2,438 research outputs found
Tradeoffs for nearest neighbors on the sphere
We consider tradeoffs between the query and update complexities for the
(approximate) nearest neighbor problem on the sphere, extending the recent
spherical filters to sparse regimes and generalizing the scheme and analysis to
account for different tradeoffs. In a nutshell, for the sparse regime the
tradeoff between the query complexity and update complexity
for data sets of size is given by the following equation in
terms of the approximation factor and the exponents and :
For small , minimizing the time for updates leads to a linear
space complexity at the cost of a query time complexity .
Balancing the query and update costs leads to optimal complexities
, matching bounds from [Andoni-Razenshteyn, 2015] and [Dubiner,
IEEE-TIT'10] and matching the asymptotic complexities of [Andoni-Razenshteyn,
STOC'15] and [Andoni-Indyk-Laarhoven-Razenshteyn-Schmidt, NIPS'15]. A
subpolynomial query time complexity can be achieved at the cost of a
space complexity of the order , matching the bound
of [Andoni-Indyk-Patrascu, FOCS'06] and
[Panigrahy-Talwar-Wieder, FOCS'10] and improving upon results of
[Indyk-Motwani, STOC'98] and [Kushilevitz-Ostrovsky-Rabani, STOC'98].
For large , minimizing the update complexity results in a query complexity
of , improving upon the related exponent for large of
[Kapralov, PODS'15] by a factor , and matching the bound
of [Panigrahy-Talwar-Wieder, FOCS'08]. Balancing the costs leads to optimal
complexities , while a minimum query time complexity can be
achieved with update complexity , improving upon the
previous best exponents of Kapralov by a factor .Comment: 16 pages, 1 table, 2 figures. Mostly subsumed by arXiv:1608.03580
[cs.DS] (along with arXiv:1605.02701 [cs.DS]
Coding for Errors and Erasures in Random Network Coding
The problem of error-control in random linear network coding is considered. A
``noncoherent'' or ``channel oblivious'' model is assumed where neither
transmitter nor receiver is assumed to have knowledge of the channel transfer
characteristic. Motivated by the property that linear network coding is
vector-space preserving, information transmission is modelled as the injection
into the network of a basis for a vector space and the collection by the
receiver of a basis for a vector space . A metric on the projective geometry
associated with the packet space is introduced, and it is shown that a minimum
distance decoder for this metric achieves correct decoding if the dimension of
the space is sufficiently large. If the dimension of each codeword
is restricted to a fixed integer, the code forms a subset of a finite-field
Grassmannian, or, equivalently, a subset of the vertices of the corresponding
Grassmann graph. Sphere-packing and sphere-covering bounds as well as a
generalization of the Singleton bound are provided for such codes. Finally, a
Reed-Solomon-like code construction, related to Gabidulin's construction of
maximum rank-distance codes, is described and a Sudan-style ``list-1'' minimum
distance decoding algorithm is provided.Comment: This revised paper contains some minor changes and clarification
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