Noise robustness of persistent homology on greyscale images, across filtrations and signatures

Topological data analysis is a recent and fast growing field that approaches the analysis of datasets using techniques from (algebraic) topology. Its main tool, persistent homology (PH), has seen a notable increase in applications in the last decade. Often cited as the most favourable property of PH and the main reason for practical success are the stability theorems that give theoretical results about noise robustness, since real data is typically contaminated with noise or measurement errors. However, little attention has been paid to what these stability theorems mean in practice. To gain some insight into this question, we evaluate the noise robustness of PH on the MNIST dataset of greyscale images. More precisely, we investigate to what extent PH changes under typical forms of image noise, and quantify the loss of performance in classifying the MNIST handwritten digits when noise is added to the data. The results show that the sensitivity to noise of PH is influenced by the choice of filtrations and persistence signatures (respectively the input and output of PH), and in particular, that PH features are often not robust to noise in a classification task.

Margins of Postural Stability in Parkinson’s Disease: An Application of Control Theory

Background: Postural instability is a restrictive feature in Parkinson’s disease (PD), usually assessed by clinical or laboratory tests. However, the exact quantification of postural stability, using stability theorems that take into account the human dynamics, is still lacking. We investigated the feasibility of control theory, Nyquist stability criterion (Gain Margin – GM –, and Phase Margin – PM –), in discriminating postural instability in PD; as well as the effects of a balance-training program. Methods: Center-of-pressure (COP) data of 40 PD patients before and after a 4-week balance-training program, and 20 healthy control subjects (HCs) (Study1); as well as COP data of 20 other PD patients at four time points during a 6-week balance-training program (Study2), collected in two earlier studies, were used. COP was recorded in four tasks, two on rigid surface and two on foam, eyes-open and closed. A postural control model (an inverted-pendulum with PID controller and time delay) was fitted to the COP data, to subject-specifically identify the model parameters; thereby calculating |GM| and PM for each subject in each task.Results: Patients had smaller margin of stability (|GM|, PM) compared to HCs. Particularly, patients, unlike HCs, showed drastic drop in PM on foam. Clinical outcomes and margin of stability improved in patients after balance training. |GM| improved early at week 4, followed by a plateau in the rest of the training. In contrast, PM improved late (week 6), in a relatively continuous-progression form. Conclusions: Using fundamental stability theorems is a promising technique for standardized quantification of postural stability in various tasks.

POWER SERIES PROOFS FOR LOCAL STABILITIES OF KÄHLER AND BALANCED STRUCTURES WITH MILD -LEMMA

By use of a natural map introduced recently by the first and third authors from the space of pure-type complex differential forms on a complex manifold to the corresponding one on the small differentiable deformation of this manifold, we will give a power series proof for Kodaira–Spencer’s local stability theorem of Kähler structures. We also obtain two new local stability theorems, one of balanced structures on an n-dimensional balanced manifold with the $(n-1,n)$ th mild $\partial \overline {\partial }$ -lemma by power series method and the other one on p-Kähler structures with the deformation invariance of $(p,p)$ -Bott–Chern numbers.

Nonlinear and Stability Analysis of a Ship with General Roll-Damping Using an Asymptotic Perturbation Method

In this work, the response of a ship rolling in regular beam waves is studied. The model is one degree of freedom model for nonlinear ship dynamics. The model consists of the terms containing inertia, damping, restoring forces, and external forces. The asymptotic perturbation method is used to study the primary resonance phenomena. The effects of various parameters are studied on the stability of steady states. It is shown that the variation of bifurcation parameters affects the bending of the bifurcation curve. The slope stability theorems are also presented.

The persistent homology of a sampled map: from a viewpoint of quiver representations

This paper is intended to introduce a filtration analysis of sampled maps based on persistent homology, providing a new method for reconstructing the underlying maps. The key idea is to extend the definition of homology induced maps of correspondences using the framework of quiver representations. Our definition of homology induced maps is given by most persistent direct summands of representations. The direct summands uniquely determine a persistent homology. We provide stability theorems of this process and show that the output persistent homology of the sampled map is the same as that of the underlying map if the sample is sufficiently dense. Compared to existing methods using eigenspace functors, our filtration analysis represents an important advantage that no prior information related to the eigenvalues of the underlying map is required. Some numerical examples are given to demonstrate the effectiveness of our method.

On Generalizations of Sampling Theorem and Stability Theorem in Shift-Invariant Subspaces of Lebesgue and Wiener Amalgam Spaces with Mixed-Norms

In this paper, we establish generalized sampling theorems, generalized stability theorems and new inequalities in the setting of shift-invariant subspaces of Lebesgue and Wiener amalgam spaces with mixed-norms. A convergence theorem of general iteration algorithms for sampling in some shift-invariant subspaces of Lp→(Rd) are also given.