Abstract
In approximate query searching (AQS), the given query point (\({\bar{\textbf{q}}}'\)) can be seen as a noise (\({{\bar{\eta }}}\)) corrupted version of one of the points (\({\bar{\textbf{q}}}\)) in the existing database \({\mathcal {X}}\), i.e., \({\bar{\textbf{q}}}' = {\bar{\textbf{q}}} + {\bar{\mathbf{\eta }}}\). Thus deciding on an appropriate distance d that would return the correct match (\({\bar{\textbf{q}}}\)) entails that the chosen distance should be aware of the type of distribution of the noise. In this work, we study the suitability of Minkowski-type distances in AQS when the \({\bar{\textbf{q}}}\) is afflicted by both white and coloured noises to different extent. To this end, we employ a simple similarity search based scoring algorithm proposed in François et al. (ESANN 2005, 13th European Symposium on Artificial Neural Networks, Bruges, Belgium, April 27–29, 2005, Proceedings, pp 339–344, 2005). Our study reveals an interesting interplay of the following 3D’s in the quest for an appropriate distance: Dimensionality and Domain geometry of the data and the type of noise Distribution and has led us to explore this problem from a basic geometric perspective. Our main contribution herein is the proposal of a novel index called the Relative Contained Volume (RCV) that helps explain the performance of the considered distances.
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Data availability
The analysis of all the data generated during this study is included in this article. The MATLAB codes to generate and analyse the data are available from the corresponding author on reasonable request.
Notes
Note that \(\left( \ell _p({\bar{\textbf{x}}}, {\bar{\textbf{y}}}) \right) ^p = \Vert {\bar{\textbf{x}}} - {\bar{\textbf{y}}} \Vert _p^p = \sum _{i=1}^m |x_i - y_i \vert ^p \) is, in fact, not a metric since it does not satisfy the triangle inequality but it satisfies the other two properties of a metric and hence the choice of our terminology.
This is an expanded version of the preliminary findings presented in Singh and Jayaram (2020).
Perhaps this explains the choice of \(\sigma _i = 0.3\) in François et al. (2005) where \({\mathcal {D}} = [0,1]^m\).
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Singh, A., Jayaram, B. Minkowski-type distances in approximate query searches. Comp. Appl. Math. 43, 187 (2024). https://doi.org/10.1007/s40314-024-02704-8
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DOI: https://doi.org/10.1007/s40314-024-02704-8
Keywords
- Approximate query searching
- High-dimensional data analysis
- Minkowski distances
- Relative contained volume