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.

在近似查询搜索（AQS）中，给定的查询点（\({\bar{\textbf{q}}}'\)）可以被视为噪声（\({{\bar{\eta }}}\ ) )现有数据库\({\mathcal {X}}\ ) 中的点之一 ( \({\bar{\textbf{q}}}\) ) 的损坏版本 \({\mathcal {X}}\) ，即\({\bar{ \textbf{q}}}' = {\bar{\textbf{q}}} + {\bar{\mathbf{\eta }}}\)。因此，决定返回正确匹配的适当距离d ( \({\bar{\textbf{q}}}\) ) 意味着所选距离应该了解噪声分布的类型。在这项工作中，我们研究了当\({\bar{\textbf{q}}}\)受到不同程度的白噪声和有色噪声影响时，Minkowski 型距离在 AQS 中的适用性。为此，我们采用了 François 等人提出的基于简单相似性搜索的评分算法。 （ESANN 2005，第 13 届欧洲人工神经网络研讨会，比利时布鲁日，2005 年 4 月 27-29 日，会议记录，第 339-344 页，2005 年）。我们的研究揭示了以下 3D 在寻求适当距离时的有趣相互作用：数据的维数和域几何形状以及噪声分布的类型，并引导我们从基本几何角度探索这个问题。我们在此的主要贡献是提出了一种称为相对包含体积（RCV）的新颖索引，该索引有助于解释所考虑距离的性能。