Discrete 1D convolutions [FR]

I’m working on a project whose main bottleneck is a series of 1d discrete convolutions, basically a lot of dot products within a for loop. Similar computations also came up in the forums a couple of times (for example here).

I’m wondering if there is a way to make them more efficient. Would it be helpful to add something like a conv1d function to stan-math? Is there a way to implement conv1d(x, kernel) such that it is more efficient than:

for (i in T:n) {
    y[i] = dot_product(kernel, x[(i-T+1):i]);
}

I guess there are many subtleties here that I'm not aware of (numpy convolve has 3 modes - "full", "valid" and "same" - to handle various padding and border scenarios), just curious if this is relevant/interesting.

Start with the proposed conv1d(x, kernel) API and the existing Stan loop shown in the issue, then review the linked forum discussion for prior performance considerations. Define the supported boundary modes and establish whether the implementation is faster than direct dot products; no source file or test is named, so both the API location and validation plan must be determined.