We explore the effect of dimensionality on the nearest neighbor problem. We show that under a broad set of conditions (much broader than independent and identically distributed dimensions), as dimensionality increases, the distance to the nearest data point approaches the distance to the farthest data point. To provide a practical perspective, we present empirical results on both real and synthetic data sets that demonstrate that this effect can occur for as few as 10-15 dimensions. These results should not be interpreted to mean that high-dimensional indexing is never meaningful; we illustrate this point by identifying some high-dimensional workloads for which this effect does not occur. However, our results do emphasize that the methodology used almost universally in the database literature to evaluate high-dimensional indexing techniques is flawed, and should be modified. In particular, most such techniques proposed in the literature are not evaluated versus simple linear scan, and are evaluated over workloads for which nearest neighbor is not meaningful. Often, even the reported experiments, when analyzed carefully, show that linear scan would outperform the techniques being proposed on the workloads studied in high (10-15) dimensionality!.

When is nearest neighbor meaningful

High dimension, low sample size data are emerging in various areas of science. We find a common structure underlying many such data sets by using a non-standard type of asymptotics: the dimension tends to ∞ while the sample size is fixed. Our analysis shows a tendency for the data to lie deterministically at the vertices of a regular simplex. Essentially all the randomness in the data appears only as a random rotation of this simplex. This geometric representation is used to obtain several new statistical insights. Copyright 2005 Royal Statistical Society.

Geometric representation of high dimension, low sample size data

Suppose that we wish to estimate a vector $\mathbf{x}$ from a set of binary paired comparisons of the form "$\mathbf{x}$ is closer to $\mathbf{p}$ than to $\mathbf{q}$" for various choices of vectors $\mathbf{p}$ and $\mathbf{q}$. The problem of estimating $\mathbf{x}$ from this type of observation arises in a variety of contexts, including nonmetric multidimensional scaling, "unfolding," and ranking problems, often because it provides a powerful and flexible model of preference. We describe theoretical bounds for how well we can expect to estimate $\mathbf{x}$ under a randomized model for $\mathbf{p}$ and $\mathbf{q}$. We also present results for the case where the comparisons are noisy and subject to some degree of error. Additionally, we show that under a randomized model for $\mathbf{p}$ and $\mathbf{q}$, a suitable number of binary paired comparisons yield a stable embedding of the space of target vectors. Finally, we also show that we can achieve significant gains by adaptively changing the distribution for choosing $\mathbf{p}$ and $\mathbf{q}$.

As you like it: Localization via paired comparisons

A similarity search method that causes a computer to perform a process, the process includes: first calculating, based on a plurality of record data, each of the record data including a plurality of feature quantities, normal hyperplane data representing a normal hyperplane, the normal hyperplane being a hyperplane dividing a feature quantity space, and a distance between a pair of divided areas having symmetry, second calculating, based on the plurality of record data and the normal hyperplane data, one-way hyperplane data representing at least one one-way hyperplane, the one-way hyperplane being a hyperplane dividing the feature quantity space, and a distance between a pair of divided areas having asymmetry, and converting, based on the normal hyperplane data and the one-way hyperplane data, query data including a plurality of feature quantities and the plurality of record data into respective binary strings.

Information processing apparatus, similarity search program, and similarity search method

Locality-sensitive hashing converts high-dimensional feature vectors, such as image and speech, into bit arrays and allows high-speed similarity calculation with the Hamming distance. There is a hashing scheme that maps feature vectors to bit arrays depending on the signs of the inner products between feature vectors and the normal vectors of hyperplanes placed in the feature space. This hashing can be seen as a discretization of the feature space by hyperplanes. If labels for data are given, one can determine the hyperplanes by using learning algorithms. However, many proposed learning methods do not consider the hyperplanes' offsets. Not doing so decreases the number of partitioned regions, and the correlation between Hamming distances and Euclidean distances becomes small. In this paper, we propose a lift map that converts learning algorithms without the offsets to the ones that take into account the offsets. With this method, the learning methods without the offsets give the discretizations of spaces as if it takes into account the offsets. For the proposed method, we input several high-dimensional feature data sets and studied the relationship between the statistical characteristics of data, the number of hyperplanes, and the effect of the proposed method.

Hyperplane Arrangements and Locality-Sensitive Hashing with Lift

An information conversion method includes: first moving positions of a plurality of particles on a unit sphere according to a value of a probability density function, defining a positional vector of a particle on the unit sphere in a multidimensional space, as a normal vector of a hyperplane configured to divide a feature vector space, defining a predetermined evaluation function configured to evaluate the hyperplane, as the probability density function configured to indicate a probability of existence of a particle on the unit sphere, by a processor; and converting the feature vector to a binary string, considering a positional vector of the moved particle as a normal vector of the hyperplane, by the processor.

Information conversion method, information conversion device, and recording medium

We propose a learning method with feature selection for Locality-Sensitive Hashing. Locality-Sensitive Hashing converts feature vectors into bit arrays. These bit arrays can be used to perform similarity searches and personal authentication. The proposed method uses bit arrays longer than those used in the end for similarity and other searches and by learning selects the bits that will be used. We demonstrated this method can effectively perform optimization for cases such as fingerprint images with a large number of labels and extremely few data that share the same labels, as well as verifying that it is also effective for natural images, handwritten digits, and speech features.

Locality-Sensitive Hashing with Margin Based Feature Selection

Since Hamming distances can be calculated by bitwise computations, they can be calculated with less computational load than L2 distances. Similarity searches can therefore be performed faster in Hamming distance space. The elements of Hamming distance space are bit strings. On the other hand, the arrangement of hyperplanes induce the transformation from the feature vectors into feature bit strings. This transformation method is a type of locality-sensitive hashing that has been attracting attention as a way of performing approximate similarity searches at high speed. Supervised learning of hyperplane arrangements allows us to obtain a method that transforms them into feature bit strings reflecting the information of labels applied to higher-dimensional feature vectors. In this p aper, we propose a supervised learning method for hyperplane arrangements in feature space that uses a Markov chain Monte Carlo (MCMC) method. We consider the probability density functions used during learning, and evaluate their performance. We also consider the sampling method for learning data pairs needed in learning, and we evaluate its performance. We confirm that the accuracy of this learning method when using a suitable probability density function and sampling method is greater than the accuracy of existing learning methods.

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Yui Noma

Papers

Hyperplane Arrangements and Locality-Sensitive Hashing with Lift

Information conversion method, information conversion device, and recording medium

Locality-Sensitive Hashing with Margin Based Feature Selection

Markov Chain Monte Carlo for Arrangement of Hyperplanes in Locality-Sensitive Hashing

Locality-Sensitive Hashing with Margin Based Feature Selection