Given a probability distribution of data vectors and a finite number of feature vectors. With each time step, a data vector randomly chosen from is presented. Subsequently, the distance order of the feature vectors to the given data vector is determined. Let denote the index of the closest feature vector, the index of the second closest feature vector, and the index of the feature vector most distant to. Then each feature vector is adapted according to
with as the adaptation step size and as the so-called neighborhood range. and are reduced with increasing. After sufficiently many adaptation steps the feature vectors cover the data space with minimum representation error. The adaptation step of the neural gas can be interpreted as gradient descent on a cost function. By adapting not only the closest feature vector but all of them with a step size decreasing with increasing distance order, compared to k-means clustering a much more robust convergence of the algorithm can be achieved. The neural gas model does not delete a node and also does not create new nodes.
Variants
A number of variants of the neural gas algorithm exists in the literature so as to mitigate some of its shortcomings. More notable is perhaps Bernd Fritzke's growing neural gas, but also one should mention further elaborations such as the Growing When Required network and also the incremental growing neural gas.. A performance-oriented approach that avoids the risk of overfitting is the Plastic Neural gas model.
Growing neural gas
Fritzke describes the growing neural gas as an incremental network model that learns topological relations by using a "Hebb-like learning rule", only, unlike the neural gas, it has no parameters that change over time and it is capable of continuous learning, i.e. learning on data streams. GNG has been widely used in several domains, demonstrating its capabilities for clustering data incrementally. The GNG is initialized with two randomly positioned nodes which are initially connected with a zero age edge and whose errors are set to 0. Since the in the GNG input data is presented sequentially one by one, the following steps are followed at each iteration:
It is calculated the errors between the two closest nodes to the current input data.
The error of the winner node is respectively accumulated.
The winner node and its topological neighbors are moving towards the current input by different fractions of their respective errors.
The age of all edges connected to the winner node are incremented.
If the winner node and the second-winner are connected by an edge, such an edge is set to 0. If case there are created, an edge is created between them.
If there are edges with an age larger than a threshold, they are removed. Nodes without connections are eliminated.
If the current iteration is an integer multiple of a predefined frequency-creation threshold, a new node is inserted between the node with the largest error and its topological neighbor presenting the highest error. The link between the former and the latter nodes is eliminated and the new node is connected to both of them. The error of the new node is initialized as the updated error of the node which had the largest error.
The accumulated error of all nodes is decreased by a given factor.
If the stopping criterion is not met, the algorithm takes a following input. The criterion might be a given number of epochs, i.e., a pre-set number of times where all data is presented, or the reach of a maximum number of nodes.
Incremental growing neural gas
Another neural gas variant inspired in the GNG algorithm is the incremental growing neural gas. The authors propose the main advantage of this algorithm to be "learning new data without degrading the previously trained network and forgetting the old input data."
Growing when required
Having a network with a growing set of nodes, like the one implemented by the GNG algorithm was seen as a great advantage, however some limitation on the learning was seen by the introduction of the parameter λ, in which the network would only be able to grow when iterations were a multiple of this parameter. The proposal to mitigate this problem was a new algorithm, the Growing When Required network, which would have the network grow more quickly, by adding nodes as quickly as possible whenever the network identified that the existing nodes would not describe the input well enough.
Plastic neural gas
The ability to only grow a network may quickly introduce overfitting; on the other hand, removing nodes on the basis of age only, as in the GNG model, does not ensure that the removed nodes are actually useless, because removal depends on a model parameter that should be carefully tuned to the "memory length" of the stream of input data. The "Plastic Neural Gas" model solves this problem by making decisions to add or remove nodes using an unsupervised version of cross-validation, which controls an equivalent notion of "generalization ability" for the unsupervised setting.
Implementations
To find the ranking of the feature vectors, the neural gas algorithm involves sorting, which is a procedure that does not lend itself easily to parallelization or implementation in analog hardware. However, implementations in both parallel software and analog hardware were actually designed.
