Using the softmax operation for pooling is not a common practice in convolutional neural networks $($CNNs$)$ for several reasons:

Normalization Issue:

Softmax operation is primarily used for converting raw scores or logits into probabilities by normalizing them. It ensures that the output values sum up to $1$ across classes.

However, in the context of pooling, the objective is typically to downsample the feature map while retaining important information, rather than converting to probabilities.

Loss of Information:

Softmax pooling would involve applying softmax to each pooling window independently, which can lead to loss of important information.

Softmax tends to amplify differences between scores, which may not be desirable for pooling where the goal is to preserve significant features.

Non$-$Maximum Preservation:

Softmax pooling does not directly capture the maximum activation within each window, unlike max$-$pooling.

Max$-$pooling is popular because it helps in preserving the most prominent features by selecting the maximum value, which is crucial for tasks like object detection or recognition.

Computational Complexity:

Softmax involves exponentiation and normalization operations, which can be computationally intensive compared to simpler pooling operations like max$-$pooling or average pooling.

This increased complexity may not be justified for pooling operations where the main goal is downsampling and feature extraction.

Training Stability:

Softmax pooling can introduce additional challenges during training, such as vanishing gradients or unstable optimization due to the normalization and exponential operations.

Overall, while softmax is valuable for tasks like classification where converting scores to probabilities is necessary, it is not well$-$suited for pooling operations in CNNs where downsampling and feature preservation are the primary objectives. Max$-$pooling and average pooling are more popular choices for pooling layers due to their simplicity, effectiveness in feature preservation, and computational efficiency.