3,773 research outputs found
Lower Bounds for Sparse Recovery
We consider the following k-sparse recovery problem: design an m x n matrix
A, such that for any signal x, given Ax we can efficiently recover x'
satisfying
||x-x'||_1 <= C min_{k-sparse} x"} ||x-x"||_1.
It is known that there exist matrices A with this property that have only O(k
log (n/k)) rows.
In this paper we show that this bound is tight. Our bound holds even for the
more general /randomized/ version of the problem, where A is a random variable
and the recovery algorithm is required to work for any fixed x with constant
probability (over A).Comment: 11 pages. Appeared at SODA 201
Compressive Mining: Fast and Optimal Data Mining in the Compressed Domain
Real-world data typically contain repeated and periodic patterns. This
suggests that they can be effectively represented and compressed using only a
few coefficients of an appropriate basis (e.g., Fourier, Wavelets, etc.).
However, distance estimation when the data are represented using different sets
of coefficients is still a largely unexplored area. This work studies the
optimization problems related to obtaining the \emph{tightest} lower/upper
bound on Euclidean distances when each data object is potentially compressed
using a different set of orthonormal coefficients. Our technique leads to
tighter distance estimates, which translates into more accurate search,
learning and mining operations \textit{directly} in the compressed domain.
We formulate the problem of estimating lower/upper distance bounds as an
optimization problem. We establish the properties of optimal solutions, and
leverage the theoretical analysis to develop a fast algorithm to obtain an
\emph{exact} solution to the problem. The suggested solution provides the
tightest estimation of the -norm or the correlation. We show that typical
data-analysis operations, such as k-NN search or k-Means clustering, can
operate more accurately using the proposed compression and distance
reconstruction technique. We compare it with many other prevalent compression
and reconstruction techniques, including random projections and PCA-based
techniques. We highlight a surprising result, namely that when the data are
highly sparse in some basis, our technique may even outperform PCA-based
compression.
The contributions of this work are generic as our methodology is applicable
to any sequential or high-dimensional data as well as to any orthogonal data
transformation used for the underlying data compression scheme.Comment: 25 pages, 20 figures, accepted in VLD
Building Confidential and Efficient Query Services in the Cloud with RASP Data Perturbation
With the wide deployment of public cloud computing infrastructures, using
clouds to host data query services has become an appealing solution for the
advantages on scalability and cost-saving. However, some data might be
sensitive that the data owner does not want to move to the cloud unless the
data confidentiality and query privacy are guaranteed. On the other hand, a
secured query service should still provide efficient query processing and
significantly reduce the in-house workload to fully realize the benefits of
cloud computing. We propose the RASP data perturbation method to provide secure
and efficient range query and kNN query services for protected data in the
cloud. The RASP data perturbation method combines order preserving encryption,
dimensionality expansion, random noise injection, and random projection, to
provide strong resilience to attacks on the perturbed data and queries. It also
preserves multidimensional ranges, which allows existing indexing techniques to
be applied to speedup range query processing. The kNN-R algorithm is designed
to work with the RASP range query algorithm to process the kNN queries. We have
carefully analyzed the attacks on data and queries under a precisely defined
threat model and realistic security assumptions. Extensive experiments have
been conducted to show the advantages of this approach on efficiency and
security.Comment: 18 pages, to appear in IEEE TKDE, accepted in December 201
On the Manhattan-distance between points on space-filling mesh-indexings
Indexing schemes based on space filling curves like the Hilbert
curve are a powerful tool for building efficient parallel
algorithms on mesh-connected computers. The main reason is that
they are locality-preserving, i.e., the Manhattan-distance between
processors grows only slowly with increasing index differences.
We present a simple and easy-to-verify proof that the Manhattan-
distance of any indices i and j is bounded by 3*sqrt
Practical and Optimal LSH for Angular Distance
We show the existence of a Locality-Sensitive Hashing (LSH) family for the
angular distance that yields an approximate Near Neighbor Search algorithm with
the asymptotically optimal running time exponent. Unlike earlier algorithms
with this property (e.g., Spherical LSH [Andoni, Indyk, Nguyen, Razenshteyn
2014], [Andoni, Razenshteyn 2015]), our algorithm is also practical, improving
upon the well-studied hyperplane LSH [Charikar, 2002] in practice. We also
introduce a multiprobe version of this algorithm, and conduct experimental
evaluation on real and synthetic data sets.
We complement the above positive results with a fine-grained lower bound for
the quality of any LSH family for angular distance. Our lower bound implies
that the above LSH family exhibits a trade-off between evaluation time and
quality that is close to optimal for a natural class of LSH functions.Comment: 22 pages, an extended abstract is to appear in the proceedings of the
29th Annual Conference on Neural Information Processing Systems (NIPS 2015
An LSH Index for Computing Kendall's Tau over Top-k Lists
We consider the problem of similarity search within a set of top-k lists
under the Kendall's Tau distance function. This distance describes how related
two rankings are in terms of concordantly and discordantly ordered items. As
top-k lists are usually very short compared to the global domain of possible
items to be ranked, creating an inverted index to look up overlapping lists is
possible but does not capture tight enough the similarity measure. In this
work, we investigate locality sensitive hashing schemes for the Kendall's Tau
distance and evaluate the proposed methods using two real-world datasets.Comment: 6 pages, 8 subfigures, presented in Seventeenth International
Workshop on the Web and Databases (WebDB 2014) co-located with ACM SIGMOD201
Maximum Inner-Product Search using Tree Data-structures
The problem of {\em efficiently} finding the best match for a query in a
given set with respect to the Euclidean distance or the cosine similarity has
been extensively studied in literature. However, a closely related problem of
efficiently finding the best match with respect to the inner product has never
been explored in the general setting to the best of our knowledge. In this
paper we consider this general problem and contrast it with the existing
best-match algorithms. First, we propose a general branch-and-bound algorithm
using a tree data structure. Subsequently, we present a dual-tree algorithm for
the case where there are multiple queries. Finally we present a new data
structure for increasing the efficiency of the dual-tree algorithm. These
branch-and-bound algorithms involve novel bounds suited for the purpose of
best-matching with inner products. We evaluate our proposed algorithms on a
variety of data sets from various applications, and exhibit up to five orders
of magnitude improvement in query time over the naive search technique.Comment: Under submission in KDD 201
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