[FIX] unstable TorchBackend.sqrtm() around repeated eigenvalues (#774)

* stable matrix sqrt using closed-form diff

* torch matrix sqrt gradcheck

* pre commit

* edit releases md

---------

Co-authored-by: Samuel Boïté <samuel.boite@polytechnique.edu>

Author: samuelbx

RELEASES.md

@@ -5,6 +5,7 @@
 #### Closed issues
 - Fix deprecated JAX function in `ot.backend.JaxBackend` (PR #771, Issue #770)
 - Add test for build from source (PR #772, Issue #764)
+- Stable `ot.TorchBackend.sqrtm` around repeated eigvals (PR #774, Issue #773)
 
 ## 0.9.6.post1
 

ot/backend.py

@@ -1938,6 +1938,7 @@ def __init__(self):
             self.rng_cuda_ = torch.Generator("cpu")
 
         from torch.autograd import Function
+        from torch.autograd.function import once_differentiable
 
         # define a function that takes inputs val and grads
         # ad returns a val tensor with proper gradients
@@ -1952,7 +1953,31 @@ def backward(ctx, grad_output):
                 # the gradients are grad
                 return (None, None) + tuple(g * grad_output for g in ctx.grads)
 
+        # define a differentiable SPD matrix sqrt
+        # with closed-form VJP
+        class MatrixSqrtFunction(Function):
+            @staticmethod
+            def forward(ctx, a):
+                a_sym = 0.5 * (a + a.transpose(-2, -1))
+                L, V = torch.linalg.eigh(a_sym)
+                s = L.clamp_min(0).sqrt()
+                y = (V * s.unsqueeze(-2)) @ V.transpose(-2, -1)
+                ctx.save_for_backward(s, V)
+                return y
+
+            @staticmethod
+            @once_differentiable
+            def backward(ctx, g):
+                s, V = ctx.saved_tensors
+                g_sym = 0.5 * (g + g.transpose(-2, -1))
+                ghat = V.transpose(-2, -1) @ g_sym @ V
+                d = s.unsqueeze(-1) + s.unsqueeze(-2)
+                xhat = ghat / d
+                xhat = xhat.masked_fill(d == 0, 0)
+                return V @ xhat @ V.transpose(-2, -1)
+
         self.ValFunction = ValFunction
+        self.MatrixSqrtFunction = MatrixSqrtFunction
 
     def _to_numpy(self, a):
         if isinstance(a, float) or isinstance(a, int) or isinstance(a, np.ndarray):
@@ -2395,12 +2420,7 @@ def pinv(self, a, hermitian=False):
         return torch.linalg.pinv(a, hermitian=hermitian)
 
     def sqrtm(self, a):
-        L, V = torch.linalg.eigh(a)
-        L = torch.sqrt(L)
-        # Q[...] = V[...] @ diag(L[...])
-        Q = torch.einsum("...jk,...k->...jk", V, L)
-        # R[...] = Q[...] @ V[...].T
-        return torch.einsum("...jk,...kl->...jl", Q, torch.transpose(V, -1, -2))
+        return self.MatrixSqrtFunction.apply(a)
 
     def eigh(self, a):
         return torch.linalg.eigh(a)

test/test_backend.py

@@ -822,6 +822,19 @@ def fun(a, b, d):
             assert nx.allclose(dl_db, b)
 
 
+def test_sqrtm_backward_torch():
+    if not torch:
+        pytest.skip("Torch not available")
+    nx = ot.backend.TorchBackend()
+    torch.manual_seed(42)
+    d = 5
+    A = torch.randn(d, d, dtype=torch.float64, device="cpu")
+    A = A @ A.T
+    A.requires_grad_(True)
+    func = lambda x: nx.sqrtm(x).sum()
+    assert torch.autograd.gradcheck(func, (A,), atol=1e-4, rtol=1e-4)
+
+
 def test_get_backend_none():
     a, b = np.zeros((2, 3)), None
     nx = get_backend(a, b)