Local Testing for Membership in Lattices

Motivated by the structural analogies between point lattices and linear error-correcting codes, and by the mature theory on locally testable codes, we initiate a systematic study of local testing for membership in lattices. Testing membership in lattices is also motivated in practice, by applications to integer programming, error detection in lattice-based communication, and cryptography. Apart from establishing the conceptual foundations of lattice testing, our results include the following: 1. We demonstrate upper and lower bounds on the query complexity of local testing for the well-known family of code formula lattices. Furthermore, we instantiate our results with code formula lattices constructed from Reed-Muller codes, and obtain nearly-tight bounds. 2. We show that in order to achieve low query complexity, it is sufficient to design one-sided non-adaptive canonical tests. This result is akin to, and based on an analogous result for error-correcting codes due to Ben-Sasson et al. (SIAM J. Computing 35(1) pp1-21)

Low-lying isovector monopole resonances

The mass difference between the even-even isobaric nuclei having the valence nucleons on the same degenerate level is attributed to a Josephson-type interaction between pairs of protons and pairs of neutrons. This interaction can be understood as an isospin symmetry-breaking mean field for a four-particle interaction separable in the two particles-two holes channel. The strength of this mean field is estimated within an o(5) algebraic model, by using the experimental value of the inertial parameter for the collective isorotation induced by the breaking of the isospin symmetry. In superfluid nuclei, the presumed interaction between the proton and neutron condensates leads to coupled oscillations of the BCS gauge angles, which should appear in the excitation spectrum as low-lying isovector monopole resonances.Comment: 16 pages/LaTex + 1 PostScript figure; related to cond-mat/9904242, math-ph/000500

Lattice-based locality sensitive hashing is optimal

Locality sensitive hashing (LSH) was introduced by Indyk and Motwani (STOC ‘98) to give the first sublinear time algorithm for the c-approximate nearest neighbor (ANN) problem using only polynomial space. At a high level, an LSH family hashes “nearby” points to the same bucket and “far away” points to different buckets. The quality of measure of an LSH family is its LSH exponent, which helps determine both query time and space usage. In a seminal work, Andoni and Indyk (FOCS ‘06) constructed an LSH family based on random ball partitionings of space that achieves an LSH exponent of 1/c2 for the ℓ2 norm, which was later shown to be optimal by Motwani, Naor and Panigrahy (SIDMA ‘07) and O’Donnell, Wu and Zhou (TOCT ‘14). Although optimal in the LSH exponent, the ball partitioning approach is computationally expensive. So, in the same work, Andoni and Indyk proposed a simpler and more practical hashing scheme based on Euclidean lattices and provided computational results using the 24-dimensional Leech lattice. However, no theoretical analysis of the scheme was given, thus leaving open the question of finding the exponent of lattice based LSH. In this work, we resolve this question by showing the existence of lattices achieving the optimal LSH exponent of 1/c2 using techniques from the geometry of numbers. At a more conceptual level, our results show that optimal LSH space partitions can have periodic structure. Understanding the extent to which additional structure can be imposed on these partitions, e.g. to yield low space and query complexity, remains an important open problem

Lattice-based locality sensitive hashing is optimal

Locality sensitive hashing (LSH) was introduced by Indyk and Motwani (STOC ‘98) to give the first sublinear time algorithm for the c-approximate nearest neighbor (ANN) problem using only polynomial space. At a high level, an LSH family hashes “nearby” points to the same bucket and “far away” points to different buckets. The quality of measure of an LSH family is its LSH exponent, which helps determine both query time and space usage. In a seminal work, Andoni and Indyk (FOCS ‘06) constructed an LSH family based on random ball partitionings of space that achieves an LSH exponent of 1/c2 for the l2 norm, which was later shown to be optimal by Motwani, Naor and Panigrahy (SIDMA ‘07) and O’Donnell, Wu and Zhou (TOCT ‘14). Although optimal in the LSH exponent, the ball partitioning approach is computationally expensive. So, in the same work, Andoni and Indyk proposed a simpler and more practical hashing scheme based on Euclidean lattices and provided computational results using the 24-dimensional Leech lattice. However, no theoretical analysis of the scheme was given, thus leaving open the question of finding the exponent of lattice based LSH. In this work, we resolve this question by showing the existence of lattices achieving the optimal LSH exponent of 1/c2 using techniques from the geometry of numbers. At a more conceptual level, our results show that optimal LSH space partitions can have periodic structure. Understanding the extent to which additional structure can be imposed on these partitions, e.g. to yield low space and query complexity, remains an important open problem

Augmenting graphs to minimize the diameter

We study the problem of augmenting a weighted graph by inserting edges of bounded total cost while minimizing the diameter of the augmented graph. Our main result is an FPT 4-approximation algorithm for the problem.Comment: 15 pages, 3 figure

Interleukin 1ß, tumour necrosis factor-alpha and interleukin 1 receptor antagonist in newly diagnosed insulin-dependent diabetes mellitus: comparison to long-standing diabetes and healthy individuals

Contains fulltext : 4899.pdf (publisher's version ) (Open Access

3D T2w fetal body MRI:automated organ volumetry, growth charts and population-averaged atlas

Structural fetal body MRI provides true 3D information required for volumetry of fetal organs. However, current clinical and research practice primarily relies on manual slice-wise segmentation of raw T2-weighted stacks, which is time consuming, subject to inter- and intra-observer bias and affected by motion-corruption. Furthermore, there are no existing standard guidelines defining a universal approach to parcellation of fetal organs. This work produces the first parcellation protocol of the fetal body organs for motion-corrected 3D fetal body MRI. It includes 10 organ ROIs relevant to fetal quantitative volumetry studies. We also introduce the first population-averaged T2w MRI atlas of the fetal body. The protocol was used as a basis for training of a neural network for automated organ segmentation. It showed robust performance for different gestational ages. This solution minimises the need for manual editing and significantly reduces time. The general feasibility of the proposed pipeline was also assessed by analysis of organ growth charts created from automated parcellations of 91 normal control 3T MRI datasets that showed expected increase in volumetry during 22-38 weeks gestational age range. In addition, the results of comparison between 60 normal and 12 fetal growth restriction datasets revealed significant differences in organ volumes.</p