Group synchronization asks to recover group elements from their pairwise measurements. It has found numerous applications across various scientific disciplines. In this work, we focus on orthogonal and permutation group synchronization which are widely used in computer vision such as object matching and structure from motion. Among many available approaches, the spectral methods have enjoyed great popularity due to their efficiency and convenience. We will study the performance guarantees of the spectral methods in solving these two synchronization problems by investigating how well the computed eigenvectors approximate each group element individually. We establish our theory by applying the recent popular~\emph{leave-one-out} technique and derive a~\emph{block-wise} performance bound for the recovery of each group element via eigenvectors. In particular, for orthogonal group synchronization, we obtain a near-optimal performance bound for the group recovery in presence of additive Gaussian noise. For permutation group synchronization under random corruption, we show that the widely-used two-step procedure (spectral method plus rounding) can recover all the group elements exactly if the SNR (signal-to-noise ratio) is close to the information theoretical limit. Our numerical experiments confirm our theory and indicate a sharp phase transition for the exact group recovery.
Group synchronization aims to recover the group elements from their noisy pairwise measurements. It has found many applications in community detection, clock synchronization, and joint alignment problem. This paper focuses on the orthogonal group synchronization which is often used in cryo-EM and computer vision. However, it is generally NP-hard to retrieve the group elements by finding the least squares estimator. In this work, we first study the semidefinite programming (SDP) relaxation of the orthogonal group synchronization and its tightness, i.e., the SDP estimator is exactly equal to the…
We present a general technique for obtaining permutation polynomials over a finite field from permutations of a subfield. By applying this technique to the simplest classes of permutation polynomials on the subfield, we obtain several new families of permutation polynomials. Some of these have the additional property that both f(x) and f(x)+x induce permutations of the field, which has combinatorial consequences. We use some of our permutation polynomials to exhibit complete sets of mutually orthogonal latin squares. In addition, we solve the open problem from a recent paper by Wu and Lin, and…
NS-RGS: Newton-Schulz based Riemannian gradient method for orthogonal group synchronization
School of Mathematical Sciences, Beihang University（北京航空航天大学数学科学学院）；Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong（香港中文大学系统工程与工程管理学系）
Given the noisy pairwise measurements among a set of unknown group elements, how to recover them efficiently and robustly? This problem, known as group synchronization, has drawn tremendous attention in the scientific community. In this work, we focus on orthogonal group synchronization that has found many applications, including computer vision, robotics, and cryo-electron microscopy. One commonly used approach is the least squares estimation that requires solving a highly nonconvex optimization program. The past few years have witnessed considerable advances in tackling this challenging prob…
OFDM (Orthogonal Frequency Division Multiplexing)provides the promising physical layer for 4G and 3GPP LTE Systems in terms of efficient use of bandwidth and data rates. This paper highlights the implementation of OFDM in Digital Video Broadcasting-Terrestrial (DVB-T). It mainly focuses on the timing offset problem present in OFDM systems and its proposed solution using Cyclic Prefix (CP) as a modified SC (Schmidl and COX) algorithm. It also highlights the timing synchronization as well as performance comparison through bit error rate. Synchronization issues in OFDM are important and can lead …
We study compositional inverses of permutation polynomials, complete mappings, mutually orthogonal Latin squares, and bent vectorial functions. Recently it was obtained in [33] the compositional inverses of linearized permutation binomials over finite fields. It was also noted in [29] that computing inverses of bijections of subspaces have applications in determining the compositional inverses of certain permutation classes related to linearized polynomials. In this paper we obtain compositional inverses of a class of linearized binomials permuting the kernel of the trace map. As an applicatio…
The discussion on using zero padding (ZP) instead of a cyclic prefix (CP) for enhancing channel estimation and equalization performance is a recurring topic in waveform design for future wireless systems that high spectral efficiency and location awareness are the key factors. This is particularly true for orthogonal signals, such as orthogonal frequency-division multiplexing (OFDM). ZP-OFDM is appealing for joint communications and sensing (JCS) in 6G networks because it takes the advantage of both OFDM and pulse radar. In term of communication, ZP-OFDM compared to CP-OFDM, has higher power e…
In this study, the hybrid Cramer-Rao bound (CRB) is developed for target localization, to establish the sensitivity of the estimation mean-square error (MSE) to the level of phase synchronization mismatch in coherent Multiple-Input Multiple-Output (MIMO) radar systems with widely distributed antennas. The lower bound on the MSE is derived for the joint estimation of the vector of unknown parameters, consisting of the target location and the mismatch of the allegedly known system parameters, i.e., phase offsets at the radars. Synchronization errors are modeled as being random and Gaussian. A cl…
Existing Orthogonal Frequency-Division Multiplexing (OFDM) variants based on cyclic prefix (CP) allow for efficient time synchronization, but suffer from lower power efficiency compared to zero-padded (ZP)-OFDM. Because of its power efficiency, ZP-OFDM is considered as an appealing solution for the emerging low-power wireless systems. However, in the absence of CP, time synchronization in ZP-OFDM is a very challenging task. In this paper, the non-data-aided (NDA) maximum-likelihood (ML) time synchronization for ZP-OFDM is analytically derived. We show that the optimal NDA-ML synchronization al…
The synchronization problem over the special orthogonal group $SO(d)$ consists of estimating a set of unknown rotations $R_1,R_2,...,R_n$ from noisy measurements of a subset of their pairwise ratios $R_{i}^{-1}R_{j}$. The problem has found applications in computer vision, computer graphics, and sensor network localization, among others. Its least squares solution can be approximated by either spectral relaxation or semidefinite programming followed by a rounding procedure, analogous to the approximation algorithms of \textsc{Max-Cut}. The contribution of this paper is three-fold: First, we int…
We give new lower bounds for $M(n,d)$, for various positive integers $n$ and $d$ with $n>d$, where $M(n,d)$ is the largest number of permutations on $n$ symbols with pairwise Hamming distance at least $d$. Large sets of permutations on $n$ symbols with pairwise Hamming distance $d$ is a necessary component of constructing error correcting permutation codes, which have been proposed for power-line communications. Our technique, {\em partition and extension}, is universally applicable to constructing such sets for all $n$ and all $d$, $d<n$. We describe three new techniques, {\em sequential part…
This paper investigates a joint source-channel secrecy problem for the Shannon cipher broadcast system. We suppose list secrecy is applied, i.e., a wiretapper is allowed to produce a list of reconstruction sequences and the secrecy is measured by the minimum distortion over the entire list. For discrete communication cases, we propose a permutation-based uncoded scheme, which cascades a random permutation with a symbol-by-symbol mapping. Using this scheme, we derive an inner bound for the admissible region of secret key rate, list rate, wiretapper distortion, and distortions of legitimate user…
