leaspy.utils.linalg

Functions

compute_orthonormal_basis(dgamma_t0, G_metric, *[, ...])

Householder decomposition, adapted for a non-Euclidean inner product defined by:

Module Contents

compute_orthonormal_basis(dgamma_t0, G_metric, *, strip_col=0)[source]

Householder decomposition, adapted for a non-Euclidean inner product defined by: (1) \(< x, y >Metric(p) = < x, G(p) y >Eucl = xT G(p) y\), where: \(G(p)\) is the symmetric positive-definite (SPD) matrix defining the metric at point p.

The Euclidean case is the special case where G is the identity matrix. Product-metric is a special case where G(p) is a diagonal matrix (identified to a vector) whose components are all > 0.

It is used to compute and set in-place the orthonormal_basis attribute given the time-derivative of the geodesic at initial time and the G_metric. The first component of the full orthonormal basis is a vector collinear G_metric x dgamma_t0 that we get rid of.

The orthonormal basis we construct is always orthonormal for the Euclidean canonical inner product. But all (but first) vectors of it lie in the sub-space orthogonal (for canonical inner product) to G_metric * dgamma_t0 which is the same thing that being orthogonal to dgamma_t0 for the inner product implied by the metric.

[We could do otherwise if we’d like a full orthonormal basis, w.r.t. the non-Euclidean inner product. But it’d imply to compute G^(-1/2) & G^(1/2) which may be computationally costly in case we don’t have direct access to them (for the special case of product-metric it is easy - just the component-wise inverse (sqrt’ed) of diagonal) TODO are there any advantages/drawbacks of one method over the other except this one? TODO are there any biases between features when only considering Euclidean orthonormal basis?]

Parameters:
dgamma_t0torch.FloatTensor 1D

Time-derivative of the geodesic at initial time. It may also be a vector collinear to it without any change to the result.

G_metricscalar, torch.FloatTensor 0D, 1D or 2D-square
The G(p) defining the metric as referred in equation (1) just before :

  • If 0D (scalar): G is proportional to the identity matrix

  • If 1D (vector): G is a diagonal matrix (diagonal components > 0)

  • If 2D (square matrix): G is general (SPD)

strip_colint in 0..model_dimension-1 (default 0)

Which column of the basis should be the one collinear to dgamma_t0 (that we get rid of)

Returns:
torch.Tensor of shape (dimension, dimension - 1)
Raises:
LeaspyModelInputError

if incoherent metric G_metric

Parameters: