零位势（q=0）奇异Sturm-Liouville问题及奇异慢特征分析
Singular Sturm-Liouville Problems with Zero Potential (q=0) and Singular Slow Feature Analysis
摘要：
一个Sturm-Liouville问题（$λwy=(ry')'+qy$）是奇异的，如果它的区域是无界的，或者如果$r$或$w$在边界处消失.那么，就很难判断正规的斯特姆-刘维尔理论的深刻结果是否适用。现有的标准往往难以适用，例如，因为它们是根据解函数制定的。我们研究的特殊情况下，潜在的$q$是零Neumann边界条件下，并给出简单而明确的标准，仅在系数函数，以评估是否适用于经常情况下的各种属性。具体地，这些性质是谱的离散性（BD）、自伴性、振荡（第$i$解具有$i$零）以及第$i$特征值等于第$i$解的SFA δ值（总能量）。我们进一步证明了每个解决方案的驻点严格交错与它的零（在奇异或定期的情况下，无论边界条件，为零电位或如果$q < λw$无处不在）。如果$\frac{r}{w}$是有界的和有界的变化，该准则简化为要求$\frac{|w'|}{w} \to \infty$在奇异边界点。这项研究的动机是慢特征分析（SFA），一种数据处理算法，从高维输入信号中提取最慢的不相关信号，并在计算机视觉，计算神经科学和盲声源分离方面取得了显着的成功。来自[Sprekeler等人，已知对于一类重要的场景（统计独立输入），SFA的解析公式简化为具有零势和Neumann边界条件的Sturm-Liouville问题。到目前为止，数学SFA理论只考虑了正则情况，除了一个特殊的情况，这是解决的埃尔米特多项式。这项工作将SFA理论推广到奇异情况，即开放空间场景。
英文摘要：
A Sturm-Liouville problem ($λwy=(ry')'+qy$) is singular if its domain is unbounded or if $r$ or $w$ vanish at the boundary. Then it is difficult to tell whether profound results from regular Sturm-Liouville theory apply. Existing criteria are often difficult to apply, e.g. because they are formulated in terms of the solution function. We study the special case that the potential $q$ is zero under Neumann boundary conditions and give simple and explicit criteria, solely in terms of the coefficient functions, to assess whether various properties of the regular case apply. Specifically, these properties are discreteness of the spectrum (BD), self-adjointness, oscillation ($i$th solution has $i$ zeros) and that the $i$th eigenvalue equals the SFA delta value (the total energy) of the $i$th solution. We further prove that stationary points of each solution strictly interlace with its zeros (in singular or regular case, regardless of the boundary condition, for zero potential or if $q < λw$ everywhere). If $\frac{r}{w}$ is bounded and of bounded variation, the criterion simplifies to requiring $\frac{|w'|}{w} \to \infty$ at singular boundary points. This research is motivated by Slow Feature Analysis (SFA), a data processing algorithm that extracts the slowest uncorrelated signals from a high-dimensional input signal and has notable success in computer vision, computational neuroscience and blind source separation. From [Sprekeler et al., 2014] it is known that for an important class of scenarios (statistically independent input), an analytic formulation of SFA reduces to a Sturm-Liouville problem with zero potential and Neumann boundary conditions. So far, the mathematical SFA theory has only considered the regular case, except for a special case that is solved by Hermite Polynomials. This work generalizes SFA theory to the singular case, i.e. open-space scenarios.
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