Bounding p-norm expression using p-norm inequality

In summary, the problem statement states the need to show that the expression ||w||_p^2+||u||_p^2-2(||u||_p^p)^{\frac{2}{p}-1}\Sigma_i(u(i)^{p-1}w(i)) can be bounded as a function of ||w-u||_p^2 for p\in[2,\infty). The work done so far includes recognizing that the expressions are equal for p=2 and suspecting that ||w||_p^2+||u||_p^2-2(||u||_p^p)^{\frac{2}{p}-1}\Sigma_i(u(i)^{p-1
  • #1
ENgez
75
0
problem statement:

need to show:
[tex]||w||_p^2+||u||_p^2-2(||u||_p^p)^{\frac{2}{p}-1}\Sigma_i(u(i)^{p-1}w(i)) [/tex]

can be bounded as a function of

[tex] ||w-u||_p^2 [/tex]

where [tex] p\in[2,\infty) [/tex]

work done:

the expressions are equal for p=2, and i suspect that

[tex] ||w||_p^2+||u||_p^2-2(||u||_p^p)^{\frac{2}{p}-1}\Sigma_i(u(i)^{p-1}w(i)) \leq||w-u||_p^2 [/tex]

but i get stuck here. Is there some kind of p-norm inequality I can apply here?Thank you!
 
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  • #2
Forum rules require you to show your work. From what I see, you just took the problem statement and re-wrote it with "I suspect that" it holds as an upper bound with a scalar multiple of one.

What kind of work can you show to at least justify your suspicion?
 
Last edited:

What is a bounding p-norm expression?

A bounding p-norm expression is a mathematical expression that describes the distance or magnitude between two points in a multi-dimensional space. It is commonly used in optimization and machine learning algorithms to measure the similarity or dissimilarity between data points.

What is the difference between a p-norm and an Lp norm?

A p-norm and an Lp norm are essentially the same thing. The "p" in p-norm stands for a specific value of p, while Lp norm refers to the general case where p can take on any real value.

How do you calculate a bounding p-norm expression?

The formula for a bounding p-norm expression is:
(|x1-y1|^p + |x2-y2|^p + ... + |xn-yn|^p)^(1/p)
where x1, x2, ..., xn and y1, y2, ..., yn are the coordinates of the two points in the multi-dimensional space.

What is the significance of p in a bounding p-norm expression?

The value of p in a bounding p-norm expression determines how the distance between two points is calculated. A smaller value of p (closer to 0) gives more weight to the coordinates with larger differences, while a larger value of p (closer to infinity) gives more weight to the coordinates with smaller differences.

What are some applications of bounding p-norm expressions?

Bounding p-norm expressions are commonly used in clustering algorithms, feature selection, and data mining. They are also used in machine learning for tasks such as classification, regression, and dimensionality reduction.

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