# Detecting approximate nearest neighbors

Aaron Lun1

1Cancer Research UK Cambridge Institute, Cambridge, United Kingdom

#### Revised: 2 December 2018

#### Package

BiocNeighbors 1.6.0

# 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,] 2472 8785 4480 2287 7154 5087  306 9725  355  3394
    ## [2,] 5679 9999 7159 1622 8671 1447 3588 9629 1197   297
    ## [3,] 9862 8733 1684 7877 4205 1388 3570 1146 3504    10
    ## [4,] 8520 1375 7533 5937 9217 1923 6576 3922 6534  3078
    ## [5,] 7475 7014 2667 1272 2384 2656 6914 7399 5816  3233
    ## [6,] 1076 1939 6881 6542 7726 8623 8700  120 6332  2766

    head(fout$distance)

    ##           [,1]      [,2]      [,3]      [,4]      [,5]     [,6]      [,7]
    ## [1,] 1.0160896 1.0305198 1.0539002 1.0731078 1.0773553 1.079474 1.0834223
    ## [2,] 0.9352278 0.9359603 0.9668155 0.9747400 0.9870442 1.019455 1.0362035
    ## [3,] 0.8528845 0.9201739 0.9223238 0.9334182 0.9353036 0.959167 0.9967918
    ## [4,] 0.8827361 0.9544235 0.9728699 1.0183616 1.0454656 1.067823 1.0709544
    ## [5,] 0.8144282 0.8833866 0.9234329 0.9283065 0.9300396 1.002293 1.0083969
    ## [6,] 0.9313099 0.9518151 0.9607627 0.9678324 1.0065429 1.022031 1.0428864
    ##          [,8]     [,9]    [,10]
    ## [1,] 1.084278 1.114447 1.116229
    ## [2,] 1.055618 1.071051 1.079966
    ## [3,] 1.001344 1.013241 1.029634
    ## [4,] 1.079332 1.102296 1.107838
    ## [5,] 1.029544 1.035694 1.036367
    ## [6,] 1.058502 1.072493 1.084400

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,] 9284 9570 1140 6935 7038
    ## [2,] 9412 1858 5012 9139 2021
    ## [3,] 4960  657 9308 8547  798
    ## [4,] 9404 4924 1916 7117 5605
    ## [5,]  163  807 2996 2809 9403
    ## [6,] 5270 2339 7438 4378 9005

    head(qout$distance)

    ##           [,1]      [,2]      [,3]      [,4]     [,5]
    ## [1,] 0.9304292 0.9459063 1.0207646 1.0232471 1.025561
    ## [2,] 0.8957567 0.9605144 1.0573766 1.0687405 1.092705
    ## [3,] 0.9340398 0.9353478 0.9844260 1.0137274 1.057676
    ## [4,] 0.9156544 0.9845881 1.0024112 1.0515140 1.055231
    ## [5,] 0.9132901 0.9353450 0.9453691 1.0402807 1.040792
    ## [6,] 0.8896392 0.8957989 0.9428604 0.9937068 1.045079

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/RtmpmRlYuW/file5485745454c1.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 4.0.0 (2020-04-24)
    ## Platform: x86_64-pc-linux-gnu (64-bit)
    ## Running under: Ubuntu 18.04.4 LTS
    ##
    ## Matrix products: default
    ## BLAS:   /home/biocbuild/bbs-3.11-bioc/R/lib/libRblas.so
    ## LAPACK: /home/biocbuild/bbs-3.11-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_1.6.0 knitr_1.28          BiocStyle_2.16.0
    ##
    ## loaded via a namespace (and not attached):
    ##  [1] Rcpp_1.0.4.6        bookdown_0.18       lattice_0.20-41
    ##  [4] digest_0.6.25       grid_4.0.0          stats4_4.0.0
    ##  [7] magrittr_1.5        evaluate_0.14       rlang_0.4.5
    ## [10] stringi_1.4.6       S4Vectors_0.26.0    Matrix_1.2-18
    ## [13] rmarkdown_2.1       BiocParallel_1.22.0 tools_4.0.0
    ## [16] stringr_1.4.0       parallel_4.0.0      xfun_0.13
    ## [19] yaml_2.2.1          compiler_4.0.0      BiocGenerics_0.34.0
    ## [22] BiocManager_1.30.10 htmltools_0.4.0