-
James C. Bezdek,
Richard J. Hathaway,
R. E. Howard,
C. A. Wilson,
and M. P. Windham.
Local Convergence Analysis of a Grouped Variable Version of Coordinate Descent.
JOTA,
54(3):471--477,
1987.
Keywords:
Clustering.
| Abstract: |
Let $F(x,y)$ be a function of the vector variables $x\in\RR^n$ and $y\in\RR^m$. One possible scheme for minimizing $F(x,y)$ is to successively alternate minimizations in one vector variable while holding the other fixed. Local convergence analysis is done for this vector (grouped variable) version of coordinate descent, and assuming certain regularity conditions, it is shown that such an approach is locally convergent to a minimizer and that the rate of convergence in each vector variable is linear. Examples where the algorithm is useful in clustering and mixture density decomposition are given, and global convergence properties are briefly discussed. |
-
James C. Bezdek,
Richard J. Hathaway,
Michael J. Sabin,
and William T. Tucker.
Convergence Theory for Fuzzy c-Means: Counterexamples and Repairs.
SMC,
17(5):873--877,
1987.
Keywords:
Clustering,
Fuzzy c-Means,
Sequential/Temporal Data.
| Abstract: |
First, a new counterexample to the original incorrect convergence theorem for the fuzzy c-means (FCM) clustering algorithms which was published in 1980 is provided. The importance of this counterexample is that it establishes the existence of saddle points of the FCM objective function at locations {\sl other than} the geometric centroid of fuzzy partition space. The presentation is augmented by a summary of the counterexamples previously discussed by Tucker. Second, the correct theorem is stated without proof: every FCM iterate sequence converges, at least along a subsequence, to either a local minimum or saddle point of the FCM objective function. Tucker's counterexamples and the corrected theory appear elsewhere. The purpose in restarting them here is to caution interested readers not to further propagate the original incorrect convergence statement. |
-
Kenneth D. Forbus.
Interpreting Observations of Physical Systems.
SMC,
17(3):350--359,
1987.
| Abstract: |
An unsolved problem in creating diagnostic expert systems is generating a qualitative understanding of how the system is behaving from raw data, especially numerical data taken across time. Yet automating this critical step is necessary for building the next generation of expert systems. The theory described provides a means of interpreting observations made of a physical system across time in terms of qualitative theories. Importantly, the theory is {\sl ontology-independent} as well as domain-independent in that it only requires a qualitative description of the domain capable of supporting envisioning and domain-specific techniques for providing an initial qualitative description of numerical measurements. The theory is illustrated step by step with two extended examples, one involving qualitative process theory and the other involving a qualitative state vector representation of motion. The performance of an implementation of the theory is also illustrated. |
|