Consider the following dataset: X=[-\pi,-0.5\pi,0,0.5\pi,\pi] with corresponding labels y=[1,-1,-1,-1,1]. Which of the following transformations would make the data linearly separable? A. \phi(x)=(x,cos( x)) B. \phi(x)=(x,sin(x)) C. \phi(x)=(x,cos(0.5 x)) D. \phi(x)=(x,sin(0.5 x))多项选择题
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Question

Consider the following dataset: X=[-\pi,-0.5\pi,0,0.5\pi,\pi] with corresponding labels y=[1,-1,-1,-1,1]. Which of the following transformations would make the data linearly separable? A. \phi(x)=(x,cos( x)) B. \phi(x)=(x,sin(x)) C. \phi(x)=(x,cos(0.5 x)) D. \phi(x)=(x,sin(0.5 x))多项选择题

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我们收录了全球超50000道真实原题与详细解析,现在登录,立即获得答案。

BlackTom AI · Accepted Answer

We are given a small dataset with inputs X = [-π, -0.5π, 0, 0.5π, π] and labels y = [1, -1, -1, -1, 1]. We want to know which feature mappings φ(x) would make the data linearly separable in the new feature space.

Option A: φ(x) = (x, cos(x))
- Intuition: Adding cos(x) as a nonlinear feature can bend the decision boundary in a way that separates the endpoints 1 (at -π) and 1 (at π) from the -1 class in the middle. Since cos(x) is symmetric and takes the same value at ±π (cos(±π) = -1) and is -1 at ±π, while x takes extreme values at the ends, the combination can produce a separating hyperplane in the augmented 2D space. In short, cos(x) injects a periodic, smooth nonlinearity that can provide a separating surface for this symmetric interval.
- Why this helps: The two positive examples lie at the ends; a cos(x) term helps create a decision boundary that can flip between ends and middle when paired with x.

Option B: φ(x) = (x, sin(x))
- Intuition: Adding sin(x) introduces another nonlinear feature. However, sin(x) is zero at x = 0 and peaks at x = ±π/2, with symmetric values elsewhere. The given labels place +1 at the ends and -1 near the center, so a boundary based on x and sin(x) may fail to separate because the sin term does not necessarily create a monotonic separation between the end points and the center for this specific arrangement.
- Why this might fail: Depending on the exact weights learned, the sin(x) feature may not provide the required contrast between the end points and central points, leaving some class regions inseparable by a linear decision boundary in (x, sin(x)).

Option C: φ(x) = (x, cos(0.5x))
- Intuition: Using cos(0.5x) stretches the cosine feature, changing its frequency so that the nonlinearity aligns differently with x’s positions. Since the data are symmetric around 0 with opposite labels at the ends versus the center, a suitably scaled cos(0.5x) term can create a decision boundary that cleanly separates the ends (y = +1) from the inner points (y = -1).
- Why this helps: The modified frequency can yield a separable arrangement by adjusting how the cosine term grows and falls across the interval, providing a linearly separable configuration in the augmented space when paired with x.

Option D: φ(x) = (x, sin(0.5x))
- Intuition: Similar to B, but with a stretched sine term. The sine feature may not line up with the required end-versus-center separation for this particular labeling. The combination (x, sin(0.5x)) could still fail to produce a linear separator because the sine term oscillates but doesn’t necessarily create a consistent margin that distinguishes the endpoints from the middle given the labeled pattern.
- Why this might fail: The sine term may produce a nonlinear boundary that still intersects the middle region in a way that cannot be separated by a single hyperplane in (x, sin(0.5x)) space for this specific dataset.

Summary of the reasoning for each option:
- A: Potentially separable due to cos(x) providing a useful nonlinear augmentation that differentiates end points from center when combined with x.
- B: Not guaranteed to separate; sin(x) may not align with the end-vs-center labeling to yield a linear separator in the augmented feature space.
- C: Also potentially separable because cos(0.5x) offers a different nonlinearity frequency that can enable a linear boundary to separate the classes in the augmented space.
- D: Similar concerns as B; sin(0.5x) may not reliably yield separability for this specific labeling.

In this particular problem, options A and C are the mappings that can lead to linear separability, whereas B and D do not guarantee separability given the label arrangement across the domain.

Consider the following dataset: X=[-\pi,-0.5\pi,0,0.5\pi,\pi] with corresponding labels y=[1,-1,-1,-1,1]. Which of the following transformations would make the data linearly separable? A. \phi(x)=(x,cos( x)) B. \phi(x)=(x,sin(x)) C. \phi(x)=(x,cos(0.5 x)) D. \phi(x)=(x,sin(0.5 x))多项选择题

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