Incorrect Pearson Correlation Output for Constant y_true in torchmetrics

[FrancescaDr]

🐛 Bug

When computing the Pearson correlation using torchmetrics.PearsonCorrCoef, it returns 1 for some cases where y_true is constant. Mathematically, the Pearson correlation should be undefined (or return NaN), since the standard deviation of y_true is zero. In contrast, scipy.stats.pearsonr returns0 in such cases.

To Reproduce

import torchmetrics
import torch
from scipy import stats

y_pred = torch.tensor([[-0.1816,  0.6568,  0.9788, -0.1425], [-0.4111,  0.3940,  1.4834,  0.1322]])
y_true = torch.tensor([[4.0268,  5.9401,  1.0000,  1.0000], [6.4956,  5.6684,  1.0000,  1.0000]])

pearson_corr = torchmetrics.PearsonCorrCoef(num_outputs=4)
print('Torchmetrics: ', pearson_corr(y_pred, y_true))
print('Scipy: ', stats.pearsonr(y_pred, y_true))

Returns:

Torchmetrics:  tensor([-1.0000,  1.0000,  1.0000,     nan])
Scipy:  PearsonRResult(statistic=array([-1.,  1.,  0.,  0.], dtype=float32), pvalue=array([1., 1., 1., 1.], dtype=float32))

Expected behavior

For columns where y_true is constant (e.g., y_true[:, 2] = [1, 1]), the Pearson correlation should return nan.

Environment

• TorchMetrics version (if build from source, add commit SHA): 1.6.1
• Python: 3.11.0
• Torch: 2.5.1

Others

Might be related to: scikit-learn/scikit-learn#25258?

[github-actions]

Hi! thanks for your contribution!, great first issue!

[Vidit-Ostwal]

This is some unexpected behaviour, I went through the code

In the particular case of yours, this particular condition will hit and this warning should be raised

    bound = math.sqrt(torch.finfo(var_x.dtype).eps)
    if (var_x < bound).any() or (var_y < bound).any():
        rank_zero_warn(
            "The variance of predis can cause instability in Pearictions or target is close to zero. Thson correlation"
            "coefficient, leading to wrong results. Consider re-scaling the input if possible or computing using a"
            f"larger dtype (currently using {var_x.dtype}).",
            UserWarning,
        )

I tried with a different y_true and y_pred tensor with same y_pred bring constant, and the result was expected, nan was in the output.
I think this is unique to your particular case of input tensor.