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$index)
##      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,] 5780 2400 1779 4331 7961 4190 6901 7654 3303  1148
## [2,] 8341 3607 9329 5306 7821 5704 2425  887  839  5322
## [3,] 1426 9879 9693 9950 5450 3607 9588 9247 3939  4238
## [4,] 7385 6061 8639  307 1967 1614 5670 3612 8624  7258
## [5,] 2847 8757 6124  965 8248 5423 6521 4720 8748    38
## [6,] 4426 9701 9824 9943 2247 5329 2941 8141 1305  2470
head(fout$distance)
##           [,1]      [,2]      [,3]      [,4]      [,5]      [,6]      [,7]
## [1,] 0.9641071 0.9698867 0.9821833 0.9833184 0.9839802 0.9957514 1.0138334
## [2,] 0.9468863 0.9715150 0.9829346 1.0262488 1.0303923 1.0331301 1.0387390
## [3,] 0.6952047 0.8365728 0.8719797 0.9394383 0.9423852 0.9429820 0.9443994
## [4,] 1.0528042 1.0632266 1.0704953 1.0734085 1.0852417 1.0909726 1.1148824
## [5,] 0.8764859 0.9155654 0.9884708 1.0181945 1.0237347 1.0265508 1.0302054
## [6,] 0.8501055 0.8716474 0.9017471 0.9515497 0.9560144 0.9703327 0.9743750
##           [,8]      [,9]     [,10]
## [1,] 1.0177632 1.0348117 1.0377992
## [2,] 1.0422043 1.0442326 1.0468834
## [3,] 0.9571395 0.9698406 0.9703301
## [4,] 1.1292918 1.1293928 1.1514815
## [5,] 1.0555274 1.0617686 1.0701864
## [6,] 0.9853485 0.9853705 0.9966202

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] 1426 9879 9693 9950 5450 3607 9588 9247 3939 4238

… with the following distances to those neighbors:

fout$distance[3,]
##  [1] 0.6952047 0.8365728 0.8719797 0.9394383 0.9423852 0.9429820 0.9443994
##  [8] 0.9571395 0.9698406 0.9703301

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$index)
##      [,1] [,2] [,3] [,4] [,5]
## [1,]    5  584 8632 5867 8204
## [2,] 4475 7706  120 1518 2866
## [3,] 6597 5807 1342 6550 8879
## [4,] 3246 3175  914 1778 4872
## [5,] 9531 5952 3194  746 5721
## [6,] 7517 6204 3966 9357 4084
head(qout$distance)
##           [,1]      [,2]      [,3]      [,4]      [,5]
## [1,] 1.0028418 1.1399802 1.1544500 1.1624324 1.1700077
## [2,] 0.8821578 0.8864041 0.8919386 0.9185352 0.9237819
## [3,] 0.7622327 1.0048679 1.0132730 1.0174456 1.0446646
## [4,] 0.8746881 0.9076532 0.9273415 0.9460961 0.9621586
## [5,] 0.9154788 0.9559962 0.9780533 1.0348812 1.0362163
## [6,] 0.9768172 1.0261409 1.0386026 1.0719880 1.0845750

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] 6597 5807 1342 6550 8879

… with the following distances to those neighbors:

qout$distance[3,]
## [1] 0.7622327 1.0048679 1.0132730 1.0174456 1.0446646

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,] 1426 9879 9693 9950 5450
## [2,] 7385 6061 8639  307 1967
## [3,] 2847 8757 6124  965 8248
## 
## $distance
##           [,1]      [,2]      [,3]      [,4]      [,5]
## [1,] 0.6952047 0.8365728 0.8719797 0.9394383 0.9423852
## [2,] 1.0528042 1.0632266 1.0704953 1.0734085 1.0852417
## [3,] 0.8764859 0.9155654 0.9884708 1.0181945 1.0237347

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_1.4.1 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.