Adopt original computation of FID (#355)

* Adopt exact computation of FID

Signed-off-by: Walter Hugo Lopez Pinaya <ianonimato@hotmail.com>

* Add verification of the imag part

Signed-off-by: Walter Hugo Lopez Pinaya <ianonimato@hotmail.com>

---------

Signed-off-by: Walter Hugo Lopez Pinaya <ianonimato@hotmail.com>

Author: Warvito

generative/metrics/fid.py

@@ -8,31 +8,14 @@
 # WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 # See the License for the specific language governing permissions and
 # limitations under the License.
-#
-# =========================================================================
-# Adapted from https://github.com/photosynthesis-team/piq
-# which has the following license:
-# https://github.com/photosynthesis-team/piq/blob/master/LICENSE
-#
-# Copyright 2023 photosynthesis-team. All rights reserved.
-#
-# Licensed under the Apache License, Version 2.0 (the "License");
-# you may not use this file except in compliance with the License.
-# You may obtain a copy of the License at
-#
-#     http://www.apache.org/licenses/LICENSE-2.0
-#
-# Unless required by applicable law or agreed to in writing, software
-# distributed under the License is distributed on an "AS IS" BASIS,
-# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
-# See the License for the specific language governing permissions and
-# limitations under the License.
-# =========================================================================
+
 
 from __future__ import annotations
 
+import numpy as np
 import torch
 from monai.metrics.metric import Metric
+from scipy import linalg
 
 
 class FIDMetric(Metric):
@@ -70,77 +53,53 @@ def get_fid_score(y_pred: torch.Tensor, y: torch.Tensor) -> torch.Tensor:
     return compute_frechet_distance(mu_y_pred, sigma_y_pred, mu_y, sigma_y)
 
 
-def _cov(m: torch.Tensor, rowvar: bool = True) -> torch.Tensor:
+def _cov(input_data: torch.Tensor, rowvar: bool = True) -> torch.Tensor:
     """
     Estimate a covariance matrix of the variables.
 
     Args:
-        m: A 1-D or 2-D array containing multiple variables and observations. Each row of `m` represents a variable,
+        input_data: A 1-D or 2-D array containing multiple variables and observations. Each row of `m` represents a variable,
             and each column a single observation of all those variables.
         rowvar: If rowvar is True (default), then each row represents a variable, with observations in the columns.
             Otherwise, the relationship is transposed: each column represents a variable, while the rows contain
             observations.
     """
-    if m.dim() < 2:
-        m = m.view(1, -1)
+    if input_data.dim() < 2:
+        input_data = input_data.view(1, -1)
 
-    if not rowvar and m.size(0) != 1:
-        m = m.t()
+    if not rowvar and input_data.size(0) != 1:
+        input_data = input_data.t()
 
-    fact = 1.0 / (m.size(1) - 1)
-    m = m - torch.mean(m, dim=1, keepdim=True)
-    mt = m.t()
-    return fact * m.matmul(mt).squeeze()
+    factor = 1.0 / (input_data.size(1) - 1)
+    input_data = input_data - torch.mean(input_data, dim=1, keepdim=True)
+    return factor * input_data.matmul(input_data.t()).squeeze()
 
 
-def _sqrtm_newton_schulz(matrix: torch.Tensor, num_iters: int = 100) -> tuple[torch.Tensor, torch.Tensor]:
-    """
-    Square root of matrix using Newton-Schulz Iterative method. Based on:
-    https://github.com/msubhransu/matrix-sqrt/blob/master/matrix_sqrt.py. Bechmark shown in:
-    https://github.com/photosynthesis-team/piq/issues/190#issuecomment-742039303
-
-    Args:
-        matrix: matrix or batch of matrices
-        num_iters: Number of iteration of the method
-
-    """
-    dim = matrix.size(0)
-    norm_of_matrix = matrix.norm(p="fro")
-    y_matrix = matrix.div(norm_of_matrix)
-    i_matrix = torch.eye(dim, dim, device=matrix.device, dtype=matrix.dtype)
-    z_matrix = torch.eye(dim, dim, device=matrix.device, dtype=matrix.dtype)
-
-    s_matrix = torch.empty_like(matrix)
-    error = torch.empty(1, device=matrix.device, dtype=matrix.dtype)
-
-    for _ in range(num_iters):
-        t = 0.5 * (3.0 * i_matrix - z_matrix.mm(y_matrix))
-        y_matrix = y_matrix.mm(t)
-        z_matrix = t.mm(z_matrix)
-
-        s_matrix = y_matrix * torch.sqrt(norm_of_matrix)
-
-        norm_of_matrix = torch.norm(matrix)
-        error = matrix - torch.mm(s_matrix, s_matrix)
-        error = torch.norm(error) / norm_of_matrix
-
-        if torch.isclose(error, torch.tensor([0.0], device=error.device, dtype=error.dtype), atol=1e-5):
-            break
-
-    return s_matrix, error
+def _sqrtm(input_data: torch.Tensor) -> torch.Tensor:
+    """Compute the square root of a matrix."""
+    scipy_res, _ = linalg.sqrtm(input_data.detach().cpu().numpy().astype(np.float_), disp=False)
+    return torch.from_numpy(scipy_res)
 
 
 def compute_frechet_distance(
     mu_x: torch.Tensor, sigma_x: torch.Tensor, mu_y: torch.Tensor, sigma_y: torch.Tensor, epsilon: float = 1e-6
 ) -> torch.Tensor:
     """The Frechet distance between multivariate normal distributions."""
     diff = mu_x - mu_y
-    covmean, _ = _sqrtm_newton_schulz(sigma_x.mm(sigma_y))
 
-    # If calculation produces singular product, epsilon is added to diagonal of cov estimates
+    covmean = _sqrtm(sigma_x.mm(sigma_y))
+
+    # Product might be almost singular
     if not torch.isfinite(covmean).all():
+        print(f"FID calculation produces singular product; adding {epsilon} to diagonal of covariance estimates")
         offset = torch.eye(sigma_x.size(0), device=mu_x.device, dtype=mu_x.dtype) * epsilon
-        covmean, _ = _sqrtm_newton_schulz((sigma_x + offset).mm(sigma_y + offset))
+        covmean = _sqrtm((sigma_x + offset).mm(sigma_y + offset))
+
+    # Numerical error might give slight imaginary component
+    if torch.is_complex(covmean):
+        if not torch.allclose(torch.diagonal(covmean).imag, torch.tensor(0, dtype=torch.double), atol=1e-3):
+            raise ValueError(f"Imaginary component {torch.max(torch.abs(covmean.imag))} too high.")
+        covmean = covmean.real
 
     tr_covmean = torch.trace(covmean)
     return diff.dot(diff) + torch.trace(sigma_x) + torch.trace(sigma_y) - 2 * tr_covmean

tests/test_compute_fid_metric.py

@@ -24,7 +24,7 @@ def test_results(self):
         x = torch.Tensor([[1, 2], [1, 2], [1, 2]])
         y = torch.Tensor([[2, 2], [1, 2], [1, 2]])
         results = FIDMetric()(x, y)
-        np.testing.assert_allclose(results.cpu().numpy(), 0.4433, atol=1e-4)
+        np.testing.assert_allclose(results.cpu().numpy(), 0.4444, atol=1e-4)
 
     def test_input_dimensions(self):
         with self.assertRaises(ValueError):