论文标题
零电势（q = 0）和单数慢特征分析的单数sturm-liouville问题
Singular Sturm-Liouville Problems with Zero Potential (q=0) and Singular Slow Feature Analysis
论文作者
论文摘要
如果其域是无限的，或者$ r $或$ w $在边界上消失了，则Sturm-Liouville问题（$λwy=（ry'）+qy $）是单数的。然后很难说出是否适用了常规的Sturm-Liouville理论的深刻结果。现有标准通常很难应用，例如因为它们是根据解决方案函数配制的。 我们研究了一个特殊情况，即潜在的$ Q $在Neumann边界条件下为零，并且仅根据系数功能提供简单明了的标准，以评估是否适用常规案例的各种特性。具体而言，这些属性是频谱（BD），自相关性，振荡（$ i $ th解决方案具有$ i $ zeros）的离散性，并且特征值等于$ i $ th解决方案的SFA Delta值（总能量）。我们进一步证明，每个溶液的固定点严格与其零（在奇异或常规情况下，无论边界条件如何，对于零电势或$ q <λw$到处都是）。如果$ \ frac {r} {w} $是有界变化的，则标准简化为需要$ \ frac {| w'|} {w} {w} \ to \ indular界点处的$。 这项研究是由慢速特征分析（SFA）激励的，该算法是从高维输入信号中提取最慢的信号，并且在计算机视觉，计算神经科学和盲源分离方面取得了显着成功。从[Sprekeler等人，2014年]来看，众所周知，对于一类重要的场景（统计独立输入），SFA的分析表达将减少到具有零潜力和Neumann边界条件的Sturm-Liouville问题。到目前为止，数学SFA理论仅考虑了常规案例，除了由Hermite多项式解决的特殊情况。这项工作将SFA理论推广到奇异案例，即开放空间方案。
A Sturm-Liouville problem ($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.
