给定下列数字

哪一个判定边界更好，为什么？

Given the following figure

Which decision boundary is better and why?

Question

给定下列数字

哪一个判定边界更好，为什么？

Given the following figure

Which decision boundary is better and why?

BlackTom AI · Accepted Answer

A boundary that maximizes the margin between classes is preferred: the blue line is better because it produces a larger perpendicular gap to the nearest points of each class.

1) Relevant knowledge points and concepts
- Margin: the perpendicular distance from the decision boundary to the closest training examples (support vectors).
- Support Vector Machine (SVM) principle: among linear separators that correctly separate classes, the one with the largest margin generalizes best (is more robust to noise).
- Distance from a point x to a line defined by w · x + b = 0 is |w · x + b| / ||w||; the margin equals twice the minimal such distance between the two classes divided by ||w|| (or equivalently the minimal distance on either side).

2) Step-by-step reasoning
- Inspect the figure: the blue line sits more centrally between the cluster of black pluses (top-left) and blue circles (bottom-right). The red line is shifted upward, closer to both classes’ nearest members.
- For each boundary, consider the closest positive and negative points (the support vectors). The perpendicular distances from these points to the blue line are visibly larger than the distances to the red line.
- Symbolically, for each class i and boundary B (blue or red), distance_i(B) = |w_B · x_i + b_B| / ||w_B||. The margin for B is min_i distance_i(B) (or sum of both sides). From the picture we observe min_i distance_i(blue) > min_i distance_i(red), so margin_blue > margin_red.

3) Why the correct answer is correct
- Larger margin (blue) implies better separation and better expected generalization; therefore the blue line is the better decision boundary.

4) Why the other options are incorrect
- “Red line is better because it is very close to the support vectors”: being close to support vectors reduces margin and is worse.
- “Red line is better because it has the largest margin”: false—visual inspection shows red has a smaller margin.
- “Blue line is better because it goes through the origin”: passing through the origin is irrelevant to classifier quality unless constrained; it does not guarantee better separation.

Conclusion: The blue line is the better decision boundary because it has the largest margin, matching the provided answer.

给定下列数字哪一个判定边界更好，为什么？ Given the following figure Which decision boundary is better and why?单项选择题

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给定下列数字 哪一个判定边界更好，为什么？ Given the following figure Which decision boundary is better and why?单项选择题