L-Infinity Norm

A L-Infinity Norm is a vector norm defined on the L-Infinity Space.

• AKA: Maximum Norm, Uniform Norm, Chebyshev Norm.
• Context:
  • It can be defined as [math]\displaystyle{ ||x||_\infty = max(|x_1|, |x_2|,... |x_n|) }[/math] where [math]\displaystyle{ ||x|| }[/math] is a vector norm defined on a Lp Space for when [math]\displaystyle{ p \rightarrow \infty }[/math].
• Example(s):
  • CHIRP.
• Counter-Example(s):
  • L1-Norm.
  • L2-Norm.
• See: L1-Norm, L2-Norm, Lp Space, L-Infinity Space,----

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References

2015

• (Wikipedia, 2015) ⇒ http://wikipedia.org/wiki/Lp_space
  • QUOTE: : The L^∞-norm or maximum norm (or uniform norm) is the limit of the L^p-norms for p → ∞. It turns out that this limit is equivalent to the following definition:

[math]\displaystyle{ \left\| x \right\| _\infty = \max \left\{ |x_1|, |x_2|, \dotsc, |x_n| \right\} }[/math]

1999

• (Wolfram Mathworld , 1999) ⇒ Weisstein, Eric W. "L^infty-Norm." From MathWorld -- A Wolfram Web Resource ⇒ http://mathworld.wolfram.com/L-Infinity-Norm.html
  • QUOTE: A vector norm defined for a vector

[math]\displaystyle{ x=\begin{bmatrix}x_1 \\ x_2 \\ x_n \end{bmatrix} }[/math]

with complex entries by

[math]\displaystyle{ |x|_\infty=max_{i}|x_i| }[/math]

2011

• (Wilkinson et al., 2011) ⇒ Leland Wilkinson, Anushka Anand, and Dang Nhon Tuan. (2011). “CHIRP: A New Classifier based on Composite Hypercubes on Iterated Random Projections.” In: Proceedings of the 17th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD-2011) Journal. ISBN:978-1-4503-0813-7 doi:10.1145/2020408.2020418
  • QUOTE: === 2011 ===
• (Wilkinson et al., 2011) ⇒ Leland Wilkinson, Anushka Anand, and Dang Nhon Tuan. (2011). “CHIRP: A New Classifier based on Composite Hypercubes on Iterated Random Projections.” In: Proceedings of the 17th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD-2011) Journal. ISBN:978-1-4503-0813-7 doi:10.1145/2020408.2020418
  • QUOTE: For CHIRP, we employ the L∞ or sup metric:

[math]\displaystyle{ ||x||_\infty = sup(|x_1|, |x_2|,... |x_n|) }[/math]

when we search for neighbors. In this search, we are looking for all neighbors of a point at the center of a hypercube of fixed size in a vector space. Because we are concerned with finite-dimensional vector spaces in practice, we will use max() instead of sup() from now on.

Definition 1 A hypercube description region (HDR) is the set of points less than a fixed distance from a single point (called the center) using the L∞ norm. A weighted hypercube description region is an HDR that uses the positively weighted L∞ norm:

[math]\displaystyle{ ||x||_\infty = max(w_1|x_1|, w_2|x_2|,...w_n|x_n|) }[/math]

We will assume the term HDR refers to this more general case. Our use of weights implies that different points in a high-dimensional space can have different weights defining their hypercubes.

Definition 2 A composite hypercube description region (CHDR) is the set of points inside the union of zero or more hypercube description regions.

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