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.

local and global mappings of topology representing networks

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