The paper classifies PD_4-complexes based on their fundamental group properties.
problem Understanding the structure of PD_4-complexes based on their fundamental group properties.
method Analyzing the fundamental group and its modules to classify PD_4-complexes.
result The classification of PD_4-complexes based on their fundamental group properties.
We define an order relation among oriented PD4-complexes. We show that with respect to this relation, two PD4-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 PD4-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 PD4-complexes X with fundamental group π such that c.d.π=2 and π has one end. Let β=β2(π;F2) and w=w1(X). Our main result is that (modulo two technical conditions on (π,w)) there are at most 2β orbits of k-invariants determining "strongly minimal" complexes (i.…
Abstract: Study of 2-knot groups with restrictions on normal subgroups.
problem Characterizing 2-knot groups based on their normal subgroups.
method Analyzing PD_4-complexes and using properties of π_1(X).
result Characterization of 2-knot groups based on their normal subgroups.
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 PD3-complex bounds a PD4-pair.
problem Bounding PD3-complexes with PD4-pairs. method Constructing PD4-pairs from PD3-complexes. result Every PD3-complex bounds a PD4-pair (Z,P). PD_3-groups split as HNN extensions, revealing homology class properties.
problem Understanding PD_3-groups splitting as HNN extensions.
method Analyzing PD_3-groups splitting as HNN extensions and examining homology classes.
result The Poincaré dual of the homology class of a PD_3-group in the kernel of an epimorphism is revealed.
The study of PD3-pairs extends results for aspherical 3-manifolds.
problem Understanding PD3-pairs with aspherical ambient spaces. method Attaching 1-handles to PD3-pairs with aspherical ambient space and π1-injective boundary. result There are only finitely many PD3-pairs with a specific group property. Proves Tits alternative for specific PD(3) groups.
problem Tits alternative for PD(3) groups. method Proving Tits alternative for almost coherent PD(3) groups. result Almost coherent PD(3) groups contain rank 2 free groups. Study pro-p completions of orientable PD_n groups, proving best results in three cases.
problem Understanding pro-p completions of orientable PD_n groups. method Examined four cases of orientable PD_3-groups and some PD_n groups (n≤5), providing examples and proving best results in three cases.
result Best results in three out of four cases of orientable PD_3-groups.
A new kernel for persistence diagrams using Sliced Wasserstein distance.
problem Incorporating persistence diagrams into machine learning pipelines.
method Proposes a new kernel for persistence diagrams based on the Sliced Wasserstein approximation of the Wasserstein distance, demonstrating its stability and discriminative power.
result The proposed kernel is stable and discriminative, outperforming existing kernels on various benchmarks.
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.
problem Efficiently estimating Partial Dependence functions for machine learning models.
method Proposes a new tree-based estimator, FastPD, to estimate PD functions.
result FastPD consistently estimates the desired population quantity and improves complexity from quadratic to linear.
The paper stabilizes PD term structures under forecast uncertainty using a Kalman filter with an anchored observation model.
problem Stable estimation of lifetime PDs under forecast uncertainty.
method Reformulated in state-space framework, introduced an anchored observation model.
result Asymptotic stochastic stability of error dynamics, leading to smoother projections.
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 PD3-pairs with aspherical boundaries to those with spherical boundaries.
problem Characterizing fundamental triples of PD3-pairs with boundary components of different types. method Extends Turaev and Bleile's work by relaxing the π1-injectivity hypothesis and considering pairs with spherical boundary components. result Characterization of fundamental triples for PD3-pairs with spherical boundary components and c.d.π1(P)≤2. Researchers formalize PD and PFI to relate them to data generating process.
problem Lack of theory linking PD and PFI to data generating process.
method Formalize PD and PFI as estimators of ground truth estimands, account for model variance with learner-PD and learner-PFI.
result PD and PFI estimates deviate from ground truth due to statistical biases, model variance, and Monte Carlo approximation errors.
Paper presents a method for estimating long-term PDs with incomplete data.
problem Estimating long-term PDs with limited and incomplete historical data.
method Single risk factor approach for simultaneous calibration of PDs across sub-portfolios.
result Method yields long-term PDs without requiring complete historical data.
New framework for efficient PD averaging and clustering.
problem Challenges in averaging and clustering persistence diagrams.
method Reformulate PD metrics as optimal transport problems, leveraging recent computational advances.
result Scalable computations of PD barycenters and clustering on thousands of diagrams.
Study extends Elkalla's work on subnormal subgroups to PD3-groups, but L2-Betti numbers need verification.
problem Verifying L2-Betti numbers for PD3-groups and group pairs. method Algebraic arguments extending Elkalla's work, but reliant on unproven L2-Betti number hypothesis. result Need further research on L2-Betti numbers for general PD3-groups. Scaff-PD improves fairness and robustness in federated learning with reduced communication.
problem Improving fairness and robustness in federated learning with limited communication.
method Scaff-PD uses a family of distributionally robust objectives and an accelerated primal dual algorithm with bias-corrected steps.
result Scaff-PD achieves significant gains in communication efficiency and convergence speed while maintaining fairness and robustness.
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.
problem Can non-positive definite kernels be decomposed into the difference of two positive definite kernels?
method Introduced signed measure to transform positive decomposition into measure decomposition, providing a sufficient and necessary condition.
result First random features algorithm for unbiased estimation of non-positive kernels.
We show that the orientable double covering space of an indecomposable non-orientable PD3-complex has torsion free fundamental group.
We show that if X is an indecomposable PD3-complex and π1(X)isthefundamentalgroupofareducedfinitegraphoffinitegroupsbutisnotvirtuallycyclicthenXisorientable,theunderlyinggraphisatree,alltheedgegroupsareZ/2Zandallbutatmostoneofthevertexgroupsisdihedraloforder2m…
New method converts complex PDs into stable vectors for ML.
problem Complex structure of persistence diagrams makes them hard to use in ML.
method Persistence bag-of-words (BoW) for vectorizing PDs.
result Achieves state-of-the-art performance and speed in ML.
A new kernel for comparing persistence diagrams without approximation.
problem Lack of suitable kernels for comparing persistence diagrams.
method Persistence Fisher kernel based on Fisher information geometry.
result Proposes a positive definite kernel for persistence diagrams without approximation.
Classifies pro-p PD2 pairs and builds a pro-p curve complex.
problem Classifying pro-p Poincaré duality pairs in dimension two. method Using classification of pro-p PD2 pairs to build a pro-p curve complex. result Established basic properties of the pro-p curve complex. Unreduced PDs can perform similarly to reduced PDs in machine learning tasks.
problem Ignoring much of the information in persistence diagrams in machine learning pipelines.
method Developed methods to generate topological feature vectors from unreduced boundary matrices.
result Unreduced PDs can perform on par with, and sometimes outperform, fully-reduced PDs in machine learning tasks.
Machine learning classifies Parkinson's Disease stages from walker sensors data.
problem Limited cost-effective methods for quantitatively assessing Parkinson's Disease stages.
method Machine learning applied to walker-mounted sensors data, feature selection methods compared.
result Feature selection method using ANOVA provides similar accuracy to full feature set and is clinically interpretable.
Tutorial on interpreting SPECT images for PD recognition using AI.
problem Difficulty in interpreting complex DL models for clinical use.
method Evaluation of six interpretation methods on four DCNN architectures.
result Guided backpropagation and SHAP methods are suitable for PD recognition.
Paper proposes E/PD-Control for better neural network training.
problem Training efficiency and robustness of CNNs in online data flows.
method E/PD-Control combines feedback PD controller with exponential signal.
result Better learning efficiency and robustness demonstrated experimentally.
Machine learning aids in diagnosing Parkinson's disease with higher accuracy.
problem Subjectivity in traditional PD diagnosis methods and missed early symptoms.
method Machine learning applied to various data modalities for PD and control group classification.
result Machine learning methods show high potential for improving PD diagnosis.
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.
AI framework diagnoses Parkinson's disease with 100% accuracy.
problem Expertise-demanding medical imaging procedures for Parkinson's disease diagnosis.
method End-to-end, multi-modality diagnosis framework using T1-MRI and 11C-CFT PET.
result 100% accuracy in PD/NL classification.
Method predicts motor symptoms of Parkinson's disease from daily activities.
problem Objective monitoring of Parkinson's disease symptoms.
method Multi-layer Gaussian process models for three types of movement abnormalities.
result Strong agreement between model predictions and clinical annotations.
A new algorithm PD improves stock-correlation network clustering and robustness.
problem Improving clustering and robustness of stock-correlation networks.
method Proposes a new proportional degree algorithm to filter information on a complete graph of normalised mutual information.
result The PD algorithm produces a network with better homogeneity and robustness compared to PMFG.
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.
New model forecasts stock market volatility better than existing methods.
problem Forecasting volatility in stock markets.
method Combines HAR model with path-dependent volatility models.
result HAR-PD model family outperforms basic HAR model family in volatility forecasting.
This paper explores the relationship between generalized manifolds and Poincaré duality complexes.
problem Understanding the relationship between generalized manifolds and finite Poincaré duality complexes.
method Introducing Λ-Poincaré duality complexes, constructing 2-patch spaces, and using Gromov-Hausdorff metric.
result Generalized manifolds can be recognized within an enlarged class of Λ-Poincaré duality complexes.
The study uses the Merton model to estimate PD and finds a phase transition affecting convergence speed.
problem Estimating the probability of default (PD) using limited historical data.
method Adopted the Merton model and analyzed phase transitions in default correlation.
result PD estimation converges slowly when temporal correlation decays by power law less than one.
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.
problem Vulnerability of permutation-based interpretation methods, particularly PD plots, to adversarial attacks.
method Adversarial framework to manipulate black-box models and produce deceptive PD plots.
result It is possible to hide discriminatory behaviors in machine learning models through interpretation tools like PD plots.
The study finds that asset correlations underestimated when exposure pools are not homogeneous.
problem Systematic error in estimating asset correlations from default data due to exposure pool inhomogeneity.
method Investigates the effect of exposure pool homogeneity on asset correlation estimation from default time series.
result Asset correlation is systematically underestimated when exposure pools are inhomogeneous, especially if PD is spread out.
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…