Transform

class paddle.distribution. Transform [source]

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 function $f$ . 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:

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

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

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.

Subclass typically implement follow methods:

_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: Tensor ) → Tensor forward¶

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

Useful for turning one random outcome into another.

Parameters: x (Tensor) – Input parameter, generally is a sample generated from Distribution.
Returns: Outcome of forward transformation.
Return type: Tensor

inverse ( y: Tensor ) → Tensor inverse¶

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: Tensor ) → Tensor forward_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.

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: Tensor ) → Tensor inverse_log_det_jacobian¶

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: Sequence[int] ) → Sequence[int] forward_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: Sequence[int] ) → Sequence[int] inverse_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]