Differentiate an affine graph parameter

Goal: return both Y(theta) and dY/dtheta for A(theta)=A+theta E. E is a fixed sparse direction; inputs and channel coefficients are constant. This is useful for parameter sweeps and sensitivity analysis of compatible linear graph filters.

import numpy as np
from scipy.sparse import csr_array
from k_graphfuse import compile_sensitivity, evaluate_sensitivity

a = csr_array([[0., 0.5], [0.25, 0.]])
e = csr_array([[0.1, 0.], [0., -0.1]])
x = np.array([[1.], [2.]])
stages = [[[[1]], [[2]], [[1]]]]
plan = compile_sensitivity(a, e, stages, 0, 0, parameter_radius=0.25)
y, dy = evaluate_sensitivity(plan, x, theta=0.125)

The retained polynomial is I+2A+A². An independent derivative is (2E + EA(theta) + A(theta)E)X. Run python examples/sensitivity.py for this check. For GPU execution, pass this plan to prepare_cuda and evaluate using the same theta.

Budget value and derivative separately

value_tolerance controls the discarded output operator; derivative_tolerance controls the discarded derivative operator. Both are nonnegative. A zero budget retains the corresponding nonzero tail. Inspect error_bound, derivative_error_bound, degree, derivative_bound_method and parameter_radius in the returned plan.

A small value tail can hide a large derivative tail: for P(z)=1+z^100 at z=0.9, the tail value is about 2.66e-5, while its derivative is about 2.95e-3. A 1e-4 value budget alone is insufficient for the derivative. The regression tests include this case.

Keep the parameter within the compiled interval

The plan is uniform over abs(theta) <= parameter_radius. A parameter outside the interval is rejected. Exact rational endpoints and floating values can differ slightly; choose interior values or represent the endpoint consistently. Arbitrary topology changes and multiple independently varying directions require a different implementation.

The coupled Horner recurrence differentiates the retained polynomial exactly in exact arithmetic. Its certificates bound omitted terms, with a manuscript-dependent complete-inequality premise when selected. They do not cover floating-point roundoff, graph canonicalization changes or differentiation through degree selection. See the derivation and API.