lecture14-SVMs-handout-6-per

Mapdataintobererrepresentaonalspace commonkernels

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normal:
 Find α1…αN such that Q(α) =Σαi - ½ΣΣαiαjyiyjxiTxj is maximized and (1) Σαiyi = 0 (2) 0 ≤ αi ≤ C for all αi   I.e.,
compute
score:
wTx
+
b
=
ΣαiyixiTx
+
b   Can
set
confidence
threshold
t.
   Neither
slack
variables
ξi

nor
their
Lagrange
mul)pliers
appear
in
the
dual
 problem!
   Again,
xi
with
non‐zero
αi will
be
support
vectors.
   Solu)on
to
the
dual
problem
is:
 w = Σαiyixi b = yk - wTxk for any k s.t. 0 < αk < C w is not needed explicitly for classification! f(x) = ΣαiyixiTx + b Score
>
t:
yes
 Score
<
-t:
no
 Else:
don’t
know
 -1 15
 Sec. 15.2.1 Introduc)on to Informa)on Retrieval 0 1 16
 Sec. 15.2.3 Introduc)on to Informa)on Retrieval Linear
SVMs:

Summary
 Non‐linear
SVMs
   The
classifier
is
a
separa)ng hyperplane.   Datasets
that
are
linearly
separable
(with
some
noise)
work
out
great:
   The
most
“important”
training
points
are
the
support
vectors;
they
define
 the
hyperplane.
   Quadra)c
op)miza)on
algorithms
can
iden)fy
which
training
points
xi
are
 support
vectors
with
non‐zero
Lagrangian
mul)pliers
αi. x 0   But
what
are
we
going
to
do
if
the
dataset
is
just
too
hard?

 x 0   How
about
…
mapping
data
to
a
higher‐dimensional
space:
   Both
in
the
dual
formula)on
of
the
problem
and
in
the
solu)on,
training
 points
appear
only
inside
inner
products:

 x2 Find α1…αN such that Q(α) =Σαi - ½ΣΣαiαjyiyjxiTxj is maximized and (1) Σαiyi = 0 (2) 0 ≤ αi ≤ C for all αi f(x) = ΣαiyixiTx + b 0 17
 x 18
 3 Sec. 15.2.3 Introduc)on to Informa)on Retrieval Non‐linear
SVMs:

Feature
spaces
 Sec. 15.2.3 Introduc)on to Informa)on Retrieval The
“Kernel
Trick...
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This document was uploaded on 02/26/2014.

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