Compressive Sensing (CS) theory states that real-world signals can often be recovered from much fewer measurements than those suggested by the Shannon sampling theorem. Nevertheless, recoverability does not only depend on the signal, but also on the measurement scheme. The measurement matrix should behave as close as possible to an isometry for the signals of interest. Therefore the search for optimal CS measurement matrices of size @math translates into the search for a set of @math @math -dimensional vectors with minimal coherence. Best Complex Antipodal Spherical Codes (BCASCs) are known to be optimal in terms of coherence. An iterative algorithm for BCASC generation has been recently proposed that tightly approaches the theoretical lower bound on coherence. Unfortunately, the complexity of each iteration depends quadratically on @math and @math . In this work we propose a modification of the algorithm that allows reducing the quadratic complexity to linear on both @math and @math . Numerical evaluation showed that the proposed approach does not worsen the coherence of the resulting BCASCs. On the contrary, an improvement was observed for large @math . The reduction of the computational complexity paves the way for using the BCASCs as CS measurement matrices in problems with large @math . We evaluate the CS performance of the BCASCs for recovering sparse signals. The BCASCs are shown to outperform both complex random matrices and Fourier ensembles as CS measurement matrices, both in terms of coherence and sparse recovery performance, especially for low @math , which is the case of interest in CS.
In short terms, the proposed ANN approximate approach and the reference algorithm generate BCASCs of equivalent quality in terms of coherence. In other words, Tables and confirm that the reduction of the algorithm complexity has no significant effect on the quality of the obtained BCASCs. In fact, both alternatives often produce codes with equal coherence, closely approaching and eventually meeting the theoretical lower bound. Cases for which our approach meets the theoretical lower bound are @math , @math and @math , @math . Furthermore, in some cases the proposed approach outperforms the reference method, which is a remarkable fact, even when the differences are rather negligible in general. The reader might observe that the coherences obtained for the reference method often do not coincide with those given in @cite_15 . This is due to the fact that we use the variant with approximate discrete summation for calculating the integral over complex rotations, in order to enable a fair comparison to the proposed approach, while the results presented in Tables I and II of @cite_15 were obtained using numerical integration.
@cite_15
2169273233 2069278600 2109323956 2460613287
Vector sets with optimal coherence according to the Welch bound cannot exist for all pairs of dimension and cardinality. If such an optimal vector set exists, it is an equiangular tight frame and represents the solution to a Grassmannian line packing problem. Best Complex Antipodal Spherical Codes (BCASCs) are the best vector sets with respect to the coherence. By extending methods used to find best spherical codes in the real-valued Euclidean space, the proposed approach aims to find BCASCs, and thereby, a complex-valued vector set with minimal coherence. There are many applications demanding vector sets with low coherence. Examples are not limited to several techniques in wireless communication or to the field of compressed sensing. Within this contribution, existing analytical and numerical approaches for coherence optimization of complex-valued vector spaces are summarized and compared to the proposed approach. The numerically obtained coherence values improve previously reported results. The drawback of increased computational effort is addressed and a faster approximation is proposed which may be an alternative for time critical cases. We present a new efficient algorithm for the search version of the approximate Closest Vector Problem with Preprocessing (CVPP). Our algorithm achieves an approximation factor of O(n sqrt log n ), improving on the previous best of O(n^ 1.5 ) due to Lag arias, Lenstra, and Schnorr hkzbabai . We also show, somewhat surprisingly, that only O(n) vectors of preprocessing advice are sufficient to solve the problem (with the slightly worse approximation factor of O(n)). We remark that this still leaves a large gap with respect to the decisional version of CVPP, where the best known approximation factor is O(sqrt n log n ) due to Aharonov and Regev AharonovR04 . To achieve these results, we show a reduction to the same problem restricted to target points that are close to the lattice and a more efficient reduction to a harder problem, Bounded Distance Decoding with preprocessing (BDDP). Combining either reduction with the previous best-known algorithm for BDDP by Liu, Lyubashevsky, and Micciancio LiuLM06 gives our main result. In the setting of CVP without preprocessing, we also give a reduction from (1+eps)gamma approximate CVP to gamma approximate CVP where the target is at distance at most 1+1 eps times the minimum distance (the length of the shortest non-zero vector) which relies on the lattice sparsification techniques of Dadush and Kun DadushK13 . As our final and most technical contribution, we present a substantially more efficient variant of the LLM algorithm (both in terms of run-time and amount of preprocessing advice), and via an improved analysis, show that it can decode up to a distance proportional to the reciprocal of the smoothing parameter of the dual lattice MR04 . We show that this is never smaller than the LLM decoding radius, and that it can be up to an wide tilde Omega (sqrt n ) factor larger. We propose a novel algorithm to register multiple 3D point sets within a common reference frame using a manifold optimization approach. The point sets are obtained with multiple laser scanners or a mobile scanner. Unlike most prior algorithms, our approach performs an explicit optimization on the manifold of rotations, allowing us to formulate the registration problem as an unconstrained minimization on a constrained manifold. This approach exploits the Lie group structure of SO 3 and the simple representation of its associated Lie algebra so 3 in terms of R3. Our contributions are threefold. We present a new analytic method based on singular value decompositions that yields a closed-form solution for simultaneous multiview registration in the noise-free scenario. Secondly, we use this method to derive a good initial estimate of a solution in the noise-free case. This initialization step may be of use in any general iterative scheme. Finally, we present an iterative scheme based on Newton's method on SO 3 that has locally quadratic convergence. We demonstrate the efficacy of our scheme on scan data taken both from the Digital Michelangelo project and from scans extracted from models, and compare it to some of the other well known schemes for multiview registration. In all cases, our algorithm converges much faster than the other approaches, (in some cases orders of magnitude faster), and generates consistently higher quality registrations. We present a new approach for efficient approximate nearest neighbor (ANN) search in high dimensional spaces, extending the idea of Product Quantization. We propose a two level product and vector quantization tree that reduces the number of vector comparisons required during tree traversal. Our approach also includes a novel highly parallelizable re-ranking method for candidate vectors by efficiently reusing already computed intermediate values. Due to its small memory footprint during traversal the method lends itself to an efficient, parallel GPU implementation. This Product Quantization Tree (PQT) approach significantly outperforms recent state of the art methods for high dimensional nearest neighbor queries on standard reference datasets. Ours is the first work that demonstrates GPU performance superior to CPU performance on high dimensional, large scale ANN problems in time-critical real-world applications, like loop-closing in videos.