1 Abstract This paper presents a novel modification to the classical Competitive Learning (CL) by adding a dynamic branching mechanism to neural networks so that the number of neurons can be increased over time until the networks reaches a good estimation of the cluster number in a data set. The algorithm, called Branching Competitive Learning (BCL), shows a fast convergence of the synaptic vectors to cluster centroids, and more importantly, shows the ability to automatically detect cluster number in a data distribution. We illustrate the formulation of the Branching Criteria and demonstrate the efficiency of BCL for data clustering through a set of experiments.

2 Introduction Some applications of data clustering: Pattern recognition Vector quantization image coding Image database indexing Key problems of most clustering algorithms: The number of clusters must be appropriately preselected, e.g., K-mean and classical CL Sensitive to the preselected cluster number and the initialization of synaptic vectors, e.g., RPCL

3 Contributions 1Propose a neuron branching mechanism to estimate cluster number and cluster data 2Present the Branching Criteria 3Present a new way of hierarchical data clustering, i.e., multi-resolution clustering The advantages of BCL for clustering: The ability to automatically detect cluster number Fast convergence of synaptic vectors Convenience to implement multiresolution data clustering

4 The Branching Criterion There are two conditions for a neuron to spawn a new one: The Angle Criteria — Based on the angle between current moving direction and the previous moving direction of a synaptic vector: (1) The Distance Criteria — Based on the distance between the input sample and winner: (2) where: : A randomly selected sample at t, the current step : The winner in current competition, : Angle and distance thresholds

5 The Algorithm of BCL for Clustering  Initialize the first synaptic vector 2Randomly take a sample from the data set, find the winner of current competition in the synaptic vector set, i.e.,, where the is the frequency that has won the competition up to now 3If satisfies the branching criterion above, a new neuron is spawn off from : otherwise, update by

6 Illustrations of BCL Figure 1: (a) An illustration of the procedure of the BCL algorithm, where: (1) Initialization of the first synaptic vector (2) Branching points of synaptic vectors (3) Final convergence of synaptic vectors (b) An illustration of a branching point, where: and

7 Experiments We conduct three sets of experiments: I The first set of experiments examines the ability of BCL to detect cluster number automatically II The second set of experiments shows a multiresolution clustering in BCL scheme III The third set of experiments compares the performance of BCL and RPCL The experimental environment: Pentium II PC with 128 Meg of internal memory running on Windows98 Implement BCL and RPCL algorithm in Visual C++6.0

8 Experiment I (1) Cluster Number Detection (a) (b) Figure 2: The learning and branching traces on a data set, which contains four Gaussian clusters (1,000 samples) with  =0.5 and centered at (-2, 0), (2, 0), (0, -2), and (0, 2) respectively. The first synaptic vector is initialized by point (4, 4) in (a) or by point (0, 0) in (b).

9 Experiment I (2) (c) (d) Figure 3: The learning and branching traces on a data set, which contains four overlapping Gaussian clusters (1,000 samples) with  =0.5 and centered at (-1.5, 0), (1.5, 0), (0, -1.5), and (0, 1.5) res- pectively. The first synaptic vector is initialized by point (3, 3) in (c) or by point (0, 0) in (d).

10 Experiment II (1) Multiresolution Data Clustering Figure 4: A multiresolution data set, where: (a) A view of the data set in a small resolution (b) A view of the data set in a large resolution Data clustering and cluster number detection are resolution-dependent

11 Experiment II (2) Figure 5: The learning and branching traces on the multiresolution data set, where: (c) Presents the learning trace in level 1 (d) Presents the learning trace in 2-level BCL

12 Experiment II (3) Figure 6: The error between the centroids of data clusters and the estimated cluster centroids.

13 Experiment III (1) Comparison of BCL and RPCL Two measures used for comparing BCL and RPCL: The average accuracy of data clustering The average speed or the average running time of the algorithms Table I. 5-Dimensional data sets Data Set Distribution Sample Cluster Overlapping No. Numb. Numb. 1 Uniform 2, none 2 Uniform 2, small 3 Gaussian 2, small 4 Gaussian 2, larger

14 Experiment III (2) Experimental results (over 20 trials) Table II. Average Accuracy RPCL BCL Data Set % % Data Set % % Data Set % 97.16% Data Set % 96.27% Table II. Average Speed (s) RPCL BCL Data Set Data Set Data Set Data Set

15 Discussion & Conclusion Discussion: Robust to various initial conditions The can be seen as a resolution control How to choose ? The advantages of BCL for clustering: The ability to automatically detect cluster number Fast convergence of synaptic vectors Convenience to implement multiresolution data clustering