# Detecting approximate nearest neighbors

Aaron Lun1

1Cancer Research UK Cambridge Institute, Cambridge, United Kingdom

#### Revised: 2 December 2018

#### Package

BiocNeighbors 1.12.0

# 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,] 9439 4415 4050 1824 7530 1765 1809 9042 4664  2563
    ## [2,] 2790 3423 3782 8726  625 2779 7927 7627 2499  9672
    ## [3,] 2457 7743  221 1509 7170 4612  357 1104 7804  4335
    ## [4,] 4969 3827 7582  434 4839 8877 8875 5642 8399  6484
    ## [5,] 3024 1233 9286 4642 7428 2218 2799 4303 3709  4587
    ## [6,]  970 8731 5495 8513 3823 1449  582 1072 4587  5913

    head(fout$distance)

    ##           [,1]      [,2]      [,3]      [,4]     [,5]     [,6]     [,7]
    ## [1,] 0.9488623 0.9526768 0.9918012 1.0155424 1.018635 1.038617 1.060869
    ## [2,] 0.9491687 0.9837804 0.9908242 1.0480871 1.055283 1.068011 1.078402
    ## [3,] 0.9902376 1.0116328 1.0365927 1.0736140 1.083189 1.086472 1.097179
    ## [4,] 0.9866520 1.0091648 1.0289248 1.0347098 1.054599 1.058922 1.071430
    ## [5,] 0.9620222 0.9739777 0.9899967 0.9961459 1.009494 1.019439 1.021883
    ## [6,] 0.9976924 1.0015095 1.0361325 1.0463392 1.058199 1.059819 1.070786
    ##          [,8]     [,9]    [,10]
    ## [1,] 1.061591 1.061877 1.079348
    ## [2,] 1.106313 1.106321 1.107904
    ## [3,] 1.113934 1.127635 1.131233
    ## [4,] 1.075089 1.079372 1.095900
    ## [5,] 1.036463 1.037183 1.043278
    ## [6,] 1.081499 1.115437 1.120747

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,] 2637  872 1732 1461 4400
    ## [2,] 1826  156 3000 8512 2001
    ## [3,]  960  524 3618 2276 5273
    ## [4,]   84 2483 4965 1349 5243
    ## [5,] 8038 8264 4906 2087 1823
    ## [6,] 9085 1358 3541 8958 6405

    head(qout$distance)

    ##           [,1]      [,2]      [,3]      [,4]      [,5]
    ## [1,] 0.9116162 0.9471505 0.9820164 0.9885749 1.0134385
    ## [2,] 0.9471223 1.0264651 1.0276965 1.0382000 1.0838469
    ## [3,] 0.9862803 1.0282627 1.0421263 1.0499210 1.0595300
    ## [4,] 0.7931599 0.8482375 0.8868514 0.9529580 0.9833588
    ## [5,] 0.9192116 0.9710984 0.9776919 0.9884884 1.0078245
    ## [6,] 0.9002737 1.0350211 1.0597970 1.0611564 1.0645769

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/RtmpBdikDR/file2cc07639387048.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 4.1.1 (2021-08-10)
    ## Platform: x86_64-pc-linux-gnu (64-bit)
    ## Running under: Ubuntu 20.04.3 LTS
    ##
    ## Matrix products: default
    ## BLAS:   /home/biocbuild/bbs-3.14-bioc/R/lib/libRblas.so
    ## LAPACK: /home/biocbuild/bbs-3.14-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_1.12.0 knitr_1.36           BiocStyle_2.22.0
    ##
    ## loaded via a namespace (and not attached):
    ##  [1] Rcpp_1.0.7          magrittr_2.0.1      BiocGenerics_0.40.0
    ##  [4] BiocParallel_1.28.0 lattice_0.20-45     R6_2.5.1
    ##  [7] rlang_0.4.12        fastmap_1.1.0       stringr_1.4.0
    ## [10] tools_4.1.1         parallel_4.1.1      grid_4.1.1
    ## [13] xfun_0.27           jquerylib_0.1.4     htmltools_0.5.2
    ## [16] yaml_2.2.1          digest_0.6.28       bookdown_0.24
    ## [19] Matrix_1.3-4        BiocManager_1.30.16 S4Vectors_0.32.0
    ## [22] sass_0.4.0          evaluate_0.14       rmarkdown_2.11
    ## [25] stringi_1.7.5       compiler_4.1.1      bslib_0.3.1
    ## [28] stats4_4.1.1        jsonlite_1.7.2