396 research outputs found
Lower Bounds on Time-Space Trade-Offs for Approximate Near Neighbors
We show tight lower bounds for the entire trade-off between space and query
time for the Approximate Near Neighbor search problem. Our lower bounds hold in
a restricted model of computation, which captures all hashing-based approaches.
In articular, our lower bound matches the upper bound recently shown in
[Laarhoven 2015] for the random instance on a Euclidean sphere (which we show
in fact extends to the entire space using the techniques from
[Andoni, Razenshteyn 2015]).
We also show tight, unconditional cell-probe lower bounds for one and two
probes, improving upon the best known bounds from [Panigrahy, Talwar, Wieder
2010]. In particular, this is the first space lower bound (for any static data
structure) for two probes which is not polynomially smaller than for one probe.
To show the result for two probes, we establish and exploit a connection to
locally-decodable codes.Comment: 47 pages, 2 figures; v2: substantially revised introduction, lots of
small corrections; subsumed by arXiv:1608.03580 [cs.DS] (along with
arXiv:1511.07527 [cs.DS]
Optimal Hashing-based Time-Space Trade-offs for Approximate Near Neighbors
[See the paper for the full abstract.]
We show tight upper and lower bounds for time-space trade-offs for the
-Approximate Near Neighbor Search problem. For the -dimensional Euclidean
space and -point datasets, we develop a data structure with space and query time for
every such that: \begin{equation} c^2 \sqrt{\rho_q} +
(c^2 - 1) \sqrt{\rho_u} = \sqrt{2c^2 - 1}. \end{equation}
This is the first data structure that achieves sublinear query time and
near-linear space for every approximation factor , improving upon
[Kapralov, PODS 2015]. The data structure is a culmination of a long line of
work on the problem for all space regimes; it builds on Spherical
Locality-Sensitive Filtering [Becker, Ducas, Gama, Laarhoven, SODA 2016] and
data-dependent hashing [Andoni, Indyk, Nguyen, Razenshteyn, SODA 2014] [Andoni,
Razenshteyn, STOC 2015].
Our matching lower bounds are of two types: conditional and unconditional.
First, we prove tightness of the whole above trade-off in a restricted model of
computation, which captures all known hashing-based approaches. We then show
unconditional cell-probe lower bounds for one and two probes that match the
above trade-off for , improving upon the best known lower bounds
from [Panigrahy, Talwar, Wieder, FOCS 2010]. In particular, this is the first
space lower bound (for any static data structure) for two probes which is not
polynomially smaller than the one-probe bound. To show the result for two
probes, we establish and exploit a connection to locally-decodable codes.Comment: 62 pages, 5 figures; a merger of arXiv:1511.07527 [cs.DS] and
arXiv:1605.02701 [cs.DS], which subsumes both of the preprints. New version
contains more elaborated proofs and fixed some typo
Hybrid Cloud-Based Privacy Preserving Clustering as Service for Enterprise Big Data
Clustering as service is being offered by many cloud service providers. It helps enterprises to learn hidden patterns and learn knowledge from large, big data generated by enterprises. Though it brings lot of value to enterprises, it also exposes the data to various security and privacy threats. Privacy preserving clustering is being proposed a solution to address this problem. But the privacy preserving clustering as outsourced service model involves too much overhead on querying user, lacks adaptivity to incremental data and involves frequent interaction between service provider and the querying user. There is also a lack of personalization to clustering by the querying user. This work “Locality Sensitive Hashing for Transformed Dataset (LSHTD)” proposes a hybrid cloud-based clustering as service model for streaming data that address the problems in the existing model such as privacy preserving k-means clustering outsourcing under multiple keys (PPCOM) and secure nearest neighbor clustering (SNNC) models, The solution combines hybrid cloud, LSHTD clustering algorithm as outsourced service model. Through experiments, the proposed solution is able is found to reduce the computation cost by 23% and communication cost by 6% and able to provide better clustering accuracy with ARI greater than 4.59% compared to existing works
Hardness of Approximate Nearest Neighbor Search
We prove conditional near-quadratic running time lower bounds for approximate
Bichromatic Closest Pair with Euclidean, Manhattan, Hamming, or edit distance.
Specifically, unless the Strong Exponential Time Hypothesis (SETH) is false,
for every there exists a constant such that computing a
-approximation to the Bichromatic Closest Pair requires
time. In particular, this implies a near-linear query time for
Approximate Nearest Neighbor search with polynomial preprocessing time.
Our reduction uses the Distributed PCP framework of [ARW'17], but obtains
improved efficiency using Algebraic Geometry (AG) codes. Efficient PCPs from AG
codes have been constructed in other settings before [BKKMS'16, BCGRS'17], but
our construction is the first to yield new hardness results
Accelerating Frank-Wolfe Algorithm using Low-Dimensional and Adaptive Data Structures
In this paper, we study the problem of speeding up a type of optimization
algorithms called Frank-Wolfe, a conditional gradient method. We develop and
employ two novel inner product search data structures, improving the prior
fastest algorithm in [Shrivastava, Song and Xu, NeurIPS 2021].
* The first data structure uses low-dimensional random projection to reduce
the problem to a lower dimension, then uses efficient inner product data
structure. It has preprocessing time and
per iteration cost for small constant .
* The second data structure leverages the recent development in adaptive
inner product search data structure that can output estimations to all inner
products. It has preprocessing time and per iteration cost
.
The first algorithm improves the state-of-the-art (with preprocessing time
and per iteration cost ) in all
cases, while the second one provides an even faster preprocessing time and is
suitable when the number of iterations is small
Optimal hashing-based time-space trade-offs for approximate near neighbors
[See the paper for the full abstract.] We show tight upper and lower bounds for time-space trade-offs for the -Approximate Near Neighbor Search problem. For the -dimensional Euclidean space and -point datasets, we develop a data structure with space and query time for every such that: \begin{equation} c^2 \sqrt{\rho_q} + (c^2 - 1) \sqrt{\rho_u} = \sqrt{2c^2 - 1}. \end{equation} This is the first data structure that achieves sublinear query time and near-linear space for every approximation factor c > 1, improving upon [Kapralov, PODS 2015]. The data structure is a culmination of a long line of work on the problem for all space regimes; it builds on Spherical Locality-Sensitive Filtering [Becker, Ducas, Gama, Laarhoven, SODA 2016] and data-dependent hashing [Andoni, Indyk, Nguyen, Razenshteyn, SODA 2014] [Andoni, Razenshteyn, STOC 2015]. Our matching lower bounds are of two types: conditional and unconditional. First, we prove tightness of the whole above trade-off in a restricted model of computation, which captures all known hashing-based approaches. We then show unconditional cell-probe lower bounds for one and two probes that match the above trade-off for , improving upon the best known lower bounds from [Panigrahy, Talwar, Wieder, FOCS 2010]. In particular, this is the first space lower bound (for any static data structure) for two probes which is not polynomially smaller than the one-probe bound. To show the result for two probes, we establish and exploit a connection to locally-decodable codes
Lower bounds on time-space trade-offs for approximate near neighbors
We show tight lower bounds for the entire trade-off between space and query time for the Approximate Near Neighbor search problem. Our lower bounds hold in a restricted model of computation, which captures all hashing-based approaches. In articular, our lower bound matches the upper bound recently shown in [Laarhoven 2015] for the random instance on a Euclidean sphere (which we show in fact extends to the entire space using the techniques from [Andoni, Razenshteyn 2015]). We also show tight, unconditional cell-probe lower bounds for one and two probes, improving upon the best known bounds from [Panigrahy, Talwar, Wieder 2010]. In particular, this is the first space lower bound (for any static data structure) for two probes which is not polynomially smaller than for one probe. To show the result for two probes, we establish and exploit a connection to locally-decodable codes
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