# Detecting approximate nearest neighbors

Aaron Lun1

1Cancer Research UK Cambridge Institute, Cambridge, United Kingdom

#### Revised: 2 December 2018

#### Package

BiocNeighbors 1.18.0

# 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,] 9662 3037  299 4397  726 5922 1325 7163  672  1337
    ## [2,] 1488 9290 7952 3676 4691 9474 3924 7058 4831  3837
    ## [3,]  844 4022 3508 6331 1039 7570 9114 7037 4939  3209
    ## [4,] 4885 5480 2471 3059 4728 9445 7564 9800 1160  4288
    ## [5,] 2498  123 7127 3073 4679 2648 1293 8873 4212  6237
    ## [6,] 4128 3969 1070 9264 5318 7792 2124  465 9227   262

    head(fout$distance)

    ##           [,1]      [,2]      [,3]      [,4]      [,5]      [,6]      [,7]
    ## [1,] 1.0018635 1.0734630 1.1102037 1.1157204 1.1295308 1.1385425 1.1708193
    ## [2,] 0.8835352 0.9777583 1.0060213 1.0081369 1.0122799 1.0259368 1.0492104
    ## [3,] 0.8123723 0.8647476 0.9139982 0.9534140 0.9876873 0.9924435 0.9940631
    ## [4,] 0.9842315 0.9888017 1.0033739 1.0249933 1.0479329 1.0561618 1.0610654
    ## [5,] 0.8673671 0.9623174 0.9827742 0.9977949 1.0064403 1.0284088 1.0384618
    ## [6,] 0.7851849 0.8384334 0.9159828 0.9715101 0.9834490 1.0078323 1.0196633
    ##           [,8]      [,9]    [,10]
    ## [1,] 1.1721100 1.1844047 1.185606
    ## [2,] 1.0585682 1.0604984 1.065688
    ## [3,] 0.9975992 0.9996915 1.002198
    ## [4,] 1.0628065 1.0649843 1.067426
    ## [5,] 1.0565158 1.0652058 1.067776
    ## [6,] 1.0235919 1.0277270 1.033206

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,] 3650 8442   28 3378 3349
    ## [2,]  328 7740 6927 2418 3180
    ## [3,] 4141 5745 2902 1041 5609
    ## [4,]  139  427 2301  422 4046
    ## [5,] 1557 7397 6540 5908 3096
    ## [6,] 8041 6676  871 7056 2647

    head(qout$distance)

    ##           [,1]      [,2]      [,3]      [,4]      [,5]
    ## [1,] 0.9686017 1.2081473 1.2145671 1.2264786 1.2347609
    ## [2,] 0.9217723 0.9357488 0.9521836 0.9589323 0.9655992
    ## [3,] 0.9635603 0.9915902 1.0060973 1.0188990 1.0341231
    ## [4,] 0.7947047 0.8804924 0.9119638 0.9358485 0.9468299
    ## [5,] 0.9376368 0.9625379 0.9931554 1.0463696 1.0515987
    ## [6,] 0.9765031 1.0734611 1.0976894 1.0983497 1.1001519

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/RtmpvaDkk3/file30a4cf51edb153.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 4.3.0 RC (2023-04-13 r84269)
    ## Platform: x86_64-pc-linux-gnu (64-bit)
    ## Running under: Ubuntu 22.04.2 LTS
    ##
    ## Matrix products: default
    ## BLAS:   /home/biocbuild/bbs-3.17-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_1.18.0 knitr_1.42           BiocStyle_2.28.0
    ##
    ## loaded via a namespace (and not attached):
    ##  [1] cli_3.6.1           rlang_1.1.0         xfun_0.39
    ##  [4] jsonlite_1.8.4      S4Vectors_0.38.0    htmltools_0.5.5
    ##  [7] stats4_4.3.0        sass_0.4.5          rmarkdown_2.21
    ## [10] grid_4.3.0          evaluate_0.20       jquerylib_0.1.4
    ## [13] fastmap_1.1.1       yaml_2.3.7          bookdown_0.33
    ## [16] BiocManager_1.30.20 compiler_4.3.0      codetools_0.2-19
    ## [19] Rcpp_1.0.10         BiocParallel_1.34.0 lattice_0.21-8
    ## [22] digest_0.6.31       R6_2.5.1            parallel_4.3.0
    ## [25] bslib_0.4.2         Matrix_1.5-4        tools_4.3.0
    ## [28] BiocGenerics_0.46.0 cachem_1.0.7