VC Dimension: Definition and Examples
The VC dimension (Vapnik–Chervonenkis dimension) is a fundamental concept in statistical learning theory that quantifies the capacity of a hypothesis class. Specifically, the VC dimension is the largest number of points that can be shattered by a hypothesis class, where shattering means that for every possible labeling of these points, there exists a hypothesis in the class that correctly classifies them. In other words, the VC dimension captures the expressive power of a set of functions by describing how complex a set of patterns it can realize.
Understanding the VC dimension helps you reason about the tradeoff between model complexity and generalization. A higher VC dimension means a hypothesis class can fit more complex patterns, but it can also increase the risk of overfitting. Conversely, a low VC dimension may limit the model’s ability to fit data but can improve generalization.
Suppose your hypothesis class consists of all threshold functions on the real line: each function classifies a real number as positive if it is greater than some threshold and negative otherwise.
- Consider two points, x1 and x2;
- No matter how you label these two points, you can always find a threshold that separates them accordingly;
- For three points, there exists at least one labeling (the middle point labeled differently from the other two) that cannot be realized by any threshold.
Thus, the VC dimension of threshold functions on the real line is 2.
Now consider the class of sets defined by intervals on the real line: each hypothesis is an interval, and a point is classified as positive if it lies within the interval.
- For any three points, you can label them in all possible ways using an interval;
- With four points, there is at least one labeling (alternating positive and negative labels) that cannot be realized by a single interval.
Therefore, the VC dimension of intervals on the real line is 3.
Consider the class of linear classifiers (lines) in the plane.
- For three non-collinear points, any labeling can be realized by a line;
- For four points in general position (not all on the same line or forming a convex quadrilateral), there is at least one labeling that cannot be realized by a single line.
Thus, the VC dimension of linear classifiers in two dimensions is 3.
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The VC dimension (Vapnik–Chervonenkis dimension) is a fundamental concept in statistical learning theory that quantifies the capacity of a hypothesis class. Specifically, the VC dimension is the largest number of points that can be shattered by a hypothesis class, where shattering means that for every possible labeling of these points, there exists a hypothesis in the class that correctly classifies them. In other words, the VC dimension captures the expressive power of a set of functions by describing how complex a set of patterns it can realize.
Understanding the VC dimension helps you reason about the tradeoff between model complexity and generalization. A higher VC dimension means a hypothesis class can fit more complex patterns, but it can also increase the risk of overfitting. Conversely, a low VC dimension may limit the model’s ability to fit data but can improve generalization.
Suppose your hypothesis class consists of all threshold functions on the real line: each function classifies a real number as positive if it is greater than some threshold and negative otherwise.
- Consider two points, x1 and x2;
- No matter how you label these two points, you can always find a threshold that separates them accordingly;
- For three points, there exists at least one labeling (the middle point labeled differently from the other two) that cannot be realized by any threshold.
Thus, the VC dimension of threshold functions on the real line is 2.
Now consider the class of sets defined by intervals on the real line: each hypothesis is an interval, and a point is classified as positive if it lies within the interval.
- For any three points, you can label them in all possible ways using an interval;
- With four points, there is at least one labeling (alternating positive and negative labels) that cannot be realized by a single interval.
Therefore, the VC dimension of intervals on the real line is 3.
Consider the class of linear classifiers (lines) in the plane.
- For three non-collinear points, any labeling can be realized by a line;
- For four points in general position (not all on the same line or forming a convex quadrilateral), there is at least one labeling that cannot be realized by a single line.
Thus, the VC dimension of linear classifiers in two dimensions is 3.
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