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Triangular Linear Bijector¤

distreqx.bijectors.TriangularLinear(distreqx.bijectors.AbstractLinearBijector) ¤

A linear bijector whose weight matrix is triangular.

The bijector is defined as \(f(x) = Ax\) where \(A\) is a \(D \times D\) triangular matrix.

The Jacobian determinant can be computed in \(O(D)\) as follows:

\[\log|\det J(x)| = \log|\det A| = \sum \log|\text{diag}(A)|\]

The inverse is computed in \(O(D^2)\) by solving the triangular system \(Ax = y\).

Note

The bijector is invertible if and only if all diagonal elements of \(A\) are non-zero. It is the responsibility of the user to make sure that this is the case; the class will make no attempt to verify that the bijector is invertible.

__init__(matrix: Array, is_lower: bool = True) ¤

Initializes a TriangularLinear bijector.

Arguments:

  • matrix: a square matrix whose triangular part defines A. Can also be a batch of matrices. Whether A is the lower or upper triangular part of matrix is determined by is_lower.
  • is_lower: if True, A is set to the lower triangular part of matrix. If False, A is set to the upper triangular part of matrix.