piecewise-linear function

Our definition of the indicator function in this work is

\[ 1_{a,b}\left(x\right):=\begin{cases} 1 & \text{if }x\in\left[a,b\right]\subset\left[-1,1\right],\\ 0 & \text{otherwise.} \end{cases} \]

Given a compression parameter \(c\in\left(0,2\right)\), non-overlapping intervals in \(\left(-1,1\right)\subset\mathbb{R}\) as parameterized by the set of endpoints \(\left\{ c_{l},d_{l}\right\}\), our piece-wise linear map \(\left[-1,1\right]\rightarrow\left[-1,1\right]\) is of the form

\[ f\left(x\right):=\begin{cases} \sum_{l}f_{l}\left(x\right) & \text{if }x\in\left[-1,1\right],\\ 1 & \text{if }x>s,\\ -1 & \text{otherwise}. \end{cases} \]

The linear function on interval \(l\) is

\(f_{l}\left(x\right):=\left(m_{l}x+b_{l}\right)1_{a_{l},b_{l}}\left(x\right).\)

The choice of \(\left\{ m_{l},b_{l}\right\}\) is such that:

\(f\) is continuous.
\(\sum_{l}\left(b_{l}-a_{l}\right)=c\).
\(m_{l}=\frac{d_{l}-c_{l}}{b_{l}-a_{l}}\).

The parameters \(\left\{ c_{l},d_{l}\right\}\) define the \(l\)-th interval in the range of \(f\), while \(\left\{ a_{l},b_{l}\right\}\) define the domain. The compression \(c\) characterizes the slope of the linears in the transform \(f\).

Implementation

The function definition of getpiecewiselines() is

getpiecewiselines(intervals_y_st::Vector{T},
    intervals_y_fin::Vector{T},
    domain_proportion::T;
    lb = -one(T),
    ub = one(T))::Tuple{Piecewise2DLineType{T}, T} where T <: Real

The compression parameter \(c\) is specified by the inputs domain_proportion, ub, and lb, via the following:

c = domain_proportion*(ub-lb)

The elements in intervals_y_st are the elements of \(\left\{ c_{l}\right\}\), and the elements in intervals_y_fin are the elements of \(\left\{ d_{l}\right\}\). We allow these endpoints to also include the boundaries -1 and 1, and readjust accordingly in our implementation.

See the piecewise-linear example for a guide.

Generalize to other intervals

To generalize the domain and range of this map from \(\left[-1,1\right]\) to \(\left[-s,s\right]\), we use \(\frac{x}{s}\) as the input to the map \(f\), and rescale the output by \(s\), e.g. \(sf\left(\frac{x}{s}\right)\) is the final output. The object scale throughout our documentation and eaxmple guide is \(s\) for this purpose.

Two-segment case

When we only consider two-segment piecewise-linear functions, we can consider one of the segments as the focus interval. We could fit a range of these functions where the focus interval gradually shifts and collectively covers the entire domain/range.

This is useful in creating a family of parametric transport maps (each is a two-segment piecewise-linear function) that each transform a uniform distribution on an interval to a two-piece-uniform distribution on the same interval. The interpretation of the focus interval is that it focuses mass to that interval in the destination distribution (i.e. the two-piece-uniform distribution). In this package, the exported function createendopiewiselines1() creates such a family of two-segment piecewise-linear functions given some information about the focus interval and the domain/range interval.

One can get a smooth approximate transport map to each two-segment piecewise-linear function by fitting the parameters of the smooth transport map to each member of the family of two-segment piecewise-linear functions.

Logistic-logit function

Given the real numbers \(a\) and \(b\), our definition of a logistic-logit function \(f\) is a composition of the logistic and logit functions, given by the following:

\[ \begin{align*} f:\left(0,1\right) & \rightarrow\left(0,1\right)\\ x & \mapsto\frac{1}{1+e^{\left(-a\ln\left(\frac{x}{1-x}-b\right)\right)}} \end{align*} \]

f is monotonically increasing if \(a>0\), monotonically decreasing if \(a<0\), and constant if \(a=0\).

Its inverse is given by

\[ \begin{align*} f^{-1}:\left(0,1\right) & \rightarrow\left(0,1\right)\\ y & \mapsto\frac{e^{b}}{e^{b}+\left(\frac{1}{y}-1\right)^{\frac{1}{a}}} \end{align*} \]

The logistic-logit fit example goes through the process of creating a family of two-segment piecewise-linear functions, fit each to a logistic-logit function, and plot its action on samples drawn from the uniform distribution over the specified interval.