The co-localization problem is a model that simultaneously localizes objects of the same class within a series of images or videos. In \cite{joulin2014efficient}, authors present new variants of the Frank-Wolfe algorithm (aka conditional gradient) that increase the efficiency in solving the image and video co-localization problems. The authors show the efficiency of their methods with the rate of decrease in a value called the Wolfe gap in each iteration of the algorithm. In this project, inspired by the conditional gradient sliding algorithm (CGS) \cite{CGS:Lan}, We propose algorithms for solving such problems and demonstrate the efficiency of the proposed algorithms through numerical experiments. The efficiency of these methods with respect to the Wolfe gap is compared with implementing them on the YouTube-Objects dataset for videos.

翻译：共定位问题是一种在同一系列图像或视频中同时定位同类目标的模型。在文献\cite{joulin2014efficient}中，作者提出了Frank-Wolfe算法（亦称条件梯度法）的新型变体，用以提升图像与视频共定位问题的求解效率。作者通过每次算法迭代中称为Wolfe间隙的数值下降速率，展示了其方法的有效性。在本项目中，受条件梯度滑动算法（CGS）\cite{CGS:Lan}启发，我们提出了针对此类问题的求解算法，并通过数值实验验证了所提算法的有效性。通过将这些方法应用于YouTube-Objects视频数据集，我们比较了其相对于Wolfe间隙的收敛效率。