发表机构
GSI and RISIS, GSEM, Université de Genève; Institute of Mathematics, EPFL; Department of Mathematics, King’s College London; RISIS, GSEM, Université de Genève（日内瓦大学; 洛桑联邦理工学院; 伦敦国王学院; 日内瓦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对曲线响应预测中的非线性几何与未知对齐问题，提出弹性核岭回归，在形状空间中进行惩罚条件弗雷歇均值估计，并开发快速对齐算法，在语音学数据上验证了其恢复缺失帧和语音反演的潜力。
AI 中文摘要
预测整条曲线的形状需要考虑非线性几何以及曲线之间未知的对齐关系。我们开发了弹性核岭回归，这是一种针对平面曲线响应、协变量为标量、多元或函数型数据的非参数方法。利用平方根速度表示，我们在一个对平移、旋转、缩放和重参数化不变的形状空间中，构建了惩罚条件弗雷歇均值估计。向量值再生核希尔伯特空间通过球面指数映射提供了灵活的非线性连接。我们使用交替算法将每条观测曲线对齐到其当前拟合值，并更新回归函数。为此，我们提供了一种新的欧几里得拟牛顿求解器，利用重参数化目标的尺度不变性；在数值比较中，该求解器在保持精度的同时加速了对齐。模拟实验证明了在回归中估计对齐以及尊重球面几何的益处。将该方法应用于实时磁共振成像记录中的声道轮廓，该方法能够恢复缺失帧，并从降采样数据中重建形状轨迹。在同一记录上的语音反演概念验证表明，该方法能够从声学特征预测舌头形状，展示了其潜力。该方法已在R包sphereg2中实现。
英文摘要
Predicting the shapes of entire curves requires accounting for nonlinear geometry and unknown alignments between curves. We develop elastic kernel ridge regression, a nonparametric method for planar curve responses with scalar, multivariate, or functional covariates. Using square-root velocity representations, we formulate penalized conditional Fréchet mean estimation in a shape space invariant to translation, rotation, scaling, and reparametrization. A vector-valued reproducing kernel Hilbert space provides a flexible nonlinear link through the spherical exponential map. We use an alternating algorithm to align each observed curve to its current fitted value, and updates the regression function. To facilitate this, we provide a new Euclidean quasi-Newton solver that exploits scale invariance of the reparametrization objective; this accelerates alignment while retaining accuracy in a numerical comparison. Simulations demonstrate the benefits of estimating alignment within the regression and respecting spherical geometry. Applied to vocal tract contours from a real-time magnetic resonance imaging recording, the method recovers missing frames and reconstructs shape trajectories from downsampled data. A speech inversion proof of concept on the same recording predicts tongue shapes from acoustic features, illustrating the method's potential. The method is implemented in the \texttt{R} package \texttt{sphereg2}.
