# Detecting approximate nearest neighbors

Aaron Lun1

1Cancer Research UK Cambridge Institute, Cambridge, United Kingdom

#### Revised: 2 December 2018

#### Package

BiocNeighbors 1.8.2

# 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,] 2654 2795 2372 3507 1828 5941 7404 2050 3322  5761
    ## [2,] 4471  528 1194 2271 4080 7943  497 1395 3512  4952
    ## [3,] 9730 8952 4478 7756 9266 3040 8816 1609  245  7683
    ## [4,] 3598 6189 2173 4587 7112 1415 3829 5403 2849  7702
    ## [5,] 7061  965 2834 7312  242 9055 3232 5683 4227  3314
    ## [6,] 1421 2326 8207 8878 5156 8131  721  642 8684  3586

    head(fout$distance)

    ##           [,1]      [,2]      [,3]      [,4]      [,5]      [,6]      [,7]
    ## [1,] 0.8012792 0.8576069 0.9046848 0.9490177 0.9916587 1.0011322 1.0193053
    ## [2,] 0.7700940 0.8116404 0.9214732 0.9281875 0.9308216 0.9314354 0.9315361
    ## [3,] 0.9155341 0.9325950 1.0045272 1.0122676 1.0171826 1.0219508 1.0568852
    ## [4,] 0.9674190 1.0256823 1.0293052 1.0404011 1.0442752 1.0458566 1.0477467
    ## [5,] 0.7588240 0.9408486 0.9848236 0.9884602 0.9973440 0.9977134 1.0008332
    ## [6,] 1.1229731 1.1668361 1.1895242 1.1918395 1.2040933 1.2074112 1.2119087
    ##           [,8]      [,9]    [,10]
    ## [1,] 1.0411603 1.0498405 1.076182
    ## [2,] 0.9545985 0.9920003 1.013403
    ## [3,] 1.0721554 1.0794981 1.085272
    ## [4,] 1.0519229 1.0652701 1.093188
    ## [5,] 1.0118767 1.0177152 1.024742
    ## [6,] 1.2119669 1.2142785 1.224844

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,] 2854 9175 5590 9116 3424
    ## [2,] 1427 8457 9117 5034 4801
    ## [3,] 7847 3334 9282 8705 7149
    ## [4,] 7792 5069 7402 3808 7300
    ## [5,] 5718 7055  741 1945  468
    ## [6,] 7550 8877 8009 5407 5133

    head(qout$distance)

    ##           [,1]      [,2]      [,3]      [,4]      [,5]
    ## [1,] 0.7150112 0.9551283 0.9579080 0.9610922 1.0176965
    ## [2,] 0.8281745 0.9279171 0.9591281 0.9603502 1.0136175
    ## [3,] 0.9463708 0.9597221 0.9659172 0.9703273 0.9726027
    ## [4,] 0.7495880 0.8711673 0.9315405 0.9349653 0.9385618
    ## [5,] 0.9025577 1.0188659 1.0495381 1.0741827 1.0832633
    ## [6,] 0.7759522 1.0424271 1.0444030 1.0837094 1.0912106

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] "C:\\Users\\biocbuild\\bbs-3.12-bioc\\tmpdir\\RtmpKexI9l\\file24d85f35434c.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 4.0.3 (2020-10-10)
    ## Platform: x86_64-w64-mingw32/x64 (64-bit)
    ## Running under: Windows Server 2012 R2 x64 (build 9600)
    ##
    ## Matrix products: default
    ##
    ## locale:
    ## [1] LC_COLLATE=C
    ## [2] LC_CTYPE=English_United States.1252
    ## [3] LC_MONETARY=English_United States.1252
    ## [4] LC_NUMERIC=C
    ## [5] LC_TIME=English_United States.1252
    ##
    ## attached base packages:
    ## [1] stats     graphics  grDevices utils     datasets  methods   base
    ##
    ## other attached packages:
    ## [1] BiocNeighbors_1.8.2 knitr_1.30          BiocStyle_2.18.1
    ##
    ## loaded via a namespace (and not attached):
    ##  [1] Rcpp_1.0.5          bookdown_0.21       lattice_0.20-41
    ##  [4] digest_0.6.27       grid_4.0.3          stats4_4.0.3
    ##  [7] magrittr_2.0.1      evaluate_0.14       rlang_0.4.9
    ## [10] stringi_1.5.3       S4Vectors_0.28.0    Matrix_1.2-18
    ## [13] rmarkdown_2.5       BiocParallel_1.24.1 tools_4.0.3
    ## [16] stringr_1.4.0       parallel_4.0.3      xfun_0.19
    ## [19] yaml_2.2.1          compiler_4.0.3      BiocGenerics_0.36.0
    ## [22] BiocManager_1.30.10 htmltools_0.5.0