LostTech.TensorFlow : API Documentation

`LinearOperator` acting like a nested block circulant matrix.

This operator acts like a block circulant matrix `A` with shape `[B1,...,Bb, N, N]` for some `b >= 0`. The first `b` indices index a batch member. For every batch index `(i1,...,ib)`, `A[i1,...,ib, : :]` is an `N x N` matrix. This matrix `A` is not materialized, but for purposes of broadcasting this shape will be relevant.

#### Description in terms of block circulant matrices

If `A` is nested block circulant, with block sizes `N0, N1, N2` (`N0 * N1 * N2 = N`): `A` has a block structure, composed of `N0 x N0` blocks, with each block an `N1 x N1` block circulant matrix.

For example, with `W`, `X`, `Y`, `Z` each block circulant,

``` A = |W Z Y X| |X W Z Y| |Y X W Z| |Z Y X W| ```

Note that `A` itself will not in general be circulant.

#### Description in terms of the frequency spectrum

There is an equivalent description in terms of the [batch] spectrum `H` and Fourier transforms. Here we consider `A.shape = [N, N]` and ignore batch dimensions.

If `H.shape = [N0, N1, N2]`, (`N0 * N1 * N2 = N`): Loosely speaking, matrix multiplication is equal to the action of a Fourier multiplier: `A u = IDFT3[ H DFT3[u] ]`. Precisely speaking, given `[N, R]` matrix `u`, let `DFT3[u]` be the `[N0, N1, N2, R]` `Tensor` defined by re-shaping `u` to `[N0, N1, N2, R]` and taking a three dimensional DFT across the first three dimensions. Let `IDFT3` be the inverse of `DFT3`. Matrix multiplication may be expressed columnwise:

```(A u)_r = IDFT3[ H * (DFT3[u])_r ]```

#### Operator properties deduced from the spectrum.

* This operator is positive definite if and only if `Real{H} > 0`.

A general property of Fourier transforms is the correspondence between Hermitian functions and real valued transforms.

Suppose `H.shape = [B1,...,Bb, N0, N1, N2]`, we say that `H` is a Hermitian spectrum if, with `%` meaning modulus division,

``` H[..., n0 % N0, n1 % N1, n2 % N2] = ComplexConjugate[ H[..., (-n0) % N0, (-n1) % N1, (-n2) % N2] ]. ```

* This operator corresponds to a real matrix if and only if `H` is Hermitian. * This operator is self-adjoint if and only if `H` is real.

See e.g. "Discrete-Time Signal Processing", Oppenheim and Schafer.

### Examples

See `LinearOperatorCirculant` and `LinearOperatorCirculant2D` for examples.

#### Performance

Suppose `operator` is a `LinearOperatorCirculant` of shape `[N, N]`, and `x.shape = [N, R]`. Then

* `operator.matmul(x)` is `O(R*N*Log[N])` * `operator.solve(x)` is `O(R*N*Log[N])` * `operator.determinant()` involves a size `N` `reduce_prod`.

If instead `operator` and `x` have shape `[B1,...,Bb, N, N]` and `[B1,...,Bb, N, R]`, every operation increases in complexity by `B1*...*Bb`.

#### Matrix property hints

This `LinearOperator` is initialized with boolean flags of the form `is_X`, for `X = non_singular, self_adjoint, positive_definite, square`. These have the following meaning * If `is_X == True`, callers should expect the operator to have the property `X`. This is a promise that should be fulfilled, but is *not* a runtime assert. For example, finite floating point precision may result in these promises being violated. * If `is_X == False`, callers should expect the operator to not have `X`. * If `is_X == None` (the default), callers should have no expectation either way.