# 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,] 2058 8583 2729 4354 5508 6397 7357 3262 4010  8446
    ## [2,] 4318 2397 9828 5949 9518 5283 8595 2716 6275  4289
    ## [3,] 7427  112 7725  188 6884 5177 6281 7469 3075  4396
    ## [4,] 9918 4779 4346 4730 2564 6460 8494 5580  110  7868
    ## [5,] 8850 6098 6622 9433 8338 7286 3181 5043 6061  2445
    ## [6,] 2896  824 1144 7969 2335  552 4366  156  177  7693

    head(fout$distance)

    ##           [,1]      [,2]      [,3]      [,4]      [,5]      [,6]      [,7]
    ## [1,] 0.8414909 0.8698932 0.9355368 0.9541349 0.9717639 0.9991492 1.0364299
    ## [2,] 0.8837594 0.9832281 1.0007690 1.0060079 1.0102490 1.0112789 1.0447117
    ## [3,] 0.9199062 0.9470513 1.0197800 1.0204711 1.0504456 1.0727901 1.0770069
    ## [4,] 0.7379470 0.8063428 0.8113312 0.8561726 0.8616189 0.8751896 0.8967044
    ## [5,] 0.9775219 1.0697929 1.0967655 1.1031532 1.1066418 1.1163386 1.1320385
    ## [6,] 0.9128596 0.9900898 0.9991747 1.0269202 1.0584425 1.0603285 1.0850970
    ##           [,8]      [,9]     [,10]
    ## [1,] 1.0367655 1.0437986 1.0546287
    ## [2,] 1.0605900 1.0618711 1.0711055
    ## [3,] 1.1224986 1.1257536 1.1304235
    ## [4,] 0.9294451 0.9345718 0.9384831
    ## [5,] 1.1427239 1.1543937 1.1700542
    ## [6,] 1.0867983 1.1067376 1.1250880

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,] 9215 4436 6180 9154 2566
    ## [2,] 1552 1741 1767 3431 1836
    ## [3,] 3886 1074 5290 3572 7506
    ## [4,]  188 6120   10 8273 6874
    ## [5,] 4147 7623 2084 5674 8640
    ## [6,] 1523 3040  250 9290 6846

    head(qout$distance)

    ##           [,1]      [,2]      [,3]      [,4]      [,5]
    ## [1,] 0.9506123 1.0106703 1.0128031 1.0335578 1.0427752
    ## [2,] 0.8683564 0.8903276 0.9439428 1.0131069 1.0594602
    ## [3,] 0.9215837 0.9470725 1.0127544 1.0229198 1.0231688
    ## [4,] 0.8339254 0.8486065 0.9568959 0.9866970 1.0023173
    ## [5,] 0.8708979 0.9105967 0.9178559 0.9603201 0.9961723
    ## [6,] 0.9792755 0.9906029 1.0066026 1.0443240 1.0502740

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()`1 1 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] "D:\\biocbuild\\bbs-3.13-bioc\\tmpdir\\RtmpawNBJG\\file3a4865a17989.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 RC (2021-05-10 r80283)
    ## Platform: x86_64-w64-mingw32/x64 (64-bit)
    ## Running under: Windows Server x64 (build 17763)
    ##
    ## Matrix products: default
    ##
    ## locale:
    ## [1] LC_COLLATE=C
    ## [2] LC_CTYPE=English_United States.1252
    ## [3] LC_MONETARY=English_United States.1252
    ## [4] LC_NUMERIC=C
    ## [5] LC_TIME=English_United States.1252
    ##
    ## 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