Related papers: Obtuse Lattice Bases
For a left vector space V over a totally ordered division ring F, let Co(V) denote the lattice of convex subsets of V. We prove that every lattice L can be embedded into Co(V) for some left F-vector space V. Furthermore, if L is finite…
The diverse world of machine learning applications has given rise to a plethora of algorithms and optimization methods, finely tuned to the specific regression or classification task at hand. We reduce the complexity of algorithm design for…
We introduce a framework generalizing lattice reduction algorithms to module lattices in order to practically and efficiently solve the $\gamma$-Hermite Module-SVP problem over arbitrary cyclotomic fields. The core idea is to exploit the…
This paper describes the formal verification of NP-hardness reduction functions of two key problems relevant in algebraic lattice theory: the closest vector problem and the shortest vector problem, both in the infinity norm. The…
Recently, lattice-reduction-aided detectors have been proposed for multiple-input multiple-output (MIMO) systems to give performance with full diversity like maximum likelihood receiver, and yet with complexity similar to linear receivers.…
For a positive integer $s$, a lattice $L$ is said to be $s$-integrable if $\sqrt{s}\cdot L$ is isometric to a sublattice of $\mathbb{Z}^n$ for some integer $n$. Conway and Sloane found two minimal non $2$-integrable lattices of rank $12$…
For a given matrix subspace, how can we find a basis that consists of low-rank matrices? This is a generalization of the sparse vector problem. It turns out that when the subspace is spanned by rank-1 matrices, the matrices can be obtained…
The credit on {\it reduction theory} goes back to the work of Lagrange, Gauss, Hermite, Korkin, Zolotarev, and Minkowski. Modern reduction theory is voluminous and includes the work of A. Lenstra, H. Lenstra and L. Lovasz who created the…
In applications throughout science and engineering one is often faced with the challenge of solving an ill-posed inverse problem, where the number of available measurements is smaller than the dimension of the model to be estimated. However…
In this article we introduce theory and algorithms for learning discrete representations that take on a lattice that is embedded in an Euclidean space. Lattice representations possess an interesting combination of properties: a) they can be…
Lattices form a compact representation of multiple hypotheses generated from an automatic speech recognition system and have been shown to improve performance of downstream tasks like spoken language understanding and speech translation,…
We give a polynomial time Turing reduction from the $\gamma^2\sqrt{n}$-approximate closest vector problem on a lattice of dimension $n$ to a $\gamma$-approximate oracle for the shortest vector problem. This is an improvement over a…
Inverse optimization (Inverse optimal control) is the task of imputing a cost function such that given test points (trajectories) are (nearly) optimal with respect to the discovered cost. Prior methods in inverse optimization assume that…
The various ways to reduce number of vectors describing condition of particles for high energy physics problems are presented. In particular decomposition of any vector with respect to the basis, consisting of any four linearly independent…
We introduce the Vector Fitting algorithm for the creation of reduced-order models from the sampled response of a linear time-invariant system. This data-driven approach to reduction is particularly useful when the system under modeling is…
Lattices and partially ordered sets have played an increasingly important role in coding theory, providing combinatorial frameworks for studying structural and algebraic properties of error-correcting codes. Motivated by recent works…
The problem of finding the sparsest vector (direction) in a low dimensional subspace can be considered as a homogeneous variant of the sparse recovery problem, which finds applications in robust subspace recovery, dictionary learning,…
The Coxeter lattices, which we denote $A_{n/m}$, are a family of lattices containing many of the important lattices in low dimensions. This includes $A_n$, $E_7$, $E_8$ and their duals $A_n^*$, $E_7^*$ and $E_8^*$. We consider the problem…
The component-by-component construction is the standard method of finding good lattice rules or polynomial lattice rules for numerical integration. Several authors have reported that in numerical experiments the generating vector sometimes…
Finding the shortest vectors in a lattice is an NP-hard problem, so low-dimensional results also play an essential role in lattice reduction theory. Using Ryskov's result for the admissible centerings and Tammela's result for determining…