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Register a new userNearest Neighbor Search for Hyperbolic Embeddings
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Added by
Xian Wu
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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Embedding into hyperbolic space is emerging as an effective representation technique for datasets that exhibit hierarchical structure. This development motivates the need for algorithms that are able to effectively extract knowledge and insights from datapoints embedded in negatively curved spaces. We focus on the problem of nearest neighbor search, a fundamental problem in data analysis. We present efficient algorithmic solutions that build upon established methods for nearest neighbor search in Euclidean space, allowing for easy adoption and integration with existing systems. We prove theoretical guarantees for our techniques and our experiments demonstrate the effectiveness of our approach on real datasets over competing algorithms.
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Approximate nearest neighbor algorithms are used to speed up nearest neighbor search in a wide array of applications. However, current indexing methods feature several hyperparameters that need to be tuned to reach an acceptable accuracy--speed trade-off. A grid search in the parameter space is often impractically slow due to a time-consuming index-building procedure. Therefore, we propose an algorithm for automatically tuning the hyperparameters of indexing methods based on randomized space-partitioning trees. In particular, we present results using randomized k-d trees, random projection trees and randomized PCA trees. The tuning algorithm adds minimal overhead to the index-building process but is able to find the optimal hyperparameters accurately. We demonstrate that the algorithm is significantly faster than existing approaches, and that the indexing methods used are competitive with the state-of-the-art methods in query time while being faster to build.
A recent series of papers by Andoni, Naor, Nikolov, Razenshteyn, and Waingarten (STOC 2018, FOCS 2018) has given approximate near neighbour search (NNS) data structures for a wide class of distance metrics, including all norms. In particular, these data structures achieve approximation on the order of $p$ for $ell_p^d$ norms with space complexity nearly linear in the dataset size $n$ and polynomial in the dimension $d$, and query time sub-linear in $n$ and polynomial in $d$. The main shortcoming is the exponential in $d$ pre-processing time required for their construction. In this paper, we describe a more direct framework for constructing NNS data structures for general norms. More specifically, we show via an algorithmic reduction that an efficient NNS data structure for a given metric is implied by an efficient average distortion embedding of it into $ell_1$ or into Euclidean space. In particular, the resulting data structures require only polynomial pre-processing time, as long as the embedding can be computed in polynomial time. As a concrete instantiation of this framework, we give an NNS data structure for $ell_p$ with efficient pre-processing that matches the approximation factor, space and query complexity of the aforementioned data structure of Andoni et al. On the way, we resolve a question of Naor (Analysis and Geometry in Metric Spaces, 2014) and provide an explicit, efficiently computable embedding of $ell_p$, for $p ge 2$, into $ell_2$ with (quadratic) average distortion on the order of $p$. We expect our approach to pave the way for constructing efficient NNS data structures for all norms.
High-dimensional Nearest Neighbor (NN) search is central in multimedia search systems. Product Quantization (PQ) is a widespread NN search technique which has a high performance and good scalability. PQ compresses high-dimensional vectors into compact codes thanks to a combination of quantizers. Large databases can, therefore, be stored entirely in RAM, enabling fast responses to NN queries. In almost all cases, PQ uses 8-bit quantizers as they offer low response times. In this paper, we advocate the use of 16-bit quantizers. Compared to 8-bit quantizers, 16-bit quantizers boost accuracy but they increase response time by a factor of 3 to 10. We propose a novel approach that allows 16-bit quantizers to offer the same response time as 8-bit quantizers, while still providing a boost of accuracy. Our approach builds on two key ideas: (i) the construction of derived codebooks that allow a fast and approximate distance evaluation, and (ii) a two-pass NN search procedure which builds a candidate set using the derived codebooks, and then refines it using 16-bit quantizers. On 1 billion SIFT vectors, with an inverted index, our approach offers a Recall@100 of 0.85 in 5.2 ms. By contrast, 16-bit quantizers alone offer a Recall@100 of 0.85 in 39 ms, and 8-bit quantizers a Recall@100 of 0.82 in 3.8 ms.
A triplet comparison oracle on a set $S$ takes an object $x in S$ and for any pair ${y, z} subset S setminus {x}$ declares which of $y$ and $z$ is more similar to $x$. Partitioned Local Depth (PaLD) supplies a principled non-parametric partitioning of $S$ under such triplet comparisons but needs $O(n^2 log{n})$ oracle calls and $O(n^3)$ post-processing steps. We introduce Partitioned Nearest Neighbors Local Depth (PaNNLD), a computationally tractable variant of PaLD leveraging the $K$-nearest neighbors digraph on $S$. PaNNLD needs only $O(n K log{n})$ oracle calls, by replacing an oracle call by a coin flip when neither $y$ nor $z$ is adjacent to $x$ in the undirected version of the $K$-nearest neighbors digraph. By averaging over randomizations, PaNNLD subsequently requires (at best) only $O(n K^2)$ post-processing steps. Concentration of measure shows that the probability of randomization-induced error $delta$ in PaNNLD is no more than $2 e^{-delta^2 K^2}$.
Fast approximate nearest neighbor (NN) search in large databases is becoming popular. Several powerful learning-based formulations have been proposed recently. However, not much attention has been paid to a more fundamental question: how difficult is (approximate) nearest neighbor search in a given data set? And which data properties affect the difficulty of nearest neighbor search and how? This paper introduces the first concrete measure called Relative Contrast that can be used to evaluate the influence of several crucial data characteristics such as dimensionality, sparsity, and database size simultaneously in arbitrary normed metric spaces. Moreover, we present a theoretical analysis to prove how the difficulty measure (relative contrast) determines/affects the complexity of Local Sensitive Hashing, a popular approximate NN search method. Relative contrast also provides an explanation for a family of heuristic hashing algorithms with good practical performance based on PCA. Finally, we show that most of the previous works in measuring NN search meaningfulness/difficulty can be derived as special asymptotic cases for dense vectors of the proposed measure.
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