Related papers: On the KZ Reduction
Zeroth-order (ZO) optimization has long been favored for its biological plausibility and its capacity to handle non-differentiable objectives, yet its computational complexity has historically limited its application in deep neural…
Leveraging algorithmic stability to derive sharp generalization bounds is a classic and powerful approach in learning theory. Since Vapnik and Chervonenkis [1974] first formalized the idea for analyzing SVMs, it has been utilized to study…
We study the growth rate of the hard squares lattice gas, equivalent to the number of independent sets on the square lattice, and two related models - non-attacking kings and read-write isolated memory. We use an assortment of techniques…
We describe an approach for finding upper bounds on an ODE dynamical system's maximal Lyapunov exponent among all trajectories in a specified set. A minimization problem is formulated whose infimum is equal to the maximal Lyapunov exponent,…
We design a sublinear-time approximation algorithm for quadratic function minimization problems with a better error bound than the previous algorithm by Hayashi and Yoshida (NIPS'16). Our approximation algorithm can be modified to handle…
In a capacitated directed graph, it is known that the set of all min-cuts forms a distributive lattice [1], [2]. Here, we describe this lattice as a regular predicate whose forbidden elements can be advanced in constant parallel time after…
In this paper, we propose a new algorithm for recovery of low-rank matrices from compressed linear measurements. The underlying idea of this algorithm is to closely approximate the rank function with a smooth function of singular values,…
The classical low rank approximation problem is to find a rank $k$ matrix $UV$ (where $U$ has $k$ columns and $V$ has $k$ rows) that minimizes the Frobenius norm of $A - UV$. Although this problem can be solved efficiently, we study an…
The shortest vector problem (SVP) over ideal lattices is closely related to the Ring-LWE problem, which is widely used to build post-quantum cryptosystems. Power-of-two cyclotomic fields are frequently adopted to instantiate Ring-LWE. Pan…
Lempel-Ziv (LZ77 or, briefly, LZ) is one of the most effective and widely-used compressors for repetitive texts. However, the existing efficient methods computing the exact LZ parsing have to use linear or close to linear space to index the…
Let $k$ and $n$ be positive integers. Define $R(n,k)$ to be the minimum positive value of $$ | e_i \sqrt{s_1} + e_2 \sqrt{s_2} + ... + e_k \sqrt{s_k} -t | $$ where $ s_1, s_2, ..., s_k$ are positive integers no larger than $n$, $t$ is an…
In this work, we study the solution of shortest vector problems (SVPs) arising in terms of learning with error problems (LWEs). LWEs are linear systems of equations over a modular ring, where a perturbation vector is added to the right-hand…
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.…
We propose a novel iterative framework for minimizing a proper lower semicontinuous Kurdyka-{\L}ojasiewicz (KL) function $\Phi$. It comprises a Zhang-Hager (ZH-type) nonmonotone decrease condition and a relative error condition. Hence, the…
Recent work have shown that the quantization for matrix multiplication problem can be optimally solved by quantizing each column in each matrix using a nested lattice code, and then multiplying the de-quantized matrices. It was further…
The closest vector problem (CVP) and shortest (nonzero) vector problem (SVP) are the core algorithmic problems on Euclidean lattices. They are central to the applications of lattices in many problems of communications and cryptography.…
We prove upper bounds on the order of convergence of lattice based algorithms for numerical integration in function spaces of dominating mixed smoothness on the unit cube with homogeneous boundary condition. More precisely, we study…
The Lipschitz constant is an important quantity that arises in analysing the convergence of gradient-based optimization methods. It is generally unclear how to estimate the Lipschitz constant of a complex model. Thus, this paper studies an…
Vector perturbation (VP) precoding is a promising technique for multiuser communication systems operating in the downlink. In this work, we introduce a hybrid framework to improve the performance of lattice reduction (LR) aided precoding in…
A lattice is a set of all the integer linear combinations of certain linearly independent vectors. One of the most important concepts on lattice is the successive minima which is of vital importance from both theoretical and practical…