Related papers: Optimization and Identification of Lattice Quantiz…
New lattice quantizers with lower normalized second moments than previously reported are constructed in 13 and 14 dimensions and conjectured to be optimal. Our construction combines an initial numerical optimization with a subsequent…
The optimal lattice quantizer is the lattice which minimizes the (dimensionless) second moment $G$. In dimensions $1$ to $8$, it has been proven that the optimal lattice quantizer is one of the classical lattices, or there is good evidence…
A lattice quantizer approximates an arbitrary real-valued source vector with a vector taken from a specific discrete lattice. The quantization error is the difference between the source vector and the lattice vector. In a classic 1996…
In practical applications, lattice quantizers leverage discrete lattice points to approximate arbitrary points in the lattice. An effective lattice quantizer significantly enhances both the accuracy and efficiency of these approximations.…
40 years ago, Conway and Sloane proposed using the highly symmetrical Coxeter-Todd lattice $K_{12}$ for quantization, and estimated its second moment. Since then, all published lists identify $K_{12}$ as the best 12-dimensional lattice…
This paper investigates low-dimensional quantizers from the perspective of complex lattices. We adopt Eisenstein integers and Gaussian integers to define checkerboard lattices $\mathcal{E}_{m}$ and $\mathcal{G}_{m}$. By explicitly linking…
We consider the minimization of theta functions $\theta\_\Lambda(\alpha)=\sum\_{p\in\Lambda}e^{-\pi\alpha|p|^2}$ amongst lattices $\Lambda\subset \mathbb R^d$, by reducing the dimension of the problem, following as a motivation the case…
We present an algorithm for the exact computer-aided construction of the Voronoi cells of lattices with known symmetry group. Our algorithm scales better than linearly with the total number of faces and is applicable to dimensions beyond…
We analyze Newton's method with lazy Hessian updates for solving general possibly non-convex optimization problems. We propose to reuse a previously seen Hessian for several iterations while computing new gradients at each step of the…
Image editing is a common task across a wide range of domains, from personal use to professional applications. Despite advances in computer vision, current tools still demand significant manual effort for editing tasks that require…
This work presents a novel lattice-based methodology for incorporating multidimensional constraints into continuous decision variables within a genetic algorithm (GA) framework. The proposed approach consolidates established transcription…
This paper introduces a heuristic topology optimization framework for thin-walled, 2D extruded lattice structures subject to complex high-speed loading. The proposed framework optimizes the wall thickness distribution in the lattice cross…
The main objective of this thesis is a classification project for integral lattices. Using Kneser's neighbour method we have developed the computer program tn to classify complete genera of integral lattices. Main results are detailed…
We describe algorithms which address two classical problems in lattice geometry: the lattice covering and the simultaneous lattice packing-covering problem. Theoretically our algorithms solve the two problems in any fixed dimension d in the…
Computing the theta series of an arbitrary lattice, and more specifically a related quantity known as the flatness factor, has been recently shown to be important for lattice code design in various wireless communication setups. However,…
Quadratic form reduction and lattice reduction are fundamental tools in computational number theory and in computer science, especially in cryptography. The celebrated Lenstra-Lenstra-Lov\'asz reduction algorithm (so-called LLL) has been…
A lattice is a partially-ordered set in which every pair of elements has a unique meet (greatest lower bound) and join (least upper bound). We present new data structures for lattices that are simple, efficient, and nearly optimal in terms…
We propose a lattice-theoretic framework for modulo sampling of multidimensional bandlimited signals. Standard modulo analog-to-digital converters (ADCs) fold the signal component-wise into a square domain, reducing the recovery problem to…
Computer simulations serve as powerful tools for scientists and engineers to gain insights into complex systems. Less costly than physical experiments, computer experiments sometimes involve large number of trials. Conventional design…
In a previous paper I showed how the ideal SLAC derivative and second-derivative operators for an infinite lattice can be obtained in simple closed form in position space, and implemented very efficiently in a stochastic fashion for…