Self-organizing maps are a topology-preserving embedding of high-dimensional data to two-dimensional surfaces such as a plane or a torus. Artificial neurons are arranged in a grid -- each neuron is assigned a weight vector with identical number of dimensions as the data to be embedded. An iterative procedure adjusts the weight vectors in each step: first a neuron closest to a data point is sought, then its weight vector and its neighbours' weight vectors are pulled closer to the data point's. After a few iterations, the topology of the data set arises in the grid in an unsupervised fashion.
If the number of nodes k in the grid is much smaller than the number of data points n, SOM reduces to a clustering algorithm. Multiple data points will share a best matching neuron in the grid.
Emergent self-organizing maps (ESOMs) are a variant in which k>n. In this arrangement, a data point will not only have a unique best matching neuron, but the neuron's neighbourhood will also `belong' to the data point. Clustering structure will still show: some areas of the map will be denser than others.
The extra information comes from the neighbourhoods of best matching neurons. These interim neurons provide a smooth chain of transitions between neurons that are assigned a data point. It is almost as if they interpolate the space between data points. If a data point sits in isolation far from other ones, it shows in the map, as its neighbourhood will be sparse. These gaps gain meaning depending on the nature of the data.
Self-organizing maps are notoriously slow to train, and ESOMs are even more so. Somoclu is a high-performance implementation that accelerates computations on multicore CPUs, GPUs, and even on multiple nodes (Wittek, 2013).
Apart from training time, additional problems surface with ESOMs. The codebook of the map -- the data structure that holds the weight vector for each neuron in the grid -- grows quadratically in the number of dimensions. We can distribute the data to a large number of nodes to save memory, but distributing the codebook is not possible: each node must have a copy of the full codebook. This puts constraints on how big the codebook can be. For the same reason, calculations with very high-dimensional data cannot be accelerated by GPUs, and the GPU memory is much more limited in size than the main memory.
Memory problems persists with sparse data: while sparse structure will drastically reduce memory requirements for holding the data, the codebook will always be a dense structure.
Training the map
We constructed a term-document vector space from the Reuters-21578 collection. We filtered the terms space to remove words that occurred less than three times or more often than the 90% most frequent words. The resulting sparse documents space had 12,347 features.
Given the fuzzy nature of a neural network, Somoclu saves memory by storing matrix entries in 4-byte floats. Even with the savings, a map with 500x300 nodes takes nearly 13 GBytes of RAM. We trained the map with the following settings:
$ somoclu -k 2 -x 500 -y 300 -m 1 -s 2 data/reuters-nolabels.svm data/reuters
The parameter -k sets the sparse CPU kernel. The dimensions of the neuron grid are passed in the parameters -x and -y. We calculated a toroid map with with the -m 1 parameter. Finally, we saved all interim data structures with -s 2.
We observed a curious phenomenon in the training process. The first six epoch took the same time each, nearly five hours (see the figure below). Then a sharp rise occurred, epoch seven took over twenty-five hours. The last two epoch took over two days each.
We thought it was a bug, but careful debugging showed that sparse kernels will always slow down with large maps. The explanation lies in the nature of floating point operations. Initially, the weight vectors have random values, then they are pulled closer and closer to the sparse data vectors. The vast majority of the weight vector entries will have values very close to zero. We suspect that these fine-precision floating operations cause the slow down.
The first three iterations resulted in maps so meaningless they were discarded. Structures began to emerge from epoch six. The figure below plots the U-matrix after epoch seven. Neighbourhoods are taking shape, but the central part, where most of the cluster are, is still blur. Bear in mind that this is a toroid map.
The video below shows the U-matrices from epochs four to nine. The final map -- looking like a honey comb -- clearly identifies hot clusters, and more sparsely populated topical areas.
This work was by supported by the European Commission Seventh Framework Programme under Grant Agreement Number FP7-601138 PERICLES and by the AWS in Education Machine Learning Grant award.