# Detecting approximate nearest neighbors

Aaron Lun1

1Cancer Research UK Cambridge Institute, Cambridge, United Kingdom

#### Revised: 2 December 2018

#### Package

BiocNeighbors 1.20.1

# 1 Introduction

The BiocNeighbors package provides several algorithms for approximate neighbor searches:

- The Annoy (Approximate Nearest Neighbors Oh Yeah) method uses C++ code from the RcppAnnoy package. It works by building a tree where a random hyperplane partitions a group of points into two child groups at each internal node. This is repeated to construct a forest of trees where the number of trees determines the accuracy of the search. Given a query data point, we identify all points in the same leaf node for each tree. We then take the union of leaf node sets across trees and search them exactly for the nearest neighbors.
- The HNSW (Hierarchical Navigable Small Worlds) method uses C++ code from the RcppHNSW package. It works by building a series of nagivable small world graphs containing links between points across the entire data set. The algorithm walks through the graphs where each step is chosen to move closer to a given query point. Different graphs contain links of different lengths, yielding a hierarchy where earlier steps are large and later steps are small. The accuracy of the search is determined by the connectivity of the graphs and the size of the intermediate list of potential neighbors.

These methods complement the exact algorithms described previously. Again, it is straightforward to switch from one algorithm to another by simply changing the `BNPARAM` argument in `findKNN` and `queryKNN`.

# 2 Identifying nearest neighbors

We perform the k-nearest neighbors search with the Annoy algorithm by specifying `BNPARAM=AnnoyParam()`.

    nobs <- 10000
    ndim <- 20
    data <- matrix(runif(nobs*ndim), ncol=ndim)

    fout <- findKNN(data, k=10, BNPARAM=AnnoyParam())
    head(fout$index)

    ##      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
    ## [1,] 3365  387 4429 3465 9376 6313 2631  589 1196  5023
    ## [2,]  797 1399 8666 3960 6249 9065 3824 6220 7205  6688
    ## [3,] 3475 9791 8731 7414 6006 8139 9495 8093  986  2631
    ## [4,] 9765 2480 7507 8983 5599 6721 2587 6637 9028  4711
    ## [5,] 5483 7628 3967 5853 4548 8975 3147 9958 5897  9196
    ## [6,] 9439 2005 4096 1539 6159 5570 4068  430 9253  1052

    head(fout$distance)

    ##           [,1]      [,2]      [,3]      [,4]     [,5]      [,6]      [,7]
    ## [1,] 0.8092533 0.9486768 0.9561160 0.9899128 1.001937 1.0039014 1.0290518
    ## [2,] 0.9459962 0.9881017 1.0045476 1.0061874 1.012161 1.0141041 1.0566691
    ## [3,] 1.0371169 1.1213335 1.1217155 1.1229984 1.125019 1.1251875 1.1469314
    ## [4,] 0.9693286 0.9813520 0.9984147 1.0018961 1.019948 1.0381206 1.0564381
    ## [5,] 0.8171108 0.8676689 0.9020651 0.9142418 0.963010 0.9866728 0.9891987
    ## [6,] 1.0201658 1.0376300 1.0569617 1.0641049 1.067901 1.0690782 1.0736088
    ##           [,8]      [,9]     [,10]
    ## [1,] 1.0356061 1.0424124 1.0511163
    ## [2,] 1.0612347 1.0883250 1.0936329
    ## [3,] 1.1523470 1.1568710 1.2112980
    ## [4,] 1.0602041 1.0615220 1.0905735
    ## [5,] 0.9909155 0.9919524 0.9941895
    ## [6,] 1.0899429 1.0903524 1.1001512

We can also identify the k-nearest neighbors in one dataset based on query points in another dataset.

    nquery <- 1000
    ndim <- 20
    query <- matrix(runif(nquery*ndim), ncol=ndim)

    qout <- queryKNN(data, query, k=5, BNPARAM=AnnoyParam())
    head(qout$index)

    ##      [,1] [,2] [,3] [,4] [,5]
    ## [1,] 4207 1925 9813 6743 5739
    ## [2,] 3594 1155 4905 7474 3241
    ## [3,] 3107 3576 5033 4005 8632
    ## [4,]  386 6812 3247 3232 2080
    ## [5,] 1848 2506 2118 8294 1904
    ## [6,] 5432 5558 7992 9020 1708

    head(qout$distance)

    ##           [,1]      [,2]      [,3]      [,4]      [,5]
    ## [1,] 1.1086656 1.1229142 1.1396971 1.2005340 1.2044449
    ## [2,] 0.9193776 0.9433740 0.9795136 0.9867874 0.9871328
    ## [3,] 0.9664251 1.0471556 1.0674046 1.1045169 1.1432260
    ## [4,] 0.8694986 0.9339198 0.9925212 1.0182252 1.0460166
    ## [5,] 1.0522285 1.0948354 1.0979621 1.1414605 1.1561661
    ## [6,] 0.8329313 0.8960790 0.9602339 0.9741856 0.9886567

It is similarly easy to use the HNSW algorithm by setting `BNPARAM=HnswParam()`.

# 3 Further options

Most of the options described for the exact methods are also applicable here. For example:

- `subset` to identify neighbors for a subset of points.
- `get.distance` to avoid retrieving distances when unnecessary.
- `BPPARAM` to parallelize the calculations across multiple workers.
- `BNINDEX` to build the forest once for a given data set and re-use it across calls.

The use of a pre-built `BNINDEX` is illustrated below:

    pre <- buildIndex(data, BNPARAM=AnnoyParam())
    out1 <- findKNN(BNINDEX=pre, k=5)
    out2 <- queryKNN(BNINDEX=pre, query=query, k=2)

Both Annoy and HNSW perform searches based on the Euclidean distance by default. Searching by Manhattan distance is done by simply setting `distance="Manhattan"` in `AnnoyParam()` or `HnswParam()`.

Users are referred to the documentation of each function for specific details on the available arguments.

# 4 Saving the index files

Both Annoy and HNSW generate indexing structures - a forest of trees and series of graphs, respectively - that are saved to file when calling `buildIndex()`. By default, this file is located in `tempdir()`11 1 On HPC file systems, you can change `TEMPDIR` to a location that is more amenable to concurrent access. and will be removed when the session finishes.

    AnnoyIndex_path(pre)

    ## [1] "/tmp/RtmppMkKua/file16de5a52146b73.idx"

If the index is to persist across sessions, the path of the index file can be directly specified in `buildIndex`. This can be used to construct an index object directly using the relevant constructors, e.g., `AnnoyIndex()`, `HnswIndex()`. However, it becomes the responsibility of the user to clean up any temporary indexing files after calculations are complete.

# 5 Session information

    sessionInfo()

    ## R version 4.3.2 Patched (2023-11-13 r85521)
    ## Platform: x86_64-pc-linux-gnu (64-bit)
    ## Running under: Ubuntu 22.04.3 LTS
    ##
    ## Matrix products: default
    ## BLAS:   /home/biocbuild/bbs-3.18-bioc/R/lib/libRblas.so
    ## LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.10.0
    ##
    ## locale:
    ##  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C
    ##  [3] LC_TIME=en_GB              LC_COLLATE=C
    ##  [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8
    ##  [7] LC_PAPER=en_US.UTF-8       LC_NAME=C
    ##  [9] LC_ADDRESS=C               LC_TELEPHONE=C
    ## [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
    ##
    ## time zone: America/New_York
    ## tzcode source: system (glibc)
    ##
    ## attached base packages:
    ## [1] stats     graphics  grDevices utils     datasets  methods   base
    ##
    ## other attached packages:
    ## [1] BiocNeighbors_1.20.1 knitr_1.45           BiocStyle_2.30.0
    ##
    ## loaded via a namespace (and not attached):
    ##  [1] cli_3.6.2           rlang_1.1.2         xfun_0.41
    ##  [4] jsonlite_1.8.8      S4Vectors_0.40.2    htmltools_0.5.7
    ##  [7] stats4_4.3.2        sass_0.4.8          rmarkdown_2.25
    ## [10] grid_4.3.2          evaluate_0.23       jquerylib_0.1.4
    ## [13] fastmap_1.1.1       yaml_2.3.8          lifecycle_1.0.4
    ## [16] bookdown_0.37       BiocManager_1.30.22 compiler_4.3.2
    ## [19] codetools_0.2-19    Rcpp_1.0.11         BiocParallel_1.36.0
    ## [22] lattice_0.22-5      digest_0.6.33       R6_2.5.1
    ## [25] parallel_4.3.2      bslib_0.6.1         Matrix_1.6-4
    ## [28] tools_4.3.2         BiocGenerics_0.48.1 cachem_1.0.8