You will be shown an excerpt from a computer science scientific research paper, followed by a question about the paper. Please answer the question. Do not include any text in your response other than the answer.

Context: Clipping pixels with small contribution: Instead of clipping pixels with small norms, we can try something slightly smarter and clip pixels with small contributions to the activation. One way of computing a pixel’s contribution to an activation is to measure how much the activation increases or decreases when the pixel is set to zero; that is, to compute the contribution as |a_{i}(\mathbf{x})-a_{i}(\mathbf{x}_{-j})|, where \mathbf{x}_{-j} is \mathbf{x} but with the j^{th} pixel set to zero. This approach is straightforward but prohibitively slow, requiring a forward pass for every pixel. Instead, we approximate this process by linearizing a_{i}(\mathbf{x}) around \mathbf{x}, in which case the contribution of each dimension of \mathbf{x} can be estimated as the elementwise product of \mathbf{x} and the gradient. We then sum over all three channels and take the absolute value, computing \left|\sum_{c}\mathbf{x}\circ\nabla_{\mathbf{x}}a_{i}(\mathbf{x})\right|. We use the absolute value to find pixels with small contribution in either direction, positive or negative. While we could choose to keep the pixel transitions where setting the pixel to zero would result in a large activation increase, these shifts are already handled by gradient ascent, and here we prefer to clip only the pixels that are deemed not to matter, not to take large gradient steps outside the region where the linear approximation is most valid. We define this r_{\theta}(\mathbf{x}) as the operation that sets pixels with contribution under the \theta_{\mathrm{c\_pct}} percentile to zero.

Question: How did the authors compute the contributions of the pixels in order to clip the pixels with smaller contributions?
Calculating absolute difference between some neuron activation of an input and the activation for same input without certain pixel can be considered a way of measuring the contribution of that pixel in the total response of the neuron. To ensure faster computation, we can estimate activation near the input with 1st order approximation (linear) and hence this leads  to total contribution estimated as the element wise product of the activation gradient and the input x (each element in this product shows how this pixel affects the total response ), we repeat this product for all different channels ,sum them all, and take absolute value  to find pixels with small contribution in either direction, positive or negative to get rid of.