arXiv research

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

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.

This paper is a synthesis and extension of three earlier papers on $PD_4$-complexes $X$ with fundamental group $π$ such that $c.d.π=2$ and $π$ has one end. Our goal is to show that the homotopy types of such complexes are determined by $π$, the Stiefel-Whitney classes and the equivariant intersection pairing on $π_2(X)…

We consider the homotopy types of $PD_4$-complexes $X$ with fundamental group $π$ such that $c.d.π=2$ and $π$ has one end. Let $β=β_2(π;F_2)$ and $w=w_1(X)$. Our main result is that (modulo two technical conditions on $(π,w)$) there are at most $2^β$ orbits of $k$-invariants determining "strongly minimal" complexes (i.…

If $X$ is an orientable, strongly minimal $PD_4$-complex and $π_1(X)$ has one end then it has no nontrivial locally-finite normal subgroup. Hence if $π$ is a 2-knot group then (a) if $π$ is virtually solvable then either $π$ has two ends or $π\congΦ$, with presentation $\langle{a,t}|ta=a^2t\rangle$, or $π$ is torsion-f…

We consider two types of minimal Poincaré $4$-complexes. One is defined with respect to the degree $1$-map order. This idea was already present in our previous papers, and more systematically studied later by Hillman. The second type of minimal Poincaré $4$-complexes were introduced by Hambleton, Kreck and Teichner. It…

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

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

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

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

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 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…

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.

We use the Dold-Whitney theorem classifying $SO(3)$-bundles over a 4-complex to give a mod 4 obstruction to a 2-component link of trivial linking number being slice. It turns out that this coincides with the reduction of the Sato-Levine invariant.

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

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.

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

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…

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

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

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 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…

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…

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.

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

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

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…

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…

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…

In this note, we prove that the $\pd$- and $\barpd$-operators introduced by Gualtieri for a generalized complex structure coincide with the $\bdees$- and $\bdel$-operators introduced by Alekseev-Xu for Evens-Lu-Weinstein modules of a Lie bialgebroid.

Convolutional neural networks (CNNs) are commonly used for image classification tasks, raising the challenge of their application on data flows. During their training, adaptation is often performed by tuning the learning rate. Usual learning rate strategies are time-based i.e. monotonously decreasing. In this paper, we…

New model forecasts stock market volatility better than existing methods.

The study uses the Merton model to estimate PD and finds a phase transition affecting convergence speed.

The risk of a credit portfolio depends crucially on correlations between the probability of default (PD) in different economic sectors. Often, PD correlations have to be estimated from relatively short time series of default rates, and the resulting estimation error hinders the detection of a signal. We present statist…

This paper proposes a simple technical approach for the analytical derivation of Point-in-Time PD (probability of default) forecasts, with minimal data requirements. The inputs required are the current and future Through-the-Cycle PDs of the obligors, their last known default rates, and a measurement of the systematic …

Paper exposes vulnerabilities in interpreting machine learning models using adversarial attacks on PD plots.

New method extends knot theory to non-bipartite knots, revealing PDs.