Implementation:Kornia Kornia Welsch Loss

Knowledge Sources	Kornia
Domains	Vision, Loss_Functions
Last Updated	2026-02-09 15:00 GMT

Overview

Welsch Loss (also known as Leclerc loss) is a robust regression loss function based on a Gaussian-like weighting that heavily down-weights outliers.

Description

The Welsch loss, also known as the Leclerc loss, is a robust loss function that uses an exponential weighting to suppress the influence of outliers. It belongs to the family of robust estimators described by Barron (2019).

The mathematical formulation is:

$\text{WL}(x,y)=1-\exp (-\frac{1}{2}(x-y{)}^2)$

Where $x$ is the prediction and $y$ is the target. The loss is bounded between 0 and 1, meaning that as residuals grow large, the loss asymptotically approaches 1. This provides a very smooth and gradual transition from quadratic behavior near zero to complete saturation for large residuals.

Compared to Geman-McClure loss which has a hard saturation, Welsch loss provides a smoother exponential decay, making it less aggressive but still highly robust.

Usage

Import this loss for robust regression tasks in computer vision where data may contain outliers, such as optical flow estimation, depth prediction, and image alignment. Its smooth behavior makes it particularly suitable when gradual outlier suppression is preferred over hard rejection.

Code Reference

Source Location

• Repository: Kornia
• File: kornia/losses/welsch.py
• Lines: 1-133

Signature

def welsch_loss(
    img1: torch.Tensor,
    img2: torch.Tensor,
    reduction: str = "none",
) -> torch.Tensor: ...

class WelschLoss(nn.Module):
    def __init__(self, reduction: str = "none") -> None: ...
    def forward(self, img1: torch.Tensor, img2: torch.Tensor) -> torch.Tensor: ...

Import

from kornia.losses import WelschLoss
from kornia.losses import welsch_loss

I/O Contract

Inputs

Name	Type	Required	Description
img1	torch.Tensor	Yes	Predicted tensor with arbitrary shape (*)
img2	torch.Tensor	Yes	Target tensor with the same shape as img1
reduction	str	No	Reduction mode: 'none' (default), 'mean', or 'sum'

Outputs

Name	Type	Description
loss	torch.Tensor	Computed Welsch loss; element-wise for 'none' (bounded in [0, 1]), scalar for 'mean'/'sum'

Usage Examples

import torch
from kornia.losses import WelschLoss

# Create sample tensors
img1 = torch.randn(2, 3, 32, 32, requires_grad=True)
img2 = torch.randn(2, 3, 32, 32)

# Using the module API
criterion = WelschLoss(reduction="mean")
output = criterion(img1, img2)
output.backward()

# Using the functional API
from kornia.losses import welsch_loss
output = welsch_loss(img1, img2, reduction="sum")

Related Pages

• Principle:Kornia_Kornia_Robust_Loss_Functions