arXiv research

The study of $PD_3$-pairs extends results for aspherical 3-manifolds.

The paper stabilizes PD term structures under forecast uncertainty using a Kalman filter with an anchored observation model.

We define partial differential (PD in the following), i.e., field theoretic analogues of Hamiltonian systems on abstract symplectic manifolds and study their main properties, namely, PD Hamilton equations, PD Noether theorem, PD Poisson bracket, etc.. Unlike in standard multisymplectic approach to Hamiltonian field the…

Persistence diagrams (PDs) play a key role in topological data analysis (TDA), in which they are routinely used to describe topological properties of complicated shapes. PDs enjoy strong stability properties and have proven their utility in various learning contexts. They do not, however, live in a space naturally endo…

We show that the orientable double covering space of an indecomposable non-orientable $PD_3$-complex has torsion free fundamental group.

A closed 4-manifold (or, more generally, a finite $PD_4$-space) has a finitely dominated infinite regular covering space if and only if either its universal covering space is finitely dominated or it is finitely covered by the mapping torus of a self homotopy equivalence of a $PD_3$-complex.

We show that if $X$ is an indecomposable $PD_3$-complex and $π_1(X) is the fundamental group of a reduced finite graph of finite groups but is not virtually cyclic then$X$is orientable, the underlying graph is a tree, all the edge groups are$Z/2Z$and all but at most one of the vertex groups is dihedral of order$2m…

The primary purpose of this paper concerns the relation of (compact) generalized manifolds to finite Poincaré duality complexes (PD complexes). The problem is that an arbitrary generalized manifold $X$ is always an ENR space, but it is not necessarily a complex. Moreover, finite PD complexes require the Poincaré dualit…

Paper solves open question about non-positive kernels by decomposing them into PD kernels.

Every $PD_3$-complex bounds a $PD_4$-pair.

New groups prevent certain geometric actions on spaces.

PD_3-groups split as HNN extensions, revealing homology class properties.

Our previous exploration of the $\cE_g^{PD}$-geometry has shown that the field is promising. Namely, the $\cE_g^{PD}$-approach is amenable to development of novel trends in relativistic and metric differential geometry and can particularly be effective in context of the Finslerian or Minkowskian Geometries. The main po…

Study pro-$p$ completions of orientable PD_n groups, proving best results in three cases.

The paper classifies PD_4-complexes based on their fundamental group properties.

The intention with this paper is to provide all the estimation concepts and techniques that are needed to implement a two-phases approach to the parametric estimation of probability of default (PD) curves. In the first phase of this approach, a raw PD curve is estimated based on parameters that reflect discriminatory p…

A new tree-based estimator, FastPD, efficiently estimates PD functions for machine learning models.

Defines coarse cohomology of space complements, proving new duality results.

In this paper we formalize a combinatorial object for describing link diagrams called a Planar Diagram Code. PD-codes are used by the KnotTheory Mathematica package developed by Bar-Natan, et al. We present the set of PD-codes as a stand alone object and discuss its relationship with link diagrams. We give an explicit …

Extends characterization of $PD_3$-pairs with aspherical boundaries to those with spherical boundaries.

Researchers formalize PD and PFI to relate them to data generating process.

Paper presents a method for estimating long-term PDs with incomplete data.

We prove the Tits alternative for an almost coherent $PD(3)$ group which is not virtually properly locally cyclic. In particular, we show that an almost coherent $PD(3)$ group which cannot be generated by fewer than four elements always contains a rank 2 free group.

A model learns symptom-drug relations for PD patients.

Study extends Elkalla's work on subnormal subgroups to $PD_3$-groups, but $L^2$-Betti numbers need verification.

Topological data analysis is becoming a popular way to study high dimensional feature spaces without any contextual clues or assumptions. This paper concerns itself with one popular topological feature, which is the number of $d-$dimensional holes in the dataset, also known as the Betti$-d$ number. The persistence of t…

We define an order relation among oriented $PD_4$-complexes. We show that with respect to this relation, two $PD_4$-complexes over the same complex are homotopy equivalent if and only if there is an isometry between the second homology groups. We also consider minimal objects of this relation.

Scaff-PD improves fairness and robustness in federated learning with reduced communication.

Persistence diagrams (PDs) are now routinely used to summarize the underlying topology of complex data. Despite several appealing properties, incorporating PDs in learning pipelines can be challenging because their natural geometry is not Hilbertian. Indeed, this was recently exemplified in a string of papers which sho…

If the probability of default parameters (PDs) fed as input into a credit portfolio model are estimated as through-the-cycle (TTC) PDs stressed market conditions have little impact on the results of the capital calculations conducted with the model. At first glance, this is totally different if the PDs are estimated as…

PD curve calibration refers to the transformation of a set of rating grade level probabilities of default (PDs) to another average PD level that is determined by a change of the underlying portfolio-wide PD. This paper presents a framework that allows to explore a variety of calibration approaches and the conditions un…

New algorithm samples constrained distributions efficiently.

Algebraic topology methods have recently played an important role for statistical analysis with complicated geometric structured data such as shapes, linked twist maps, and material data. Among them, \textit{persistent homology} is a well-known tool to extract robust topological features, and outputs as \textit{persist…

We consider the representation power of siamese-style similarity functions used in neural network-based graph embedding. The inner product similarity (IPS) with feature vectors computed via neural networks is commonly used for representing the strength of association between two nodes. However, only a little work has b…

Many attempts have been made in recent decades to integrate machine learning (ML) and topological data analysis. A prominent problem in applying persistent homology to ML tasks is finding a vector representation of a persistence diagram (PD), which is a summary diagram for representing topological features. From the pe…

Background: Parkinson's disease (PD) is a prevalent long-term neurodegenerative disease. Though the diagnostic criteria of PD are relatively well defined, the current medical imaging diagnostic procedures are expertise-demanding, and thus call for a higher-integrated AI-based diagnostic algorithm. Methods: In this pape…

Multidimensional scaling (MDS) is a class of projective algorithms traditionally used in Euclidean space to produce two- or three-dimensional visualizations of datasets of multidimensional points or point distances. More recently however, several authors have pointed out that for certain datasets, hyperbolic target spa…

The assessment of Parkinson's disease (PD) poses a significant challenge as it is influenced by various factors which lead to a complex and fluctuating symptom manifestation. Thus, a frequent and objective PD assessment is highly valuable for effective health management of people with Parkinson's disease (PwP). Here, w…

Unreduced PDs can perform similarly to reduced PDs in machine learning tasks.

Estimates and quantizes expected persistence diagrams for efficient analysis.

Machine learning classifies Parkinson's Disease stages from walker sensors data.

Machine learning aids in diagnosing Parkinson's disease with higher accuracy.

Many datasets can be viewed as a noisy sampling of an underlying space, and tools from topological data analysis can characterize this structure for the purpose of knowledge discovery. One such tool is persistent homology, which provides a multiscale description of the homological features within a dataset. A useful re…

We show that there are two homotopy types of PD_3-complexes with fundamental group S_3*_{Z/2Z}S_3, and give explicit constructions for each, which differ only in the attachment of the top cell.

Persistent homology (PH) is a rigorous mathematical theory that provides a robust descriptor of data in the form of persistence diagrams (PDs). PDs exhibit, however, complex structure and are difficult to integrate in today's machine learning workflows. This paper introduces persistence bag-of-words: a novel and stable…

We study discrete group actions on coarse Poincare duality spaces, e.g. acyclic simplicial complexes which admit free cocompact group actions by Poincare duality groups. When G is an (n-1) dimensional duality group and X is a coarse Poincare duality space of formal dimension n, then a free simplicial action of G on X d…

In the past few years, there are several researches on Parkinson's disease (PD) recognition using single-photon emission computed tomography (SPECT) images with deep learning (DL) approach. However, the DL model's complexity usually results in difficult model interpretation when used in clinical. Even though there are …

Persistent homology (PH) is a rigorous mathematical theory that provides a robust descriptor of data in the form of persistence diagrams (PDs) which are 2D multisets of points. Their variable size makes them, however, difficult to combine with typical machine learning workflows. In this paper we introduce persistence c…