# Detecting approximate nearest neighbors

Aaron Lun1

1Cancer Research UK Cambridge Institute, Cambridge, United Kingdom

#### Revised: 2 December 2018

#### Package

BiocNeighbors 1.16.0

# 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,] 2052 5309 6282 2557 2168 8263 9205 7528 8789  3740
    ## [2,] 5736 9615 3647 1358 9782 1680 4137 5847 9136  6982
    ## [3,] 6651 5430 4595 8265 7810 8425 6423  330 2549   570
    ## [4,] 3696 3778 3165 1608 1901 3711 8224  577 8295  1629
    ## [5,] 4287  548 5115 3675 7887 8367 6388 4526 2628  6315
    ## [6,] 7735 7322 8125 6150 3509 5365 4657 3303 9131  6261

    head(fout$distance)

    ##           [,1]      [,2]     [,3]      [,4]      [,5]      [,6]      [,7]
    ## [1,] 0.8627000 0.9854210 1.001156 1.0017806 1.0360787 1.0406851 1.0484282
    ## [2,] 1.0128206 1.0780077 1.078687 1.1209522 1.1439979 1.1514652 1.1527506
    ## [3,] 0.7901211 0.8264780 0.863410 0.8977429 0.9352114 0.9389573 0.9490262
    ## [4,] 0.8244431 0.9287865 1.008486 1.0160102 1.0175763 1.0424658 1.0495710
    ## [5,] 0.9664762 0.9750541 0.992348 1.0133016 1.0293305 1.0478083 1.0694299
    ## [6,] 0.9233503 1.0199189 1.053357 1.0571043 1.0590004 1.0592493 1.0667665
    ##           [,8]      [,9]     [,10]
    ## [1,] 1.0540121 1.0542150 1.0659641
    ## [2,] 1.1559690 1.1592537 1.1608156
    ## [3,] 0.9617434 0.9673256 0.9715796
    ## [4,] 1.1036350 1.1150218 1.1169617
    ## [5,] 1.0697476 1.0756302 1.0765107
    ## [6,] 1.0774070 1.0920420 1.0928895

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,] 5275 9661 1609 9234 4824
    ## [2,]  643 4035 6056 2402 7981
    ## [3,] 2980 8271 3716 3533 1564
    ## [4,]  600 1649 3742 4262 4470
    ## [5,] 2453 3329 8097  736 1528
    ## [6,] 1261 1086 9393 2514 1234

    head(qout$distance)

    ##           [,1]      [,2]      [,3]     [,4]     [,5]
    ## [1,] 0.8067096 0.9223074 1.0235654 1.025655 1.069929
    ## [2,] 0.9043778 1.0257226 1.0618618 1.087329 1.103766
    ## [3,] 0.9051384 1.0243276 1.0300729 1.038151 1.059343
    ## [4,] 0.8496410 0.9191397 0.9824249 1.003406 1.033886
    ## [5,] 0.9433628 0.9715790 1.0286065 1.082799 1.109918
    ## [6,] 0.8933529 0.9920318 1.0104198 1.017549 1.040444

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] "/tmp/RtmpgdJPve/file842a243d0995.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 4.2.1 (2022-06-23)
    ## Platform: x86_64-pc-linux-gnu (64-bit)
    ## Running under: Ubuntu 20.04.5 LTS
    ##
    ## Matrix products: default
    ## BLAS:   /home/biocbuild/bbs-3.16-bioc/R/lib/libRblas.so
    ## LAPACK: /home/biocbuild/bbs-3.16-bioc/R/lib/libRlapack.so
    ##
    ## 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
    ##
    ## attached base packages:
    ## [1] stats     graphics  grDevices utils     datasets  methods   base
    ##
    ## other attached packages:
    ## [1] BiocNeighbors_1.16.0 knitr_1.40           BiocStyle_2.26.0
    ##
    ## loaded via a namespace (and not attached):
    ##  [1] Rcpp_1.0.9          magrittr_2.0.3      BiocGenerics_0.44.0
    ##  [4] BiocParallel_1.32.0 lattice_0.20-45     R6_2.5.1
    ##  [7] rlang_1.0.6         fastmap_1.1.0       stringr_1.4.1
    ## [10] tools_4.2.1         parallel_4.2.1      grid_4.2.1
    ## [13] xfun_0.34           cli_3.4.1           jquerylib_0.1.4
    ## [16] htmltools_0.5.3     yaml_2.3.6          digest_0.6.30
    ## [19] bookdown_0.29       Matrix_1.5-1        BiocManager_1.30.19
    ## [22] S4Vectors_0.36.0    sass_0.4.2          codetools_0.2-18
    ## [25] cachem_1.0.6        evaluate_0.17       rmarkdown_2.17
    ## [28] stringi_1.7.8       compiler_4.2.1      bslib_0.4.0
    ## [31] stats4_4.2.1        jsonlite_1.8.3