torchcubicspline

Interpolating natural cubic splines using PyTorch. Includes support for:

• Batching
• GPU support and backpropagation via PyTorch
• Support for missing values (represent them as NaN)
• Evaluating the first derivative of the spline

Installation

pip install git+https://github.com/patrick-kidger/torchcubicspline.git

Example

Simple example:

import torch
from torchcubicspline import(natural_cubic_spline_coeffs, 
                             NaturalCubicSpline)

length, channels = 7, 3
t = torch.linspace(0, 1, length)
x = torch.rand(length, channels)
coeffs = natural_cubic_spline_coeffs(t, x)
spline = NaturalCubicSpline(coeffs)
point = torch.tensor(0.4)
out = spline.evaluate(point)

With multiple batch and evaluation dimensions:

import torch
from torchcubicspline import(natural_cubic_spline_coeffs, 
                             NaturalCubicSpline)

t = torch.linspace(0, 1, 7)
# (2, 1) are batch dimensions. 7 is the time dimension
# (of the same length as t). 3 is the channel dimension.
x = torch.rand(2, 1, 7, 3)
coeffs = natural_cubic_spline_coeffs(t, x)
# coeffs is a tuple of tensors

# ...at this point you can save the coeffs, put them
# through PyTorch's Datasets and DataLoaders, etc...

spline = NaturalCubicSpline(coeffs)

point = torch.tensor(0.4)
# will be a tensor of shape (2, 1, 3), corresponding to
# batch, batch, and channel dimensions
out = spline.derivative(point)

point = torch.tensor([[0.4, 0.5]])
# will be a tensor of shape (2, 1, 1, 2, 3), corresponding to
# batch, batch, time, time and channel dimensions
out = spline.derivative(point)

Functionality

Functionality is provided via the natural_cubic_spline_coeffs function and NaturalCubicSpline class.

natural_cubic_spline_coeffs takes an increasing sequence of times represented by a tensor t of shape (length,) and some corresponding observations x of shape (..., length, channels), where ... are batch dimensions, and each (length, channels) slice represents a sequence of length points, each point with channels many values.

Then calling

coeffs = natural_cubic_spline_coeffs(t, x)
spline = NaturalCubicSpline(coeffs)

produces an instance spline such that

spline.evaluate(t[i]) == x[..., i, :]

for all i.

Why is there a function and a class?

The slow bit is done during natural_cubic_spline_coeffs. The fast bit is NaturalCubicSpline. The returned coeffs are a tuple of PyTorch tensors, so you can take this opportunity to save or load them, push them through torch.utils.data.Dataset or torch.utils.data.DataLoader, etc.

Derivatives

The derivative of the spline at a point may be calculated via spline.derivative. (Not be confused with backpropagation, which is also supported through both spline.evaluate and spline.derivative.)

Missing values

Support for missing values is done by setting that element of x to NaN. In particular this allows for batching elements with different observation times: take times to be the observation times of all elements in the batch, and just set each element to have a missing observation NaN at the times of the observations of the other batch elements.

Limitations

If possible, you should cache the coefficients returned by natural_cubic_spline_coeffs. In particular if there are missing values then the computation can be quite slow.

Any issues?

Any issues or questions - open an issue to let me know. :)