# Transform¶

Base class for the transformations of random variables.

Transform can be used to represent any differentiable and injective function from the subset of $$R^n$$ to subset of $$R^m$$, generally used for transforming a random sample generated by Distribution instance.

Suppose $$X$$ is a K-dimensional random variable with probability density function $$p_X(x)$$. A new random variable $$Y = f(X)$$ may be defined by transforming $$X$$ with a suitably well-behaved funciton $$f$$. It suffices for what follows to note that if f is one-to-one and its inverse $$f^{-1}$$ have a well-defined Jacobian, then the density of $$Y$$ is

$p_Y(y) = p_X(f^{-1}(y)) |det J_{f^{-1}}(y)|$

where det is the matrix determinant operation and $$J_{f^{-1}}(y)$$ is the Jacobian matrix of $$f^{-1}$$ evaluated at $$y$$. Taking $$x = f^{-1}(y)$$, the Jacobian matrix is defined by

$\begin{split}J(y) = \begin{bmatrix} {\frac{\partial x_1}{\partial y_1}} &{\frac{\partial x_1}{\partial y_2}} &{\cdots} &{\frac{\partial x_1}{\partial y_K}} \\ {\frac{\partial x_2}{\partial y_1}} &{\frac{\partial x_2} {\partial y_2}}&{\cdots} &{\frac{\partial x_2}{\partial y_K}} \\ {\vdots} &{\vdots} &{\ddots} &{\vdots}\\ {\frac{\partial x_K}{\partial y_1}} &{\frac{\partial x_K}{\partial y_2}} &{\cdots} &{\frac{\partial x_K}{\partial y_K}} \end{bmatrix}\end{split}$

A Transform can be characterized by three operations:

1. forward Forward implements $$x \rightarrow f(x)$$, and is used to convert one random outcome into another.

2. inverse Undoes the transformation $$y \rightarrow f^{-1}(y)$$.

3. log_det_jacobian The log of the absolute value of the determinant of the matrix of all first-order partial derivatives of the inverse function.

• _forward

• _inverse

• _forward_log_det_jacobian

• _inverse_log_det_jacobian (optional)

If the transform changes the shape of the input, you must also implemented:

• _forward_shape

• _inverse_shape

forward ( x )

Forward transformation with mapping $$y = f(x)$$.

Useful for turning one random outcome into another.

Parameters

x (Tensos) – Input parameter, generally is a sample generated from Distribution.

Returns

Outcome of forward transformation.

Return type

Tensor

inverse ( y )

Inverse transformation $$x = f^{-1}(y)$$. It’s useful for “reversing” a transformation to compute one probability in terms of another.

Parameters

y (Tensor) – Input parameter for inverse transformation.

Returns

Outcome of inverse transform.

Return type

Tensor

forward_log_det_jacobian ( x )

The log of the absolute value of the determinant of the matrix of all first-order partial derivatives of the inverse function.

Parameters

x (Tensor) – Input tensor, generally is a sample generated from Distribution

Returns

The log of the absolute value of Jacobian determinant.

Return type

Tensor

inverse_log_det_jacobian ( y )

Compute $$log|det J_{f^{-1}}(y)|$$. Note that forward_log_det_jacobian is the negative of this function, evaluated at $$f^{-1}(y)$$.

Parameters

y (Tensor) – The input to the inverse Jacobian determinant evaluation.

Returns

The value of $$log|det J_{f^{-1}}(y)|$$.

Return type

Tensor

forward_shape ( shape )

Infer the shape of forward transformation.

Parameters

shape (Sequence[int]) – The input shape.

Returns

The output shape.

Return type

Sequence[int]

inverse_shape ( shape )

Infer the shape of inverse transformation.

Parameters

shape (Sequence[int]) – The input shape of inverse transformation.

Returns

The output shape of inverse transformation.

Return type

Sequence[int]