75,848 research outputs found
A Fast Algorithm for the Inversion of Quasiseparable Vandermonde-like Matrices
The results on Vandermonde-like matrices were introduced as a generalization
of polynomial Vandermonde matrices, and the displacement structure of these
matrices was used to derive an inversion formula. In this paper we first
present a fast Gaussian elimination algorithm for the polynomial
Vandermonde-like matrices. Later we use the said algorithm to derive fast
inversion algorithms for quasiseparable, semiseparable and well-free
Vandermonde-like matrices having complexity. To do so we
identify structures of displacement operators in terms of generators and the
recurrence relations(2-term and 3-term) between the columns of the basis
transformation matrices for quasiseparable, semiseparable and well-free
polynomials. Finally we present an algorithm to compute the
inversion of quasiseparable Vandermonde-like matrices
Sharp Quantum vs. Classical Query Complexity Separations
We obtain the strongest separation between quantum and classical query
complexity known to date -- specifically, we define a black-box problem that
requires exponentially many queries in the classical bounded-error case, but
can be solved exactly in the quantum case with a single query (and a polynomial
number of auxiliary operations). The problem is simple to define and the
quantum algorithm solving it is also simple when described in terms of certain
quantum Fourier transforms (QFTs) that have natural properties with respect to
the algebraic structures of finite fields. These QFTs may be of independent
interest, and we also investigate generalizations of them to noncommutative
finite rings.Comment: 13 pages, change in title, improvements in presentation, and minor
corrections. To appear in Algorithmic
A quantum genetic algorithm with quantum crossover and mutation operations
In the context of evolutionary quantum computing in the literal meaning, a
quantum crossover operation has not been introduced so far. Here, we introduce
a novel quantum genetic algorithm which has a quantum crossover procedure
performing crossovers among all chromosomes in parallel for each generation. A
complexity analysis shows that a quadratic speedup is achieved over its
classical counterpart in the dominant factor of the run time to handle each
generation.Comment: 21 pages, 1 table, v2: typos corrected, minor modifications in
sections 3.5 and 4, v3: minor revision, title changed (original title:
Semiclassical genetic algorithm with quantum crossover and mutation
operations), v4: minor revision, v5: minor grammatical corrections, to appear
in QI
Large-Scale MIMO Detection for 3GPP LTE: Algorithms and FPGA Implementations
Large-scale (or massive) multiple-input multiple-output (MIMO) is expected to
be one of the key technologies in next-generation multi-user cellular systems,
based on the upcoming 3GPP LTE Release 12 standard, for example. In this work,
we propose - to the best of our knowledge - the first VLSI design enabling
high-throughput data detection in single-carrier frequency-division multiple
access (SC-FDMA)-based large-scale MIMO systems. We propose a new approximate
matrix inversion algorithm relying on a Neumann series expansion, which
substantially reduces the complexity of linear data detection. We analyze the
associated error, and we compare its performance and complexity to those of an
exact linear detector. We present corresponding VLSI architectures, which
perform exact and approximate soft-output detection for large-scale MIMO
systems with various antenna/user configurations. Reference implementation
results for a Xilinx Virtex-7 XC7VX980T FPGA show that our designs are able to
achieve more than 600 Mb/s for a 128 antenna, 8 user 3GPP LTE-based large-scale
MIMO system. We finally provide a performance/complexity trade-off comparison
using the presented FPGA designs, which reveals that the detector circuit of
choice is determined by the ratio between BS antennas and users, as well as the
desired error-rate performance.Comment: To appear in the IEEE Journal of Selected Topics in Signal Processin
Fast linear algebra is stable
In an earlier paper, we showed that a large class of fast recursive matrix
multiplication algorithms is stable in a normwise sense, and that in fact if
multiplication of -by- matrices can be done by any algorithm in
operations for any , then it can be done
stably in operations for any . Here we extend
this result to show that essentially all standard linear algebra operations,
including LU decomposition, QR decomposition, linear equation solving, matrix
inversion, solving least squares problems, (generalized) eigenvalue problems
and the singular value decomposition can also be done stably (in a normwise
sense) in operations.Comment: 26 pages; final version; to appear in Numerische Mathemati
Efficient Higher Order Derivatives of Objective Functions Composed of Matrix Operations
This paper is concerned with the efficient evaluation of higher-order
derivatives of functions that are composed of matrix operations. I.e., we
want to compute the -th derivative tensor , where is given as an algorithm that
consists of many matrix operations. We propose a method that is a combination
of two well-known techniques from Algorithmic Differentiation (AD): univariate
Taylor propagation on scalars (UTPS) and first-order forward and reverse on
matrices. The combination leads to a technique that we would like to call
univariate Taylor propagation on matrices (UTPM). The method inherits many
desirable properties: It is easy to implement, it is very efficient and it
returns not only but yields in the process also the derivatives
for . As performance test we compute the gradient
% and the Hessian by a combination of forward
and reverse mode of f(X) = \trace (X^{-1}) in the reverse mode of AD for . We observe a speedup of about 100 compared to
UTPS. Due to the nature of the method, the memory footprint is also small and
therefore can be used to differentiate functions that are not accessible by
standard methods due to limited physical memory
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