1. Building Adaptive Basis Function with Continuous Self-Organizing Map
Advisor：Dr. Hsu
Graduate：Ching-Lung Chen
Author：MARCOS M. CAMPOS
GAIL A. CARPENTER

2. Outline Motivation Objective Introduction The CSOM algorithm
CSOM interpolation
Basis Functions
Example
Relate work
Conclusions
Personal opinion

3. Motivation
The SOM discrete coding scheme often yield poor performance when used for function approximation：it can produce only a piecewise-constant approximation, with precision limited a priori by the number of coding nodes.

4. Objective
To overcome this limitation, this paper propose CSOM(Continuous Self-Organizing Map), a four-layer feedforward neural network.

5. Introduction (1/3)

6. Introduction (2/3)
The main innovation is the use of a distributed SOM to implement a continuous, topology-preserving coordinate transformation from the input space to a regular lattice.(Figure 2)

7. Introduction (3/3)

8. The CSOM algorithm (1/5)
In CSOM, an interpolation step maps inputs to a new coordinate system in a continuous fashion.
A node in the BF layer performs a filtering of the distributed activity at the CSOM layer.
The distributed activation of the CSOM layer is represented by a point X* = in a continuous S-dimensional grid.

9. The CSOM algorithm (2/5) X X
The discrete receptive field of the BF unit is approximated by a continuous function ,
is the vector of CSOM grid coordinates of the center of the receptive field of the unit
is a vector of parameters that define the shape of the function. ) . , ( * d y X r X r d

10. The CSOM algorithm (3/5)

11. The CSOM algorithm (4/5)

12. CSOM Parameters

13. CSOM variables

14. The CSOM algorithm (5/5)
How X* is computed(step 4), the key difference between the CSOM algorithm and the traditional SOM algorithm is the use of a normalized activity vector(y) to drive the map adaptation(step 11)
In the traditional SOM algorithm the coordinate use BMU, in CSOM use the piecewise-linear fashion which developed by Goppert and Rosenstiel.

15. CSOM interpolation (1/5)
In this step, the neighbors of the winning node define a local linear system (L) that is used to decompose the input vector A.
The coordinate thus obtained in turn define coordinates in another local linear system (P).

16. CSOM interpolation (2/5)

17. CSOM interpolation (3/5)

18. CSOM interpolation (4/5)
left
right

19. CSOM interpolation (5/5)

20. Basis Functions (1/2)
Step 5 of the CSOM algorithm specifies a gaussian radial basis function (RBF).
The network may also be implemented with a variety of alternative basis functions (Figure 5)
A key property of the radial basis function created by CSOM is that they are adapted to the input distribution (Figure 6)

21. Alternative basis functions

22. Adapted radial basis functions

23. Basis Functions (2/2)
CSOM defines the basis function in grid coordinates. Because node positions do not change in this coordinate system, the standard deviation of the basis function can be specified as a constant in the model.
For gaussian basis functions and an integer CSOM grid, is set equal to d S
4247 .

24. Examples
This section illustrates CSOM’s capabilities with three function approximation tasks.
In the first two tasks the CSOM grid has the same dimensionality as the input space (S=M).
The third task illustrates how CSOM accomplishes dimensionality reduction.

25. 1-D function approximation
Training set consists of 7,000 input points, testing set consists 3,000 observations.

26. Simulation parameters.

27. 1-D summarizes performance

28. 2-D function approximation

29. 2-D summarizes performance

30. Map learned and receptive fields

31. Inverse kinematics of a two-join arm

32. Dimensionality reduction

33. Final map configuration

34. Detail easier map with 10 units

35. Non-normalized BF activity levels

36. Related Work(1/2)
Goppert and Rosenstiel propose a way to interpolate the predictions of a SOM after the map has been learned with the traditional SOM algorithm.
CSOM completely integrates self-organization and smooth prediction.
The two approaches different for selecting neighbors and basis vectors for the local basis system.

37. Related Work(2/2)
CSOM focuses on creating a continuous map that uses the affine transformation to define a continuous variable(feature),in the grid coordinates, that can then be used to define basis function.

38. Conclusions(1/2)
Due to the continuous nature of the mapping, CSOM outperforms the SOM in function approximation tasks.
CSOM also implements a new approach for building basis functions adapted to the distribution of the input data.

39. Conclusions(2/2)
The main idea proposed here is the mapping of the input onto a grid space in a continuous fashion and the specification of the basis function in the grid space coordinates instead of the input space coordinates.

40. Personal Opinion
This can be improved by adapting the concept introduced here to work with other approaches to topological map construction that use a variable topology.