime per iteration OCap

Goat improve to 0cm per iteration

key observation0 can be represented as

a EE pi locascalar n variables insteadof p

Proof by inductionAt iteration 00 0 IE O 46

supposeat iteration t O Eh pi 19amNext iteration o y z yet or la Dolce

Epi 19am

0 E patacy Gollum 4cal9new pi

represent 0 C IRP implicitly by 13 Elkworks better p n

Update rule for p 01 Block

Bi pit Nyc OTQue p allaDT

path y E B Gcn 5 love

pied y Ep Ca loca D

atb La bObservations

LOCK OC x 7 As can be precomputed

Often Collum Olsen can be computed fasterthan OcpI

Iid P Ad

25 71 2 EdEdgin z.cziid

l

yXd

10K lolz It 71 12u t.IE eXiXjZi2jtd

EEnXeXsXk2c32qIt CX 27 t Lx

2X 273

Can compute 2041 0677 in old tune

Klee 2 QU 01677 kernelkn IrdXIRD IR

Algocompute kCx x Collie localfor all j e I n

Setp OOP

pi p x y Ep Mx a

preprocessing O n'dEach iteration 0cm per iteration

prediction Gwenn compute 00h seal teh training00cal EE pi love

TOllie example

EE pi Kcnc na new example

OCnd time

Deeper Observation

TheAlgoonly depends on KC e

design of features design of kernel functions

what kernels are valid 7

Flo St Kae 2 local OG

Necessary conditionindata points x 2dm

Kernelmatrix KEIR mKy K za za

matrix K is posceivesemidefinite 4K Z OKE o es 2T K2 70 Ky LolCid locumIT Kz Zi Ky 3 locale 4Cn dad

2 Ollie T01 41 2

E zi deck deck 2g

Ee C ai dela 75 30

K Z O

Th m Mercer K is a valid kernel fn ie Kca2 a Caiolaiff forany ncos and any x Xthe corresponding kernel matrix Kst Kyi Kae red

is positive semidefinite

Other kernels

Mn 2 XTZ t CT Cal lolz

Kfk 2 LxTz ct

Gaussian kernel Kca 2 exp 42221122

Kcr 2 LOCH 01677Infinite dimensional

Xi axe Xi Xi

Protein sequence classificationsequencerof amino acids CA T

AAAAAAAB

204 dem vector

TTT T

Histograms take men and seem up

SV Ms for classification

I n.IE o

O o x wtf n 16 030

SVMgooey n for nowyil E E L I

Warmupifeud w b s t

if y L win c b 70

y z I wt k cb s O

Many such w b

Newgoal Among all w b satisfyingFund web St mw.abx.gyy.my

dirt Cail boundary4

n wheels o

WTH'tbF f N U on positive sidewhats o

max mon y what bw b LEI n

Scale invariant W b 400W toob

wlog we want to fund w bSA mm y Wta b I

iE l n

max I men 11WhAWHz

men11412

St fi y Luta t b 71

men 11Wh men ENKESt the y Ewtn cb 31

Facts optimal soli w dig n for somedisco

The h in is theoptimizerofwk EYi Iz 7,5 y t Cn n T

SA A 70 Edy O

Kernelize replaceLsd HD wah kfnih.nl

Fixing linearlyYep

r dIFssumption

Joo

yy wTn tb I 3 it

y tt t

min lzltwlk CE.is