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Implementation

HNSW is an algorithm that creates layers of NSW graphs where the top layer is least refined and the "zero layer" is the most refined. The graph nodes are items from the search set in all cases and M edges are chosen by finding the M nearest-neighbors according to the graph's ANN search. The authors of HNSW probabalistically select which items from the search set to include in each graph layer and below to maintain global connectivity. This is effectively combining the concept of skip lists with NSW.

The authors define some parameters:

  • M is the number of nearest-neighbors to connect a new entry to when it is inserted.
    • Choose this depending on the dimensionality and recall rate.
    • The paper provides no formula, just some graphs and guidance, so this number might need to be fine-tuned.
    • The paper suggests an M between 5 and 48.
    • Benchmark your dataset to find the optimal M.
  • Mmax0 is the maximum M for the "zero" layer.
    • It should be set to M * 2, which the paper claims is relatively empirically optimal, but makes no mathematical claim.
      • See the graphs in figure 6 to see the empirical evidence.
    • In the code of this crate, this is the parameter M0.
  • Mmax is the maximum M for every non-zero layer.
    • In the code of this crate, this is the parameter M.
  • mL is a parameter the controls the random selection of the max layer an insertion will appear on.
    • This parameter is chosen to probabalistically approximate a skip list with an average of one element of overlap between layers.
    • The formula for the maximum layer l that an insertion appears on is -ln(unif(0..1)) * mL.
    • mL is chosen to be 1 / ln(M).
    • The chosen mL seems to empirically be ideal according to the paper's findings, so this is not exposed to the user.
  • ef is the number of nearest neighbors to keep in a priority queue while searching.
    • This is the parameter controlled to change the recall rate.
    • This is also reffered to as ef in this crate.
  • efConstruction is the ef to use when searching for nearest neighbors when inserting.
    • This parameter is different from ef so that the quality of the graph can be improved (closer to the Delaunay graph).
      • Improving the quality of the graph means faster query time at the same recall, but slower graph construction.
    • This may need to be fine-tuned for your needs.
      • The authors used a value of 100 on a 10 million SIFT dataset to get good quality.
      • A value of 40 on a 200 million SIFT dataset still gets virtually the same results as 500.
    • Increasing this beyond a certian point does practically nothing at a high cost.
    • See figure 10 for some data.
    • I would set this to about 400 if insertion performance is not a concern.
      • If insertion performance is a concern, benchmark it on your dataset.