Add forward AD for torch.linalg.eigh #62163
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IvanYashchuk
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module: autograd
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@albanD, CI fails with |
lezcano
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A few notes on forward mode AD for maps with constrained inputs.
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| auto F = eigenvalues.unsqueeze(-2) - eigenvalues.unsqueeze(-1); | ||
| F.diagonal(/*offset=*/0, /*dim1=*/-2, /*dim2=*/-1).fill_(INFINITY); | ||
| F = F.pow(-1); |
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It's better to divide by F than to compute the inverse explicitly an then multiply.
| // https://ora.ox.ac.uk/objects/uuid:8d0c0a29-c92b-4153-a1d2-38b276e93124 | ||
| // Section 3.1 Eigenvalues and eigenvectors | ||
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| auto tmp = at::matmul(at::matmul(eigenvectors.transpose(-2, -1).conj(), input_tangent), eigenvectors); |
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I believe that the input tangent needs to be symmetrised as: const auto in = 0.5 * (input_tangent + input_tangent.transpose(-2, -1).conj()).
Denote Her(n) the n x n Hermitian matrices and U(n) the unitary matrices. We have that eigh : Her(n) -> R^n x U(n). This means that the differential of eigh goes from the tangent to Her(n) to the tangent of R^n times the tangent of U(n) (at the output matrices). Now, the tangent of Her(n) is Her(n) itself as Her(n) is just a linear subspace. As such, the input needs to be symmetric for this function to make sense.
Computing (A + A^H)/2 happens to be the orthogonal projection of a matrix onto the space Her(n). The theorem that formalises all this is the one that says that the differential of a map on an embedded manifold is the differential of the map on the total space restricted to the tangent space of the embedded manifold.
I wrote all this down in a comment in eigh_backward. In general it would be interesting that the way we compute forward / backward are somewhat equivalent (i.e. one is "taking the transpose" of the other).
| F.diagonal(/*offset=*/0, /*dim1=*/-2, /*dim2=*/-1).fill_(INFINITY); | ||
| F = F.pow(-1); | ||
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| auto tmp = at::matmul(at::matmul(eigenvectors.transpose(-2, -1).conj(), input_tangent), eigenvectors); |
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input_tangent should be projected on to the Hermitian matrices (see comment above).
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Codecov Report
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| auto hermitian_tangent = 0.5*(input_tangent + input_tangent.transpose(-2, -1).conj()); | ||
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| auto tmp = at::matmul(at::matmul(eigenvectors.transpose(-2, -1).conj(), hermitian_tangent), eigenvectors); | ||
| auto eigenvectors_tangent = at::matmul(eigenvectors, tmp.div(E)); |
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nit you can return directly here to avoid the extra assignment.
| decorators=[skipCUDAIfNoMagma, skipCUDAIfRocm, skipCPUIfNoLapack], | ||
| skips=( | ||
| # Gradcheck for complex hangs for this function | ||
| SkipInfo('TestGradients', 'test_forward_mode_AD', dtypes=complex_types()),), |
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Do we have an issue open tracking this? Do we want one if not?
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There is no open issue about it. I added a NotImplementedError for complex inputs for now, we'll enable it later.
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LGTM modulo @albanD's comments.
The only point is that it might be worth to land this after the support for several outputs in AD mode as, at the moment, this AD is doing twice the same computations. Once for eigvals and another one for eigvectors.
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Now that this is ready, I think we can just add this to master. And merge the implementation when the multi-output support is added. |
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@albanD has imported this pull request. If you are a Facebook employee, you can view this diff on Phabricator. |
6 participants
This PR adds forward mode differentiation for
torch.linalg.eighand a few other functions required for tests to pass.For some reason running tests for
torch.linalg.eigvalshand complextorch.linalg.eighhangs. These tests are skipped for now.cc @ezyang @albanD @zou3519 @gqchen @pearu @nikitaved @soulitzer @lezcano @Varal7 @jianyuh @mruberry @heitorschueroff @walterddr @IvanYashchuk @xwang233