pyrand.InterpolateSchatten(kind=’ext’)#

class pyrand.InterpolateSchatten(A, B=None, p=2, options={}, verbose=False, kind='ext')

Evaluates Schatten norm (or anti-norm) of an affine matrix function (no interpolation).

See also

This page describes only the ext method. For other kinds, see pyrand.InterpolateSchatten().

This class does not interpolate. Rather, it only returns the exact function value by calling pyrand.schatten(). The output of this function could be used as a benchmark to test the other interpolation methods.

Parameters:

Anumpy.ndarray, scipy.sparse matrix

A square matrix. Matrix can be dense or sparse.

Bnumpy.ndarray, scipy.sparse matrix, default=None

A square matrix of the same type and size as A.

pfloat, default=2

The order \(p\) in the Schatten \(p\)-norm which can be real positive, negative or zero.

optionsdict, default={}

The Schatten norm is computed using pyrand.schatten() function which itself calls either of

pyrand.logdet() (if \(p=0\))
pyrand.trace() (if \(p>0\))
pyrand.traceinv() (if \(p < 0\)).

The options passes a dictionary of arguments to the above functions.

verbosebool, default=False

If True, it prints some information about the computation process.

See also

pyrand.InterpolateTrace
pyrand.InterpolateLogdet
pyrand.schatten

Notes

Schatten Norm:

In this class, the Schatten \(p\)-norm of the matrix \(\mathbf{A}\) is defined by

(1)#\[\begin{split}\Vert \mathbf{A} \Vert_p = \begin{cases} \left| \mathrm{det}(\mathbf{A}) \right|^{\frac{1}{n}}, & p=0, \\ \left| \frac{1}{n} \mathrm{trace}(\mathbf{A}^{p}) \right|^{\frac{1}{p}}, & p \neq 0, \end{cases}\end{split}\]

where \(n\) is the size of the matrix. When \(p \geq 0\), the above definition is the Schatten norm, and when \(p < 0\), the above is the Schatten anti-norm.

Note

Conventionally, the Schatten norm is defined without the normalizing factor \(\frac{1}{n}\) in (1). However, this factor is justified by the continuity granted by

(2)#\[\lim_{p \to 0} \Vert \mathbf{A} \Vert_p = \Vert \mathbf{A} \Vert_0.\]

See [1] (Section 2) and the examples in pyrand.schatten() for details.

Affine Matrix Function:

This class evaluates the one-parameter matrix function:

\[\tau_p: t \mapsto \| \mathbf{A} + t \mathbf{B} \|_p,\]

where \(t\) is a real parameter.

References

[1]

Ameli, S., and Shadden. S. C. (2022). Interpolating Log-Determinant and Trace of the Powers of Matrix \(\mathbf{A} + t\mathbf{B}\). arXiv: 2009.07385 [math.NA].

Examples

Basic Usage:

Evaluate the Schatten 2-norm of the affine matrix function \(\mathbf{A} + t \mathbf{I}\):

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

>>> # Initialize interpolator object
>>> from pyrand import InterpolateSchatten
>>> f = InterpolateSchatten(A, p=2, kind='ext')

>>> # Evaluate at inquiry point t = 0.4
>>> t = 4e-1
>>> f(t)
1.7175340160001527

Alternatively, call pyrand.InterpolateSchatten.eval() to evaluate at points t:

>>> # This is the same as f(t)
>>> f.eval(t)
1.7175340160001527

Passing Options:

The above examples, the internal computation is passed to pyrand.trace() function since \(p=2\) is positive. You can pass arguments to the latter function using options argument. To do so, create a dictionary with the keys as the name of the argument. For instance, to use pyrand.trace(method=’slq’) method with min_num_samples=20 and max_num_samples=100, create the following dictionary:

>>> # Specify arguments as a dictionary
>>> options = {
...     'method': 'slq',
...     'min_num_samples': 20,
...     'max_num_samples': 100
... }

>>> # Pass the options to the interpolator
>>> f = InterpolateSchatten(A, B, options=options, kind='ext')
>>> f(t)
1.7047510802581667

Evaluate on Range of Points:

Evaluate an array of inquiry points t_array:

>>> # Initialize interpolator object
>>> from pyrand import InterpolateSchatten
>>> f = InterpolateSchatten(A, kind='ext')

>>> # Interpolate at an array of points
>>> import numpy
>>> t_array = numpy.logspace(-2, 1, 1000)
>>> norm_array = f(t_array)

Plotting:

To plot the function, call pyrand.InterpolateSchatten.plot() method. To compare with the true function values, pass compare=True argument.

>>> f.plot(t_array, compare=True)

../_images/interpolate_schatten_ext_eig.png

Since the ext method exactly evaluates the function (without interpolation), the error of the result is zero, as shown on the right-hand side plot.

Attributes:

kindstr: Method of interpolation. For this class, kind is ext.
verbosebool: Verbosity of the computation process
nint: Size of the matrix
qint: Number of interpolant points. For this class, q is zero.
pfloat: Order of Schatten \(p\)-norm

Methods

__call__	See `pyrand.InterpolateSchatten.__call__()`.
eval	See `pyrand.InterpolateSchatten.eval()`.
interpolate	See `pyrand.InterpolateSchatten.interpolate()`.
bound	See `pyrand.InterpolateSchatten.bound()`.
upper_bound	See `pyrand.InterpolateSchatten.upper_bound()`.
plot	See `pyrand.InterpolateSchatten.plot()`.