The hashing trick in VW hashes multiple same words into one integer, not the same as reformer, but similar to how reformer puts similar vectors together.
With VW's ngram/skipgram features, you get the same kind of effect - similar strings hash into the same hash.
So locality sensitive hashing = (is around about the same thing as) ngram/skipgram on strings plus hashing trick.
Locality sensitive hashing is a way to put similar vectors into the same buckets - by omission etc. It does this by hashing, but the intent is to approximate nearest neigbours.
skipgram/ngrams turn features into other features by omission etc, and so makes similar things the same. The hashing trick then reduces memory usage.
So yes, you're right the hashing in locality sensitive hashing is different in intent, but my point is, that both these approaches are designed to be more memory and compute efficient.
And vowpal's feature interactions give you transformer layers.
Add up all these together, and they have about the same net effect.
Yes, obviously locality sensitive hashing is a form of hashing. The fact that it's locality sensitive is important for this application, but you'd rather ignore that and insist on labeling them as the same thing just because they're both hashing.
Simhash algorithm, the LSH i knew about (which i mistakenly thought is LSH) works exactly like VW. It is ngrams + hash.