Aspect	Convolution	Cross-Correlation
Kernel Flipping	Yes, the kernel is flipped both horizontally and vertically before applying.	No, the kernel is used as-is without flipping.
Operation.	Reflects mathematical convolution, incorporating a flip to maintain certain theoretical properties.	Similar to convolution but without the kernel flip, simplifying computation.
Use in Theory	Essential in signal processing for properties like time-invariance.	Not traditionally defined as a separate operation in mathematical theory.
Use in Practice	In deep learning, often referred to but not actually used in standard CNNs.	Predominantly used in CNNs for tasks like image and signal processing.
Efficiency	The flipping step adds computational complexity.	More computationally efficient as it skips the flipping step.
Pattern Detection	Designed to detect features and patterns in the input data by considering the spatial relationship in a theoretically rigorous way.	Effectively detects features and patterns without the theoretical flipping, leveraging spatial relationships directly.

Conclusion

In the context of CNNs, although the term “convolution” is widely used, the operation practically implemented is cross-correlation. This choice is driven by cross-correlation’s computational efficiency and its direct applicability to feature detection without compromising the network’s learning capability. The distinction, while important from a theoretical perspective, does not significantly impact the practical outcomes in deep learning applications. CNNs continue to efficiently learn and detect patterns using cross-correlation, achieving state-of-the-art results in various tasks such as image classification, object detection, and beyond.