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$index)
##      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,] 8196 8062 5454 8991 3695 4456 2351 9153 5492  9385
## [2,] 6855 9974 5145 4374 8669 3509 4197  202 6575  1654
## [3,] 9193 6357 8573  158  492 9272 1625 1463 9650  3545
## [4,] 3469 7175 5310 2072 3011 4354  209 9976 7209  7787
## [5,] 7848 5991 4298 5257 2899 5531 9457 7889 7311   817
## [6,] 3380 1305 2584 7206 2499 4583 8138 4804 4424  8307
head(fout$distance)
##           [,1]      [,2]      [,3]      [,4]      [,5]      [,6]      [,7]
## [1,] 1.0830825 1.0840079 1.0925842 1.0987089 1.1154806 1.1241577 1.1262541
## [2,] 0.9137818 0.9213725 0.9620821 0.9645489 0.9741699 0.9805689 1.0019530
## [3,] 0.9392383 1.0427644 1.0489440 1.0529705 1.0552776 1.0620763 1.0695114
## [4,] 0.8683043 0.9229621 0.9329104 0.9740085 0.9743233 0.9752475 0.9981356
## [5,] 0.9067922 0.9277674 0.9633644 0.9788265 0.9911254 1.0010642 1.0015802
## [6,] 0.9334378 0.9856910 1.0331202 1.0651609 1.0849952 1.0851405 1.0859730
##           [,8]     [,9]    [,10]
## [1,] 1.1301843 1.136466 1.148723
## [2,] 1.0048953 1.015604 1.018154
## [3,] 1.0813986 1.083556 1.083570
## [4,] 0.9993105 1.005160 1.032564
## [5,] 1.0300116 1.034624 1.036723
## [6,] 1.0905224 1.105057 1.114984

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] 9193 6357 8573  158  492 9272 1625 1463 9650 3545

… with the following distances to those neighbors:

fout$distance[3,]
##  [1] 0.9392383 1.0427644 1.0489440 1.0529705 1.0552776 1.0620763 1.0695114
##  [8] 1.0813986 1.0835555 1.0835697

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$index)
##      [,1] [,2] [,3] [,4] [,5]
## [1,] 2827 6806 4510 9524 7143
## [2,] 6475 9164 9218 5324 7940
## [3,] 2358 7492 6582 4371 3681
## [4,] 6770 1509 1962 8259 7997
## [5,] 1328  110 1088 9042 7418
## [6,] 1630 4159  500 7568 5137
head(qout$distance)
##           [,1]      [,2]      [,3]      [,4]      [,5]
## [1,] 1.0060549 1.0099957 1.0240274 1.0491598 1.0495019
## [2,] 0.7291079 0.9566622 0.9714249 0.9815270 0.9829133
## [3,] 0.9315287 0.9371423 0.9820317 1.0882948 1.0984504
## [4,] 0.9080521 0.9415054 0.9447102 0.9626292 0.9783061
## [5,] 0.8181125 0.8218580 0.8521564 0.9436925 1.0071166
## [6,] 0.9300248 0.9488922 0.9656993 0.9785690 0.9957626

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] 2358 7492 6582 4371 3681

… with the following distances to those neighbors:

qout$distance[3,]
## [1] 0.9315287 0.9371423 0.9820317 1.0882948 1.0984504

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,] 9193 6357 8573  158  492
## [2,] 3469 7175 5310 2072 3011
## [3,] 7848 5991 4298 5257 2899
## 
## $distance
##           [,1]      [,2]      [,3]      [,4]      [,5]
## [1,] 0.9392383 1.0427644 1.0489440 1.0529705 1.0552776
## [2,] 0.8683043 0.9229621 0.9329104 0.9740085 0.9743233
## [3,] 0.9067922 0.9277674 0.9633644 0.9788265 0.9911254

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)

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 3.6.0 (2019-04-26)
## 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_1.18.0 BiocNeighbors_1.2.0 knitr_1.22         
## [4] BiocStyle_2.12.0   
## 
## loaded via a namespace (and not attached):
##  [1] Rcpp_1.0.1          bookdown_0.9        digest_0.6.18      
##  [4] stats4_3.6.0        magrittr_1.5        evaluate_0.13      
##  [7] stringi_1.4.3       S4Vectors_0.22.0    rmarkdown_1.12     
## [10] tools_3.6.0         stringr_1.4.0       parallel_3.6.0     
## [13] xfun_0.6            yaml_2.2.0          compiler_3.6.0     
## [16] BiocGenerics_0.30.0 BiocManager_1.30.4  htmltools_0.3.6

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.