Question at position 2 What are support vectors?The only data points that determine the location of the maximum-margin hyperplaneThe data points that are easiest to classifyThe data points that are farthest from the maximum-margin hyperplane

Question

Question at position 2 What are support vectors?The only data points that determine the location of the maximum-margin hyperplaneThe data points that are easiest to classifyThe data points that are farthest from the maximum-margin hyperplane

BlackTom AI · Accepted Answer

Support vectors are the training examples that lie exactly on the margin boundaries of the maximum-margin hyperplane, and therefore they alone determine that hyperplane’s position.

1) Relevant concepts:
- Support Vector Machine (SVM) finds a hyperplane w·x + b = 0 that separates classes with maximum margin.
- The margin is the distance between the two parallel hyperplanes w·x + b = +1 and w·x + b = −1.
- Support vectors are the data points that satisfy y_i(w·x_i + b) = 1 (lie on these margin boundaries).
- Karush–Kuhn–Tucker (KKT) conditions and Lagrange multipliers α_i show which points influence the solution.

2) Step-by-step reasoning:
- SVM optimization (hard-margin) minimizes 1/2 ||w||^2 subject to y_i(w·x_i + b) ≥ 1 for all i.
- Form Lagrangian: L = 1/2 ||w||^2 − Σ_i α_i [y_i(w·x_i + b) − 1], with α_i ≥ 0.
- Stationarity gives w = Σ_i α_i y_i x_i. Only examples with α_i > 0 contribute to w.
- Complementary slackness (KKT) requires α_i [y_i(w·x_i + b) − 1] = 0. Thus α_i > 0 only when y_i(w·x_i + b) = 1, i.e., the point lies on the margin.
- Therefore the support vectors (those with α_i > 0) determine w and b and hence the location of the maximum-margin hyperplane. Points strictly outside the margin have α_i = 0 and do not affect the solution.

3) Intermediate math shown above (Lagrangian, stationarity w = Σ α_i y_i x_i, complementary slackness).

4) Why the correct answer is correct:
Because only the points that sit on the margin boundaries enter the representation of the weight vector w and set the offset b; they fix the margin width and hyperplane location.

5) Why the other options are incorrect:
- “The data points that are easiest to classify” is false: support vectors are actually the hardest (closest to the decision boundary), not the easiest.
- “The data points that are farthest from the maximum-margin hyperplane” is false: those farthest away are well inside class regions and have no effect (α_i = 0).

Conclusion: The correct answer is “The only data points that determine the location of the maximum-margin hyperplane,” which matches the provided answer.

Question at position 2 What are support vectors?The only data points that determine the location of the maximum-margin hyperplaneThe data points that are easiest to classifyThe data points that are farthest from the maximum-margin hyperplane单项选择题

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