Margins and Support Vectors

Margin and Support Vectors

The margin lines are the score levels just outside the separator. Points that touch those lines exactly are support vectors, while the remaining shown points sit beyond the margin.

highlighted = computed this step

The margin lines

The margin lines are w·x+b=plus-or-minus 1. For this w and b they are x+y=0 and x+y=4.

w⋅x+b=±1w\cdot x+b=\pm 1
Margins and support vectorsThe margin lines and support vectors are recomputed from w and b.SVM margin geometryExact hyperplane and margin lines for the displayed separator.SV+ASV-B+C-Ddash: w·x+b=-1solid: w·x+b=0dash: w·x+b=1

Why support vectors are special

The points (2, 2) and (0, 0) touch the margin exactly. Their functional margins are 1 and 1.

yi(w⋅xi+b)=1y_i(w\cdot x_i+b)=1
Margins and support vectorsThe margin lines and support vectors are recomputed from w and b.SVM margin geometryExact hyperplane and margin lines for the displayed separator.SV+ASV-B+C-Ddash: w·x+b=-1solid: w·x+b=0dash: w·x+b=1

Summary

All displayed points satisfy y·(w·x+b)>=1. The exact margin values are 1, 1, 2, 2.

[1,1,2,2][1,1,2,2]
Margins and support vectorsThe margin lines and support vectors are recomputed from w and b.SVM margin geometryExact hyperplane and margin lines for the displayed separator.SV+ASV-B+C-Ddash: w·x+b=-1solid: w·x+b=0dash: w·x+b=1