Local and global mappings of topology representing networks
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Transcript of Local and global mappings of topology representing networks
Intelligent Database Systems Lab
國立雲林科技大學National Yunlin University of Science and Technology
Local and global mappings of topology representing networks
Agnes Vathy-Fogarassy , Janos Abonyi
InS, Vol.179, 2009, pp. 3791–3803.
Presenter : Wei-Shen Tai
2009/10/13
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Outline
Introduction Vector quantization Competitive Hebbian Learning
Topology representing network based mapping algorithms Neural Gas (NG), Topology Representing Network (TRN) Mapping vs. Dimension Reduction
Analysis of the Topology Representing Network based mapping methods Distance preservation and neighborhood preservation
Conclusion Comments
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Motivation
Combine vector quantization and mapping methods in order to visualize the data structure in a low-dimensional vector space.
Vector quantization Vector quantization & mapping3-D data structure
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Objective
Topology Representing Network Map (TRN Map) TRN obtains the graph of Topology Representing Network. MDS based on graph (geodesic) distances to visualize representing
node in 2-dimension vector space.
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Vector Quantization and Competitive Hebbian Learning
Vector Quantization A large set of points (vectors) are divided into groups. Each
group is represented by its centroid point, as in k-means and some other clustering algorithms.
Competitive Hebbian Learning For each input signal x connects the two closest (measured by
Euclidean distance) centers by an edge.
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NG & TRN
Neural Gas Neighborhood ranking reference vector wj .for input xi
Update wj according to the distance ranking.
Topology Representing Network NG was used for clustering purpose in conjunction with the
Hebbian learning.
Input cluster
Reference vector
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TRN Map
1. Normalize the input data set X.
2. Create the Topology Representing Network of X by the use of the TRN algorithm .
3. If M(D) is not connected, connect these subgraphs.** this process is necessary for building a full connected graph.
4. Calculate the geodesic distances between all pairs wi;wj M(D).
5. Map the graph M(D) into a 2-dimensional vector space with MDS based on the graph distances of M(D).
6. Create component planes for the resulting TRN Map based on the values of wi M(D).
Vector quantization
Vector quantization & mapping
3-D data structure
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Mapping vs. Dimension Reduction
Mapping Represents the data structure of input data in a map with lower
dimension. However, it cannot guarantee the consistency between data space and map space, such as CGS, NG and SOM.
Dimension Reduction Attributes of inputs are transformed into fewer representative variables
by statistical function or the characteristic of geodesic distance can be preserved by objective function. Those methods can fully present the original data structure in coordinates, such as PCA, SM and MDS.
Dimension Reduction can be regarded as a mapping method.
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Mapping quality
Distance preservation MDS stress function
Sammon stress function
Neighborhood preservation Trustworthiness (data)
k=3, green and blue Continuity (map)
k=3, gray and navy
map
data
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Analysis of TRN mapping methods
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Analysis of TRN mapping methods
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Conclusions
Mapping based on the TRN MDS is a global reconstruction technique, hence it
is less sensitive to the number k-nearest neighbors and the number of codebook vectors.
Metric mapping based algorithms minimize the stress functions directly, hence their performance is the best in distance perseveration.
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Comments Advantage
This paper provides four quality index to evaluate distance preservation and neighborhood preservation.
Drawback MDS can apply metric (distance) and non-metric (ranking) to preserve the
pairwise distance and rank ordering among data objects. Nevertheless, the mapping result of original data set via MDS is not compared to the other methods in this paper.
(Neighborhood Preservation) NP based methods should outperform than (Distance Preservation) DP NP based methods in two neighborhood preservation index. However, it seems unreasonable that a different result happened in optical recognition of handwritten digits.
Application Dimension reduction and visualization for high- dimension data.