The paper identifies all link projections with isolate-region number one.
problem Determining link projections with a specific isolate-region number.
method Analyzing link projections to find isolated regions and their cardinality.
result All link projections with isolate-region number one are identified.
Study flows with isolated non-saddle sets and their region of influence.
problem Understanding the complexity of isolated non-saddle sets in flows.
method Cohomological analysis and construction of flows with prescribed structures.
result Construct flows with specific structures for their region of influence.
The paper finds singular isoperimetric regions in high-dimensional spaces.
problem Existence of singular isoperimetric regions in high-dimensional manifolds.
method Construction of specific manifolds with singular isoperimetric regions.
result Construction of manifolds with singular isoperimetric regions.
Heavy-tailed distributions are frequently used to enhance the robustness of regression and classification methods to outliers in output space. Often, however, we are confronted with "outliers" in input space, which are isolated observations in sparsely populated regions. We show that heavy-tailed stochastic processes (…
One-hot CNN (convolutional neural network) has been shown to be effective for text categorization (Johnson & Zhang, 2015). We view it as a special case of a general framework which jointly trains a linear model with a non-linear feature generator consisting of `text region embedding + pooling'. Under this framework, we…
Theorem proves topological censorship for universes with positive cosmological constant.
problem Proving topological censorship for spacetimes with positive cosmological constant.
method Developed a new theorem assuming eventual isolation of black hole collections.
result Regions near black hole collections have trivial fundamental group.
Develops a machine learning method for parameter estimation in branching processes models.
problem Parameter evaluation for unevenly distributed sparse and dense regions in stochastic datasets.
method Approximate Bayesian computation based on Isolation Kernel mapping and maxima weighted kernel.
result Effective parameter estimation for cancer cell evolution models using personal data.
Automatic detection of anomalies in space- and time-varying measurements is an important tool in several fields, e.g., fraud detection, climate analysis, or healthcare monitoring. We present an algorithm for detecting anomalous regions in multivariate spatio-temporal time-series, which allows for spotting the interesti…
The large-scale structure of the universe is comprised of virialized blob-like clusters, linear filaments, sheet-like walls and huge near empty three-dimensional voids. Characterizing the large scale universe is essential to our understanding of the formation and evolution of galaxies. The density range of clusters, wa…
The study extends Lp-spectrum analysis to warped products and Kleinian groups.
problem Extending Lp-spectrum analysis to new types of manifolds. method Generalized to warped products and certain quotients of hyperbolic space.
result Proves the Lp-spectrum contains a parabolic region for specific manifolds. Study global geometry of dynamical systems with entire vector fields.
problem Understanding the global structure of equilibria and their basins.
method Step-by-step analysis of basins of centers, nodes, and foci; introduction of global elliptic sectors.
result Characterization of heteroclinic regions connecting equilibria.
We present a method to desingularize a compact G_2 manifold with isolated conical singularities by cutting out a neighbourhood of each singular point and glueing in an asymptotically conical G_2 manifold. Controlling the error on the overlap glueing region enables us to use a result of Joyce to conclude that the result…
Study how neural networks optimize to stable linearly connected regions.
problem Understanding how neural networks converge to stable solutions under different training conditions.
method Investigate the stability of neural networks to SGD noise and apply it to iterative magnitude pruning.
result Subnetworks that reach full accuracy must be stable to SGD noise, either at initialization or early in training.
We show that discrete synaptic weights can be efficiently used for learning in large scale neural systems, and lead to unanticipated computational performance. We focus on the representative case of learning random patterns with binary synapses in single layer networks. The standard statistical analysis shows that this…
Study reveals properties of local minima in ReLU networks.
problem Understanding the loss landscape of neural networks.
method Theoretical analysis of one-hidden-layer ReLU networks.
result All differentiable local minima are global within certain regions.
Study examines dependence of extreme electricity prices in Australian markets.
problem Understanding and managing risks of extreme price outcomes in Australian electricity markets.
method Examined extremal dependence using extremograms for 5-minute and 30-minute price data.
result Persistence and dependence of extreme prices are influenced by market structure and renewable energy share.
Study identifies high-density anomalies in normal data regions.
problem Detecting anomalies in normal data regions.
method Introduces non-parametric algorithmic frameworks for unsupervised detection.
result IPP framework yields the best detection results.
Stochasticity and limited precision of synaptic weights in neural network models are key aspects of both biological and hardware modeling of learning processes. Here we show that a neural network model with stochastic binary weights naturally gives prominence to exponentially rare dense regions of solutions with a numb…
Study of G2-structures with isolated singularities and bounded torsion.
problem Understanding G2-structures with special torsion and isolated singularities. method Revisiting known examples, describing symmetries, and analyzing collapsing of circle fibres.
result Collapsing circle fibres at isolated points cannot produce G2-structures with bounded torsion. New stability and isolation results for Einstein manifolds.
problem Stability and isolation of Einstein manifolds.
method Conditions on Weyl tensor for AH and ALE manifolds, Bochner tensor for Kähler and Sasaki manifolds.
result Established new stability criteria and isolation results for various types of Einstein manifolds.
Map quandle orders to actions, characterize isolated orders, and prove no isolated right orders.
problem Characterizing isolated orders on quandles and proving the absence of certain orders.
method Construct a continuous map from orders to actions, use strong rigidity, and analyze specific cases.
result No isolated right orders on free quandles, except for specific cases.
This paper presents a new insight into improving the performance of Stochastic Neighbour Embedding (t-SNE) by using Isolation kernel instead of Gaussian kernel. Isolation kernel outperforms Gaussian kernel in two aspects. First, the use of Isolation kernel in t-SNE overcomes the drawback of misrepresenting some structu…
Defines Perelman's functionals on manifolds with non-isolated conical singularities.
problem Defining functionals on manifolds with non-isolated conical singularities.
method Starting from a spectral point of view for the Perelman's λ-functional, defining the spectrum of Schrödinger operator and proving the existence of discrete eigenvalues.
result Proves the existence of the infimum of W-functional and obtains asymptotic behavior of eigenfunctions.
iMondrian forest combines isolation forest and Mondrian forest for better anomaly detection.
problem Anomaly detection in batch and online settings.
method Hybrid of isolation forest and Mondrian forest, using depth in Mondrian forest structure.
result iMondrian forest outperforms existing methods in batch and online settings.
We investigate the face numbers of simplicial complexes with Buchsbaum vertex links, especially pseudomanifolds with isolated singularities. This includes deriving Dehn-Sommerville relations for pseudomanifolds with isolated singularities and establishing lower bound theorems when the singularities are also homological…
We explore the geometry of nonpositively curved spaces with isolated flats, and its consequences for groups that act properly discontinuously, cocompactly, and isometrically on such spaces. We prove that the geometric boundary of the space is an invariant of the group up to equivariant homeomorphism. We also prove that…
New bounds on trajectory safety in training models with Langevin Dynamics.
problem Bounding the probability of a model's trajectory staying away from a designated failure region.
method Analyzes Langevin dynamics on smooth, strongly convex loss landscapes, introducing shape-free and local relaxation bounds.
result The in-set probability relaxes to the static value after a burn-in time of order d, using only the global spectral gap of the loss.
The Poincaré-Hopf theorem is extended to projective varieties with isolated singularities.
problem Extending the Poincaré-Hopf theorem to varieties with isolated singularities.
method Using generalizations of the Poincaré-Hopf index.
result A Poincaré-Hopf type theorem for projective varieties with isolated singularities.
Quantifies how geodesic planes isolate in hyperbolic 3-manifolds.
problem Understanding isolation properties of geodesic planes in hyperbolic 3-manifolds.
method Quantitative estimates of geodesic planes in frame bundles, using tight areas and densities.
result Polynomial estimates of isolation properties with degree given by modified critical exponents.
Study dynamics and topology of flows near non-saddle sets or W-sets.
problem Understanding the dynamics and topology of flows near specific invariant sets.
method Cohomological relations and global properties analysis.
result Dynamical classification of surfaces and robustness of non-saddle-sets.
We define two types of local indices of a vector field at an isolated zero on the boundary, and prove Poincare-Hopf-type index theorems for certain vector fields on a compact smooth manifold which have only isolated zeros.
Study reveals GAGA phenomenon in Poisson cohomology for plane structures with isolated singularities.
problem Understanding Poisson cohomology for plane structures with isolated singularities.
method Determined Gerstenhaber algebra structure over Poisson cohomology groups.
result GAGA type phenomenon observed in Poisson cohomology.
Proves conditions for positive scalar curvature on certain manifolds with conical singularities.
problem Conditions for positive scalar curvature on manifolds with isolated conical singularities.
method Analyzes isolated conical singularities and uses Geroch type results.
result No metric with positive scalar curvature on X#Tn with isolated conical singularity. We study the Euler obstruction of essentially isolated determinantal singularities (EIDS). The EIDS were defined by W. Ebeling and S. Gusein-Zade, as a generalization of isolated singularity. We obtain some formulas to calculate the Euler obstruction for the determinantal varieties with singular set an ICIS.
The paper builds complex hyperbolic 2-manifolds with isolated singularities.
problem Finding compact complex hyperbolic 2-manifolds with non-free actions and isolated fixed points.
method Constructs specific examples for each prime p with Z/pZ action. result General examples for p=2 related to complex hyperbolic lattices conjugacy separability. The Poincaré-Hopf theorem is extended to projective varieties with isolated singularities.
problem Extending the Poincaré-Hopf theorem to projective varieties with isolated singularities.
method Using generalized Poincaré-Hopf indices for a projective variety with isolated determinantal singularities.
result A Poincaré-Hopf type theorem is proven for projective varieties with isolated singularities.
Robert Bryant (Theorie des varietes minimales et applications, 1988, 154: 321-347) proved that an isolated singularity of a conformal metric of positive constant curvature on a Riemann surface is a conical one. Using Complex Analysis, we find all of the local models for an isolated singularity of a flat metric whose ar…
Study on solutions near isolated singularities in 6D Yamabe equation.
problem Behavior of solutions near isolated singularities in 6D Yamabe equation.
method Analyze asymptotic behavior of local solutions in non-conformally flat metrics.
result Solutions are asymptotically close to Fowler solutions in 6D.
A generalized semitoric system F:=(J,H): M --> R^2 on a symplectic 4-manifold is an integrable system whose essential properties are that F is a proper map, its set of regular values is connected, J generates an S^1-action and is not necessarily proper. These systems can exhibit focus-focus singularities, which corresp…
Two new scoring methods improve anomaly detection in Isolation Forest.
problem Anomaly and outlier detection in data.
method Generalized score function and volume-based scoring for individual trees.
result Significant improvement in anomaly detection on 34 datasets.
We study solutions to conformally invariant equations with isolated singularties.
Local data coverage governs memorization in diffusion models.
problem Memorization in diffusion models
method Derive a theoretical criterion based on local data coverage
result Predicts memorization based on density of training data in neighborhood and dataset size
The purpose of this erratum is to correct the proof of Theorem A.0.1 in the appendix to our article ``Hadamard spaces with isolated flats'' math.GR/0411232, which was jointly authored by Mohamad Hindawi, Hruska and Kleiner. In that appendix, many of the results of math.GR/0411232 about CAT(0) spaces with isolated flats…
New isolated geometric triangulations found in once-punctured torus bundles.
problem Identifying isolated geometric triangulations in 3-manifolds.
method Examining ideal triangulations and their moves to find isolated geometric ones.
result Infinite family of once-punctured torus bundles with isolated geometric triangulations.
Model predicts insolvency risks in banks due to liquidity and credit risks.
problem Determining insolvency regions in banks due to non-linear interaction between liquidity and credit risks.
method Developed a continuous-time structural dynamic model integrating Basel III requirements into a stochastic optimal control framework. Used Hamilton-Jacobi-Bellman (HJB) equation to solve for insolvency boundary. Derived surrogate analytical approximation for real-time monitoring.
result Calibrated model reveals significant non-linear threshold effects and accelerates insolvency transition.
Proposes methods to improve interpretability of Isolation Forest for anomaly detection.
problem Lack of interpretability in Isolation Forest.
method Defines feature importance scores and unsupervised feature selection methods.
result Improves interpretability of Isolation Forest for anomaly detection.
Score-based methods fail with isolated components and incorrect mixing proportions.
problem Score-based methods struggle with distributions having isolated components and incorrect mixing proportions.
method Score-based methods, including score matching, are used but fail in the presence of isolated components and incorrect mixing proportions.
result Score-based methods cannot discover isolated components or identify correct mixing proportions.
ABIForest improves anomaly detection using attention weights.
problem Anomaly detection in datasets.
method Attention mechanism integrated into Isolation Forest.
result ABIForest outperforms standard Isolation Forest on synthetic and real datasets.