1,947 research outputs found
Reconstruction of Integers from Pairwise Distances
Given a set of integers, one can easily construct the set of their pairwise
distances. We consider the inverse problem: given a set of pairwise distances,
find the integer set which realizes the pairwise distance set. This problem
arises in a lot of fields in engineering and applied physics, and has
confounded researchers for over 60 years. It is one of the few fundamental
problems that are neither known to be NP-hard nor solvable by polynomial-time
algorithms. Whether unique recovery is possible also remains an open question.
In many practical applications where this problem occurs, the integer set is
naturally sparse (i.e., the integers are sufficiently spaced), a property which
has not been explored. In this work, we exploit the sparse nature of the
integer set and develop a polynomial-time algorithm which provably recovers the
set of integers (up to linear shift and reversal) from the set of their
pairwise distances with arbitrarily high probability if the sparsity is
O(n^{1/2-\eps}). Numerical simulations verify the effectiveness of the
proposed algorithm.Comment: 14 pages, 4 figures, submitted to ICASSP 201
Sparse PCA: Optimal rates and adaptive estimation
Principal component analysis (PCA) is one of the most commonly used
statistical procedures with a wide range of applications. This paper considers
both minimax and adaptive estimation of the principal subspace in the high
dimensional setting. Under mild technical conditions, we first establish the
optimal rates of convergence for estimating the principal subspace which are
sharp with respect to all the parameters, thus providing a complete
characterization of the difficulty of the estimation problem in term of the
convergence rate. The lower bound is obtained by calculating the local metric
entropy and an application of Fano's lemma. The rate optimal estimator is
constructed using aggregation, which, however, might not be computationally
feasible. We then introduce an adaptive procedure for estimating the principal
subspace which is fully data driven and can be computed efficiently. It is
shown that the estimator attains the optimal rates of convergence
simultaneously over a large collection of the parameter spaces. A key idea in
our construction is a reduction scheme which reduces the sparse PCA problem to
a high-dimensional multivariate regression problem. This method is potentially
also useful for other related problems.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1178 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Acceleration Techniques for Sparse Recovery Based Plane-wave Decomposition of a Sound Field
Plane-wave decomposition by sparse recovery is a reliable and accurate technique for plane-wave decomposition which can be used for source localization, beamforming, etc. In this work, we introduce techniques to accelerate the plane-wave decomposition by sparse recovery. The method consists of two main algorithms which are spherical Fourier transformation (SFT) and sparse recovery. Comparing the two algorithms, the sparse recovery is the most computationally intensive. We implement the SFT on an FPGA and the sparse recovery on a multithreaded computing platform. Then the multithreaded computing platform could be fully utilized for the sparse recovery. On the other hand, implementing the SFT on an FPGA helps to flexibly integrate the microphones and improve the portability of the microphone array. For implementing the SFT on an FPGA, we develop a scalable FPGA design model that enables the quick design of the SFT architecture on FPGAs. The model considers the number of microphones, the number of SFT channels and the cost of the FPGA and provides the design of a resource optimized and cost-effective FPGA architecture as the output. Then we investigate the performance of the sparse recovery algorithm executed on various multithreaded computing platforms (i.e., chip-multiprocessor, multiprocessor, GPU, manycore). Finally, we investigate the influence of modifying the dictionary size on the computational performance and the accuracy of the sparse recovery algorithms. We introduce novel sparse-recovery techniques which use non-uniform dictionaries to improve the performance of the sparse recovery on a parallel architecture
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