p-norm ball (Ballp)

Ballp{N<:AbstractFloat, VN<:AbstractVector{N}} <: AbstractCentrallySymmetric{N}

Type that represents a ball in the p-norm, for $1 ≤ p ≤ ∞$.

It is defined as the set

\[\mathcal{B}_p^n(c, r) = \{ x ∈ \mathbb{R}^n : ‖ x - c ‖_p ≤ r \},\]

where $c ∈ \mathbb{R}^n$ is its center and $r ∈ \mathbb{R}_+$ its radius. Here $‖ ⋅ ‖_p$ for $1 ≤ p ≤ ∞$ denotes the vector $p$-norm, defined as $‖ x ‖_p = \left( \sum\limits_{i=1}^n |x_i|^p \right)^{1/p}$ for any $x ∈ \mathbb{R}^n$.

Fields

• p – norm as a real scalar
• center – center of the ball as a real vector
• radius – radius of the ball as a scalar ($≥ 0$)

Notes

The special cases $p=1$, $p=2$ and $p=∞$ fall back to the specialized types Ball1, Ball2 and BallInf, respectively.

Examples

A five-dimensional ball in the $p=3/2$ norm centered at the origin of radius 0.5:

julia> B = Ballp(3/2, zeros(5), 0.5)
Ballp{Float64,Array{Float64,1}}(1.5, [0.0, 0.0, 0.0, 0.0, 0.0], 0.5)

julia> dim(B)
5

We evaluate the support vector in direction $[1,2,…,5]$:

julia> σ([1., 2, 3, 4, 5], B)
5-element Array{Float64,1}:
 0.013516004434607558
 0.05406401773843023
 0.12164403991146802
 0.21625607095372093
 0.33790011086518895

σ(d::AbstractVector, B::Ballp)

Return the support vector of a Ballp in a given direction.

Input

• d – direction
• B – ball in the p-norm

Output

The support vector in the given direction. If the direction has norm zero, the center of the ball is returned.

Algorithm

The support vector of the unit ball in the $p$-norm along direction $d$ is:

\[σ_{\mathcal{B}_p^n(0, 1)}(d) = \dfrac{\tilde{v}}{‖\tilde{v}‖_q},\]

where $\tilde{v}_i = \frac{|d_i|^q}{d_i}$ if $d_i ≠ 0$ and $tilde{v}_i = 0$ otherwise, for all $i=1,…,n$, and $q$ is the conjugate number of $p$. By the affine transformation $x = r\tilde{x} + c$, one obtains that the support vector of $\mathcal{B}_p^n(c, r)$ is

\[σ_{\mathcal{B}_p^n(c, r)}(d) = \dfrac{v}{‖v‖_q},\]

where $v_i = c_i + r\frac{|d_i|^q}{d_i}$ if $d_i ≠ 0$ and $v_i = 0$ otherwise, for all $i = 1, …, n$.

∈(x::AbstractVector, B::Ballp)

Check whether a given point is contained in a ball in the p-norm.

Input

• x – point/vector
• B – ball in the p-norm

Output

true iff $x ∈ B$.

Notes

This implementation is worst-case optimized, i.e., it is optimistic and first computes (see below) the whole sum before comparing to the radius. In applications where the point is typically far away from the ball, a fail-fast implementation with interleaved comparisons could be more efficient.

Algorithm

Let $B$ be an $n$-dimensional ball in the p-norm with radius $r$ and let $c_i$ and $x_i$ be the ball's center and the vector $x$ in dimension $i$, respectively. Then $x ∈ B$ iff $\left( ∑_{i=1}^n |c_i - x_i|^p \right)^{1/p} ≤ r$.

Examples

julia> B = Ballp(1.5, [1., 1.], 1.)
Ballp{Float64,Array{Float64,1}}(1.5, [1.0, 1.0], 1.0)

julia> [.5, -.5] ∈ B
false

julia> [.5, 1.5] ∈ B
true

center(B::Ballp)

Return the center of a ball in the p-norm.

Input

• B – ball in the p-norm

Output

The center of the ball in the p-norm.

rand(::Type{Ballp}; [N]::Type{<:Real}=Float64, [dim]::Int=2,
     [rng]::AbstractRNG=GLOBAL_RNG, [seed]::Union{Int, Nothing}=nothing)

Create a random ball in the p-norm.

Input

• Ballp – type for dispatch
• N – (optional, default: Float64) numeric type
• dim – (optional, default: 2) dimension
• rng – (optional, default: GLOBAL_RNG) random number generator
• seed – (optional, default: nothing) seed for reseeding

Output

A random ball in the p-norm.

Algorithm

The center and radius are normally distributed with mean 0 and standard deviation 1. Additionally, the radius is nonnegative. The p-norm is a normally distributed number ≥ 1 with mean 1 and standard deviation 1.

translate(B::Ballp, v::AbstractVector)

Translate (i.e., shift) a ball in the p-norm by a given vector.

Input

• B – ball in the p-norm
• v – translation vector

Output

A translated ball in the p- norm.

Algorithm

We add the vector to the center of the ball.