jax_privacy.matrix_factorization.toeplitz.inverse_as_streaming_matrix
- jax_privacy.matrix_factorization.toeplitz.inverse_as_streaming_matrix(coef, column_normalize_for_n=None)[source]
Create $C^{-1}$ as a StreamingMatrix object.
If column_normalize_for_n is None, the returned object represents $C^{-1}$, the inverse of an arbitrarily large banded Toeplitz matrix $C$ with coefficients coef.
If column_normalize_for_n is finite, the returned object represents $C^{-1}$, the inverse of a banded matrix $C$ of the given size n x n, formed by taking the banded Toeplitz matrix with coefficients coef and re-scaling each column so it has L2 norm 1.0. We recommend setting column_normalize_for_n in centralized training scenarios where n is known in advance.
- Formal Guarantees:
The maximum L2 norm of the strategy matrix $C$ is || coef ||_2 if column_normalize_for_n is None, and 1.0 if it is not None.
The strategy matrix $C$ corresponding to the returned noising matrix $C^{-1}$ is b-banded, where b = coef.size.
This implementation is based on Algorithm 9 from https://arxiv.org/abs/2306.08153.
- Parameters:
coef (
Array) – The Toeplitz coefficients of the strategy.column_normalize_for_n (
int|None) – If given, the returned object represents the inverse of a column-normalized banded Toeplitz matrix of the given size. Otherwise, it reprsents the inverse of an ordinary banded Toeplitz matrix. If not None, the supplied coef must have an L2 norm of 1.0 (otherwise, column normalization could change the sensitivity of the implied $C$ matrix).
- Return type:
- Returns:
A StreamingMatrix object representing $C^{-1}$.