Abstract
Given a finite dataset in a metric space, we investigate the definition of a representative sample. Such a definition is important in data analysis strategies to seed algorithms (such as \(k\)-means) and for pivot-based data indexing techniques. We discuss the geometrical and statistical facets of such a definition.
We propose the Hubness Half Space Partitioning (HubHSP) strategy as an effective sampling heuristic that combines both geometric and statistical constraints. We show that the HubHSP sampling strategy is sound and stable in non-uniform high-dimensional regimes and compares favorably with classical sampling techniques.
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Notes
- 1.
A Dirichlet domain is the generalization of a Voronoi region for high-dimensional spaces. Here, we look at subsets of data from \(\mathcal {X} \) closer to a given point in \(\mathcal {Y} \) than to any other point in \(\mathcal {Y} \).
- 2.
Here, we allow \(x_j\in \mathcal {L} _j\) since generically \(\mathcal {Y} \subseteq \mathcal {X} \).
- 3.
Here, centrality relates mainly to notion of degree centrality.
References
Amato, G., Esuli, A., Falchi, F.: A comparison of pivot selection techniques for permutation-based indexing. Inf. Syst. 52, 176–188 (2015). https://doi.org/10.1016/j.is.2015.01.010
Arthur, D., Vassilvitskii, S.: K-means++: the advantages of careful seeding. In: Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007, pp. 1027–1035. Society for Industrial and Applied Mathematics, USA (2007)
Bustos, B., Navarro, G., Chávez, E.: Pivot selection techniques for proximity searching in metric spaces. Pattern Recogn. Lett. 24, 2357–2366 (2003)
Chavez, E., et al.: Half-space proximal: a new local test for extracting a bounded dilation spanner of a unit disk graph. In: Anderson, J.H., Prencipe, G., Wattenhofer, R. (eds.) OPODIS 2005. LNCS, vol. 3974, pp. 235–245. Springer, Heidelberg (2006). https://doi.org/10.1007/11795490_19
Chávez, E., Navarro, G., Baeza-Yates, R., Marroquín, J.L.: Searching in metric spaces. ACM Comput. Surv. 33(3), 273–321 (2001)
Dasgupta, S., Long, P.M.: Performance guarantees for hierarchical clustering. J. Comput. Syst. Sci. 70, 555–569 (2005). Farthest First Traversal for Pivot Selection
Hoyos, A., Ruiz, U., Marchand-Maillet, S., Chávez, E.: Indexability-based dataset partitioning. In: Amato, G., Gennaro, C., Oria, V., Radovanović, M. (eds.) SISAP 2019. LNCS, vol. 11807, pp. 143–150. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-32047-8_13
Jégou, H., Douze, M., Schmid, C.: Product quantization for nearest neighbor search. IEEE Trans. Pattern Anal. Mach. Intell. 33(1), 117–128 (2011)
Marchand-Maillet, S., Pedreira, O., Chávez, E.: Structural intrinsic dimensionality. In: Reyes, N., et al. (eds.) SISAP 2021. LNCS, vol. 13058, pp. 173–185. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-89657-7_14
Ruiz, G., Chávez, E., Ruiz, U., Tellez, E.S.: Extreme pivots: a pivot selection strategy for faster metric search. Knowl. Inf. Syst. 62(6), 2349–2382 (2020). https://doi.org/10.1007/s10115-019-01423-5
Terrell, G.R., Scott, D.W.: Variable kernel density estimation. Ann. Stat. 20(3), 1236–1265 (1992)
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This work is partly funded by the Swiss National Science Foundation under grant number 207509 “Structural Intrinsic Dimensionality”.
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Annexes
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HubHSP Projection. The HSP selects its neighbors based on increasing distance after discarding half-planes. Since the neighbors selected by the HubHSP can occur in random order of their distance values from the central point \(x_i\), it is critical to consider them as projected over a common sphere centered at \(x_i\).
The most canonical choice is the sphere \(C_i\) including the first neighbor \(x_l\) of \(x_i\). Note \(\rho _i=d(x_l,x_i)\) its radius (the distance between \(x_i\) and its closest neighbor), then a point \(x_j\) is projected as \(\tilde{x}_j\) onto \(C_i\) by:
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Marchand-Maillet, S., Chávez, E. (2022). HubHSP Graph: Effective Data Sampling for Pivot-Based Representation Strategies. In: Skopal, T., Falchi, F., Lokoč, J., Sapino, M.L., Bartolini, I., Patella, M. (eds) Similarity Search and Applications. SISAP 2022. Lecture Notes in Computer Science, vol 13590. Springer, Cham. https://doi.org/10.1007/978-3-031-17849-8_13
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