The usual pattern of usage for approximate nearest neighbors methods is:

- Build some kind of index with input data.
- Query that index with new data to find the nearest neighbors of your query.

This is how e.g. Annoy
and hnswlib work. If you
want just the k-nearest neighbors of the data you used to build the
index in step 1, then you can just pass that data as the query in step
2, but `rnndescent`

provides some specialized functions for
this case that are slightly more efficient, see for example
`nnd_knn`

and `rpf_knn`

. Nonetheless, querying the
index with the original data can produce a more accurate result. See the
hubness vignette for an example of that.

Below we will see some of the options that `rnndescent`

has for querying an index.

For convenience, I will use all the even rows of the
`iris`

data to build an index, and search using the odd
rows:

```
iris_even <- iris[seq_len(nrow(iris)) %% 2 == 0, ]
iris_odd <- iris[seq_len(nrow(iris)) %% 2 == 1, ]
```

## Brute Force

If your dataset is small enough, you can just use brute force to find the neighbors. No index to build, no worry about how approximate the results are:

```
brute_nbrs <- brute_force_knn_query(
query = iris_odd,
reference = iris_even,
k = 15
)
```

The format of `brute_nbrs`

is the usual k-nearest
neighbors graph format, a list of two matrices, both of dimension
`(nrow(iris_odd), k)`

. The first matrix, `idx`

contains the indices of the nearest neighbors, and the second matrix,
`dist`

contains the distances to those neighbors (here I’ll
just show the first five results per row):

```
lapply(brute_nbrs, function(m) {
head(m[, 1:5])
})
#> $idx
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 9 20 14 4 25
#> [2,] 24 2 23 15 1
#> [3,] 19 9 4 20 14
#> [4,] 24 6 15 2 19
#> [5,] 2 7 24 23 15
#> [6,] 14 10 3 16 11
#>
#> $dist
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.1000000 0.1414213 0.1414213 0.1732050 0.2236068
#> [2,] 0.1414213 0.2449490 0.2645753 0.3000001 0.3000002
#> [3,] 0.1414213 0.1732050 0.2236066 0.2449488 0.2449488
#> [4,] 0.2236068 0.3000002 0.3162278 0.3316627 0.4123106
#> [5,] 0.2999998 0.3464101 0.3605550 0.4242641 0.4690414
#> [6,] 0.2828429 0.3316626 0.3464102 0.3605551 0.3605553
```

## Random Projection Forests

If you build a random projection forest with `rpf_build`

,
you can query it with `rpf_knn_query`

:

```
rpf_index <- rpf_build(iris_even)
rpf_nbrs <- rpf_knn_query(
query = iris_odd,
reference = iris_even,
forest = rpf_index,
k = 15
)
```

See the Random Partition Forests vignette for more.

## Graph Search

See (Dobson et al. 2023) for an overview of graph search algorithms, which can be described as a greedy beam search over a graph: to find the nearest neighbors, you start at a candidate in the graph, find the distance from that candidate to the query point, and update the neighbor list of your query accordingly. If the candidate made it into the neighbor list of the query, this seems like a promising direction to go in, so add the candidate’s neighbors to the list of candidates to explore. Repeat this until such a time as you run out of candidates. You may want to explore the neighbors of the candidate even if it doesn’t make it onto the current neighbor list, if its distance is sufficiently small. How much tolerance you have for this controls how much back-tracking you do and hence how much exploration and the amount of time you spend in the search.

`graph_knn_query`

implements this search. At the very
least you must provide a `reference_graph`

to search, the
`reference`

data that built the `reference_graph`

(so we can calculate distances), `k`

the number of neighbors
you want, and of course the `query`

data:

```
graph_nbrs <- graph_knn_query(
query = iris_odd,
reference = iris_even,
reference_graph = rpf_nbrs,
k = 15
)
```

If you aren’t using the `metric = "euclidean"`

, you should
also provide the same `metric`

that you used to build the
`reference_graph`

. The default `metric`

is always
`"euclidean"`

for any function in `rnndescent`

so
it’s not provided in the examples here.

There are some other parameters you will want to tweak in any real world case that merit some deeper discussion.

###
`n_threads`

`n_threads`

controls how many threads to use in the
search. Be aware that `graph_knn_query`

is designed for
*batch* parallelism, and each thread will be responsible for
searching a subset of the `query`

points. This means that in
a streaming context, where queries to search are likely to arrive one at
a time, you won’t get any speed up from using multiple threads.

###
`epsilon`

`epsilon`

controls how much exploration of the neighbors
of a candidate to do, as suggested by (Iwasaki
and Miyazaki 2018). The default value is `0.1`

, which
is also the default of the NGT
library. The larger the value, the more back-tracking is permitted. The
exact meaning of the value is related to how large a distance is
considered “close enough” the current neighbor list of the query to be
worth exploring.

`epsilon = 0.1`

means that the query-candidate distance is
allowed to be 10% larger than the largest distance in the neighbor list.
If you set `epsilon = 0.2`

, for example, then the
query-candidate distance is allowed to be 20% higher than the largest
distance in the neighbor list and so on. If you set
`epsilon = 0`

then you get a pure greedy search.

It’s hard to give a general rule for what value to set, because it’s
highly dependent on the distribution of distances in the dataset and
that is determined by the distance metric and the dimensionality of the
data itself. I recommend leaving this as the default, and only modifying
it if you find that the search is unreasonably slow (in which case make
`epsilon`

smaller) or unreasonably inaccurate (in which case
make `epsilon`

larger). Yes, not very helpful I know. In the
benchmarking done in (Dobson et al. 2023)
using a similar back-tracking method, `epsilon = 0.25`

was
the maximum value used and in (Wang et al.
2021) `epsilon = 0.1`

was used.

###
`init`

This controls how the search is initialized. If you don’t provide
this, then `k`

random neighbors per item in
`query`

will be generated for you.

#### Neighbor Graph Input

You may provide your own input for this. It should be in the neighbor
graph format, i.e. a list of two matrices, `idx`

and
`dist`

, as described above. Make sure that the
`dist`

matrix contains the distances using the same
`metric`

you will use in the search.

#### Neighbor Indices Only

In fact, the `dist`

matrix is optional. If you only
provide the `idx`

matrix, then the `dist`

matrix
will be calculated for you. If the `dist`

matrix is already
available to you and it was generated by `rnndescent`

then
there is no reason *not* to use it, but you could have neighbors
that come from:

- another nearest neighbor package and for some reason you don’t have the distance
- or you have indices from a different metric that you nonetheless believe are a good guess for the “real” metric.

A case where this might be worth experimenting with could be if you
can cheaply binarize your input data, i.e. convert it to 0/1 then to
`FALSE`

/`TRUE`

: you could then use the
`hamming`

metric or another binary-specialized metric on that
input data. Even a brute force search can be very fast on this data.
This could be a good way to get a good guess for the real data.

This is a very contrived example with `iris`

, but let’s do
it anyway:

```
numeric_iris <- iris[, sapply(iris, is.numeric)]
logical_iris <- sweep(numeric_iris, 2, colMeans(numeric_iris), ">")
logical_iris_even <- logical_iris[seq_len(nrow(logical_iris)) %% 2 == 0, ]
logical_iris_odd <- logical_iris[seq_len(nrow(logical_iris)) %% 2 == 1, ]
head(logical_iris_even)
#> Sepal.Length Sepal.Width Petal.Length Petal.Width
#> [1,] FALSE FALSE FALSE FALSE
#> [2,] FALSE TRUE FALSE FALSE
#> [3,] FALSE TRUE FALSE FALSE
#> [4,] FALSE TRUE FALSE FALSE
#> [5,] FALSE TRUE FALSE FALSE
#> [6,] FALSE TRUE FALSE FALSE
```

Do a brute force search on the binarized data:

```
iris_logical_brute_nbrs <- brute_force_knn_query(
query = logical_iris_odd,
reference = logical_iris_even,
k = 15,
metric = "hamming"
)
```

Then pass the indices of the brute force search to
`graph_knn_query`

, which will generate the Euclidean
distances for you:

```
graph_nbrs <- graph_knn_query(
query = iris_odd,
reference = iris_even,
reference_graph = rpf_nbrs,
init = iris_logical_brute_nbrs$idx,
k = 15
)
```

Whether this is worth doing all depends on whether the time taken to
binarize the data followed by the initial search on the binary data (it
doesn’t have to be brute force) gives you a good enough guess to save
time in the “real” search with `graph_knn_query`

.

#### Forest initialization

If you have previously built an RP Forest with the data you may also
use that to initialize the query. We can re-use `rpf_index`

here.

```
forest_init_nbrs <- graph_knn_query(
query = iris_odd,
reference = iris_even,
reference_graph = rpf_nbrs,
init = rpf_index,
k = 15
)
```

In general, the RP forest initialization is likely to be a better
initial guess than random, but in terms of a speed/accuracy trade-off,
using a large forest may not be the best choice. You may want to use
`rpf_filter`

to reduce the size of the forest before using it
as an initial guess. In the PyNNDescent Python
package that `rnndescent`

is based on, only one tree is used
for initializing query results.

## Preparing the Search Graph

In all the examples so far, we have used the k-nearest neighbors
graph as the `reference_graph`

input to
`graph_knn_query`

. Is this actually a good idea? Probably
not! There is no guarantee that all the items in the original dataset
can actually be reached via the k-nearest neighbors graph. Some nodes
just aren’t very popular and may not be in the neighbor list of
*any* other item. That means you can never reach them via the
k-nearest neighbors graph, no matter how thoroughly you search it.

We can solve this problem by reversing all the edges in the graph and
adding them to the graph. So if you can get to item `i`

from
item `j`

, you can now get to item `j`

from item
`i`

. This solves one problem but adds some more which is that
just like some items are very unpopular, other items might be very
popular and appear often in the neighbor list of other items. Having a
large number of these edges in the graph can make the search very slow.
We therefore need to prune some of these edges.

`prepare_search_graph`

is a function that will take a
k-nearest neighbor graph and add edges to it to make it more useful for
a search. The procedure is based on the process described in (Harwood and Drummond 2016) and consists of:

- Reversing all the edges in the graph.
- “Diversifying” the graph by “occlusion pruning”. This considers triplets of points, and removes long edges which are probably redundant. For an item $i$ with neighbors $p$ and $q$ if the distances $d_{pq} \lt d_{ip}$ i.e. the neighbors are closer to each other than they are to $i$, then it is said that $p$ occludes $q$ and we don’t need both edges $i \rightarrow p$ and $i \rightarrow q$ – it’s likely that $q$ is in the neighbor list of $p$ or vice versa, so it’s unlikely that we are doing any harm by getting rid of $i \rightarrow p$.
- After occlusion pruning, if any item still has an excessive number of edges, the longest edges are removed until the number of edges is below the threshold.

To control all this pruning the following parameters are available:

### Diversification Probability

`diversify_prob`

is the probability of a neighbor being
removed if it is found to be an “occlusion”. This should take a value
between `0`

(no diversification) and `1`

(remove
as many edges as possible). The default is `1.0`

.

The DiskAnn/Vamana
method’s pruning algorithm is almost identical but instead of a
probability, uses a related parameter called `alpha`

, which
acts in the opposite direction: increasing `alpha`

increases
the density of the graph. Why am I telling you this? The pbbsbench
implementation of PyNNDescent uses `alpha`

instead of
`diversify_prob`

and in the accompanying paper (Dobson et al. 2023) they mention that the use
of `alpha`

yields “modest improvements” – from context this
seems to mean relative to using `diversify_prob = 1.0`

. I
can’t give an exact mapping between the two values unfortunately.

### Degree Pruning

`pruning_degree_multiplier`

controls how many edges to
remove after the occlusion pruning relative to the number of neighbors
in the original nearest neighbor graph. The default is `1.5`

which means to allow as many as 50% more edge than the original graph.
So if the input graph was for `k = 15`

, each item in the
search graph will have at most `15 * 1.5 = 22`

edges.

Let’s see how this works on the `iris`

neighbors:

```
set.seed(42)
iris_search_graph <- prepare_search_graph(
data = iris_even,
graph = rpf_nbrs,
diversify_prob = 0.1,
pruning_degree_multiplier = 1.5
)
```

Because the returned search graph can contain different number of
edges per item, the neighbor graph format isn’t suitable. Instead you
get back a sparse matrix, specifically a `dgCMatrix`

. Here’s
a histogram of how the edges are distributed:

```
search_graph_edges <- diff(iris_search_graph@p)
hist(search_graph_edges,
main = "Distribution of search graph edges", xlab = "# edges"
)
```

```
range(search_graph_edges)
#> [1] 11 22
```

So most items have around about `k = 15`

edges just like
the nearest neighbor graph. But some have have the maximum number of
edges and few have only 10 edges.

```
search_nbrs <- graph_knn_query(
query = iris_odd,
reference = iris_even,
reference_graph = iris_search_graph,
init = rpf_index,
k = 15
)
```

## References

*arXiv Preprint arXiv:2305.04359*.

*Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition*, 5713–22.

*arXiv Preprint arXiv:1810.07355*.

*arXiv Preprint arXiv:2101.12631*.