# 1 Introduction

The BiocNeighbors package provides an implementation of the k-means for k-nearest neighbors (KMKNN) algorithm, as described by Wang (2012). For a dataset with $$N$$ points, the pre-training is done as follows:

1. Apply k-means clustering to all points, partitioning the data into $$\sqrt{N}$$ clusters.
2. Compute the distance from each data point to its cluster center.
3. Store the cluster identities and distances.

For each query point, identification of the nearest neighbors is done as follows:

1. Start with a threshold distance $$d$$ to the current kth-nearest neighbor (this can be set with arbitrary points).
2. Compute the distance from the query to each cluster center.
3. For any given cluster center, apply the triangle inequality on the query-center distance, the center-point distances and $$d$$. Only compute query-point distances for points where the triangle inequality holds.
4. Update $$d$$ with the new closest kth-nearest neighbor and repeat for the next cluster.

The pre-clustering arranges the points in a manner that effectively reduces the search space, even in high-dimensional data. Note that, while kmeans itself is random, the k-nearest neighbors result is fully deterministic1 Except in the presence of ties, see ?findKNN for details..

The algorithm is implemented in a combination of R and C++, derived from code in cydar (Lun, Richard, and Marioni 2017). We observe 2-5-fold speed-ups in 20- to 50-dimensional data, compared to KD-trees in FNN and RANN (see https://github.com/LTLA/OkNN2018 for timings). This is consistent with results from Wang (2012).

# 2 Identifying k-nearest neighbors

The most obvious application is to perform a k-nearest neighbors search. We’ll mock up an example here with a hypercube of points, for which we want to identify the 10 nearest neighbors for each point.

nobs <- 10000
ndim <- 20
data <- matrix(runif(nobs*ndim), ncol=ndim)

The findKNN() method expects a numeric matrix as input with data points as the rows and variables/dimensions as the columns. We indicate that we want to use the KMKNN algorithm by setting BNPARAM=KmknnParam() (which is also the default, so this is not strictly necessary here).

fout <- findKNN(data, k=10, BNPARAM=KmknnParam())
head(fout$index) ## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] ## [1,] 8348 2079 4736 3502 9414 572 2442 1267 7704 836 ## [2,] 2337 4921 8189 6943 1764 4972 4591 2667 7761 3753 ## [3,] 5538 7759 8677 666 9359 6286 2244 4550 5 5858 ## [4,] 233 6731 2933 2571 5413 9063 993 3348 9286 3006 ## [5,] 8482 3114 8329 7454 5157 3850 7096 7214 1537 6223 ## [6,] 6759 4456 986 5793 9699 1628 3969 6168 2692 2962 head(fout$distance)
##           [,1]      [,2]      [,3]      [,4]      [,5]      [,6]     [,7]
## [1,] 0.9389646 1.0246597 1.0809231 1.0848316 1.0854004 1.1159382 1.127869
## [2,] 0.9756834 1.0439384 1.0904817 1.0982062 1.1187872 1.1229278 1.127581
## [3,] 0.9672927 0.9799397 1.0203636 1.0643326 1.0750760 1.0783592 1.081310
## [4,] 0.9633158 1.0318064 1.0653143 1.0826126 1.1119344 1.1153617 1.117624
## [5,] 0.8818073 0.8873177 0.9171854 0.9613600 0.9915028 0.9915082 1.013779
## [6,] 0.8738909 0.9602641 0.9647539 0.9789634 0.9963525 1.0135661 1.016573
##          [,8]     [,9]    [,10]
## [1,] 1.129126 1.156162 1.157357
## [2,] 1.139209 1.140677 1.155761
## [3,] 1.105882 1.111450 1.122835
## [4,] 1.131882 1.132397 1.135052
## [5,] 1.016158 1.021197 1.022092
## [6,] 1.034331 1.046464 1.046562

Each row of the index matrix corresponds to a point in data and contains the row indices in data that are its nearest neighbors. For example, the 3rd point in data has the following nearest neighbors:

fout$index[3,] ## [1] 5538 7759 8677 666 9359 6286 2244 4550 5 5858 … with the following distances to those neighbors: fout$distance[3,]
##  [1] 0.9672927 0.9799397 1.0203636 1.0643326 1.0750760 1.0783592 1.0813100
##  [8] 1.1058818 1.1114497 1.1228352

Note that the reported neighbors are sorted by distance.

# 3 Querying k-nearest neighbors

Another application is to identify the k-nearest neighbors in one dataset based on query points in another dataset. Again, we mock up a small data set:

nquery <- 1000
ndim <- 20
query <- matrix(runif(nquery*ndim), ncol=ndim)

We then use the queryKNN() function to identify the 5 nearest neighbors in data for each point in query.

qout <- queryKNN(data, query, k=5, BNPARAM=KmknnParam())
head(qout$index) ## [,1] [,2] [,3] [,4] [,5] ## [1,] 1595 2232 7850 3846 5146 ## [2,] 1868 776 971 5400 901 ## [3,] 4493 8490 6240 4826 9763 ## [4,] 9987 9241 5364 7506 9294 ## [5,] 1305 878 755 3071 9608 ## [6,] 8043 3931 8989 1913 3324 head(qout$distance)
##           [,1]      [,2]      [,3]      [,4]      [,5]
## [1,] 0.9267549 1.0948798 1.1076541 1.1148193 1.1215079
## [2,] 0.8081274 0.8638353 0.8764378 0.8878907 0.8881987
## [3,] 0.8871820 0.9110784 0.9861469 0.9915133 1.0194031
## [4,] 0.9987595 1.0064124 1.0105680 1.0153516 1.0988825
## [5,] 0.8459096 0.8743964 0.9751048 0.9978839 1.0123009
## [6,] 0.9226638 1.0071283 1.0274564 1.0449529 1.0450088

Each row of the index matrix contains the row indices in data that are the nearest neighbors of a point in query. For example, the 3rd point in query has the following nearest neighbors in data:

qout$index[3,] ## [1] 4493 8490 6240 4826 9763 … with the following distances to those neighbors: qout$distance[3,]
## [1] 0.8871820 0.9110784 0.9861469 0.9915133 1.0194031

Again, the reported neighbors are sorted by distance.

# 4 Further options

Users can perform the search for a subset of query points using the subset= argument. This yields the same result as but is more efficient than performing the search for all points and subsetting the output.

findKNN(data, k=5, subset=3:5)
## $index ## [,1] [,2] [,3] [,4] [,5] ## [1,] 5538 7759 8677 666 9359 ## [2,] 233 6731 2933 2571 5413 ## [3,] 8482 3114 8329 7454 5157 ## ##$distance
##           [,1]      [,2]      [,3]     [,4]      [,5]
## [1,] 0.9672927 0.9799397 1.0203636 1.064333 1.0750760
## [2,] 0.9633158 1.0318064 1.0653143 1.082613 1.1119344
## [3,] 0.8818073 0.8873177 0.9171854 0.961360 0.9915028

If only the indices are of interest, users can set get.distance=FALSE to avoid returning the matrix of distances. This will save some time and memory.

names(findKNN(data, k=2, get.distance=FALSE))
## [1] "index"

It is also simple to speed up functions by parallelizing the calculations with the BiocParallel framework.

out <- findKNN(data, k=10, BPPARAM=MulticoreParam(3))

For multiple queries to a constant data, the pre-clustering can be performed in a separate step with buildNNIndex(). The result can then be passed to multiple calls, avoiding the overhead of repeated clustering2 The algorithm type is automatically determined when BNINDEX is specified, so there is no need to also specify BNPARAM in the later functions..

pre <- buildNNIndex(data, BNPARAM=KmknnParam())
out1 <- findKNN(BNINDEX=pre, k=5)
out2 <- queryKNN(BNINDEX=pre, query=query, k=2)

Advanced users may also be interested in the raw.index= argument, which returns indices directly to the precomputed object rather than to data. This may be useful inside package functions where it may be more convenient to work on a common precomputed object.

# 5 Session information

sessionInfo()
## R version 3.5.1 Patched (2018-07-12 r74967)
## Platform: x86_64-pc-linux-gnu (64-bit)
## Running under: Ubuntu 16.04.5 LTS
##
## Matrix products: default
## BLAS: /home/biocbuild/bbs-3.8-bioc/R/lib/libRblas.so
## LAPACK: /home/biocbuild/bbs-3.8-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
## [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_1.0.0 BiocParallel_1.16.0 knitr_1.20
## [4] BiocStyle_2.10.0
##
## loaded via a namespace (and not attached):
##  [1] Rcpp_0.12.19        bookdown_0.7        digest_0.6.18
##  [4] rprojroot_1.3-2     backports_1.1.2     stats4_3.5.1
##  [7] magrittr_1.5        evaluate_0.12       stringi_1.2.4
## [10] S4Vectors_0.20.0    rmarkdown_1.10      tools_3.5.1
## [13] stringr_1.3.1       parallel_3.5.1      xfun_0.4
## [16] yaml_2.2.0          compiler_3.5.1      BiocGenerics_0.28.0
## [19] BiocManager_1.30.3  htmltools_0.3.6

# References

Lun, A. T. L., A. C. Richard, and J. C. Marioni. 2017. “Testing for differential abundance in mass cytometry data.” Nat. Methods 14 (7):707–9.

Wang, X. 2012. “A Fast Exact k-Nearest Neighbors Algorithm for High Dimensional Search Using k-Means Clustering and Triangle Inequality.” Proc Int Jt Conf Neural Netw 43 (6):2351–8.