Margins and Support Vectors

Which Points Are Support Vectors

Support-vector status is exact bookkeeping on the displayed rows: compute each signed margin, mark equality to one, and leave larger margins outside the setting set.

highlighted = computed this step

Compute signed margins

For each displayed row, compute y times (w dot x plus b). The exact signed margins are 1, 1, 2, and 2.

yi(w⋅xi+b)=[1,1,2,2]y_i(w\cdot x_i+b)=[1,1,2,2]
Which Points Are Support VectorsExact signed margins identify the displayed support-vector rows.support vectors from exact signed marginssupport vector iff exact signed margin equals 1rowpointyw·x+by(w·x+b)support?A(2,2)+111yes: margin = 1B(0,0)-1-11yes: margin = 1C(3,3)+122no: margin > 1D(-1,-1)-1-22no: margin > 1displayed support vectors: A, Bdisplayed-row bookkeeping only

Margin equal to one

A displayed row is a support vector here when its signed margin equals 1. Rows A and B have margin 1, so they are marked as support vectors.

support vector  ⟺  yi(w⋅xi+b)=1\text{support vector}\iff y_i(w\cdot x_i+b)=1
Which Points Are Support VectorsExact signed margins identify the displayed support-vector rows.support vectors from exact signed marginssupport vector iff exact signed margin equals 1rowpointyw·x+by(w·x+b)support?A(2,2)+111yes: margin = 1B(0,0)-1-11yes: margin = 1C(3,3)+122no: margin > 1D(-1,-1)-1-22no: margin > 1displayed support vectors: A, Bdisplayed-row bookkeeping only

Margin greater than one

Rows C and D have signed margin 2. Since 2 is greater than 1, those rows sit beyond the margin and do not set it. This is displayed-row bookkeeping only.

C,D: 2>1C,D:\ 2>1
Which Points Are Support VectorsExact signed margins identify the displayed support-vector rows.support vectors from exact signed marginssupport vector iff exact signed margin equals 1rowpointyw·x+by(w·x+b)support?A(2,2)+111yes: margin = 1B(0,0)-1-11yes: margin = 1C(3,3)+122no: margin > 1D(-1,-1)-1-22no: margin > 1displayed support vectors: A, Bdisplayed-row bookkeeping only