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That makes sense, but how do you efficiently evaluate the Gaussian kernel based approach (“operator-based data structures (OBDS)”)? Presumably you want to do it in a way that keeps a dynamically updating data structure instead of computing a low rank approximation to the kernel etc? In my understanding the upside of the kNN based approaches are fast querying and ability to dynamically insert additional vectors..?

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Thank you for the thoughtful comment. Your questions are valid given the title, which I used to make the post more accessible to a general HN audience. To clarify: the core distinction here is not kernelization vs kNN, but field evaluation vs point selection (or selection vs superposition as retrieval semantics). The kernel is just a concrete example.

FAISS implements selection (argmax ⟨q,v⟩), so vectors are discrete atoms and deletion must be structural. The weighted formulation represents a field: vectors act as sources whose influence superposes into a potential. Retrieval evaluates that field (or follows its gradient), not a point identity. In this regime, deletion is algebraic (append -v for cancellation), evaluation is sparse/local, and no index rebuild is required.

The paper goes into this in more detail.

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