# Detecting approximate nearest neighbors

Aaron Lun1

1Cancer Research UK Cambridge Institute, Cambridge, United Kingdom

#### Revised: 2 December 2018

#### Package

BiocNeighbors 1.22.0

# 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,] 2848 4521 3928 6364 7001 4703 1666 2480 1078  3972
    ## [2,] 6407 8051 1606 8266 9804 3901  979 9696 9847  8034
    ## [3,] 3268 6710 9705 6519 8288 3434 7658 3646 3511   750
    ## [4,]  644 5238 3916 8231 2047 2580 5973 7882 8380  8753
    ## [5,] 5831 8211 5357 1487 4543 5926 8326 3097 9270   321
    ## [6,] 2642 5718 8416 9595  353  917 2331 6665 9137  7356

    head(fout$distance)

    ##           [,1]      [,2]      [,3]      [,4]      [,5]      [,6]      [,7]
    ## [1,] 0.8405916 0.8798988 0.8870191 0.9181291 0.9606939 0.9607217 0.9621375
    ## [2,] 0.9910530 1.0142183 1.0160263 1.0423205 1.0513763 1.0766540 1.0766710
    ## [3,] 0.9836815 0.9932737 0.9981568 1.0231274 1.0285779 1.0355318 1.0406004
    ## [4,] 0.9610632 0.9665034 0.9712912 0.9984006 1.0192084 1.0245547 1.0364628
    ## [5,] 0.8947265 0.9010723 1.0180677 1.0241910 1.0603987 1.0669070 1.0714582
    ## [6,] 0.9048750 0.9258313 0.9498504 0.9655181 1.0334547 1.0394921 1.0454750
    ##           [,8]      [,9]     [,10]
    ## [1,] 0.9811712 0.9916555 0.9966623
    ## [2,] 1.0969291 1.0970849 1.1040561
    ## [3,] 1.0470090 1.0584370 1.0684750
    ## [4,] 1.0380205 1.0724257 1.0753788
    ## [5,] 1.0744734 1.0889053 1.1060251
    ## [6,] 1.0577204 1.0620967 1.1146837

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,] 9802  478 2593 1906 2152
    ## [2,] 9295  429 6912 5393 6771
    ## [3,] 5647  692  806 6929 7735
    ## [4,] 3412 2906 5580 6854 8117
    ## [5,] 7189 4875 7230 9791 5891
    ## [6,] 7096 3292 9337  862 2863

    head(qout$distance)

    ##           [,1]      [,2]      [,3]      [,4]      [,5]
    ## [1,] 1.0188084 1.0249070 1.0424213 1.0674503 1.0863097
    ## [2,] 0.9499870 1.0097837 1.0343519 1.0417926 1.0602092
    ## [3,] 0.8892059 0.9806697 1.0053345 1.0672708 1.0726628
    ## [4,] 0.8908395 0.9013096 0.9373750 0.9587816 0.9697466
    ## [5,] 0.8676614 0.9245446 0.9404544 1.0141488 1.0142174
    ## [6,] 0.9886166 1.0187125 1.0624171 1.0642477 1.0776553

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()`1 1 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] "F:\\biocbuild\\bbs-3.19-bioc\\tmpdir\\Rtmp0s84m9\\file31a05b7c20b2.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 4.4.0 beta (2024-04-15 r86425 ucrt)
    ## Platform: x86_64-w64-mingw32/x64
    ## Running under: Windows Server 2022 x64 (build 20348)
    ##
    ## Matrix products: default
    ##
    ##
    ## locale:
    ## [1] LC_COLLATE=C
    ## [2] LC_CTYPE=English_United States.utf8
    ## [3] LC_MONETARY=English_United States.utf8
    ## [4] LC_NUMERIC=C
    ## [5] LC_TIME=English_United States.utf8
    ##
    ## time zone: America/New_York
    ## tzcode source: internal
    ##
    ## attached base packages:
    ## [1] stats     graphics  grDevices utils     datasets  methods   base
    ##
    ## other attached packages:
    ## [1] BiocNeighbors_1.22.0 knitr_1.46           BiocStyle_2.32.0
    ##
    ## loaded via a namespace (and not attached):
    ##  [1] cli_3.6.2           rlang_1.1.3         xfun_0.43
    ##  [4] jsonlite_1.8.8      S4Vectors_0.42.0    htmltools_0.5.8.1
    ##  [7] stats4_4.4.0        sass_0.4.9          rmarkdown_2.26
    ## [10] grid_4.4.0          evaluate_0.23       jquerylib_0.1.4
    ## [13] fastmap_1.1.1       yaml_2.3.8          lifecycle_1.0.4
    ## [16] bookdown_0.39       BiocManager_1.30.22 compiler_4.4.0
    ## [19] codetools_0.2-20    Rcpp_1.0.12         BiocParallel_1.38.0
    ## [22] lattice_0.22-6      digest_0.6.35       R6_2.5.1
    ## [25] parallel_4.4.0      bslib_0.7.0         Matrix_1.7-0
    ## [28] tools_4.4.0         BiocGenerics_0.50.0 cachem_1.0.8