Replace torch.det() with manual implementation for 3x3 matrix

Summary:
# Background
There is an unstable error during training (it can happen after several minutes or after several hours).
The error is connected to `torch.det()` function in  `_check_valid_rotation_matrix()`.

if I remove the function `torch.det()` in `_check_valid_rotation_matrix()` or remove the whole functions `_check_valid_rotation_matrix()` the error is disappeared (D29555876).

# Solution
Replace `torch.det()` with manual implementation for 3x3 matrix.

Reviewed By: patricklabatut

Differential Revision: D29655924

fbshipit-source-id: 41bde1119274a705ab849751ece28873d2c45155

pytorch3d/common/workaround.py

@@ -0,0 +1,31 @@
+# Copyright (c) Facebook, Inc. and its affiliates.
+# All rights reserved.
+#
+# This source code is licensed under the BSD-style license found in the
+# LICENSE file in the root directory of this source tree.
+
+
+import torch
+
+
+def _safe_det_3x3(t: torch.Tensor):
+    """
+    Fast determinant calculation for a batch of 3x3 matrices.
+
+    Note, result of this function might not be the same as `torch.det()`.
+    The differences might be in the last significant digit.
+
+    Args:
+        t: Tensor of shape (N, 3, 3).
+
+    Returns:
+        Tensor of shape (N) with determinants.
+    """
+
+    det = (
+        t[..., 0, 0] * (t[..., 1, 1] * t[..., 2, 2] - t[..., 1, 2] * t[..., 2, 1])
+        - t[..., 0, 1] * (t[..., 1, 0] * t[..., 2, 2] - t[..., 2, 0] * t[..., 1, 2])
+        + t[..., 0, 2] * (t[..., 1, 0] * t[..., 2, 1] - t[..., 2, 0] * t[..., 1, 1])
+    )
+
+    return det

pytorch3d/transforms/transform3d.py

@@ -11,6 +11,7 @@ from typing import List, Optional, Union
 import torch
 
 from ..common.types import Device, get_device, make_device
+from ..common.workaround import _safe_det_3x3
 from .rotation_conversions import _axis_angle_rotation
 
 
@@ -774,7 +775,7 @@ def _check_valid_rotation_matrix(R, tol: float = 1e-7):
     eye = torch.eye(3, dtype=R.dtype, device=R.device)
     eye = eye.view(1, 3, 3).expand(N, -1, -1)
     orthogonal = torch.allclose(R.bmm(R.transpose(1, 2)), eye, atol=tol)
-    det_R = torch.det(R)
+    det_R = _safe_det_3x3(R)
     no_distortion = torch.allclose(det_R, torch.ones_like(det_R))
     if not (orthogonal and no_distortion):
         msg = "R is not a valid rotation matrix"