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$index)
##      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,] 3731 8193 8715 5101  711 9204 1135 2972 6793  1775
## [2,] 4811 8193 8891 7985 4050 6157 5341 3824 7244  5625
## [3,] 6846 8962 3097 6675 7936 8069 8800 8157 8786  3964
## [4,] 9821 2201 5118 6937 9310 5379 3665 9321 7683   792
## [5,] 7039 6058 1976   74 6614 9370 4828 2472 9208  4929
## [6,] 4403 7676 6566 1123 1060 1340 9133 2079  859  1330
head(fout$distance)
##           [,1]      [,2]      [,3]      [,4]      [,5]      [,6]      [,7]
## [1,] 0.8877962 0.9304677 0.9474476 0.9536876 1.0181418 1.0427188 1.0646312
## [2,] 0.7734972 0.8820828 0.9029303 0.9325645 0.9325774 0.9345082 0.9684734
## [3,] 0.8620950 0.8731361 0.8939240 0.9472417 0.9658871 0.9909066 0.9930843
## [4,] 0.9427948 0.9476987 0.9939279 1.0230269 1.0268361 1.0443186 1.0574298
## [5,] 0.9981762 1.0152140 1.0420699 1.0472745 1.0641663 1.0873278 1.1004294
## [6,] 0.9230868 0.9355691 0.9402517 0.9776362 1.0129484 1.0157912 1.0170590
##          [,8]      [,9]     [,10]
## [1,] 1.067713 1.0749696 1.0790763
## [2,] 0.978591 0.9788287 0.9788691
## [3,] 1.000988 1.0153002 1.0204145
## [4,] 1.076196 1.1035483 1.1167119
## [5,] 1.103175 1.1060766 1.1095735
## [6,] 1.018815 1.0316504 1.0521335

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] 6846 8962 3097 6675 7936 8069 8800 8157 8786 3964

… with the following distances to those neighbors:

fout$distance[3,]
##  [1] 0.8620950 0.8731361 0.8939240 0.9472417 0.9658871 0.9909066 0.9930843
##  [8] 1.0009880 1.0153002 1.0204145

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$index)
##      [,1] [,2] [,3] [,4] [,5]
## [1,] 7112 4909 8180 4689 9334
## [2,] 2375 6015 4717 1141 6207
## [3,] 1991 4335  397 6297 3539
## [4,] 2896 2085 8052  779  119
## [5,] 5758 5470 6605 3937 8776
## [6,] 6793 7596 6274 8882 6115
head(qout$distance)
##           [,1]      [,2]      [,3]      [,4]      [,5]
## [1,] 0.9912301 1.0527817 1.0680408 1.0955668 1.1148675
## [2,] 0.8987992 0.9267031 0.9283958 0.9473312 0.9627777
## [3,] 0.9735692 0.9914094 1.0068135 1.0471135 1.0555359
## [4,] 0.8664562 0.9515824 1.0290090 1.0332125 1.0390702
## [5,] 0.9695840 1.0482527 1.0823021 1.0933350 1.1015033
## [6,] 0.8759028 0.9320623 0.9325132 0.9352203 0.9575112

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] 1991 4335  397 6297 3539

… with the following distances to those neighbors:

qout$distance[3,]
## [1] 0.9735692 0.9914094 1.0068135 1.0471135 1.0555359

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,] 6846 8962 3097 6675 7936
## [2,] 9821 2201 5118 6937 9310
## [3,] 7039 6058 1976   74 6614
## 
## $distance
##           [,1]      [,2]      [,3]      [,4]      [,5]
## [1,] 0.8620950 0.8731361 0.8939240 0.9472417 0.9658871
## [2,] 0.9427948 0.9476987 0.9939279 1.0230269 1.0268361
## [3,] 0.9981762 1.0152140 1.0420699 1.0472745 1.0641663

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-w64-mingw32/x64 (64-bit)
## Running under: Windows Server 2012 R2 x64 (build 9600)
## 
## Matrix products: default
## 
## locale:
## [1] LC_COLLATE=C                          
## [2] LC_CTYPE=English_United States.1252   
## [3] LC_MONETARY=English_United States.1252
## [4] LC_NUMERIC=C                          
## [5] LC_TIME=English_United States.1252    
## 
## 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.