We propose a paradigm for interpretable Manifold Learning for scientific data analysis, whereby we parametrize a manifold with $d$ smooth functions from a scientist-provided dictionary of meaningful, domain-related functions. When such a parametrization exists, we provide an algorithm for finding it based on sparse non-linear regression in the manifold tangent bundle, bypassing more standard manifold learning algorithms. We also discuss conditions for the existence of such parameterizations in function space and for successful recovery from finite samples. We demonstrate our method with experimental results from a real scientific domain.
翻译:我们提出一个可解释的数学学习模式,用于科学数据分析,通过这一模式,我们从科学家提供的有意义、与域有关的功能字典中以美元平滑的功能对一个元件进行模拟。当存在这种对称时,我们提供一种算法,根据多层相切的捆绑中稀少的非线性回归,绕过更标准的多重学习算法,找到它。我们还讨论在功能空间中存在这种参数化的条件和从有限样本中成功回收的条件。我们用一个真正的科学领域的实验结果来展示我们的方法。