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$index)
##      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,] 3471 9597 6464 1523  739 3467 6083 5928 8217  4251
## [2,] 1295   91 7913 2910 9595 1744 7896 1427 8003  9929
## [3,] 9441 1934 2339 1915 3488 6020 2848 4402 3963  9024
## [4,] 1182 5211  811 1262 6731 5202 6977 7171  311  8911
## [5,] 1758 3774 4030 8039 6074 4825 3655  900 3276  4645
## [6,] 8731 6885  771 5097 9203 9990 9744 5781 9116  8644
head(fout$distance)
##           [,1]      [,2]      [,3]     [,4]      [,5]      [,6]      [,7]
## [1,] 0.9268582 1.0194071 1.0388245 1.066455 1.0665949 1.0735655 1.0824382
## [2,] 0.9048371 0.9359871 1.0120881 1.043779 1.0444235 1.0455441 1.0539717
## [3,] 0.8742062 1.0089846 1.0463352 1.092934 1.0957071 1.1094109 1.1122402
## [4,] 1.0222398 1.0787940 1.1068635 1.151365 1.1552857 1.1567531 1.1812896
## [5,] 0.8097950 0.8529465 0.8786228 0.898006 0.9218992 0.9421305 0.9433766
## [6,] 0.8107696 0.9445785 0.9817260 0.996435 1.0598795 1.0600409 1.0772381
##           [,8]      [,9]     [,10]
## [1,] 1.0861026 1.0911286 1.0939449
## [2,] 1.0543530 1.0643257 1.0665777
## [3,] 1.1136279 1.1211136 1.1215756
## [4,] 1.1840725 1.1974826 1.2041063
## [5,] 0.9455532 0.9602737 0.9887302
## [6,] 1.0894124 1.0899001 1.1018226

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] 9441 1934 2339 1915 3488 6020 2848 4402 3963 9024

… with the following distances to those neighbors:

fout$distance[3,]
##  [1] 0.8742062 1.0089846 1.0463352 1.0929340 1.0957071 1.1094109 1.1122402
##  [8] 1.1136279 1.1211136 1.1215756

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$index)
##      [,1] [,2] [,3] [,4] [,5]
## [1,] 9606    4  811 6674 6268
## [2,]  579   85 1794 5640 1477
## [3,] 6376 1290 9318 9619 9845
## [4,] 8142 1163 8498 9618 4912
## [5,] 1086  718 8134 2105 5030
## [6,] 2871 7543 2875 9281 9353
head(qout$distance)
##           [,1]      [,2]      [,3]      [,4]      [,5]
## [1,] 1.0499570 1.0873210 1.1693635 1.1906433 1.1906587
## [2,] 0.7563082 0.8060312 0.9013444 0.9136694 0.9237472
## [3,] 0.9370101 0.9899433 0.9916329 1.0248902 1.0273259
## [4,] 0.9029047 0.9092907 1.0669237 1.0870684 1.0878579
## [5,] 0.8753091 0.9623679 0.9792136 0.9888376 0.9939094
## [6,] 0.9021625 0.9179247 0.9275851 0.9527184 0.9609630

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] 6376 1290 9318 9619 9845

… with the following distances to those neighbors:

qout$distance[3,]
## [1] 0.9370101 0.9899433 0.9916329 1.0248902 1.0273259

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,] 9441 1934 2339 1915 3488
## [2,] 1182 5211  811 1262 6731
## [3,] 1758 3774 4030 8039 6074
## 
## $distance
##           [,1]      [,2]      [,3]     [,4]      [,5]
## [1,] 0.8742062 1.0089846 1.0463352 1.092934 1.0957071
## [2,] 1.0222398 1.0787940 1.1068635 1.151365 1.1552857
## [3,] 0.8097950 0.8529465 0.8786228 0.898006 0.9218992

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 3.6.1 (2019-07-05)
## Platform: x86_64-pc-linux-gnu (64-bit)
## Running under: Ubuntu 18.04.3 LTS
## 
## Matrix products: default
## BLAS:   /home/biocbuild/bbs-3.10-bioc/R/lib/libRblas.so
## LAPACK: /home/biocbuild/bbs-3.10-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                 
##  [9] LC_ADDRESS=C               LC_TELEPHONE=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] BiocParallel_1.20.0 BiocNeighbors_1.4.0 knitr_1.25         
## [4] BiocStyle_2.14.0   
## 
## loaded via a namespace (and not attached):
##  [1] Rcpp_1.0.2          bookdown_0.14       lattice_0.20-38    
##  [4] digest_0.6.22       grid_3.6.1          stats4_3.6.1       
##  [7] magrittr_1.5        evaluate_0.14       rlang_0.4.1        
## [10] stringi_1.4.3       S4Vectors_0.24.0    Matrix_1.2-17      
## [13] rmarkdown_1.16      tools_3.6.1         stringr_1.4.0      
## [16] parallel_3.6.1      xfun_0.10           yaml_2.2.0         
## [19] compiler_3.6.1      BiocGenerics_0.32.0 BiocManager_1.30.9 
## [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.