1 Introduction

The BiocNeighbors package implements a few algorithms for exact nearest neighbor searching:

  • The k-means for k-nearest neighbors (KMKNN) algorithm (Wang 2012) uses k-means clustering to create an index. Within each cluster, the distance of each of that cluster’s points to the cluster center are computed and used to sort all points. Given a query point, the distance to each cluster center is determined and the triangle inequality is applied to determine which points in each cluster warrant a full distance calculation.
  • The vantage point (VP) tree algorithm (Yianilos 1993) involves constructing a tree where each node is located at a data point and is associated with a subset of neighboring points. Each node progressively partitions points into two subsets that are either closer or further to the node than a given threshold. Given a query point, the triangle inequality is applied at each node in the tree to determine if the child nodes warrant searching.

Both methods involve a component of randomness during index construction, though the k-nearest neighbors result is fully deterministic1 Except in the presence of ties, see ?findKNN for details..

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). We could use a VP tree instead by setting BNPARAM=VptreeParam().

fout <- findKNN(data, k=10, BNPARAM=KmknnParam())
head(fout$index)
##      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,] 1881 8278 3628 5229 6391 9639 6161 7671 5375  5037
## [2,] 3432 6646 4652  782 7068 5634 4408 5663 2795   751
## [3,] 9029 8006 4575    7 3232  774 1207 1112 8270  7643
## [4,]  135 2543  613 2054 5954  797 7852 1271 5963  5719
## [5,]  521  752 8056 4628 1134 5878 8624 7083 6162  9431
## [6,] 6003 4150  574 8355 5948 6947 4176 3989 5718  5707
head(fout$distance)
##           [,1]      [,2]      [,3]      [,4]      [,5]      [,6]      [,7]
## [1,] 0.8053135 0.8344659 0.8597889 0.8675684 0.8903521 0.9510813 0.9541194
## [2,] 0.9231167 0.9317375 0.9677469 0.9691288 0.9693556 0.9707015 0.9762002
## [3,] 0.9674346 1.0275486 1.0556525 1.0923730 1.1129987 1.1219975 1.1402440
## [4,] 1.0107910 1.0346099 1.0550829 1.0576011 1.0749991 1.0769960 1.0805129
## [5,] 0.9196937 0.9447228 0.9824839 0.9871147 0.9942311 1.0174962 1.0189802
## [6,] 0.9658588 0.9816334 0.9958824 1.0291708 1.0416947 1.0525788 1.0570698
##           [,8]      [,9]     [,10]
## [1,] 0.9567863 0.9773559 0.9800960
## [2,] 0.9766540 0.9814470 0.9872668
## [3,] 1.1422397 1.1547846 1.1638458
## [4,] 1.0839071 1.0840470 1.0903321
## [5,] 1.0193400 1.0205089 1.0397422
## [6,] 1.0711706 1.0814136 1.0856588

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] 9029 8006 4575    7 3232  774 1207 1112 8270 7643

… with the following distances to those neighbors:

fout$distance[3,]
##  [1] 0.9674346 1.0275486 1.0556525 1.0923730 1.1129987 1.1219975 1.1402440
##  [8] 1.1422397 1.1547846 1.1638458

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$index)
##      [,1] [,2] [,3] [,4] [,5]
## [1,] 3116 1922 7629  104   77
## [2,] 4330 8770 5493 8495 6613
## [3,]   96 2671 2717 5074 3563
## [4,] 6919 5293 7579 6739 8893
## [5,] 5898 1706  532 9060 1513
## [6,] 4944 3700  987 8835 9258
head(qout$distance)
##           [,1]      [,2]      [,3]      [,4]      [,5]
## [1,] 0.8477125 0.9520432 0.9528356 0.9533169 0.9873199
## [2,] 1.0110129 1.0158369 1.0206459 1.0561397 1.0701870
## [3,] 0.9009902 0.9609223 0.9808124 0.9998865 1.0299645
## [4,] 0.9132506 0.9161617 0.9163441 0.9603837 0.9647815
## [5,] 0.9640249 0.9984658 1.0019975 1.0360102 1.0717339
## [6,] 0.8440024 0.8859069 0.8912034 0.9447737 0.9465425

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]   96 2671 2717 5074 3563

… with the following distances to those neighbors:

qout$distance[3,]
## [1] 0.9009902 0.9609223 0.9808124 0.9998865 1.0299645

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,] 9029 8006 4575    7 3232
## [2,]  135 2543  613 2054 5954
## [3,]  521  752 8056 4628 1134
## 
## $distance
##           [,1]      [,2]      [,3]      [,4]      [,5]
## [1,] 0.9674346 1.0275486 1.0556525 1.0923730 1.1129987
## [2,] 1.0107910 1.0346099 1.0550829 1.0576011 1.0749991
## [3,] 0.9196937 0.9447228 0.9824839 0.9871147 0.9942311

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.

library(BiocParallel)
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 buildIndex(). 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 <- buildIndex(data, BNPARAM=KmknnParam())
out1 <- findKNN(BNINDEX=pre, k=5)
out2 <- queryKNN(BNINDEX=pre, query=query, k=2)

The default setting is to search on the Euclidean distance. Alternatively, we can use the Manhattan distance by setting distance="Manhattan" in the BiocNeighborParam object.

out.m <- findKNN(data, k=5, BNPARAM=KmknnParam(distance="Manhattan"))

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 3.6.2 (2019-12-12)
## Platform: x86_64-apple-darwin15.6.0 (64-bit)
## Running under: OS X El Capitan 10.11.6
## 
## Matrix products: default
## BLAS:   /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRblas.0.dylib
## LAPACK: /Library/Frameworks/R.framework/Versions/3.6/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] BiocParallel_1.20.1 BiocNeighbors_1.4.2 knitr_1.28         
## [4] BiocStyle_2.14.4   
## 
## loaded via a namespace (and not attached):
##  [1] Rcpp_1.0.3          bookdown_0.17       lattice_0.20-40    
##  [4] digest_0.6.25       grid_3.6.2          stats4_3.6.2       
##  [7] magrittr_1.5        evaluate_0.14       rlang_0.4.4        
## [10] stringi_1.4.6       S4Vectors_0.24.3    Matrix_1.2-18      
## [13] rmarkdown_2.1       tools_3.6.2         stringr_1.4.0      
## [16] parallel_3.6.2      xfun_0.12           yaml_2.2.1         
## [19] compiler_3.6.2      BiocGenerics_0.32.0 BiocManager_1.30.10
## [22] htmltools_0.4.0

References

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.

Yianilos, P. N. 1993. “Data Structures and Algorithms for Nearest Neighbor Search in General Metric Spaces.” In SODA, 93:311–21. 194.