The paper calculates critical configurations and Morse indices for polygons on circles or ellipses.
problem Finding critical configurations and their properties for polygons on circles or ellipses.
method Computing Morse indices and gradient vector fields for isolated critical points, relating to eigenvalue questions.
result Computed Morse indices and relationships to eigenvalue questions for polygons on circles or ellipses.
Paper develops a new local convexity condition for non-isolated minima in non-convex optimization.
problem Lack of theory for non-isolated minima in non-convex optimization.
method Formulates a new local convexity condition and studies SGD convergence under this condition.
result Shows SGD can converge locally under the new condition.
Theorems on the existence of vector fields with given sets of Indexes of isolated Singular points are proved for the cases of closed manifolds, pairs of manifolds, manifolds with boundary, and gradient fields. It is proved that, on a two-dimensional manifold, an index of an isolated Singular point of the gradient field…
A new method learns representations without end-to-end backpropagation.
problem Learning representations without labeled data and end-to-end backpropagation.
method Gradient-isolated modules trained using InfoNCE bound.
result Top module's representations yield competitive results on downstream tasks.
Proposes ContSup to boost local learning by supplying context between isolated modules.
problem Local learning's performance degrades with more isolated modules.
method Theoretical analysis and ContSup scheme to supply context between modules.
result Significant performance improvement with minimal overhead.
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.
The ADAM optimizer is exceedingly popular in the deep learning community. Often it works very well, sometimes it doesn't. Why? We interpret ADAM as a combination of two aspects: for each weight, the update direction is determined by the sign of stochastic gradients, whereas the update magnitude is determined by an esti…
Relative cup-length defined for non-Morse functions on manifolds.
problem Defining a lower bound on critical points of non-Morse functions.
method Using local Morse cohomology and cohomology of isolating neighborhoods.
result A lower bound on critical points stronger than absolute cup-length.
We propose a new sampler that integrates the protocol of parallel tempering with the Nosé-Hoover (NH) dynamics. The proposed method can efficiently draw representative samples from complex posterior distributions with multiple isolated modes in the presence of noise arising from stochastic gradient. It potentially faci…
In this paper we discuss Perelman's Lambda-functional, Perelman's Ricci shrinker entropy as well as the Ricci expander entropy on a class of manifolds with isolated conical singularities. On such manifolds, a singular Ricci de Turck flow preserving the isolated conical singularities exists by our previous work. We prov…
In this paper, we prove the compactness theorem for gradient Ricci solitons. Let (Mα,gα) be a sequence of compact gradient Ricci solitons of dimension n≥4, whose curvatures have uniformly bounded L2n norms, whose Ricci curvatures are uniformly bounded from below with uniformly lower bounded vol…
Paper uses DRL for smart MG energy dispatch, improving stability and performance.
problem Improving energy dispatch in IoT-driven smart MGs with DGs, PVs, and batteries.
method Formulated POMDP model, proposed FH-DDPG and FH-RDPG algorithms, compared with baseline algorithms.
result Proposed algorithms enhance MG performance and stability under uncertainty.
Study separates learning rate effects from adaptive gradient methods.
problem Understanding the impact of learning rates on neural network training.
method Introduced a 'grafting' experiment to isolate learning rate effects.
result Many existing beliefs about adaptive gradient methods may be incorrect.
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.
Improved pruning method finds winning neural network subnetworks.
problem Finding a small subnetwork that performs as well as a full neural network.
method Data-dependent pruning criterion using gradient of training loss.
result Data-dependent pruning improves existing pruning algorithms.
GMC benchmark isolates retrieval in Transformers, revealing max-margin alignment.
problem Understanding how Transformers develop match-and-copy behavior on natural data.
method Introducing Gaussian Match-and-Copy (GMC) as a minimalist benchmark.
result Gradient descent drives parameters to diverge while aligning with max-margin separator.
We introduce a natural extension of the concept of gradient Ricci soliton: the Ricci almost soliton. We provide existence and rigidity results, we deduce a-priori curvature estimates and isolation phenomena, and we investigate some topological properties. A number of differential identities involving the relevant geome…
This paper proves Liouville theorems for conformally invariant fully nonlinear equations.
problem Positive entire solutions of certain fully nonlinear equations are unique.
method Derives necessary and sufficient conditions for Liouville-type theorems.
result Enhanced understanding of solutions near isolated singularities.
Study finds all local models for flat metrics with isolated singularities.
problem Understanding isolated singularities in flat metrics on Riemann surfaces.
method Complex Analysis to find local models and polynomial growth conditions.
result An isolated singularity of a flat metric with finite area is a conical one.
Additive models, such as produced by gradient boosting, and full interaction models, such as classification and regression trees (CART), are widely used algorithms that have been investigated largely in isolation. We show that these models exist along a spectrum, revealing never-before-known connections between these t…
This paper improves t-SNE using Isolation kernel for better data representation and efficiency.
problem Misrepresentation of data structures and high computational cost in t-SNE.
method Replacing Gaussian kernel with Isolation kernel in t-SNE.
result Isolation kernel improves t-SNE's accuracy and efficiency without sacrificing quality.
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. Variational inference is increasingly being addressed with stochastic optimization. In this setting, the gradient's variance plays a crucial role in the optimization procedure, since high variance gradients lead to poor convergence. A popular approach used to reduce gradient's variance involves the use of control varia…
Study on slow convergence in geometric variational problems.
problem Slow convergence of solutions in geometric variational problems.
method Identifying necessary conditions for slowly converging solutions and characterizing their convergence rate and direction.
result Characterization of the rate and direction of convergence for slowly converging solutions.
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.
EoS selectively shapes learning, affecting some groups more than others.
problem EoS affects learning differently across the data distribution.
method Branching intervention to enter or exit EoS regime, controlled perturbation to isolate mechanisms.
result EoS redistributes learning, amplifying progress on some groups and suppressing others.
A new framework isolates exploration challenges in RL without explicit rewards.
problem Challenges in reinforcement learning, especially exploration.
method Reward-free RL framework, collecting trajectories without a reward function, then computing policies for various reward functions.
result Efficient algorithm that conducts exploration and computes near-optimal policies for multiple reward functions.
This research proposes a new distance metric using Isolation Forests.
problem Approximating spatial distance between data points.
method Isolation Forests for outlier detection, transforming separation depth into a distance metric.
result The method produces a distance metric invariant to variable scales and capable of handling non-linear relationships.
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.
DBSCAN clustering improved by using nearest neighbour-induced Isolation Similarity.
problem Improving clustering performance of DBSCAN.
method Proposed nearest neighbour method to implement Isolation Similarity.
result DBSCAN clustering performance surpassed by DP algorithm.
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…
Unified signSGD and gradient descent analysis for neural networks.
problem Performance of sign-based optimization methods in neural networks.
method Unified analysis of separable smoothness and ℓ∞-smoothness, isolating geometric properties affecting performance. result Sign-based methods are preferable over gradient descent under specific Hessian properties in deep networks.
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.
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. 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.
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.
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.
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.
Path connectedness of boundaries for certain CAT(0) groups with isolated flats.
problem Conditions for the path connectedness of boundaries of CAT(0) groups.
method Study of CAT(0) groups with isolated flats acting on CAT(0) spaces.
result Visual boundaries of CAT(0) groups with isolated flats are path connected.
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.
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…