# Detecting approximate nearest neighbors

Aaron Lun1

1Cancer Research UK Cambridge Institute, Cambridge, United Kingdom

#### Revised: 2 December 2018

#### Package

BiocNeighbors 1.8.0

# 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,] 8584 3942 1711 2359 4200 6331  561 4329 5708  3803
    ## [2,] 7924 5792 8419 3740  920 5103 5852 7777 7011  5669
    ## [3,] 9756 2517 4308 6564  822 8126 4162  497 8632  2656
    ## [4,] 2063 1679 5260 9437 3510 3585 7202 9278  716  5452
    ## [5,] 6947 5654 9142 4882 8038 2318 7935 8963  396  8464
    ## [6,] 3856 1723 2288 6001 7079 9658 8187 1679 4700  9871

    head(fout$distance)

    ##           [,1]      [,2]      [,3]      [,4]      [,5]      [,6]      [,7]
    ## [1,] 0.9785542 0.9937828 0.9974682 1.0311086 1.0537837 1.0600865 1.0702299
    ## [2,] 0.9445717 0.9981147 1.0318105 1.0446899 1.0486889 1.0545481 1.0545697
    ## [3,] 0.8897178 0.9314029 0.9426289 1.0321267 1.0487431 1.0595485 1.0612212
    ## [4,] 0.8848239 0.9776480 0.9812904 0.9933047 1.0061167 1.0202012 1.0442240
    ## [5,] 0.8590254 1.0091568 1.0164609 1.0269836 1.0304595 1.0316750 1.0341054
    ## [6,] 0.8624543 0.8903279 0.9003656 0.9090142 0.9092635 0.9480212 0.9584342
    ##           [,8]      [,9]     [,10]
    ## [1,] 1.0708619 1.0742493 1.0764540
    ## [2,] 1.0652611 1.0881760 1.0938987
    ## [3,] 1.0615385 1.0625939 1.0665069
    ## [4,] 1.0445368 1.0677571 1.0711876
    ## [5,] 1.0761930 1.0767903 1.0863526
    ## [6,] 0.9694126 0.9704188 0.9780809

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,] 9618  115 9893 3345  540
    ## [2,] 6628 9246 8889 9068 9166
    ## [3,] 4498  791 2105  113 2994
    ## [4,] 5805 5270 7929  179 9037
    ## [5,] 4913 4837 5209 4451 4154
    ## [6,] 3726 6298 8561 5285 1615

    head(qout$distance)

    ##           [,1]      [,2]      [,3]      [,4]      [,5]
    ## [1,] 0.9300495 0.9845788 1.0282300 1.0416820 1.0553656
    ## [2,] 0.9036013 0.9143441 0.9365810 0.9747946 0.9785217
    ## [3,] 0.8599398 0.9458985 0.9963076 1.0066876 1.0082071
    ## [4,] 0.9643821 0.9886798 1.0625366 1.0658094 1.0951591
    ## [5,] 0.8982796 0.9959829 1.0505439 1.0590628 1.0673671
    ## [6,] 0.9704359 0.9718943 1.0704929 1.0744665 1.1003007

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/RtmpbwZrPR/file5410fe6e783.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-pc-linux-gnu (64-bit)
    ## Running under: Ubuntu 18.04.5 LTS
    ##
    ## Matrix products: default
    ## BLAS:   /home/biocbuild/bbs-3.12-bioc/R/lib/libRblas.so
    ## LAPACK: /home/biocbuild/bbs-3.12-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_1.8.0 knitr_1.30          BiocStyle_2.18.0
    ##
    ## 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_1.5        evaluate_0.14       rlang_0.4.8
    ## [10] stringi_1.5.3       S4Vectors_0.28.0    Matrix_1.2-18
    ## [13] rmarkdown_2.5       BiocParallel_1.24.0 tools_4.0.3
    ## [16] stringr_1.4.0       parallel_4.0.3      xfun_0.18
    ## [19] yaml_2.2.1          compiler_4.0.3      BiocGenerics_0.36.0
    ## [22] BiocManager_1.30.10 htmltools_0.5.0