Unsupervised nearest neighbors is the foundation of many other learning methods, notably manifold learning and spectral clustering. Supervised neighbors-based learning comes in two flavors: classification for data with discrete labels, and regression for data with continuous labels. The principle behind nearest neighbor methods is to find a predefined number of training samples closest in distance to the new point, and predict the label from these. The number of samples can be a user-defined constant (k-nearest neighbor learning), or vary based on the local density of points (radius-based neighbor learning). The distance can, in general, be any metric measure: standard Euclidean distance is the most common choice. Neighbors-based methods are known as non-generalizing machine learning methods, since they simply "remember" all of its training data (possibly transformed into a fast indexing structure such as a Ball Tree or KD Tree.).

Hyvönen, Ville, Pitkänen, Teemu, Tasoulis, Sotiris, Jääsaari, Elias, Tuomainen, Risto, Wang, Liang, Corander, Jukka, Roos, Teemu

Efficient index structures for fast approximate nearest neighbor queries are required in many applications such as recommendation systems. In high-dimensional spaces, many conventional methods suffer from excessive usage of memory and slow response times. We propose a method where multiple random projection trees are combined by a novel voting scheme. The key idea is to exploit the redundancy in a large number of candidate sets obtained by independently generated random projections in order to reduce the number of expensive exact distance evaluations. The method is straightforward to implement using sparse projections which leads to a reduced memory footprint and fast index construction. Furthermore, it enables grouping of the required computations into big matrix multiplications, which leads to additional savings due to cache effects and low-level parallelization. We demonstrate by extensive experiments on a wide variety of data sets that the method is faster than existing partitioning tree or hashing based approaches, making it the fastest available technique on high accuracy levels.

Dong, Yihe, Indyk, Piotr, Razenshteyn, Ilya, Wagner, Tal

Most of the efficient sublinear-time indexing algorithms for the high-dimensional nearest neighbor search problem (NNS) are based on space partitions of the ambient space $\mathbb{R}^d$. Inspired by recent theoretical work on NNS for general metric spaces [Andoni, Naor, Nikolov, Razenshteyn, Waingarten STOC 2018, FOCS 2018], we develop a new framework for constructing such partitions that reduces the problem to balanced graph partitioning followed by supervised classification. We instantiate this general approach with the KaHIP graph partitioner [Sanders, Schulz SEA 2013] and neural networks, respectively, to obtain a new partitioning procedure called Neural Locality-Sensitive Hashing (Neural LSH). On several standard benchmarks for NNS, our experiments show that the partitions found by Neural LSH consistently outperform partitions found by quantization- and tree-based methods.

Fichtenberger, Hendrik, Rohde, Dennis

In the $k$-nearest neighborhood model ($k$-NN), we are given a set of points $P$, and we shall answer queries $q$ by returning the $k$ nearest neighbors of $q$ in $P$ according to some metric. This concept is crucial in many areas of data analysis and data processing, e.g., computer vision, document retrieval and machine learning. Many $k$-NN algorithms have been published and implemented, but often the relation between parameters and accuracy of the computed $k$-NN is not explicit. We study property testing of $k$-NN graphs in theory and evaluate it empirically: given a point set $P \subset \mathbb{R}^\delta$ and a directed graph $G=(P,E)$, is $G$ a $k$-NN graph, i.e., every point $p \in P$ has outgoing edges to its $k$ nearest neighbors, or is it $\epsilon$-far from being a $k$-NN graph? Here, $\epsilon$-far means that one has to change more than an $\epsilon$-fraction of the edges in order to make $G$ a $k$-NN graph. We develop a randomized algorithm with one-sided error that decides this question, i.e., a property tester for the $k$-NN property, with complexity $O(\sqrt{n} k^2 / \epsilon^2)$ measured in terms of the number of vertices and edges it inspects, and we prove a lower bound of $\Omega(\sqrt{n / \epsilon k})$. We evaluate our tester empirically on the $k$-NN models computed by various algorithms and show that it can be used to detect $k$-NN models with bad accuracy in significantly less time than the building time of the $k$-NN model.