Although Nesterov’s accelerated gradient method (AGM) has been studied from various perspectives, it remains unclear why the most popular forms of AGMs must handle convex and strongly convex objective functions separately. To address this inconsistency, we propose a novel unified framework for Lagrangians, ordinary differential equation (ODE) models, and algorithms. As a special case, our new simple momentum algorithm, which we call the unified AGM, seamlessly bridges the gap between the two most popular forms of Nesterov’s AGM and has a superior convergence guarantee compared to existing algorithms for non-strongly convex objective functions. This property is beneficial in practice when considering ill-conditioned $\mu$-strongly convex objective functions (with small $\mu$). Furthermore, we generalize this algorithm and the corresponding ODE model to the higher-order non-Euclidean setting. Last but not least, our unified framework is used to construct the unified AGM-G ODE, a novel ODE model for minimizing the gradient norm of strongly convex functions.

We propose a novel methodology that systematically analyzes ordinary differential equation (ODE) models for first-order optimization methods by converting the task of proving convergence rates into verifying the positive semidefiniteness of specific Hilbert-Schmidt integral operators. Our approach is based on the performance estimation problems (PEP) introduced by Drori and Teboulle. Unlike previous works on PEP, which rely on finite-dimensional linear algebra, we use tools from functional analysis. Using the proposed method, we establish convergence rates of various accelerated gradient flow models, some of which are new. As an immediate consequence of our framework, we show a correspondence between minimizing function values and minimizing gradient norms.

We propose computationally tractable accelerated first-order methods for Riemannian optimization, extending the Nesterov accelerated gradient (NAG) method. For both geodesically convex and geodesically strongly convex objective functions, our algorithms are shown to have the same iteration complexities as those for the NAG method on Euclidean spaces, under only standard assumptions. To the best of our knowledge, the proposed scheme is the first fully accelerated method for geodesically convex optimization problems. Our convergence analysis makes use of novel metric distortion lemmas as well as carefully designed potential functions. A connection with the continuous-time dynamics for modeling Riemannian acceleration in (Alimisis et al., 2020) is also identified by letting the stepsize tend to zero. We validate our theoretical results through numerical experiments.

This demonstration shows a novel approach for Webizing mobile augmented reality, which uses HTML as its content structure, and its real-life use cases. Insight is a mobile AR Web browser that executes HTML5-based AR applications. By extending physical objects and places with a uniform resource identifier (URI), we could build objects of interest for mobile AR application as document object model (DOM) elements and control their behavior and user interactions through DOM events in standard HTML documents. A new CSS media type is defined to augment virtual objects to the physical objects. In this model, we introduce the concept of PLACE, which is the model of a physical space in which the user can be located. With this approach, mobile AR applications can be seamlessly developed as common HTML documents under the current Web eco-system. The advantages of the webized mobile AR, which is able to utilize all kind of Web resources without reworking, and its high productivity due to the seamless development of AR contents in HTML documents are shown with real-life use cases in various domains, such as shopping, entertainment, education, and manufacturing.

Information presentation techniques are important in augmented reality applications as they are in the traditional desktop user interface (WIMP) and web user interface [6]. This paper introduces view management for a web-based augmented reality mobile platform. We use a webized mobile AR browser called Insight that provides separation of the application logic, including the tracking engine and AR content, so that the view management logic and contents are easy to reuse. In addition, the view management is able to accommodate in-situ context of an AR application.