Questions tagged [vc-theory]

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VC dimension of half spaces over the real line

I'm studying VC dimension and I'm having a little difficulty understanding it. I read lots of explanations, but when I come across this simple exercise I did not get a good intuition. The problem is ...
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1answer
46 views

Why is a lower bound necessary in proofs of VC-dimensions for various examples of hypotheses?

In the book "Foundations of Machine Learning" there are examples of proving the VC dimensions for various hypotheses, e.g., for axis-aligned rectangles, convex polygons, sine functions, hyperplanes, ...
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1answer
168 views

VC dimension of hypothesis space of finite union of intervals

I have the following concept: $$C = \left\{\bigcup_{i=1}^{k}(a_i, b_i): a_i, b_i \in {\Bbb R}, a_i < b_i, i=1,2,..,k\right\} $$ and was wondering how to determine the VC dimension of C?
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1answer
50 views

A question on realizable sample complexity

I came across the following exercise, and I just can't seem to crack it: Let $l$ be some loss function such that $l \leq 1$. Let $H$ be some hypothesis class, and let $A$ be a learning algorithm. ...
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2answers
691 views

Why the VC dimension to this linear hypothesis equal to 3?

I am trying hard to understand this. Here is the scenario: X = R^2 H = { h(x) = x + 10 } I need to calculate the VC dimension for the above linear separator. ...
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0answers
104 views

Intuition behind Occam's Learner Algorithm using VC-Dimension

So I'm learning about Occam's Learning algorithm and PAC-Learning where for a given hypothesis space $H$, if we want to have a model/hypothesis $h$ that has an True error of $error_D \leq \epsilon$, ...
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1answer
3k views

What is the exact definition of VC dimension?

I'm studying machine learning from Andrew Ng Stanford lectures and just came across the theory of VC dimensions. According to the lectures and what I understood, the definition of VC dimension can be ...
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0answers
218 views

How to determine the VC Dimension of axis-aligned, origin-centered ellipses?

I've been trying to determine the VC dimension of ellipses which are origin centered and axis aligned. My first approach was to find some equivalence to a threshold classifier function family of the ...
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1answer
4k views

With regards to VC-dimension, why can you shatter 3 points with circles but not 4 points?

When using VC-dimensions to estimate the capability of a binary classifier, you can find 3 points in R2 that can be shattered, e.g.: But you can not shatter any 4 points with a circle. This is ...
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2answers
13k views

How to calculate VC-dimension?

Im studying machine learning, and I would like to know how to calculate VC-dimension. For example: $h(x)=\begin{cases} 1 &\mbox{if } a\leq x \leq b \\ 0 & \mbox{else } \end{cases} $, with ...