# Detecting approximate nearest neighbors

Aaron Lun1

1Cancer Research UK Cambridge Institute, Cambridge, United Kingdom

#### Revised: 2 December 2018

#### Package

BiocNeighbors 1.4.0

# 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,] 3799  707 9765 8776 7164 1241 5042 9003 3079  9487
    ## [2,] 3639 1775 6540 8377 4170 4105 4239 3780  623  9149
    ## [3,] 3193 5702 6131 3556 2278 1467 5905 7972  751  6794
    ## [4,] 8563 4240 4950 3937 3915 1173  637 7253 5242  8764
    ## [5,] 6958 9775 2348 1723 6309 6189 8332 2306 7131  9346
    ## [6,] 3535 7392 9955  341 1448 6327 1496 1221 5703     7

    head(fout$distance)

    ##           [,1]      [,2]      [,3]      [,4]      [,5]      [,6]     [,7]
    ## [1,] 0.8796675 0.9298825 0.9407840 0.9484969 0.9696676 0.9733458 1.009521
    ## [2,] 0.9334475 0.9498138 0.9767259 1.1378182 1.1410992 1.1478661 1.162463
    ## [3,] 0.8371768 0.9287036 0.9650908 1.0288686 1.0400424 1.0506155 1.061247
    ## [4,] 0.8504387 0.9287976 0.9349621 0.9534878 0.9718963 0.9721311 0.981039
    ## [5,] 0.9797012 1.0404111 1.0508379 1.0708305 1.0896506 1.1226237 1.126803
    ## [6,] 0.9662001 1.0002621 1.0080508 1.0088716 1.0454358 1.0702065 1.075105
    ##           [,8]     [,9]    [,10]
    ## [1,] 1.0214065 1.029604 1.033413
    ## [2,] 1.1681520 1.170679 1.173934
    ## [3,] 1.0814985 1.088075 1.089289
    ## [4,] 0.9860948 0.996186 1.016510
    ## [5,] 1.1364032 1.139101 1.141199
    ## [6,] 1.0786674 1.092711 1.096879

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,] 2248 8687 7970 6662 6771
    ## [2,] 1764 2156 9678 7172 9997
    ## [3,] 6334 7578 5116 6939 2555
    ## [4,] 2546 6219 2553  295 6261
    ## [5,] 1235  779 2902 7586 9467
    ## [6,] 8396 3550  200 7478 4880

    head(qout$distance)

    ##           [,1]      [,2]      [,3]      [,4]      [,5]
    ## [1,] 0.9522341 0.9728507 0.9741805 0.9752116 0.9843085
    ## [2,] 0.9348707 0.9561337 1.0239024 1.0491139 1.0654186
    ## [3,] 0.9712541 0.9960141 1.0080669 1.0858932 1.1007937
    ## [4,] 0.9419668 1.0071369 1.0388074 1.0424126 1.0685666
    ## [5,] 0.8917747 0.9413266 1.0215584 1.0394553 1.0646684
    ## [6,] 0.8049647 0.8999624 0.9166077 0.9314021 0.9382524

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 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/Rtmp0lflDQ/file7ca41f3074fe.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 3.6.1 (2019-07-05)
    ## Platform: x86_64-pc-linux-gnu (64-bit)
    ## Running under: Ubuntu 18.04.3 LTS
    ##
    ## Matrix products: default
    ## BLAS:   /home/biocbuild/bbs-3.10-bioc/R/lib/libRblas.so
    ## LAPACK: /home/biocbuild/bbs-3.10-bioc/R/lib/libRlapack.so
    ##
    ## locale:
    ##  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C
    ##  [3] LC_TIME=en_US.UTF-8        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_1.4.0 knitr_1.25          BiocStyle_2.14.0
    ##
    ## loaded via a namespace (and not attached):
    ##  [1] Rcpp_1.0.2          bookdown_0.14       lattice_0.20-38
    ##  [4] digest_0.6.22       grid_3.6.1          stats4_3.6.1
    ##  [7] magrittr_1.5        evaluate_0.14       rlang_0.4.1
    ## [10] stringi_1.4.3       S4Vectors_0.24.0    Matrix_1.2-17
    ## [13] rmarkdown_1.16      BiocParallel_1.20.0 tools_3.6.1
    ## [16] stringr_1.4.0       parallel_3.6.1      xfun_0.10
    ## [19] yaml_2.2.0          compiler_3.6.1      BiocGenerics_0.32.0
    ## [22] BiocManager_1.30.9  htmltools_0.4.0