The DOI matches its Crossref record; 39 of the 39 references it cites were found in a registry or an index.
Least Square Projection: A Fast High-Precision Multidimensional Projection Technique and Its Application to Document Mapping
Abstract
The problem of projecting multidimensional data into lower dimensions has been pursued by many researchers due to its potential application to data analysis of various kinds. This paper presents a novel multidimensional projection technique based on least square approximations. The approximations compute the coordinates of a set of projected points based on the coordinates of a reduced number of control points with defined geometry. We name the technique Least Square Projections (LSP). From an initial projection of the control points, LSP defines the positioning of their neighboring points through a numerical solution that aims at preserving a similarity relationship between the points given by a metric in mD. In order to perform the projection, a small number of distance calculations is necessary and no repositioning of the points is required to obtain a final solution with satisfactory precision. The results show the capability of the technique to form groups of points by degree of similarity in 2D. We illustrate that capability through its application to mapping collections of textual documents from varied sources, a strategic yet difficult application. LSP is faster and more accurate than other existing high quality methods, particularly where it was mostly tested, that is, for mapping text sets.
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Cites
A Global Geometric Framework for Nonlinear Dimensionality Reduction
Tenenbaum, J. B., Silva, V. D., & Langford, J. C. (2000) · Science · doi:10.1126/science.290.5500.2319 · Cited by 13,585
Goodall, C. (1988) · Technometrics · doi:10.1080/00401706.1988.10488412 · Cited by 8,675
Jolliffe, I. T. (1986) · Springer Series in Statistics · doi:10.1007/978-1-4757-1904-8 · Cited by 8,273
Multidimensional Scaling by Optimizing Goodness of Fit to a Nonmetric Hypothesis
Kruskal, J. B. (1964) · Psychometrika · doi:10.1007/bf02289565 · Cited by 7,317
Graph drawing by force‐directed placement
Fruchterman, T. M. J., & Reingold, E. M. (1991) · Software: Practice and Experience · doi:10.1002/spe.4380211102 · Cited by 6,352
Readings in Information Visualization: Using Vision to Think
Card, S. K., Mackinlay, J. D., & Shneiderman, B. (1999) · Cited by 3,935
A Nonlinear Mapping for Data Structure Analysis
Sammon, J. W. (1969) · IEEE Transactions on Computers · doi:10.1109/t-c.1969.222678 · Cited by 3,380
A Survey of Clustering Data Mining Techniques
Berkhin, P. (2006) · Grouping Multidimensional Data · doi:10.1007/3-540-28349-8_2 · Cited by 2,804
Storing and querying ordered XML using a relational database system
Tatarinov, I., Viglas, S. D., Beyer, K., Shanmugasundaram, J., Shekita, E., & Zhang, C. (2002) · Proceedings of the 2002 ACM SIGMOD international conference on Management of data · doi:10.1145/564691.564715 · Cited by 2,235
An Introduction to the Conjugate Gradient Method Without the Agonizing Pain
Shewchuk, J. R. (1994) · Cited by 2,173
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Campello, R. J. G. B., Moulavi, D., & Sander, J. (2013) · Lecture Notes in Computer Science · doi:10.1007/978-3-642-37456-2_14 · Cited by 2,680
Visualizing high-dimensional data: advances in the past decade
Liu, S., Maljovec, D., Wang, B., Bremer, P.-T., & Pascucci, V. (2017) · IEEE Transactions on Visualization and Computer Graphics · doi:10.1109/tvcg.2016.2640960 · Cited by 427
Approximated and User Steerable tSNE for Progressive Visual Analytics
Pezzotti, N., Lelieveldt, B. P. F., Maaten, L. V. D., Hollt, T., Eisemann, E., & Vilanova, A. (2017) · IEEE Transactions on Visualization and Computer Graphics · doi:10.1109/tvcg.2016.2570755 · Cited by 308
Visualizing the Hidden Activity of Artificial Neural Networks
Rauber, P. E., Fadel, S. G., Falcao, A. X., & Telea, A. C. (2017) · IEEE Transactions on Visualization and Computer Graphics · doi:10.1109/tvcg.2016.2598838 · Cited by 280
Metadata
- Authors
- Paulovich, F. V., Nonato, L. G., Minghim, R., & Levkowitz, H.
- Venue
- IEEE Transactions on Visualization and Computer Graphics
- Published
- 2008-05
- Volume
- 14
- Issue
- 3
- Pages
- 564-575
- ISSN
- 1077-2626 (ISSN-L)
- OpenAlex
- W2128021408
- Cited by
- 316
- References
- 39
- FWCI
- 20.05
- Type
- journal article
- Language
- en
- Open access
- closed
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- Paulovich, F. V., Nonato, L. G., Minghim, R., & Levkowitz, H. (2008). Least Square Projection: A Fast High-Precision Multidimensional Projection Technique and Its Application to Document Mapping. IEEE Transactions on Visualization and Computer Graphics. https://doi.org/10.1109/tvcg.2007.70443
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Contains information from the Semantic Scholar Academic Graph Datasets, made available under the ODC Attribution License (ODC-BY 1.0).
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