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Morten Hjorth-Jensen
2021-11-25 08:34:39 +01:00
parent dd5df5a0e1
commit 9158407e7d
9 changed files with 274 additions and 251 deletions
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doc/src/week47/week47.do.txt
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@@ -52,7 +52,7 @@ The theory behind support vector machines (SVM hereafter) is based on
the mathematical description of so-called hyperplanes. Let us start
with a two-dimensional case. This will also allow us to introduce our
first SVM examples. These will be tailored to the case of two specific
classes, as displayed in the figure here based on the usage of the petal data.
classes, as displayed in the figure here based on the usage of the petal data.
We assume here that our data set can be well separated into two
domains, where a straight line does the job in the separating the two
@@ -561,7 +561,10 @@ Below we discuss how to find the optimal values of $\lambda_i$. Before we procee
Till now, the margin is strictly defined by the support vectors. This defines what is called a hard classifier, that is the margins are well defined.
Suppose now that classes overlap in feature space, as shown in the
figure here. One way to deal with this problem before we define the
figure in the "handwritten notes":"https://github.com/CompPhysics/MachineLearning/tree/master/doc/HandWrittenNotes/2021" for Thursday November 25.
One way to deal with this problem before we define the
so-called _kernel approach_, is to allow a kind of slack in the sense
that we allow some points to be on the wrong side of the margin.
@@ -658,8 +661,9 @@ space using other basis expansions such as higher-order polynomials,
wavelets, splines etc.
If our feature space is not easy to separate, as shown in the figure
here, we can achieve a better separation by introducing more complex
basis functions. The ideal would be, as shown in the next figure, to, via a specific transformation to
in the "handwritten notes":"https://github.com/CompPhysics/MachineLearning/tree/master/doc/HandWrittenNotes/2021" for Thursday November 25.
, we can achieve a better separation by introducing more complex
basis functions. The ideal would be to, via a specific transformation to
obtain a separation between the classes which is almost linear.
The change of basis, from $x\rightarrow z=\phi(x)$ leads to the same type of equations to be solved, except that