18,583 research outputs found
Approximate Nearest Neighbor Search for Low Dimensional Queries
We study the Approximate Nearest Neighbor problem for metric spaces where the
query points are constrained to lie on a subspace of low doubling dimension,
while the data is high-dimensional. We show that this problem can be solved
efficiently despite the high dimensionality of the data.Comment: 25 page
Analysis of approximate nearest neighbor searching with clustered point sets
We present an empirical analysis of data structures for approximate nearest
neighbor searching. We compare the well-known optimized kd-tree splitting
method against two alternative splitting methods. The first, called the
sliding-midpoint method, which attempts to balance the goals of producing
subdivision cells of bounded aspect ratio, while not producing any empty cells.
The second, called the minimum-ambiguity method is a query-based approach. In
addition to the data points, it is also given a training set of query points
for preprocessing. It employs a simple greedy algorithm to select the splitting
plane that minimizes the average amount of ambiguity in the choice of the
nearest neighbor for the training points. We provide an empirical analysis
comparing these two methods against the optimized kd-tree construction for a
number of synthetically generated data and query sets. We demonstrate that for
clustered data and query sets, these algorithms can provide significant
improvements over the standard kd-tree construction for approximate nearest
neighbor searching.Comment: 20 pages, 8 figures. Presented at ALENEX '99, Baltimore, MD, Jan
15-16, 199
Down the Rabbit Hole: Robust Proximity Search and Density Estimation in Sublinear Space
For a set of points in , and parameters and \eps, we present
a data structure that answers (1+\eps,k)-\ANN queries in logarithmic time.
Surprisingly, the space used by the data-structure is \Otilde (n /k); that
is, the space used is sublinear in the input size if is sufficiently large.
Our approach provides a novel way to summarize geometric data, such that
meaningful proximity queries on the data can be carried out using this sketch.
Using this, we provide a sublinear space data-structure that can estimate the
density of a point set under various measures, including:
\begin{inparaenum}[(i)]
\item sum of distances of closest points to the query point, and
\item sum of squared distances of closest points to the query point.
\end{inparaenum}
Our approach generalizes to other distance based estimation of densities of
similar flavor. We also study the problem of approximating some of these
quantities when using sampling. In particular, we show that a sample of size
\Otilde (n /k) is sufficient, in some restricted cases, to estimate the above
quantities. Remarkably, the sample size has only linear dependency on the
dimension
Bolt: Accelerated Data Mining with Fast Vector Compression
Vectors of data are at the heart of machine learning and data mining.
Recently, vector quantization methods have shown great promise in reducing both
the time and space costs of operating on vectors. We introduce a vector
quantization algorithm that can compress vectors over 12x faster than existing
techniques while also accelerating approximate vector operations such as
distance and dot product computations by up to 10x. Because it can encode over
2GB of vectors per second, it makes vector quantization cheap enough to employ
in many more circumstances. For example, using our technique to compute
approximate dot products in a nested loop can multiply matrices faster than a
state-of-the-art BLAS implementation, even when our algorithm must first
compress the matrices.
In addition to showing the above speedups, we demonstrate that our approach
can accelerate nearest neighbor search and maximum inner product search by over
100x compared to floating point operations and up to 10x compared to other
vector quantization methods. Our approximate Euclidean distance and dot product
computations are not only faster than those of related algorithms with slower
encodings, but also faster than Hamming distance computations, which have
direct hardware support on the tested platforms. We also assess the errors of
our algorithm's approximate distances and dot products, and find that it is
competitive with existing, slower vector quantization algorithms.Comment: Research track paper at KDD 201
Robust Proximity Search for Balls using Sublinear Space
Given a set of n disjoint balls b1, . . ., bn in IRd, we provide a data
structure, of near linear size, that can answer (1 \pm \epsilon)-approximate
kth-nearest neighbor queries in O(log n + 1/\epsilon^d) time, where k and
\epsilon are provided at query time. If k and \epsilon are provided in advance,
we provide a data structure to answer such queries, that requires (roughly)
O(n/k) space; that is, the data structure has sublinear space requirement if k
is sufficiently large
Probabilistic Polynomials and Hamming Nearest Neighbors
We show how to compute any symmetric Boolean function on variables over
any field (as well as the integers) with a probabilistic polynomial of degree
and error at most . The degree
dependence on and is optimal, matching a lower bound of Razborov
(1987) and Smolensky (1987) for the MAJORITY function. The proof is
constructive: a low-degree polynomial can be efficiently sampled from the
distribution.
This polynomial construction is combined with other algebraic ideas to give
the first subquadratic time algorithm for computing a (worst-case) batch of
Hamming distances in superlogarithmic dimensions, exactly. To illustrate, let
. Suppose we are given a database
of vectors in and a collection of query vectors
in the same dimension. For all , we wish to compute a
with minimum Hamming distance from . We solve this problem in randomized time. Hence, the problem is in "truly subquadratic"
time for dimensions, and in subquadratic time for . We apply the algorithm to computing pairs with maximum
inner product, closest pair in for vectors with bounded integer
entries, and pairs with maximum Jaccard coefficients.Comment: 16 pages. To appear in 56th Annual IEEE Symposium on Foundations of
Computer Science (FOCS 2015
Space Exploration via Proximity Search
We investigate what computational tasks can be performed on a point set in
, if we are only given black-box access to it via nearest-neighbor
search. This is a reasonable assumption if the underlying point set is either
provided implicitly, or it is stored in a data structure that can answer such
queries. In particular, we show the following: (A) One can compute an
approximate bi-criteria -center clustering of the point set, and more
generally compute a greedy permutation of the point set. (B) One can decide if
a query point is (approximately) inside the convex-hull of the point set.
We also investigate the problem of clustering the given point set, such that
meaningful proximity queries can be carried out on the centers of the clusters,
instead of the whole point set
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