arXiv subjects

Ioannis Psarros

Publications and source records attributed to Ioannis Psarros.

20 records · Page 2Linked to original sources

arXiv2016

Practical linear-space Approximate Near Neighbors in high dimension

The $c$-approximate Near Neighbor problem in high dimensional spaces has been mainly addressed by Locality Sensitive Hashing (LSH), which offers polynomial dependence on the dimension, query time sublinear in the size of the dataset, and subquadratic space requirement. For practical applications, linear space is typically imperative. Most previous work in the linear space regime focuses on the case that $c$ exceeds $1$ by a constant term. In a recently accepted paper, optimal bounds have been achieved for any $c>1$ \cite{ALRW17}. Towards practicality, we present a new and simple data structure using linear space and sublinear query time for any $c>1$ including $c\to 1^+$. Given an LSH family of functions for some metric space, we randomly project points to the Hamming cube of dimension $\log n$, where $n$ is the number of input points. The projected space contains strings which serve as keys for buckets containing the input points. The query algorithm simply projects the query point, then examines points which are assigned to the same or nearby vertices on the Hamming cube. We analyze in detail the query time for some standard LSH families. To illustrate our claim of practicality, we offer an open-source implementation in {\tt C++}, and report on several experiments in dimension up to 1000 and $n$ up to $10^6$. Our algorithm is one to two orders of magnitude faster than brute force search. Experiments confirm the sublinear dependence on $n$ and the linear dependence on the dimension. We have compared against state-of-the-art LSH-based library {\tt FALCONN}: our search is somewhat slower, but memory usage and preprocessing time are significantly smaller.

cs.CG↗

arXiv2016

Randomized embeddings with slack, and high-dimensional Approximate Nearest Neighbor

The approximate nearest neighbor problem ($ε$-ANN) in high dimensional Euclidean space has been mainly addressed by Locality Sensitive Hashing (LSH), which has polynomial dependence in the dimension, sublinear query time, but subquadratic space requirement. In this paper, we introduce a new definition of "low-quality" embeddings for metric spaces. It requires that, for some query point $q$, there exists an approximate nearest neighbor among the pre-images of the $k>1$ approximate nearest neighbors in the target space. Focusing on Euclidean spaces, we employ random projections in order to reduce the original problem to one in a space of dimension inversely proportional to $k$. The $k$ approximate nearest neighbors can be efficiently retrieved by a data structure such as BBD-trees. The same approach is applied to the problem of computing an approximate near neighbor, where we obtain a data structure requiring linear space, and query time in $O(d n^ρ)$, for $ρ\approx 1-ε^2/\log(1/ε)$. This directly implies a solution for $ε$-ANN, while achieving a better exponent in the query time than the method based on BBD-trees. Better bounds are obtained in the case of doubling subsets of $\ell_2$, by combining our method with $r$-nets. We implement our method in C++, and present experimental results in dimension up to $500$ and $10^6$ points, which show that performance is better than predicted by the analysis. In addition, we compare our ANN approach to E2LSH, which implements LSH, and we show that the theoretical advantages of each method are reflected on their actual performance.

cs.CG↗