Subscribe

Issue No.12 - Dec. (2011 vol.17)

pp: 2563-2571

Paulo Joia , Universidade de São Paulo (USP)

Danilo Coimbra , Universidade de São Paulo (USP)

José A. Cuminato , Universidade de São Paulo (USP)

Fernando V. Paulovich , Universidade de São Paulo (USP)

Luis G. Nonato , Universidade de São Paulo (USP)

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/TVCG.2011.220

ABSTRACT

Multidimensional projection techniques have experienced many improvements lately, mainly regarding computational times and accuracy. However, existing methods do not yet provide flexible enough mechanisms for visualization-oriented fully interactive applications. This work presents a new multidimensional projection technique designed to be more flexible and versatile than other methods. This novel approach, called Local Affine Multidimensional Projection (LAMP), relies on orthogonal mapping theory to build accurate local transformations that can be dynamically modified according to user knowledge. The accuracy, flexibility and computational efficiency of LAMP is confirmed by a comprehensive set of comparisons. LAMP's versatility is exploited in an application which seeks to correlate data that, in principle, has no connection as well as in visual exploration of textual documents.

INDEX TERMS

Multidimensional Projection, High Dimensional Data, Visual Data Mining.

CITATION

Paulo Joia, Danilo Coimbra, José A. Cuminato, Fernando V. Paulovich, Luis G. Nonato, "Local Affine Multidimensional Projection",

*IEEE Transactions on Visualization & Computer Graphics*, vol.17, no. 12, pp. 2563-2571, Dec. 2011, doi:10.1109/TVCG.2011.220REFERENCES

- [1] M. Belkin and P. Niyogi, Laplacian eigenmaps for dimensionality reduction and data representation.
Neural Comput., 15 (6): 1373–1396, 2003.- [2] L. S. Blackford, J. Demmel, J. Dongarra, I. Duff, S. Hammarling, G. Henry, M. Heroux, L. Kaufman, A. Lumsdaine, A. Petitet, R. Pozo, K. Remington, and R. C. Whaley, An updated set of basic linear algebra subprograms (BLAS).
ACM Trans. Math. Softw., 28: 135–151, June 2002.- [3] U. Brandes and C. Pich, Eigensolver methods for progressive multidimensional scaling of large data. In M. Kaufmann, and D. Wagner editors
LNCS, volume 4372, pages 42–53. 2007.- [4] M. L. Braun, J. Schaback, M. L. Jugel, and N. Oury, jBlas: Linear algebra for java, 2011.
- [5] M. M. Bronstein, A. M. Bronstein, R. Kimmel, and I. Yavneh, Multigrid multidimensional scaling. Numerical Linear Algebra with Applications, 13: 149–171, 2006.
- [6] M. Chalmers, A linear iteration time layout algorithm for visualizing high-dimensional data.
In IEEE Visualization, pages 127–ff., 1996.- [7] Y. Chen, L. Wang, M. Dong, and J. Hua, Exemplar-based visualization of large document corpus.
IEEE Trans. Vis. Comput. Graph., 15: 1161– 1168, 2009.- [8] W. Cui, Y. Wu, S. Liu, F. Wei, M. X. Zhou, and H. Qu, Context-preserving, dynamic word cloud visualization.
IEEE Computer Graphics and Applications, pages 42–53, 2010.- [9] J. Daniels, E. W. Anderson, L. G. Nonato, and C. T. Silva, Interactive vector field feature identification.
IEEE Trans. Vis. Comput. Graph., 16: 1560–1568, 2010.- [10] V. de Silva and J. B. Tenenbaum, Global versus local methods in nonlinear dimensionality reduction.
In Advances in Neural Information Processing Systems 15, pages 705–712. MIT Press, 2003.- [11] V. de Silva and J. B. Tenenbaum, Sparse multidimensional scaling using landmark points.
Technical report, Stanford, 2004.- [12] C. Faloutsos and K. Lin, FastMap: A fast algorithm for indexing, datamining and visualization of traditional and multimedia databases.
In ACM SIGMOD, pages 163–174, 1995.- [13] R. Fergus, P. Perona, and A. Zisserman, Object class recognition by un-supervised scale-invariant learning.
In CVPR, volume 2, pages 264–271, 2003.- [14] A. Frank and A. Asuncion, UCI machine learning repository, 2010.
- [15] Y. Frishman and A. Tal, Multi-level graph layout on the GPU.
IEEE Trans. Vis. Comput. Graph., 13: 1310–1319., 2007.- [16] I. Fuinaga and D. McEnnis, On-demand metadata extraction network (OMEN).
In ACM/IEEE-CS Joint Conf. on Digital Libraries, pages 346– 346, 2006.- [17] J. Gower and G. Dijksterhuis,
Procrustes Problems. Oxford University Press, 2004.- [18] S. Ingram, T. Munzner, and M. Olano, Glimmer: Multilevel MDS on the GPU.
IEEE Trans. Vis. Comp. Graph., 15 (2): 249–261, 2009.- [19] F. Jourdan and G. Melançon, Multiscale hybrid MDS.
In Information Visualisation, pages 388–393, 2004.- [20] Y. Koren, L. Carmel, and D. Harel, ACE: A fast multiscale eigenvectors computation for drawing huge graphs.
In IEEE Information Visualization, pages 137–144, 2002.- [21] J. B. Kruskal, Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis.
Psychometrika, 29: 115–129, 1964.- [22] A. Morrison, G. Ross, and M. Chalmers, A hybrid layout algorithm for sub-quadratic multidimensional scaling.
In IEEE Information Visualization, pages 152–158, 2002.- [23] F. V. Paulovich, D. M. Eler, J. Poco, C. P. Botha, R. Minghim, and L. G. Nonato, Piecewise laplacian-based projection for interactive data exploration and organization.
Computer Graphics Forum, 30 (3): 1091–1100, 2011.- [24] F. V. Paulovich, L. G. Nonato, and R. Minghim, Visual mapping of text collections through a fast high precision projection technique.
In International Conference on Information Visualization, pages 282–290, 2006.- [25] F. V. Paulovich, L. G. Nonato, R. Minghim, and H. Levkowitz, Least square projection: A fast high-precision multidimensional projection technique and its application to document mapping.
IEEE Trans. Visual. Comp. Graph., 14 (3): 564–575, 2008.- [26] F. V. Paulovich, C. T. Silva, and L. G. Nonato, Two-phase mapping for projecting massive data sets.
IEEE Trans. on Vis. Comp. Graph., 16 (6): 1281–1290, 2010.- [27] E. Pekalska, D. de Ridder, R.P.W. Duin, and M. A. Kraaijveld, A new method of generalizing Sammon mapping with application to algorithm speed-up. In M. Boasson, J. A. Kaandorp, J. F. M. Tonino, and M. G. Vosselman editors,
Annual Conf. Advanced School for Comput. Imag., pages 221–228, 1999.- [28] S. T. Roweis, and L. K. Saul, Nonlinear dimensionality reduction by locally linear embedding.
Science, 290 (5500): 2323–2326, December 2000.- [29] G. Salton, Developments in automatic text retrieval.
Science, 253: 974– 980, 1991.- [30] S. Schaefer, T. McPhail, and J. Warren, Image deformation using moving least squares.
ACM Trans. Graph., 25 (3): 533–540, 2006.- [31] P. Tan, M. Steinbach, and V. Kumar,
Introduction to Data Mining. Addison-Wesley, 2005.- [32] E. Tejada, R. Minghim, and L. G. Nonato, On improved projection techniques to support visual exploration of multidimensional data sets.
Information Visualization, 2 (4): 218–231, 2003.- [33] J. B. Tenenbaum, V. de Silva, and J. C. Langford, A global geometric framework for nonlinear dimensionality reduction.
Science, 290 (5500): 2319–2323, December 2000.- [34] W. S. Torgeson, Multidimensional scaling of similarity.
Psychometrika, 30: 379–393, 1965.- [35] D. Whalen and M. L. Norman, Competition data set and description.
In 2008 IEEE Visualization Design Contest. http://vis.computer.org/VisWeek2008/viscontests.html , 2008.- [36] J. Yang, Y.-G. Jiang, A. G. Hauptmann, and C.-W. Ngo, Evaluating bag-of-visual-words representations in scene classification.
In International Workshop on Multimedia Information Retrieval, pages 197–206, 2007. |