Intuitive Explanation of What a p norm is, for any arbitrary p>0

In summary, the p-norm of a vector in a normed vector space can be calculated by taking the p-th root of the sum of the p-th powers of its components. This norm has three special cases: p=2, which represents Euclidean distance, p=1, which represents Taxicab distance, and p=\infty, which represents Supremum distance. As p varies between these special cases, the unit circle changes shape, with outliers having a greater influence on the norm for higher values of p. For values of p<1, the unit circle becomes a non-convex "star" shape, leading to the failure of the triangle inequality.
  • #1
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Supposing [itex] V [/itex] is a normed vector space, the p-norm of [itex]{\bf x} \in V [/itex] is:
[itex]\lVert {\bf x} \rVert_p := \left(\sum_{i=1}^n |x|^p \right)^{\frac{1}{p}}[/itex]

There are 3 special cases:
[itex] p= 2: [/itex] Euclidean distance - 'as the crow flies'
[itex] p = 1: [/itex] Taxicab distance - sum the absolute value of components
[itex] p=\infty: [/itex] Supremum distance - take the maximum component (in abs value)

I would like to have an intuition for what happens when we change p to values in-between these special cases. In particular, if we interpret x to be a vector of data points, what changes about the way the norm ||.||_p describes the data vector as we change p?

Also, if we consider p<1, the l-p 'norm' does not satisfy the triangle inequality, but still tells us something about the data. Can anyone give an intuition for what happens with values of 0<p<1?

The only thing I can think of is that with higher values of p, outliers (components with higher deviation from the mean of all components) contribute more to the value of the norm. Is there anything more to it?

Thanks,

RS
 
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  • #2
One good way to think about it is to look at what happens to the unit circle as p varies. When p=∞, the unit circle is just a square. As p decreases, the outer corners begin to move inward until p=1, when they become just straight lines between the cardinal unit vectors, and the unit circle becomes a diamond. For p<1, the diagonal lines start to bulge inward, creating a four-pointed "star". This figure is not convex, which is why the triangle inequality fails for p<1.
 

FAQ: Intuitive Explanation of What a p norm is, for any arbitrary p>0

1. What is a p norm?

A p norm, also known as a p-norm or Lp norm, is a mathematical concept used to measure the size or magnitude of a vector. It is often used in various fields such as physics, engineering, and statistics.

2. How is a p norm calculated?

The p norm is calculated by taking the absolute value of each element in a vector, raising them to the power of p, adding them together, and then taking the p-th root of the result. In other words, it is the p-th root of the sum of the p-th powers of the vector elements.

3. What does the value of p represent in a p norm?

The value of p in a p norm represents the order or degree of the norm. It determines the weight given to each element in the vector and can range from 1 to infinity. A higher value of p gives more weight to larger values in the vector, while a lower value of p gives more weight to smaller values.

4. What is the difference between a p norm and other types of norms?

A p norm is a generalization of other types of norms, such as the Euclidean norm (p = 2) and the Manhattan norm (p = 1). It allows for a more flexible measurement of vector size by varying the value of p. Additionally, p norms are often used in functional analysis, while other norms are more commonly used in geometry and physics.

5. How is a p norm used in real-world applications?

P norms are used in a variety of real-world applications, such as data analysis, signal processing, and machine learning. They can be used to measure similarity between vectors, to define error and convergence criteria, and to regularize models. They are also commonly used in optimization problems to define the objective function.

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