点云上的梯度下降及其在学习算子校正中的应用
Gradient Descent on Point Clouds and Applications in Learned Operator Correction
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中文总结 AI 辅助
该研究针对点云隐式未知流形上的能量最小化问题，提出了一种保持在流形邻域内的梯度下降格式，并将其应用于逆问题的学习算子校正。
中文摘要 AI 辅助
我们考虑在由点云隐式给出的未知流形上最小化能量的问题。对于已知流形，可类比欧氏空间中的经典构造定义梯度下降格式；但当流形未知时，需在最小化能量的同时估计流形。我们定义了一种梯度下降格式，该格式始终保持在流形的邻域内，且在合适的稳定性和采样假设下，会收敛到局部极小值点的邻域，该邻域的大小会随时间步长和采样误差的消失而消失。作为示例，我们展示了该方法在逆问题中学习算子校正的应用。
英文摘要
We consider the problem of minimising an energy over an unknown manifold that is given implicitly by a point cloud. For a known manifold one can define a gradient descent scheme analogously to the classical construction in Euclidean spaces. However, when the manifold is not known one has to simultaneously estimate the manifold whilst minimising the energy. We define a gradient descent scheme which remains in a neighbourhood of the manifold and, under suitable stability and sampling assumptions, converges to a neighbourhood of a local minimiser whose size vanishes as the time step and sampling errors vanish. As an example we show the application of the methodology to learning operator corrections in inverse problems.
