# 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,] 9419 5383 5801 9819 6156 5697 1356 8024 7357  7540
    ## [2,] 4779 4255 4130 8292 1443 8055 1486 9925 8293  2982
    ## [3,] 5442 9170 6624 7997 6284 9768 6725 6781 9442  9028
    ## [4,] 4302 5048 9194 1659 7720 7617 4212 8528  814  6324
    ## [5,] 4530 5494 7118 1715 6176 8868 3942 4484 2468  2079
    ## [6,] 9039 3176 7014 9810  577 8868 2781 3591 3334  3193

    head(fout$distance)

    ##           [,1]      [,2]      [,3]      [,4]      [,5]      [,6]      [,7]
    ## [1,] 0.8132501 0.8488185 0.8631178 0.8958908 0.9000078 0.9576015 0.9660131
    ## [2,] 0.8062201 0.8723599 0.8773741 0.9117950 0.9172041 0.9524634 0.9702494
    ## [3,] 0.9617059 0.9792035 1.0102417 1.0139475 1.0299968 1.0361048 1.0634689
    ## [4,] 0.8564699 0.8584789 0.8761120 0.9338816 0.9732273 0.9996098 1.0361505
    ## [5,] 0.6855983 0.9793282 0.9934275 0.9937800 1.0161976 1.0279846 1.0353119
    ## [6,] 0.9363781 0.9726624 0.9879481 0.9955242 1.0042561 1.0085794 1.0132691
    ##           [,8]     [,9]    [,10]
    ## [1,] 0.9889613 1.009330 1.011024
    ## [2,] 0.9849720 1.010131 1.023426
    ## [3,] 1.0769291 1.076997 1.077183
    ## [4,] 1.0406028 1.056442 1.057538
    ## [5,] 1.0397735 1.040114 1.047013
    ## [6,] 1.0344353 1.043902 1.049183

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,]  554 8831 5359 5778 9117
    ## [2,] 5091 1960 5960 8102 3176
    ## [3,] 3667 3024 5256 3273 3159
    ## [4,] 6237 9451 8803 4074 9523
    ## [5,] 6377 9364 8209 8845 9010
    ## [6,] 4433 7194 8348 2378 2911

    head(qout$distance)

    ##           [,1]      [,2]      [,3]      [,4]      [,5]
    ## [1,] 0.8107834 0.9270491 1.0416028 1.0461897 1.0559363
    ## [2,] 0.7699306 0.9092005 0.9839563 0.9840844 1.0110680
    ## [3,] 0.8941807 0.8978932 0.9420072 0.9506536 0.9589225
    ## [4,] 0.8914589 0.9189705 0.9708286 1.0087608 1.0187865
    ## [5,] 1.0459340 1.0973414 1.1116436 1.1399796 1.1554955
    ## [6,] 0.8877277 0.9150562 0.9550412 0.9810067 0.9820225

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] "/tmp/RtmpJWVm2v/file7109568b367.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-apple-darwin17.0 (64-bit)
    ## Running under: macOS Mojave 10.14.6
    ##
    ## Matrix products: default
    ## BLAS:   /Library/Frameworks/R.framework/Versions/4.1/Resources/lib/libRblas.dylib
    ## LAPACK: /Library/Frameworks/R.framework/Versions/4.1/Resources/lib/libRlapack.dylib
    ##
    ## locale:
    ## [1] C/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
    ##
    ## 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