# Random Partition Forests

library(rnndescent)

The Nearest Neighbor Descent method as usually described is technically a way to optimize an existing estimate of the nearest neighbor graph. You must think of a way to initialize the graph. The obvious approach and the one used in the description of NND in (Dong, Moses, and Li 2011) is to start with a random selection of neighbors. One of the clever things about the PyNNDescent implementation is that it uses a random partition forest (Dasgupta and Freund 2008) to come up with the initial guess. Random partition forests are part of a large group of tree-based methods. These are often very fast and conceptually simple, but can be inaccurate. Much of the literature is devoted to proposals of tweaks to these methods to improve their performance, often at the expense of their simplicity and speed. PyNNDescent (and rnndescent follows its lead) avoids this because we only need to get to a decent guess of the nearest neighbor graph which we can then improve by nearest neighbor descent. As long as we don’t take substantially longer than the random initialization to come up with the guess and it’s sufficiently good, we should come out ahead.

## Random Partition Forests

Here’s a basic introduction to how random partition forests work.

### Building a Space-Partitioning Tree

First, we will consider the recipe for building a space-partitioning tree:

1. Select a dimension.
2. Select a split point along that dimension.
3. Split the data into two child nodes based on the split point.
4. Repeat steps 1-3 on each of the two groups.
5. When the number of items in a group is less than some threshold, the node is now a leaf, and stop splitting.

Variations of steps 1 and 2 determines the vast majority of the differences between the various tree-based methods.

### Building a Random Partition Tree

For a random partition tree we:

1. Select two points at random.
2. Calculate the mid-point between those two points.

This is enough to define a hyperplane in the data. This is not exactly the algorithm as described in (Dasgupta and Freund 2008), but it is how it’s done in the very similar method Annoy.

Step 3 then involves calculating which side of the hyperplane each point is on and assigning data to the child nodes on that basis.

### From Trees to Forests

A random partition forest is just a collection of random partition trees. Because of the random nature of the trees, they will all be different.

## Build a Forest

To build a forest with rnndescent, use the rpf_build function. We’ll use the iris dataset as an example, with the goal of finding the 15-nearest neighbors of each item in the dataset.

iris_forest <- rpf_build(iris, leaf_size = 15)

• metric: the type of distance calculation to use. The default is euclidean, but there are a lot to choose from. See the help text for the metric parameter in rpf_build() for details.
• n_trees: the number of trees to build. The default is to choose based on the size of the data provided, with a maximum of 32: eventually you will get diminishing returns from the number of trees in a forest.
• leaf_size: the number of items in a leaf. The splitting procedure stops when there are fewer than this number of items in a node. The default is 10 but you will want the leaf size to scale with the number of neighbors you will look for, so I have increased it to 15 for this example. The bigger this value the more accurate the search will be, but at the cost of a lot more distance calculations to carry out. Conversely, if you make it too small compared to the number of neighbors, then you may end up with not all items finding k neighbors.
• max_tree_depth: the maximum depth of the tree. If a tree reaches this depth then even if the current node size exceeds the value of leaf_size, it will stop splitting. The point of splitting a tree is that the size of each leaf should rapidly decrease as you go down the tree, and in an ideal case it would decrease by a factor of two at each level, so ideally we can process datasets that vary by many orders of magnitude while the depth of the tree only increases by a few levels. The default max_tree_depth is 200, so if you trigger this limit, the answer may not be to increase the depth. It’s more likely that there is something about the distribution of your data that prevents it from splitting well. In this case, if there’s a different metric to try that still has relevance for your data, that’s worth a try, but possibly the best solution is to abandon the tree-based approach (for example initialize nearest neighbor descent with random neighbors). If you set verbose = TRUE you will get a warning about the maximum leaf size being larger than leaf_size.
• margin: this makes a slight modification to how the assignment of data to the sides of the hyperplane is calculated. We’ll discuss this below.

The forest that is returned is just an R list, so you can save it and load it with saveRDS and readRDS without issue. But it’s not something you will want to inspect and definitely don’t modify it. It’s mainly useful for passing to other functions, like the one we will talk about next.

## Finding Nearest Neighbors

To use this to find nearest neighbors, a query point will traverse the tree from the root to a leaf, calculating the side of each hyperplane it encounters. All the items in the leaf in which it ends up are then candidates for nearest neighbors.

To query the forest we just build, we use the rpf_knn_query function. Apart from the forest itself, we also need the data we want to query (query) and the data used to build the forest (reference), because the forest doesn’t store that information. In thus case, because we are looking at the k-nearest neighbors or iris, the query and the reference are the same, but they don’t have to be. At this point, we must also specify the number of neighbors we want.

iris_query <-
rpf_knn_query(
query = iris,
reference = iris,
forest = iris_forest,
k = 15
)

The iris_query that is returned is a list with two matrices: idx contains for each row the indices of the k-nearest neighbors, and dist contains the distances.

## A Small Optimization for the k-Nearest Neighbors

You could use the querying approach mentioned above for finding the k-nearest neighbors of the data that was used in building the tree. However, the data has already been partitioned so if you want k-nearest neighbor data, there’s a more efficient way to do that: for each leaf, the k-nearest neighbors of each point in the leaf are the other members of that leaf. While usually the distance calculations take up most of the time when looking for neighbors, you do avoid having to make any tree traversals and the associated hyperplane distance calculations.

iris_knn <- rpf_knn(iris, k = 15)

This should give the same result as running rpf_build followed by rpf_knn_query (apart from the vagaries of the random number generator), but is a lot more convenient and a bit faster. You have access to the same parameters for forest building as rpf_build, e.g. leaf_size, n_trees, max_tree_depth etc.

Additionally, if you want the k-nearest neighbors and you also want the forest for future querying, if you set ret_forest = TRUE, the return value will now also contain the forest as the forest item in the list. In this example we build the forest (and get the 15-nearest neighbors) for the first 50 iris items and then query the remaining 100:

iris_knn_with_forest <-
rpf_knn(iris[1:50, ], k = 15, ret_forest = TRUE)
iris_query_virginica <-
rpf_knn_query(
query = iris[51:150, ],
reference = iris[1:50, ],
forest = iris_knn_with_forest\$forest,
k = 15
)

## Margin

The margin parameter determines how to calculate the side of the hyperplane each item in a split belongs to. The usual method (margin = "explicit") does the same thing as in PyNNDescent: the way the hyperplane is defined is to use the vector defined by the two points $$a$$ and $$b$$ as the normal vector to a plane, and then the point midway between them as the point on the plane. We then calculate the margin of a point $$x$$ (effectively the signed distance from the plane to $$x$$) as:

$\text{margin}(\mathbf{x}) = ((\mathbf{b} - \mathbf{a}) \cdot (\mathbf{x} - \frac{\mathbf{a} + \mathbf{b}}{2}))$

Taking dot products of vectors and finding mid points is all totally unexceptional if you are using a Euclidean metric. And because there is a monotonic relationship between the cosine distances and the Euclidean distance after normalization of vectors, we can define an “angular” version of this calculation that works on the normalized vectors.

But for some datasets this will be a bit weird and un-natural. Imagine a dataset of binary vectors in which you are applying e.g. the Hamming metric. The mid-point of two binary vectors is not a binary vector, and nor does it make sense to think about the geometric relationship implied by a dot product.

As an alternative to calculating the margin via an explicit creation of a hyperplane, you could instead think about how the distance between $$x$$ and $$a$$, $$d_{xa}$$ compares to the distance between $$x$$ and $$b$$, $$d_{xb}$$ and what the significance for the margin is. Remember that the vector defined by $$a$$ and $$b$$ is the normal vector to the hyperplane, so you can think of a line connecting $$a$$ and $$b$$, with the hyperplane splitting that line in two equal halves. Now imagine $$x$$ is somewhere on that line. If $$x$$ is closer to $$a$$ than $$b$$ it must be on the same side of the hyperplane as $$a$$, and vice versa. Therefore we can calculate the margin by comparing $$d_{xa}$$ and $$d_{xb}$$ and seeing which value is smaller.

Because we don’t explicitly create the hyperplane, I call this the “implicit” margin method and you can choose to generate splits this way by setting margin = "implicit". We’ll use some random binary data for this example.

binary_data <- matrix(as.logical(rbinom(1000, 1, 0.5)), ncol = 10)

Note the as.logical call: if rnndescent detects binary data in this format and you specify a metric which is appropriate for binary data (e.g. Hamming), and you use margin = "implicit" then a specialized function is called which should be much faster than the functions written only with generic floating point data in mind.

bin_knn_imp <-
rpf_knn(binary_data,
k = 15,
metric = "hamming",
margin = "implicit"
)

The following will give the same results but for large datasets is likely to be noticeably slower:

bin_knn_exp <-
rpf_knn(binary_data,
k = 15,
metric = "hamming",
margin = "explicit"
)

So if the implicit margin method is faster (and makes sense for more metrics) why would you ever want to use the explicit method? Well, the implicit method is only faster for binary data with specialized metrics. The downside of the implicit method is that determining the side of the hyperplane requires two distance calculations per point, whereas the explicit method only requires the dot product calculation, which is likely to be only as costly as a single distance calculation. So for floating point data, the explicit method is likely to be about twice as fast. That’s a lot to think about so the default setting for margin is "auto", which tries to do the right thing: if you are using binary data with a suitable metric, it will use the implicit method, otherwise it will use the explicit method and normalize the vectors to give a more “angular” approach for some metrics that put more emphasis on angle versus magnitude.

## Filtering a Forest

As mentioned at the beginning of this vignette, in rnndescent it’s expected that you would only use random partition forests as an initialization to nearest neighbor descent. In that case, keeping the entire forest for querying new data is probably unnecessary: we can keep only the “best” trees. PyNNDescent only keeps one tree for this purpose. For determining what tree is “best”, we mean the tree that reproduces the k-nearest neighbor graph most effectively. You can do this by comparing an existing k-nearest neighbor graph with that produced by a single tree. The rpf_filter function does this for you:

iris_filtered <-
rpf_filter(
nn = iris_query,
forest = iris_forest,
n_trees = 1
)

n_trees` is the number of trees to keep. Feel free to keep more if you like, although there is no extra diversification step to ensure that the trees being retained are both good at reproducing the k-nearest neighbor graph and are diverse from each other (perhaps they reproduce different parts of the neighbor graph well?). The higher quality the k-nearest-neighbor graph is, the better the filtering will work so although the example above uses the graph from the forest, you might get better results using the graph from having run nearest neighbor descent with the forest result as input.

## References

Dasgupta, Sanjoy, and Yoav Freund. 2008. “Random Projection Trees and Low Dimensional Manifolds.” In Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, 537–46.
Dong, Wei, Charikar Moses, and Kai Li. 2011. “Efficient k-Nearest Neighbor Graph Construction for Generic Similarity Measures.” In Proceedings of the 20th International Conference on World Wide Web, 577–86.