By the way. I enjoyed listening the interview with you on Practical AI podcast!
However, I'd like to add that LSH has recently been applied to other types of problems with very good results.
For example, LSH-based methods are at the core of the best sampling methods for kernel density estimation (KDE) and other problems [1,2]. This has recently led to fast approximations for the top eigenvalues of kernel matrices [3], SGD sampling [4], and probably more. LSH can also be used to construct sketches for kernel sums [5], which have applications to bioinformatics and can lead to fast neural network inference [6]. So, LSH is alive and well - just not necessarily for ANN.
(Full disclaimer: I am an author of [5] and [6])
[1]: https://arxiv.org/abs/1808.10530 [2]: http://proceedings.mlr.press/v97/siminelakis19a [3]: https://arxiv.org/abs/2102.08341 [4]: https://arxiv.org/abs/1910.14162 [5]: https://arxiv.org/abs/1912.02283 [6]: https://arxiv.org/abs/2106.11426
ScaNN is currently SOTA: https://ai.googleblog.com/2020/07/announcing-scann-efficient...
Also, most search use cases have requirements besides just the "get_k_nearest" functionality too (scoring-level filtering, business logic built into scoring unrelated to similarity, disparate query patterns that don't fit nicely into a single similarity space), so unless you only need to retrieve the K most-similar documents it might not be a good solution.
But, they are used/designed for different use cases. DR is mostly concerned with preserving distances such that
c * D( f(x), f(y) ) < d(x,y) < C*D( f(x), f(y) )
Here, f(x) is the mapping of x to a lower dimensional space (or any other object, really). d(-,-) and D(-,-) are the original distance measure and new distance measure respectively. The smaller the ratio C/c the better. Clearly, the computational overhead of applying f and D are also important.
LSH, on the other hand, is mostly concerned with probabilistically filtering and searching for nearby points efficiently. Here are some (old) class notes from a course I gave 2013https://edoliberty.github.io//classnotes/datamining2013/pdfs...
While LSH is mathematically beautiful, it is no longer considered the gold standard for ANN. There are many open source algorithms that outperform LSH on most datasets.
(I founded a vector search company: https://www.pinecone.io)