# Detecting approximate nearest neighbors

Aaron Lun1

1Cancer Research UK Cambridge Institute, Cambridge, United Kingdom

#### Revised: 2 December 2018

#### Package

BiocNeighbors 1.10.0

# 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,] 6839 3564 8970 8494 2514 3375 8041 4673 2809  6557
    ## [2,] 3762 8510 1930 1709 7576 7814 9139 3511 1450  2967
    ## [3,] 3446 3654 4728 1638 2751 2345 3632 1000 7965  4392
    ## [4,] 8817 6584 8180   59 3050 2645 1465 7738 9324  6834
    ## [5,] 9889 1561 3571    9 4544 2686 9911 6703 2909  9965
    ## [6,] 2855 5428 5531 5760  771 7536 3368 6385  670  2652

    head(fout$distance)

    ##           [,1]      [,2]      [,3]      [,4]      [,5]      [,6]      [,7]
    ## [1,] 0.9921864 0.9957834 0.9994216 1.0113053 1.0138078 1.0223548 1.0240149
    ## [2,] 0.8645985 0.9447311 0.9579430 0.9840193 0.9866839 0.9876871 0.9897641
    ## [3,] 0.9259275 0.9518557 0.9632121 0.9735848 0.9742126 0.9901797 0.9999163
    ## [4,] 0.8075849 0.8929428 0.9125759 0.9290096 0.9432879 0.9578164 0.9866946
    ## [5,] 0.9659729 0.9993480 1.0383309 1.1012558 1.1051797 1.1090480 1.1115953
    ## [6,] 0.7669217 0.9034132 0.9614732 0.9849571 0.9958620 0.9966433 1.0280401
    ##           [,8]      [,9]    [,10]
    ## [1,] 1.0260382 1.0628729 1.072670
    ## [2,] 0.9969519 0.9970498 1.003801
    ## [3,] 1.0107327 1.0114325 1.023189
    ## [4,] 0.9878797 1.0132945 1.048254
    ## [5,] 1.1497359 1.1527627 1.160382
    ## [6,] 1.0282890 1.0388981 1.039620

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,] 4718 2150 4276 4908 1207
    ## [2,] 4228 2541 8287 1392 6196
    ## [3,] 3533 7801 6299 1937 7967
    ## [4,] 9597 9572 5819 6116 5156
    ## [5,] 8190  367 8620  223 5560
    ## [6,] 4112 6999 9241 1377 9643

    head(qout$distance)

    ##           [,1]      [,2]      [,3]      [,4]      [,5]
    ## [1,] 0.8218788 0.8370792 0.8468705 0.9359561 0.9615508
    ## [2,] 0.8191825 0.8814184 0.9263154 0.9368848 0.9414537
    ## [3,] 0.7210571 0.8295944 0.8740436 0.9274628 0.9452164
    ## [4,] 0.8753332 0.8821529 0.8910749 0.9699543 1.0150330
    ## [5,] 0.8536909 0.8577128 0.8752092 0.8840870 0.8868778
    ## [6,] 1.0393667 1.0482323 1.1029489 1.1369380 1.1387308

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/RtmpfiYBOk/file2a908cf0048a2.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 4.1.0 (2021-05-18)
    ## Platform: x86_64-pc-linux-gnu (64-bit)
    ## Running under: Ubuntu 20.04.2 LTS
    ##
    ## Matrix products: default
    ## BLAS:   /home/biocbuild/bbs-3.13-bioc/R/lib/libRblas.so
    ## LAPACK: /home/biocbuild/bbs-3.13-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_1.10.0 knitr_1.33           BiocStyle_2.20.0
    ##
    ## loaded via a namespace (and not attached):
    ##  [1] Rcpp_1.0.6          magrittr_2.0.1      BiocGenerics_0.38.0
    ##  [4] BiocParallel_1.26.0 lattice_0.20-44     R6_2.5.0
    ##  [7] rlang_0.4.11        stringr_1.4.0       tools_4.1.0
    ## [10] parallel_4.1.0      grid_4.1.0          xfun_0.23
    ## [13] jquerylib_0.1.4     htmltools_0.5.1.1   yaml_2.2.1
    ## [16] digest_0.6.27       bookdown_0.22       Matrix_1.3-3
    ## [19] BiocManager_1.30.15 S4Vectors_0.30.0    sass_0.4.0
    ## [22] evaluate_0.14       rmarkdown_2.8       stringi_1.6.2
    ## [25] compiler_4.1.0      bslib_0.2.5.1       stats4_4.1.0
    ## [28] jsonlite_1.7.2