imate.InterpolateSchatten.upper_bound

InterpolateSchatten.upper_bound(t)

Upper bound of the interpolation function.

Note

This function only applies to \(p=-1\).

Parameters:

tfloat or numpy.array

An inquiry point or an array of inquiry points.

Returns:

ubfloat or numpy.array

Upper bound. If t is an array, the output is also an array of the size of t.

Raises:

ValueError

If \(p \neq -1\).

See also

imate.InterpolateSchatten.bound

Notes

A upper bound of the function \(\Vert \mathbf{A} + t \mathbf{B} \Vert_{-1}\) is (see [1])

\[\Vert \mathbf{A} + t \mathbf{B} \Vert_{-1} \leq \frac{\mathrm{trace}(\mathbf{A}) + t \mathrm{trace}(\mathbf{B})}{n}.\]

Note that the above bound is not sharp as \(t \to 0\).

References

[1]

Ameli, S., and Shadden. S. C. (2022). Interpolating Log-Determinant and Trace of the Powers of Matrix \(\mathbf{A} + t \mathbf{B}\). Statistics and Computing 32, 108. https://doi.org/10.1007/s11222-022-10173-4.

Examples

Create an interpolator object \(f\) using four interpolant points \(t_i\):

>>> # Generate sample matrices (symmetric positive-definite)
>>> from imate.sample_matrices import correlation_matrix
>>> A = correlation_matrix(size=20, scale=1e-1)
>>> B = correlation_matrix(size=20, scale=2e-2)

>>> # Initialize interpolator object
>>> from imate import InterpolateSchatten
>>> ti = [1e-2, 1e-1, 1, 1e1]
>>> f = InterpolateSchatten(A, B, p=-1, ti=ti)

Create an array t and evaluate upper bound on t. Also, interpolate the function \(f\) on the array t.

>>> # Interpolate at an array of points
>>> import numpy
>>> t = numpy.logspace(-2, 1, 1000)
>>> ub = f.upper_bound(t)
>>> interp = f.interpolate(t)

Plot the results:

>>> import matplotlib.pyplot as plt

>>> # Plot settings (optional)
>>> from imate._utilities import set_custom_theme
>>> set_custom_theme(font_scale=1.15)

>>> plt.semilogx(t, interp, color='black', label='Interpolation')
>>> plt.semilogx(t, ub, '--', color='black', label='Upper bound')
>>> plt.xlim([t[0], t[-1]])
>>> plt.ylim([0, 10])
>>> plt.xlabel('$t$')
>>> plt.ylabel('$\Vert \mathbf{A} + t \mathbf{B} \Vert_{-1}$')
>>> plt.title('Interpolation of Schatten Anti-Norm')
>>> plt.legend()
>>> plt.show()