Given data points p 0,…,p N on a closed submanifold M of ℝ n and time instants 0=t 01<⋅⋅⋅ N =1, we consider the problem of finding a curve γ on M that best approximates the data points at the given instants while being as “regular” as possible. Specifically, γ is expressed as the curve that minimizes the weighted sum of a sum-of-squares term penalizing the lack of fitting to the data points and a regularity term defined, in the first case as the mean squared velocity of the curve, and in the second case as the mean squared acceleration of the curve. In both cases, the optimization task is carried out by means of a steepest-descent algorithm on a set of curves on M. The steepest-descent direction, defined in the sense of the first-order and second-order Palais metric, respectively, is shown to admit analytical expressions involving parallel transport and covariant integral along curves. Illustrations are given in ℝ n and on the unit sphere.
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机译：给定数据点p 0 ，…，p N 在ℝ n 的闭合子流形M上并且时刻0 = t 0 < / sub> 1 <⋅⋅⋅ N = 1，我们考虑在M上找到最接近给定数据点的曲线γ的问题瞬间变得尽可能“规律”。具体而言，将γ表示为使平方和项的加权和最小的曲线，该平方和项不利于数据点的拟合，而正则项则定义为正则项，在第一种情况下为曲线的均方速度，并且第二种情况是曲线的均方加速度。在这两种情况下，优化任务都是通过最速下降算法对M上的一组曲线执行的。最速下降方向分别定义为一阶和二阶Palais度量，证明可以接受包含平行传输和沿曲线的协变积分的分析表达式。插图在ℝ n 中和单位球面上给出。
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