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对齐两个点云，均可归结为最小二乘问题并采用相应方法求解。然而，标准非最小化最小二乘求解器对异常值极为敏感。为此，研究者提出了多种鲁棒损失函数（如伪Huber、Cauchy和Geman-McClure）以降低对异常值的敏感性。近期，这些损失函数被统一为单一通用损失函数，使得能够根据残差分布自适应地确定最佳损失函数。但即使采用通用鲁棒损失函数，由于问题的非凸性，大多数非最小化求解器仍仅能在给定先验状态估计的情况下获得局部解。本文的第一项贡献在于将分级非凸性（GNC）与通用鲁棒损失函数结合，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（包括通用损失函数）均基于高斯型分布，但残差通常定义为多元误差的平方范数，呈卡方分布特性。本文的第二项贡献是在GNC框架中引入范数感知的自适应鲁棒损失函数。所提方法实现了通用损失函数的GNC形式化，使得GNC能够直接适用于更广泛的损失函数族。仿真与实验表明，与未采用GNC的方法相比，所提方法具有更强的鲁棒性，且相较于其他GNC方法展现出更快的收敛速度。