Interpolate signal using polyharmonic interpolation.

The interpolant has the form

\[f(x) = \sum_{i = 1}^n w_i \phi(||x - c_i||) + v^T x + b.\]

This is a sum of two terms: (1) a weighted sum of radial basis function (RBF) terms, with the centers \(c_1, ... c_n\), and (2) a linear term with a bias. The \(c_i\) vectors are 'training' points. In the code, b is absorbed into v by appending 1 as a final dimension to x. The coefficients w and v are estimated such that the interpolant exactly fits the value of the function at the \(c_i\) points, the vector w is orthogonal to each \(c_i\), and the vector w sums to 0. With these constraints, the coefficients can be obtained by solving a linear system.

\(\phi\) is an RBF, parametrized by an interpolation order. Using order=2 produces the well-known thin-plate spline.

We also provide the option to perform regularized interpolation. Here, the interpolant is selected to trade off between the squared loss on the training data and a certain measure of its curvature (details). Using a regularization weight greater than zero has the effect that the interpolant will no longer exactly fit the training data. However, it may be less vulnerable to overfitting, particularly for high-order interpolation.

Note the interpolation procedure is differentiable with respect to all inputs besides the order parameter.

We support dynamically-shaped inputs, where batch_size, n, and m are None at graph construction time. However, d and k must be known.

train_points [batch_size, n, d] float Tensor of n d-dimensional locations. These do not need to be regularly-spaced.
train_values [batch_size, n, k] float Tensor of n c-dimensional values evaluated at train_points.
query_points [batch_size, m, d] Tensor of m d-dimensional locations where we will output the interpolant's values.
order order of the interpolation. Common values are 1 for \(\phi(r) = r\), 2 for \(\phi(r) = r^2 * log(r)\) (thin-plate spline), or 3 for \(\phi(r) = r^3\).
regularization_weight weight placed on the regularization term. This will depend substantially on the problem, and it should always be tuned. For many problems, it is reasonable to use no regularization. If using a non-zero value, we recommend a small value like 0.001.
name name prefix for ops created by this function

[b, m, k] float Tensor of query values. We use train_points and train_values to perform polyharmonic interpolation. The query values are the values of the interpolant evaluated at the locations specified in query_points.