# Why is SVM unable to separate linearly separable data?

I have two sets of data, namely blue and yellow. I manually added a point {8, -3} to the blue data.

sampledata[center_] := BlockRandom[SeedRandom; RandomVariate[MultinormalDistribution[center, IdentityMatrix], 200]];
clusters1 = sampledata /@ {{9, 0}, {-9, 0}};
clusters1[] = Append[clusters1[], {8, -3}];

plot1 = ListPlot[clusters1, PlotStyle -> Darker@{Yellow, Blue}];
plot2 = Plot[0.2 - 0.375*x, {x, -12, 12}, PlotStyle -> Red];
Show[plot1, plot2] As you can see, the two sets are linearly separable. Thus the SVM algorithm should be able to separate all points by using just a linear kernel. Now I try below:-

c3 = Classify[<|Yellow -> clusters1[], Blue -> clusters1[]|>, Method -> {"SupportVectorMachine", "KernelType" -> "Linear"}]
Show[Plot3D[{c3[{x, y}, "Probability" -> Yellow], c3[{x, y}, "Probability" -> Blue]}, {x, -15, 15}, {y, -4, 4}, Exclusions -> None], ListPointPlot3D[Map[Append[#, 1] &, clusters1, {2}], PlotStyle -> {Yellow, Blue}]] As you can see, SVM failed to separate the points. The blue point {8, -3} is now located in the yellow region. Why would SVM be failed to separate the linearly separable points?

Many thanks!

• – Niki Estner Aug 11 '18 at 6:24
• @Niki Estner Thanks. In fact I tried something like Method -> {"SupportVectorMachine", "KernelType" -> "Linear", "L2Regularization" -> 0.5}in Classify, but got errors... – H42 Aug 11 '18 at 6:30
• I think you are using a hard margin classifier. That means that no misclassified data points are allowed and as result the margin can get arbitrary crappy in the strife to make sure that exactly all data points are classified correctly. – mathreadler Aug 11 '18 at 11:15
• Please don't use JPG for non-photographic images! – Andreas Rejbrand Aug 11 '18 at 12:30
• Wait I see now I misread your question, what I meant to say is that you have a soft margin classifier and you need to reduce the "softness" parameter. But I see you already know this from your answers. – mathreadler Aug 11 '18 at 21:43

It's not explicitly documented, but I think Mathematica is using the C-SVM variant, where a regularization parameter C basically says how "expensive" mislabeled training samples are, compared to the size of the margin. So in your case, SVM will perfer a larger margin between the yellow and the blue points. In 99% of the cases, this kind of robust behavor is exactly what you want.

If you don't want that, you can play with the (undocumented) SoftMarginParameter option. Think of this as the cost of mislabeling a training sample, compared to getting a larger separation margin:

c3 = Classify[<|Yellow -> clusters1[], Blue -> clusters1[]|>,
Method -> {"SupportVectorMachine", "KernelType" -> "Linear",
"SoftMarginParameter" -> 1000000}]

Show[ContourPlot[
c3[{x, y}, "Probability" -> Yellow], {x, -15, 15}, {y, -4, 4},
AspectRatio -> Automatic], plot1] Now all samples are classified "correctly", but if you tried the classifier on new data, it will likely perform worse, as some of the dots are much closer to the margin

• As to why it would be likely to perform worse... Consider the likelihood of the outlier point being a true member of one or another distribution. Chances of any point being as far or further from the blue cluster with its distribution are about 2 * 10^-65, while from the yellowish cluster the chances are around 7 * 10^-3. – kirma Aug 11 '18 at 7:05
• @kirma not necessarily, if we dont know the distribution the blue one could well be a sum of monomodal distributions where the second mode is really really close to the yellow cluster. For example sum of two gaussians where the big cluster has samples 100 or 1000 times as often as the "outlier". – mathreadler Aug 11 '18 at 12:20
• @kirma I think mathreadler's point is exactly that... When giving these numbers you are speculating yourself ;) – sebhofer Aug 11 '18 at 12:39
• @sebhofer Well... In this particular case we actually know that data originates, apart from one point, from specific distributions. Of course it's more complicated with real data, but quite often (multi)normal distribution is a good starting assumption for a fit, or at least it's convenient to work with. There's a tuning parameter for the task, but it might out turn into a gun pointing on ones' own foot. – kirma Aug 11 '18 at 12:44
• It is quite often in real life that data is much more complicated than monomodal. The probability distribution over locations for getting drunk for example are probably sum of different distributions over a citys bars and friends places. If you are drunk 10 times in Bar A and then 1 time at Bar B you could claim that bar B is an outlier. But maybe it's just the rare instances when you meet a friend who is rarely in town. – mathreadler Aug 11 '18 at 12:53