Margins and Support Vectors

What Max-Margin Is and Is Not

The finale keeps the exact table and plot visible while stating the boundary of the claim. This surface verifies displayed geometry; broader model claims are outside the lesson.

highlighted = computed this step

What is verified

The table verifies w·x+b values 1, -1, 2, -2 and margins 1, 1, 2, 2.

yi(w⋅xi+b)≥1y_i(w\cdot x_i+b)\ge 1
Exact SVM geometryThe table and plot verify the same displayed separator.SVM margin tablepointyw·x+by(w·x+b)support(2,2)+111yes(0,0)-1-11yes(3,3)+122no(-1,-1)-1-22now·w=1/2margin width=2/√(1/2)√ is named; no decimal margin is pinnedSVM 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

What this surface stops at

The surface displays a given exact separator and checks its geometry. The optimizer that would choose it is outside the rendered surface.

shown separator geometry only\text{shown separator geometry only}
Exact SVM geometryThe table and plot verify the same displayed separator.SVM margin tablepointyw·x+by(w·x+b)support(2,2)+111yes(0,0)-1-11yes(3,3)+122no(-1,-1)-1-22now·w=1/2margin width=2/√(1/2)√ is named; no decimal margin is pinnedSVM 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

This verifies a given exact solution's geometry; it is not the optimization or training process, not a claim about accuracy, and no claim of generalization to unseen points.

verified toy-data max-margin geometry only\text{verified toy-data max-margin geometry only}
Exact SVM geometryThe table and plot verify the same displayed separator.SVM margin tablepointyw·x+by(w·x+b)support(2,2)+111yes(0,0)-1-11yes(3,3)+122no(-1,-1)-1-22now·w=1/2margin width=2/√(1/2)√ is named; no decimal margin is pinnedSVM 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