Question3Question 3 Copy of SE Q3 On the dataset below with two features x1, x2, which of the following decision boundaries could be learned by 1-NN (K Nearest Neighbour with K=1)? (select only one option) Reset

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I will begin by noting that the answer options provided are empty in the dataset snippet, so I cannot pinpoint which specific boundary option among the given choices is marked as correct. Nevertheless, I can explain how 1-NN (K = 1) would determine decision boundaries for a two-feature (x1, x2) dataset, which will help you evaluate any concrete option.

General principle of 1-NN with K = 1:
- Each training point becomes a basis for its own Voronoi cell. A new point is labeled according to the label of its nearest neighbor in the training set.
- Consequently, the decision boundary between two classes consists of the set of points equidistant to two nearest training samples of opposite labels. In two dimensions, these boundaries are piecewise linear (perpendicular bisectors) between neighboring samples of different classes.
- When multiple points of the same class lie nearby, the boundary will follow the arrangement of perpendicular bisectors between opposing-class points, potentially creating multiple linear segments and non-smooth, stair-step-like boundaries depending on the spatial arrangement of labeled points.

Analyzing each possible option conceptually (without seeing the actual option text):
- If an option depicts a single straight line that cleanly separates all nearest-opposite-class regions with a linear boundary across the entire plane, that would correspond to a scenario where two points of opposite labels lie on opposite sides of a single bisector and no other points disrupt that region. In practice with multiple samples, 1-NN boundaries are usually more complex than a single global line.
- If an option shows a series of multiple linear segments forming a network of lines (a piecewise-linear boundary), that aligns well with how Voronoi boundaries can appear when different labeled points intersperse across the plane.
- If an option claims a nonlinear curved boundary, that would typically reflect models like 2-NN with certain distance metrics or kernel methods, but not standard 1-NN with Euclidean distance, which yields straight-line segment boundaries between adjacent region boundaries.
- If an option suggests a boundary that disregards some nearby opposite-label samples (e.g., a boundary that would misclassify a point that is very close to a nearby opposite-class neighbor), that would be inconsistent with 1-NN, since the label is determined by the single nearest point regardless of other nearby samples.

Potential pitfalls or common misconceptions:
- Believing 1-NN always yields a globally linear separator. This is false in general; the boundary is piecewise linear and can be highly non-linear depending on point arrangement.
- Thinking the boundary is determined by class means only; in reality, the arrangement of individual labeled samples directly shapes the boundaries through their pairwise bisectors.
- Assuming the boundary ignores density or distribution of points. In 1-NN, density affects where nearest neighbors lie, thus altering the boundary locally.

In summary, without the actual textual options, the correct choice should reflect a boundary that is a collection of straight-line segments forming a Voronoi-style partition around the training samples, rather than a single global line or a curved boundary. If you can supply the specific text of the four options, I can map each option to this reasoning and indicate which option correctly describes the 1-NN boundary for the given dataset.

Question3Question 3 Copy of SE Q3 On the dataset below with two features x1, x2, which of the following decision boundaries could be learned by 1-NN (K Nearest Neighbour with K=1)? (select only one option) Reset

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