Fast Exact Search in Hamming Space with Multi-Index Hashing

There is growing interest in representing image data and feature descriptors using compact binary codes for fast near neighbor search. Although binary codes are motivated by their use as direct indices (addresses) into a hash table, codes longer than 32 bits are not being used as such, as it was thought to be ineffective. We introduce a rigorous way to build multiple hash tables on binary code substrings that enables exact k-nearest neighbor search in Hamming space. The approach is storage efficient and straightforward to implement. Theoretical analysis shows that the algorithm exhibits sub-linear run-time behavior for uniformly distributed codes. Empirical results show dramatic speedups over a linear scan baseline for datasets of up to one billion codes of 64, 128, or 256 bits

A New Identity for the Least-square Solution of Overdetermined Set of Linear Equations

In this paper, we prove a new identity for the least-square solution of an over-determined set of linear equation $Ax=b$, where $A$ is an $m\times n$ full-rank matrix, $b$ is a column-vector of dimension $m$, and $m$ (the number of equations) is larger than or equal to $n$ (the dimension of the unknown vector $x$). Generally, the equations are inconsistent and there is no feasible solution for $x$ unless $b$ belongs to the column-span of $A$. In the least-square approach, a candidate solution is found as the unique $x$ that minimizes the error function $\|Ax-b\|_2$. We propose a more general approach that consist in considering all the consistent subset of the equations, finding their solutions, and taking a weighted average of them to build a candidate solution. In particular, we show that by weighting the solutions with the squared determinant of their coefficient matrix, the resulting candidate solution coincides with the least square solution

Double point self-intersection surfaces of immersions

A self-transverse immersion of a smooth manifold M^{k+2} in R^{2k+2} has a double point self-intersection set which is the image of an immersion of a smooth surface, the double point self-intersection surface. We prove that this surface may have odd Euler characteristic if and only if k is congruent to 1 modulo 4 or k+1 is a power of 2. This corrects a previously published result by Andras Szucs. The method of proof is to evaluate the Stiefel-Whitney numbers of the double point self-intersection surface. By earier work of the authors these numbers can be read off from the Hurewicz image h(\alpha ) in H_{2k+2}\Omega ^{\infty }\Sigma ^{\infty }MO(k) of the element \alpha in \pi _{2k+2}\Omega ^{\infty }\Sigma ^{\infty }MO(k) corresponding to the immersion under the Pontrjagin-Thom construction.Comment: 22 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol4/paper4.abs.htm

An algorithmic proof for the completeness of two-dimensional Ising model

We show that the two dimensional Ising model is complete, in the sense that the partition function of any lattice model on any graph is equal to the partition function of the 2D Ising model with complex coupling. The latter model has all its spin-spin coupling equal to i\pi/4 and all the parameters of the original model are contained in the local magnetic fields of the Ising model. This result has already been derived by using techniques from quantum information theory and by exploiting the universality of cluster states. Here we do not use the quantum formalism and hence make the completeness result accessible to a wide audience. Furthermore our method has the advantage of being algorithmic in nature so that by following a set of simple graphical transformations, one is able to transform any discrete lattice model to an Ising model defined on a (polynomially) larger 2D lattice.Comment: 18 pages, 15 figures, Accepted for publication in Physical Review

Hydrogen adsorption in metal-organic frameworks: the role of nuclear quantum effects

The role of nuclear quantum effects on the adsorption of molecular hydrogen in metal-organic frameworks (MOFs) has been investigated on grounds of Grand-Canonical Quantized Liquid Density-Functional Theory (GC-QLDFT) calculations. For this purpose, we have carefully validated classical H2 -host interaction potentials that are obtained by fitting Born-Oppenheimer ab initio reference data. The hydrogen adsorption has first been assessed classically using Liquid Density-Functional Theory (LDFT) and the Grand-Canonical Monte Carlo (GCMC) methods. The results have been compared against the semi-classical treatment of quantum effects by applying the Feynman-Hibbs correction to the Born-Oppenheimer-derived potentials, and by explicit treatment within the Grand-Canonical Quantized Liquid Density-Functional Theory (GC-QLDFT). The results are compared with experimental data and indicate pronounced quantum and possibly many-particle effects. After validation calculations have been carried out for IRMOF-1 (MOF-5), GC-QLDFT is applied to study the adsorption of H2 in a series of MOFs, including IRMOF-4, -6, -8, -9, -10, -12, -14, -16, -18 and MOF-177. Finally, we discuss the evolution of the H2 quantum fluid with increasing pressure and lowering temperature