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Rotation group $\mathcal{SO}(d)$ synchronization is an important inverse problem and has attracted intense attention from numerous application fields such as graph realization, computer vision, and robotics. In this paper, we focus on the…
We analyze statistical features of the ``optimization landscape'' in a random version of one of the simplest constrained optimization problems of the least-square type: finding the best approximation for the solution of an overcomplete…
We propose a concept to generate and stabilize diverse partial synchronization patterns (phase clusters) in adaptive networks which are widespread in neuro- and social sciences, as well as biology, engineering, and other disciplines. We…
Based on a local greedy numerical algorithm, we compute the topology of weighted, directed, and of unlimited extension networks of non identical Kuramoto oscillators which simultaneously satisfy 2 criteria: i) global frequency…
The Kuramoto model can be formulated as a gradient flow on a nonconvex energy landscape of the form $E(\boldsymbol{\theta}) := \frac{1}{2} \sum_{1\le i,j\le n} A_{ij}\bigl(1-\cos(\theta_i-\theta_j)\bigr).$ A fundamental question is to…
We quantify the dynamical implications of the small-world phenomenon. We consider the generic synchronization of oscillator networks of arbitrary topology, and link the linear stability of the synchronous state to an algebraic condition of…
We introduce a new paradigm, $\textit{measure synchronization}$, for synchronizing graphs with measure-valued edges. We formulate this problem as maximization of the cycle-consistency in the space of probability measures over relative…
We study the non-convex optimization landscape for maximum likelihood estimation in the discrete orbit recovery model with Gaussian noise. This model is motivated by applications in molecular microscopy and image processing, where each…
A network of coupled dynamical systems is represented by a graph whose vertices represent individual cells and whose edges represent couplings between cells. Motivated by the impact of synchronization results of the Kuramoto networks, we…
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…
Maintaining the stability of synchronization state is crucial for the functioning of many natural and artificial systems. In this study, we develop methods to optimize the synchronization stability of the Kuramoto model by minimizing the…
Designing high-performing networks requires optimizing for functionality while respecting physical, geometric, or budget constraints. Yet, mathematical and computational tools to design such systems remain limited, particularly for…
An interesting problem in synchronization is the study of coupled oscillators, wherein oscillators with different natural frequencies synchronize to a common frequency and equilibrium phase difference. In this paper, we investigate the…
The loss surface of deep neural networks has recently attracted interest in the optimization and machine learning communities as a prime example of high-dimensional non-convex problem. Some insights were recently gained using spin glass…
In this paper we develop a new framework that captures the common landscape underlying the common non-convex low-rank matrix problems including matrix sensing, matrix completion and robust PCA. In particular, we show for all above problems…
The angular synchronization problem of estimating a set of unknown angles from their known noisy pairwise differences arises in various applications. It can be reformulated as a optimization problem on graphs involving the graph Laplacian…
This paper addresses the problem of synchronizing orthogonal matrices over directed graphs. For synchronized transformations (or matrices), composite transformations over loops equal the identity. We formulate the synchronization problem as…
Synchronization in coupled oscillators networks is a remarkable phenomenon of relevance in numerous fields. For Kuramoto oscillators the loss of synchronization is determined by a trade-off between coupling strength and oscillator…
Can synchronization properties of a network of identical oscillators in the presence of noise be improved through appropriate rewiring of its connections? What are the optimal network architectures for a given total number of connections?…
We study a nonconvex optimization algorithmic approach to phase retrieval and the more general problem of semidefinite low-rank matrix sensing. Specifically, we analyze the nonconvex landscape of a quartic Burer-Monteiro factored…