Both the convolution and the cross-correlation operations are defined as the dot product between a small matrix and different parts of another typically bigger matrix (in the case of CNNs, it is an image or a feature map). Here's the usual illustration (which is applicable to both the convolution and cross-correlation).

Both the convolution and the cross-correlation operations are defined as the dot product between a small matrix and different parts of another typically bigger matrix (in the case of CNNs, it is an image or a feature map). Here's the usual illustration (of the cross-correlation, but the idea of the convolution is the same!).

TL;DR: Technically, convolutional neural networks (CNNs) can either perform the cross-correlation or convolution: it does not really matter whether you perform the cross-correlation or convolution because the kernels are learnable, so they can adapt to the cross-correlation or convolution given the data, although, in the typical diagrams, CNNs are shown to perform the cross-correlation (probably because it is conceptually simpler and it is possible that they are implemented with cross-correlations)! Moreover, in general, the kernels can or not be symmetric (although they typically won't be symmetric). In the case they are symmetric, the cross-correlation is equal to the convolution.

To understand the answer to this question, I will focus on the convolution and cross-correlation applied to 1-dimensional signals (which is the simplest case to which these operations can be applied). I will only talk about convolutions and cross-correlations for discrete and finite signals because that's the case of CNNs. Moreover, in this answer, I will assume that you are at least familiar with how the convolution (or cross-correlation?) in a CNN is performed.

Anyway, let me first remind you of the intuitive definition of the convolution and cross-correlation. Both of these operations can be thought of as the dot product between a small vector and different parts of another typically bigger vector (in the case of CNNs, it is an image or a feature map).

To be more concrete, let's suppose that we have the output of a function $f$ grouped in a vector $$f = [2, 1, 3, 5, 4] \in \mathbb{R}^{1 \times 5},$$ and the output of a kernel function also grouped in another vector $$h=[1, -1] \in \mathbb{R}^{1 \times 2}.$$ For simplicity, let's assume that we do not pad the input signal and we perform the convolution and cross-correlation with a stride of 1 (I assume that you are familiar with the concepts of padding and stride).

\begin{align} f \circledast h = g_1 &=\\ [(-1)*2 + 1*1, (-1)*1 + 1*3, (-1)*3 + 1*5, (-1)*5+1*4] &=\\ [-2 + 1, -2 + 3, -3 + 5, -5 + 4] &=\\ [-1, 1, 2, -1] \in \mathbb{R}^{1 \times 4} \end{align}

In the case of the cross-correlation, denoted as $f \otimes h = g_2$, where $\otimes$ is the cross-correlation operator, the only difference is that we do not flip the elements of the kernel before applying the element-wise multiplications, that is

\begin{align} f \otimes h = g_2 &=\\ [1*2 + (-1)*1, 1*1 + (-1)*3, 1*3 + (-1)*5, 1*5 + (-1)*4] &=\\ [2 - 1, 1 - 3, 3 - 5, 5 - 4] &=\\ [1, -2, -2, 1] \in \mathbb{R}^{1 \times 4} \end{align}

The first thing to note is that in both cases the result is a $1 \times 4$ vector. If we had convolved $f$ with a $1 \times 1$ vector, the result would have been a $1 \times 5$ vector. Recall that we assumed no padding (i.e. we don't add dummy elements to the left or right borders of $f$) and stride 1 (i.e. we shift the kernel to the right one element at a time). Similarly, if we had convolved $f$ with a $1 \times 3$, the result would have been a $1 \times 3$ vector.

The second thing to note is that $g_1$ and $g_2$ are different, so the result of the convolution is generally different than the result of the cross-correlation, given the same signals and kernels (as you probably have suspected?).

The third thing to note is that the only difference between the convolution and cross-correlation operations is that, in the first case, the kernel is flipped (along all spatial dimensions) before being applied.

(After having written this answer, I found the article Convolution vs. Cross-Correlation (2019) by Rachel Draelos, which essentially says the same thing that I am saying here, but provides more details and examples).

Short answer

Theoretically, convolutional neural networks (CNNs) can either perform the cross-correlation or convolution: it does not really matter whether they perform the cross-correlation or convolution because the kernels are learnable, so they can adapt to the cross-correlation or convolution given the data, although, in the typical diagrams, CNNs are shown to perform the cross-correlation because (in libraries like TensorFlow) they are typically implemented with cross-correlations (and cross-correlations are conceptually simpler than convolutions). Moreover, in general, the kernels can or not be symmetric (although they typically won't be symmetric). In the case they are symmetric, the cross-correlation is equal to the convolution.

Long answer

To understand the answer to this question, I will provide two examples that show the similarities and differences between the convolution and cross-correlation operations. I will focus on the convolution and cross-correlation applied to 1-dimensional discrete and finite signals (which is the simplest case to which these operations can be applied) because, essentially, CNNs process finite and discrete signals (although typically higher-dimensional ones, but this answer applies to higher-dimensional signals too). Moreover, in this answer, I will assume that you are at least familiar with how the convolution (or cross-correlation) in a CNN is performed, so that I do not have to explain these operations in detail (otherwise this answer would be even longer).

What is the convolution and cross-correlation?

Both the convolution and the cross-correlation operations are defined as the dot product between a small matrix and different parts of another typically bigger matrix (in the case of CNNs, it is an image or a feature map). Here's the usual illustration (which is applicable to both the convolution and cross-correlation).

Example 1

To be more concrete, let's suppose that we have the output of a function (or signal) $f$ grouped in a matrix $$f = [2, 1, 3, 5, 4] \in \mathbb{R}^{1 \times 5},$$ and the output of a kernel function also grouped in another matrix $$h=[1, -1] \in \mathbb{R}^{1 \times 2}.$$ For simplicity, let's assume that we do not pad the input signal and we perform the convolution and cross-correlation with a stride of 1 (I assume that you are familiar with the concepts of padding and stride).

Convolution

\begin{align} f \circledast h = g_1 &=\\ [(-1)*2 + 1*1, \\ (-1)*1 + 1*3, \\ (-1)*3 + 1*5, \\ (-1)*5+1*4] &=\\ [-2 + 1, -1 + 3, -3 + 5, -5 + 4] &=\\ [-1, 2, 2, -1] \in \mathbb{R}^{1 \times 4} \end{align}

Cross-correlation

Similarly, the cross-correlation of $f$ with $h$, denoted as $f \otimes h = g_2$, where $\otimes$ is the cross-correlation operator, is also defined as a dot product between $h$ and different parts of $f$, but without flipping the elements of the kernel before applying the element-wise multiplications, that is

\begin{align} f \otimes h = g_2 &=\\ [1*2 + (-1)*1, \\ 1*1 + (-1)*3, \\ 1*3 + (-1)*5, \\ 1*5 + (-1)*4] &=\\ [2 - 1, 1 - 3, 3 - 5, 5 - 4] &=\\ [1, -2, -2, 1] \in \mathbb{R}^{1 \times 4} \end{align}

Notes

1. The only difference between the convolution and cross-correlation operations is that, in the first case, the kernel is flipped (along all spatial dimensions) before being applied.

2. In both cases, the result is a $1 \times 4$ vector. If we had convolved $f$ with a $1 \times 1$ vector, the result would have been a $1 \times 5$ vector. Recall that we assumed no padding (i.e. we don't add dummy elements to the left or right borders of $f$) and stride 1 (i.e. we shift the kernel to the right one element at a time). Similarly, if we had convolved $f$ with a $1 \times 3$, the result would have been a $1 \times 3$ vector (as you will see from the next example).

3. The results of the convolution and cross-correlation, $g_1$ and $g_2$, are different. Specifically, one is the negated version of the other. So, the result of the convolution is generally different than the result of the cross-correlation, given the same signals and kernels (as you might have suspected).

Example 2: symmetric kernel

CNNs have learnable kernels

Do libraries implement the convolution or correlation?

In practice, certain libraries provide functions to compute both convolution and cross-correlation. For example, NumPy provides both the functions convolve and correlate to compute both the convolution and cross-correlation, respectively. If you execute the following piece of code (Python 3.7), you will get results that are consistent with my explanations above.

import numpy as np 

f = np.array([2., 1., 3., 5., 4.])

h = np.array([1., -1.])
h2 = np.array([-1., 2., 1.])

g1 = np.convolve(f, h, mode="valid")
g2 = np.correlate(f, h, mode="valid")

print("g1 =", g1) # g1 = [-1.  2.  2. -1.]
print("g2 =", g2) # g2 = [ 1. -2. -2.  1.]

However, NumPy is not really a library that provides out-of-the-box functionality to build CNNs.

On the other hand, TensorFlow's and PyTorch's functions to build the convolutional layers actually perform cross-correlations. As I said above, although it does not really matter whether CNNs perform the convolution or cross-correlation, this naming is misleading. Here's a proof that TensorFlow's tf.nn.conv1d actually implements the cross-correlation.

import tensorflow as tf # TensorFlow 2.2

f = tf.constant([2., 1., 3., 5., 4.], dtype=tf.float32)
h = tf.constant([1., -1.], dtype=tf.float32)

# Reshaping the inputs because conv1d accepts only certain shapes.
f = tf.reshape(f, [1, int(f.shape[0]), 1])
h = tf.reshape(h, [int(h.shape[0]), 1, 1])

g = tf.nn.conv1d(f, h, stride=1, padding="VALID")
print("g =", g) # [1, -2, -2, 1]

Further reading

After having written this answer, I found the article Convolution vs. Cross-Correlation (2019) by Rachel Draelos, which essentially says the same thing that I am saying here, but provides more details and examples.

To understand the answer to this question, I will focus on the convolution and cross-correlation applied to 1-dimensional signals (which is the simplest case to which these operations can be applied). For simplicity, I will only talk about convolutions and cross-correlations for discrete and finite signals (because that's also the case of CNNs). Moreover, in this answer, I will assume that you are at least familiar with how the convolution (or cross-correlation?) in a CNN is performed.

To understand the answer to this question, I will focus on the convolution and cross-correlation applied to 1-dimensional signals (which is the simplest case to which these operations can be applied). I will only talk about convolutions and cross-correlations for discrete and finite signals because that's the case of CNNs. Moreover, in this answer, I will assume that you are at least familiar with how the convolution (or cross-correlation?) in a CNN is performed.