Robustifiers

ArctanRobustifier <: AbstractRobustifierFunction

A robustifier that is based on the arctangent function. Note that robustifiers act on the squared residuals within the nonlinear least squares framework, i.e., $ρ(f_i(p)^2)$.

The formula for the robustifier is given as

\[ρ(x) = \mathrm{atan}(x)\]

and its first and second derivatives read as

\[ρ'(x) = \frac{1}{1 + x^2}\]

and

\[ρ''(x) = -\frac{2x}{(1 + x^2)^2}\]

CauchyRobustifier <: AbstractRobustifierFunction

A robustifier that is based on the Cauchy function. Note that robustifiers act on the squared residuals within the nonlinear least squares framework, i.e., $ρ(f_i(p)^2)$.

The formula for the Cauchy robustifier is given as

\[ρ(x) = \log(1 + x)\]

and its first and second derivatives read as

\[ρ'(x) = \frac{1}{1 + x}\]

and

\[ρ''(x) = -\frac{1}{(1 + x)^2}\]

ComponentwiseRobustifierFunction{F<:AbstractRobustifierFunction} <: AbstractRobustifierFunction

A robustifier to indicate that for a certain AbstractVectorGradientFunction a robustifier should be applied component wise.

Fields

• robustifier::R: the robustifier to be applied componentwise

Constructor

ComponentwiseRobustifierFunction(robustifier::AbstractRobustifierFunction)

Create a new componentwise robustifier function. This wrapper avoids “double wrapping”: calling the constructor with a componentwise robustifier returns a new componentwise robustifier around the same internal robustifier, rather than nesting another layer.

ComposedRobustifierFunction{F<:AbstractRobustifierFunction, G<:AbstractRobustifierFunction} <: AbstractRobustifierFunction

A robustifier that is the composition of two robustifier functions $ρ = ρ_1 ∘ ρ_2$.

The formulae for the first and second derivatives are

\[ρ'(x) = ρ_1'(ρ_2(x)) ⋅ ρ_2'(x)\]

and

\[ρ''(x) = ρ_1''(ρ_2(x)) ⋅ (ρ_2'(x))^2 + ρ_1'(ρ_2(x)) ⋅ ρ_2''(x)\]

Fields

• ρ1::F : the first robustifier function
• ρ2::G : the second robustifier function

Constructor

ComposedRobustifierFunction(ρ1::F, ρ2::G) where {F<:AbstractRobustifierFunction, G<:AbstractRobustifierFunction}ρ1 ∘ ρ2

HuberRobustifier <: AbstractRobustifierFunction

A robustifier that is based on the Huber function. Note that robustifiers act on the squared residuals within the nonlinear least squares framework, i.e., $ρ(f_i(p)^2)$.

The formula for the Huber robustifier is given as

\[ρ(x) = \begin{cases} x & \text{if } x ≤ 1\\\\ 2\sqrt{x} - 1 & \text{if } x > 1\end{cases}\]

that is, its first and second derivatives read as

\[ρ'(x) = \begin{cases} 1 & \text{if } x ≤ 1\\\\ \frac{1}{\sqrt{x}} & \text{if } x > 1\end{cases}\]

and

\[ρ''(x) = \begin{cases} 0 & \text{if } x ≤ 1\\\\ -\frac{1}{2 x^{3/2}} & \text{if } x > 1\end{cases}\]

If you want to use a different threshold δ > 0, use a ScaledRobustifierFunction to scale the residuals accordingly, or even use the shorthand δ ∘ HuberRobustifier().

IdentityRobustifier <: AbstractRobustifierFunction

A robustifier that is the identity function, i.e., $ρ(x) = x$.

Its first and second derivatives read as $ρ'(x) = 1$ and $ρ''(x) = 0$.

RobustifierFunction{F, G, H} <: AbstractRobustifierFunction

A struct to represent a robustifier function $ρ: ℝ → ℝ$ along with its first and second derivative $ρ'$ and $ρ''$, respectively.

Fields

• ρ::F : the robustifier function
• ρ_prime::G : the first derivative of the robustifier function
• ρ_double_prime::H : the second derivative of the robustifier function

Constructor

RobustifierFunction(ρ, ρ_prime, ρ_double_prime)

Generate a RobustifierFunction given the function ρ and its first and second derivative.

ScaledRobustifierFunction{F<:AbstractRobustifierFunction, R <: Real} <: AbstractRobustifierFunction

A robustifier function whose residuals are scaled by a real value scale $s$, i.e. it considers $ρ_s(f(p)^2) = ρ(s^2⋅f(p)^2)$ for some AbstractRobustifierFunction $ρ$. The function and its derivatives hence read as

• $ρ_s(x) = s^2 ρ(x / s^2)$
• $ρ_s'(x) = ρ'(x / s^2)$
• $ρ_s''(x) = \frac{1}{s^2} ρ''(x / s^2)$

Fields

• robustifier::F : the underlying robustifier function
• scale::R : the scaling factor s

Constructor

ScaledRobustifierFunction(robustifier::F, scale::R) where {F<:AbstractRobustifierFunction, R <: Real}scale ∘ robustifier

Generate a ScaledRobustifierFunction given a robustifier function and a scaling factor.

SoftL1Robustifier <: AbstractRobustifierFunction

A robustifier that is based on the soft $ℓ_1$ norm. Note that robustifiers act on the squared residuals within the nonlinear least squares framework, i.e., $ρ(f_i(p)^2)$.

The formula for the robustifier is given as

\[ρ(x) = 2(\sqrt{1 + x} - 1)\]

and its first and second derivatives read as

\[ρ'(x) = \frac{1}{\sqrt{1 + x}}\]

and

\[ρ''(x) = -\frac{1}{2 (1 + x)^{3/2}}.\]

TolerantRobustifier <: AbstractRobustifierFunction

A robustifier that is based on the tolerant function. Note that robustifiers act on the squared residuals within the nonlinear least squares framework, i.e., $ρ(f_i(p)^2)$.

The formula for the robustifier is given as

\[ρ_{a,b}(x) = b\log(1+ \mathrm{e}^{(x-a)/b}) - b\log(1 + \mathrm{e}^{-a/b})\]

and its first and second derivatives read as

\[ρ'_{a,b}(x) = \frac{1}{1 + \mathrm{e}^{(a - x)/b}}\]

and

\[ρ''_{a,b}(x) = \frac{1}{4b\mathrm{cosh}^2\Bigl(\frac{(a - x)}{2b}\Bigr)}.\]

TukeyRobustifier <: AbstractRobustifierFunction

A robustifier that is based on the Tukey function. Note that robustifiers act on the squared residuals within the nonlinear least squares framework, i.e., $ρ(f_i(p)^2)$.

The formula for the Tukey robustifier is given as

\[ρ(x) = \begin{cases} \frac{1}{3}(1-(1-x)^3) & \text{if } x ≤ 1\\\\ \frac{1}{3} & \text{if } x > 1\end{cases}\]

that is, its first and second derivatives read as

\[ρ'(x) = \begin{cases} (1 - x)^2 & \text{if } x ≤ 1\\\\ 0 & \text{if } x > 1\end{cases}\]

and

\[ρ''(x) = \begin{cases} -2(1 - x) & \text{if } x ≤ 1\\\\ 0 & \text{if } x > 1\end{cases}.\]