Related papers: An efficient algorithm for integer lattice reducti…
The algorithm for finding the optimal consistent approximation of an inconsistent pairwise comparisons matrix is based on a logarithmic transformation of a pairwise comparisons matrix into a vector space with the Euclidean metric.…
Subset-sum problems belong to the NP class and play an important role in both complexity theory and knapsack-based cryptosystems, which have been proved in the literature to become hardest when the so-called density approaches one. Lattice…
We consider the integer points in a unimodular cone K ordered by a lexicographic rule defined by a lattice basis. To each integer point x in K we associate a family of inequalities (lex-cuts) that defines the convex hull of the integer…
The data structure at the core of large-scale search engines is the inverted index, which is essentially a collection of sorted integer sequences called inverted lists. Because of the many documents indexed by such engines and stringent…
We propose a functional description of rewriting systems where reduction rules are represented by linear maps called reduction operators. We show that reduction operators admit a lattice structure. Using this structure we define the notion…
Cutting plane methods are a fundamental approach for solving integer linear programs (ILPs). In each iteration of such methods, additional linear constraints (cuts) are introduced to the constraint set with the aim of excluding the previous…
A lattice decoder which represents messages explicitly as a mixture of Gaussians functions is given. In order to prevent the number of functions in a mixture from growing as the decoder iterations progress, a method for replacing N Gaussian…
The weighted star discrepancy is a quantitative measure for the performance of point sets in quasi-Monte Carlo algorithms for numerical integration. We consider polynomial lattice point sets, whose generating vectors can be obtained by a…
A subgradient method is presented for solving general convex optimization problems, the main requirement being that a strictly-feasible point is known. A feasible sequence of iterates is generated, which converges to within user-specified…
Low density lattice codes (LDLC) are a family of lattice codes that can be decoded efficiently using a message-passing algorithm. In the original LDLC decoder, the message exchanged between variable nodes and check nodes are continuous…
Lattice decoders constructed with neural networks are presented. Firstly, we show how the fundamental parallelotope is used as a compact set for the approximation by a neural lattice decoder. Secondly, we introduce the notion of…
Linear algebra's main concerns are sets of vectors, linear functions, subspaces, linear systems, matrices and concepts about those, such as whether the solution of linear system exists or is unique; a set of vectors is linearly independent…
Reductions---rules that reduce input size while maintaining the ability to compute an optimal solution---are critical for developing efficient maximum independent set algorithms in both theory and practice. While several simple reductions…
In real world, our datasets often contain outliers. Moreover, the outliers can seriously affect the final machine learning result. Most existing algorithms for handling outliers take high time complexities (e.g. quadratic or cubic…
We consider a stochastic variant of the packing-type integer linear programming problem, which contains random variables in the objective vector. We are allowed to reveal each entry of the objective vector by conducting a query, and the…
Lattice rounding in Euclidean space can be viewed as finding the nearest point in the orbit of an action by a discrete group, relative to the norm inherited from the ambient space. Using this point of view, we initiate the study of…
We introduce the new concept of computation coding. Similar to how rate-distortion theory is concerned with the lossy compression of data, computation coding deals with the lossy computation of functions. Particularizing to linear…
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…
We study numerical integration of smooth functions defined over the $s$-dimensional unit cube. A recent work by Dick et al. (2019) has introduced so-called extrapolated polynomial lattice rules, which achieve the almost optimal rate of…
Lattice reduction algorithms have numerous applications in number theory, algebra, as well as in cryptanalysis. The most famous algorithm for lattice reduction is the LLL algorithm. In polynomial time it computes a reduced basis with…