jax_privacy.matrix_factorization.toeplitz.compute_banded_inverse_sensitivity_squared

jax_privacy.matrix_factorization.toeplitz.compute_banded_inverse_sensitivity_squared(n, noising_coef, min_sep, max_participations=None, use_matrix_upper_bound=False)[source]

Returns squared sensitivity for a banded inverse Toeplitz noising matrix.

This function takes Toeplitz coefficients of a lower-triangular noising matrix $C^{-1}$, computes the implied strategy coefficients for $C$, and then estimates the min-separation sensitivity of $C$.

Tightness depends on the sign and monotonicity of the implied strategy coefficients. If the strategy coefficients are positive and non-increasing, this uses the closed-form Toeplitz sensitivity computation in minsep_sensitivity_squared, which is exact.

Otherwise, the behavior depends on use_matrix_upper_bound:

  • If False, the absolute strategy coefficients are projected onto the smallest non-increasing majorant, and the resulting sequence is used to compute an upper bound. This bound is exact when the strategy coefficients are positive and decreasing, but may be looser when they are not non-increasing.

  • If True, the Toeplitz matrix formed from the absolute strategy coefficients is materialized, and the generic sensitivity upper bound from sensitivity.py is used instead. This is more computationally expensive, but gives a tighter bound when the sequence is not non-increasing, and is exact when the strategy coefficients are positive.

Parameters:

  • n (int) – Size of the implied Toeplitz matrix.

  • noising_coef (Array) – Toeplitz coefficients of the noising matrix $C^{-1}$.

  • min_sep (int) – Minimum separation between participations.

  • max_participations (int | None) – Optional cap on the number of participations.

  • use_matrix_upper_bound (bool) – Whether to use the generic matrix-based upper bound instead of the projected-coefficient upper bound when the absolute strategy coefficients are not non-increasing.

Return type:

Array

Returns:

The squared b-min-separated sensitivity of the implied strategy matrix $C$.