A Unified Complexity Metric for Nonconvex Matrix Completion and Matrix Sensing in the Rank-one Case

In this work, we develop a new complexity metric for an important class of low-rank matrix optimization problems, where the metric aims to quantify the complexity of the nonconvex optimization landscape of each problem and the success of local search methods in solving the problem. The existing literature has focused on two complexity measures. The RIP constant is commonly used to characterize the complexity of matrix sensing problems. On the other hand, the sampling rate and the incoherence are used when analyzing matrix completion problems. The proposed complexity metric has the potential to unify these two notions and also applies to a much larger class of problems. To mathematically study the properties of this metric, we focus on the rank-$1$ generalized matrix completion problem and illustrate the usefulness of the new complexity metric from three aspects. First, we show that instances with the RIP condition have a small complexity. Similarly, if the instance obeys the Bernoulli sampling model, the complexity metric will take a small value with high probability. Moreover, for a one-parameter class of instances, the complexity metric shows consistent behavior to the first two scenarios. Furthermore, we establish theoretical results to provide sufficient conditions and necessary conditions on the existence of spurious solutions in terms of the proposed complexity metric. This contrasts with the RIP and incoherence notions that fail to provide any necessary condition

An Investigation of the Constructional Design Components Affecting the Mechanical Response and Cellular Activity of Electrospun Vascular Grafts

Cardiovascular disease is anticipated to remain the leading cause of death globally. Due to the current problems connected with using autologous arteries for bypass surgery, researchers are developing tissue-engineered vascular grafts (TEVGs). The major goal of vascular tissue engineering is to construct prostheses that closely resemble native blood vessels in terms of morphological, mechanical, and biological features so that these scaffolds can satisfy the functional requirements of the native tissue. In this setting, morphology and cellular investigation are usually prioritized, while mechanical qualities are generally addressed superficially. However, producing grafts with good mechanical properties similar to native vessels is crucial for enhancing the clinical performance of vascular grafts, exposing physiological forces, and preventing graft failure caused by intimal hyperplasia, thrombosis, aneurysm, blood leakage, and occlusion. The scaffold’s design and composition play a significant role in determining its mechanical characteristics, including suturability, compliance, tensile strength, burst pressure, and blood permeability. Electrospun prostheses offer various models that can be customized to resemble the extracellular matrix. This review aims to provide a comprehensive and comparative review of recent studies on the mechanical properties of fibrous vascular grafts, emphasizing the influence of structural parameters on mechanical behavior. Additionally, this review provides an overview of permeability and cell growth in electrospun membranes for vascular grafts. This work intends to shed light on the design parameters required to maintain the mechanical stability of vascular grafts placed in the body to produce a temporary backbone and to be biodegraded when necessary, allowing an autologous vessel to take its place