jax_privacy.matrix_factorization.banded.ColumnNormalizedBanded

class jax_privacy.matrix_factorization.banded.ColumnNormalizedBanded(params)

Bases: Mapping

A column-normalized banded lower triangular n x n matrix.

This matrix class is parameterized by an arbitrary n x b matrix. C(params) is obtained by setting the first b bands of C based on params. The matrix is normalized to have sensitivity 1 under a single epoch, by dividing each column by its respective norm.

Below we show how params relates to the matrix (before column normalization):

params = [a b c]
         [d e f]
         [g h i]
         [j k -]
         [m - -]

C = [a        ]
    [b d      ]
    [c e g    ]
    [  f h j  ]
    [    i k m]

Methods

__init__
default	Construct a default instance of this object given n and bands.
from_banded_toeplitz	Construct an instance of this object from banded toeplitz coefficients.
from_tuple
get	Retrieves the corresponding layout by the string key.
inverse_as_streaming_matrix	Create $C^{-1}$ as a StreamingMatrix object.
items	D.items() -> a set-like object providing a view on D's items
keys	D.keys() -> a set-like object providing a view on D's keys
materialize
replace
to_tuple
values	D.values() -> an object providing a view on D's values

Attributes

bands
n
params

params: Array

property n: int

property bands: int

classmethod from_banded_toeplitz(n, coefs)

Construct an instance of this object from banded toeplitz coefficients.

Parameters:

• n (int) – the number of training iterations.

• coefs (Array) – an array of b toeplitz coefficients defining the strategy.

Return type:

ColumnNormalizedBanded

Returns:

A ColumnNormalizedBanded representation of the banded toeplitz matrix.

classmethod default(n, bands)

Construct a default instance of this object given n and bands.

This object is initialized by using the fixed toeplitz strategy proposed in [1; Algorithm 1], truncating to $b$ entries, and column normalizing. It can act as a useful initialization for further optimization.

[1] https://proceedings.mlr.press/v202/fichtenberger23a/fichtenberger23a.pdf

Parameters:

• n (int) – the number of training iterations.

• bands (int) – the number of bands in the strategy.

Return type:

ColumnNormalizedBanded

Returns:

A ColumnNormalizedBanded object.

materialize()

Return type:

Array

inverse_as_streaming_matrix()

Create $C^{-1}$ as a StreamingMatrix object.

Return type:

StreamingMatrix

from_tuple()

items() → a set-like object providing a view on D's items

keys() → a set-like object providing a view on D's keys

replace(**kwargs)

to_tuple()

values() → an object providing a view on D's values