Many problems in robotics, such as estimating the state from noisy sensor data or aligning two point clouds, can be posed and solved as least-squares problems. Unfortunately, vanilla nonminimal solvers for least-squares problems are notoriously sensitive to outliers. As such, various robust loss functions have been proposed to reduce the sensitivity to outliers. Examples of loss functions include pseudo-Huber, Cauchy, and Geman-McClure. Recently, these loss functions have been generalized into a single loss function that enables the best loss function to be found adaptively based on the distribution of the residuals. However, even with the generalized robust loss function, most nonminimal solvers can only be solved locally given a prior state estimate due to the nonconvexity of the problem. The first contribution of this paper is to combine graduated nonconvexity (GNC) with the generalized robust loss function to solve least-squares problems without a prior state estimate and without the need to specify a loss function. Moreover, existing loss functions, including the generalized loss function, are based on Gaussian-like distribution. However, residuals are often defined as the squared norm of a multivariate error and distributed in a Chi-like fashion. The second contribution of this paper is to apply a norm-aware adaptive robust loss function within a GNC framework. The proposed approach enables a GNC formulation of a generalized loss function such that GNC can be readily applied to a wider family of loss functions. Furthermore, simulations and experiments demonstrate that the proposed method is more robust compared to non-GNC counterparts, and yields faster convergence times compared to other GNC formulations.

翻译：机器人学中的许多问题，例如从含噪声传感器数据估计状态或配准两个点云，均可表述并求解为最小二乘问题。然而，传统的最小二乘非极小化求解器对异常值极为敏感。为此，学界提出了多种鲁棒损失函数以降低对异常值的敏感性，例如伪Huber、Cauchy和Geman-McClure损失函数。近期，这些损失函数被统一泛化为单一损失函数，使得能够根据残差分布自适应地选择最优损失函数。但即使采用这种广义鲁棒损失函数，由于问题的非凸性，大多数非极小化求解器仍需在给定先验状态估计的前提下进行局部求解。本文的第一个贡献在于将渐进非凸性（GNC）与广义鲁棒损失函数相结合，从而无需先验状态估计且无需指定损失函数即可求解最小二乘问题。此外，现有损失函数（包括广义损失函数）均基于类高斯分布假设，但残差通常被定义为多元误差的平方范数，并以类卡方分布形式存在。本文的第二个贡献是在GNC框架中应用范数感知的自适应鲁棒损失函数。所提方法实现了广义损失函数的GNC形式化，使得GNC能够直接应用于更广泛的损失函数族。仿真与实验结果表明，相较于非GNC方法，所提方法具有更强的鲁棒性；相较于其他GNC形式化方案，其收敛速度更快。