In a Support Vector Machine (SVM), what are ‘support vectors’?

Question

BlackTom AI · Accepted Answer

Support vectors are the data points closest to the decision boundary that define the maximum margin.

Relevant concepts:
- A Support Vector Machine constructs a decision boundary (hyperplane) that separates classes while maximizing the margin, the distance between the hyperplane and the nearest data points of each class.
- The optimization is: minimize (1/2)||w||^2 subject to y_i(w · x_i + b) ≥ 1 for all i.
- The dual formulation uses Lagrange multipliers α_i: maximize Σα_i − (1/2)ΣΣα_iα_j y_i y_j (x_i · x_j), subject to α_i ≥ 0 and Σα_i y_i = 0. Points with α_i > 0 are support vectors.

Step-by-step reasoning:
1. The primal SVM objective enforces a margin of at least 1 for correctly classified points; the algorithm minimizes the weight norm to maximize that margin.
2. In the dual, only those training points that lie on or inside the margin have nonzero Lagrange multipliers α_i. These are the support vectors.
3. The optimal weight vector w can be written as w = Σα_i y_i x_i, so only support vectors (α_i > 0) contribute to w and thus to the position and orientation of the decision boundary.
4. Because these points are the closest to the hyperplane, moving any support vector would change the optimal margin and the decision boundary; points farther away do not affect the solution.

Why the correct answer is correct:
- It directly identifies that support vectors are the nearest points that determine the maximum-margin hyperplane and are characterized by nonzero α_i in the dual optimization.

Why other options are incorrect:
- “The hyperplanes that separate all data points” is wrong: support vectors are points, not hyperplanes.
- “The kernel functions that project data into higher dimensions” is wrong: kernels are functions used in SVMs, not data points.
- “The weights assigned to each input feature” is wrong: weights (components of w) are derived from support vectors, but are not the support vectors themselves.

Conclusion: The correct answer is “The data points closest to the decision boundary that define the maximum margin,” matching the provided answer.

In a Support Vector Machine (SVM), what are ‘support vectors’?单项选择题

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