# Detecting approximate nearest neighbors

Aaron Lun1

1Cancer Research UK Cambridge Institute, Cambridge, United Kingdom

#### Revised: 2 December 2018

#### Package

BiocNeighbors 1.20.2

# 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,] 9170 2187 3043 4787  861 2532 5533 2606 9158  4469
    ## [2,] 3827 4027 5679 2543 1496 8751 5169 2427 4364  5374
    ## [3,] 2490 6220  491  782 1402 8024 5193  241 4212  3119
    ## [4,] 7890 3473 6915 8848 6597 7120 4462 6592  427  2074
    ## [5,] 4081 4380 8101  928 1315 5265 9589 3443 1797  6275
    ## [6,] 8776 8308 4594  866 8936 9907 2138 3964 2636  1667

    head(fout$distance)

    ##           [,1]      [,2]      [,3]      [,4]      [,5]      [,6]      [,7]
    ## [1,] 0.8146991 0.8668554 0.8736681 0.8850469 0.9021078 0.9033343 0.9171228
    ## [2,] 0.8051909 0.8937845 0.8978544 0.9111407 0.9298306 0.9401582 0.9517565
    ## [3,] 0.8022693 0.8939355 0.9111685 0.9647251 0.9694588 0.9939378 0.9946436
    ## [4,] 0.6883357 0.9274211 0.9429976 0.9458513 0.9663146 0.9871839 0.9910587
    ## [5,] 0.9946769 1.0624418 1.0891799 1.1080674 1.1111146 1.1336514 1.1581675
    ## [6,] 0.7206981 0.8502777 0.8793322 0.9103470 0.9132983 0.9172049 0.9481169
    ##           [,8]      [,9]     [,10]
    ## [1,] 0.9353919 0.9707053 0.9714199
    ## [2,] 0.9609740 0.9773840 0.9798102
    ## [3,] 1.0049042 1.0062886 1.0068802
    ## [4,] 1.0054924 1.0091718 1.0177615
    ## [5,] 1.1589582 1.1643794 1.1648592
    ## [6,] 0.9489189 0.9538189 0.9592605

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,]  862 6302 1223 4677 5413
    ## [2,] 5172  732 8591 7410 3946
    ## [3,] 4982 4212 2830 3798 2292
    ## [4,] 4808 4269 3125 5776 8491
    ## [5,] 5411 8148  462 7649 1857
    ## [6,] 7683 9486   50 5077 1361

    head(qout$distance)

    ##           [,1]      [,2]      [,3]      [,4]      [,5]
    ## [1,] 0.9219522 1.0506451 1.0899365 1.1152922 1.1193435
    ## [2,] 0.9985842 1.0723281 1.0783435 1.0823529 1.0984902
    ## [3,] 0.9523321 0.9549680 0.9740916 0.9771364 0.9997572
    ## [4,] 0.7610670 0.9169366 0.9436059 0.9979706 1.0366374
    ## [5,] 0.9318078 0.9839750 0.9871143 1.0033323 1.0038940
    ## [6,] 0.8195763 0.9064996 0.9649357 0.9751717 0.9966074

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/Rtmp3mVRlN/file11b4df45400394.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 4.3.2 Patched (2023-11-13 r85521)
    ## Platform: x86_64-pc-linux-gnu (64-bit)
    ## Running under: Ubuntu 22.04.3 LTS
    ##
    ## Matrix products: default
    ## BLAS:   /home/biocbuild/bbs-3.18-bioc/R/lib/libRblas.so
    ## LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.10.0
    ##
    ## 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
    ##
    ## time zone: America/New_York
    ## tzcode source: system (glibc)
    ##
    ## attached base packages:
    ## [1] stats     graphics  grDevices utils     datasets  methods   base
    ##
    ## other attached packages:
    ## [1] BiocNeighbors_1.20.2 knitr_1.45           BiocStyle_2.30.0
    ##
    ## loaded via a namespace (and not attached):
    ##  [1] cli_3.6.2           rlang_1.1.2         xfun_0.41
    ##  [4] jsonlite_1.8.8      S4Vectors_0.40.2    htmltools_0.5.7
    ##  [7] stats4_4.3.2        sass_0.4.8          rmarkdown_2.25
    ## [10] grid_4.3.2          evaluate_0.23       jquerylib_0.1.4
    ## [13] fastmap_1.1.1       yaml_2.3.8          lifecycle_1.0.4
    ## [16] bookdown_0.37       BiocManager_1.30.22 compiler_4.3.2
    ## [19] codetools_0.2-19    Rcpp_1.0.11         BiocParallel_1.36.0
    ## [22] lattice_0.22-5      digest_0.6.33       R6_2.5.1
    ## [25] parallel_4.3.2      bslib_0.6.1         Matrix_1.6-4
    ## [28] tools_4.3.2         BiocGenerics_0.48.1 cachem_1.0.8