# Questions tagged [vc-dimension]

The VC dimension (for Vapnik–Chervonenkis dimension) is a measure of the capacity (complexity, expressive power, richness, or flexibility) of a statistical classification algorithm, defined as the cardinality of the largest set of points that the algorithm can shatter.

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### How can the VC-dimension of Turing machine be finite?

The VC-dimension of a hypothesis class $\mathcal{H}$ is defined to be the size of the maximal set $C$ such that $\mathcal{H}$ cannot shutter. This paper shows that the VC-dimension of the set of all ...
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### VC dimension of finite unions of one-sided intervals

What is the VC dimension of $k$ finite unions of one-sided intervals: If we take 3 one-sided intervals like $(-\infty, a_1]$, $(-\infty, a_2]$ and $(-\infty, a_3]$, I think union of these ...
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### VC Dimension of A Set of Hypothesis

I am confused about what does a VC dimension of a set of hypothesis means. I have two hypothesis, say $H_1$ with VC dimension of $x$, and $H_2$ of VC dimension of $y$. Does this automatically mean ...
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### The VC dimension when the samples are fixed

The VC dimension is usually used in the following way. There is a space of hypotheses. There is an unknown probability distribution. We sample some training-samples from this distribution. We find the ...
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### VC dimension of monotone disjunctions of length k over n variables?

There are of course $n \choose k$ monotone disjunctions which bounds the VC dimension at $\log_2 {n \choose k}$. I'm wondering if this is bound at $k \log_2 n$? (Possibly follows from combinatorial ...
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### Sample Complexity for Real-Valued PAC-Learnable Functions

Can anyone shed some light on how the VC Dimension affects the sample complexity bounds of infinite hypothesis classes with real-valued outputs in PAC learning, or how to calculate the sample ...
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### VC Dimension Calculation for Intervals

As i See in ML Course a VC dimension calculation is very theoretical. What is the VC-dimension of intervals in R? The target function is specifieed by an interval, and labels any example positive ...
0answers
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### If one hypothesis class is a proper subset of another, what is the relation of their VC dimensions?

Assume two hypotheses classes $H_A\subset H_B$ defined over the same instance space $\delta$. Assume also $VC(H_A)=d$, does this mean $VC(H_B)\geq d$ ? where $VC$ is the VC dimension. We can use the ...
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### Why is the VC dimension different on intervals and half intervals?

As I read this lecture for being familiar with VC dimension we find on p. 8: VC(half intervals in $\mathbb{R}$ ) = 1 .... no subset of size 2 can be shattered VC(intervals in $\mathbb{R}$ )...
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### VC dimension of 1-NN classifier for discrete metric space?

We know VC dimension of 1-NN classifier is infinite for continuous metric space. Is there any proof of VC dimension of 1-NN classifier if the metric space is discrete?
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### VC dimension of complement

Let $C\subseteq 2^X$ be a concept class over $X$ and let $\bar{C}:=\{X\setminus c\mid c\in C\}$ be the complement. Show that $VCdim(C)=VCdim(\bar{C})$. Proof: Let $d:=VC_{dim}(C)$, then there exists ...
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