Geometrically solves Schrödinger flow on sphere.
problem Solving periodic Cauchy problem for Schrödinger flow on sphere.
method Explicit geometric algorithm for construction of solutions.
result Explicit geometric algorithm for solving Schrödinger flow on sphere.
New algorithm converges geometrically fast with different step-sizes.
problem Distributed optimization with varying step-sizes.
method Adapt-Then-Combine (ATC) variation of DIGing algorithm.
result Geometric convergence with uncoordinated step-sizes.
The paper analyzes a geometrical algorithm for statistical inference with convergence guarantees.
problem Statistical inference on nonparametric cases.
method Derives a bound for learning rate to ensure local convergence of a geometrical projection algorithm.
result Specific forms of the bound are calculated for m-mixture and e-mixture estimation problems.
New statistical measures assess group separability in low-dimensional geometrical spaces.
problem Lack of statistical measures to evaluate group separability in low-dimensional geometrical spaces.
method Proposed three statistical measures (PSI-ROC, PSI-PR, PSI-P) based on Projection Separability rationale.
result Statistical-based measures outperform traditional cluster validity indices in evaluating group separability.
New geometric SDEs and discretizations on Riemannian manifolds with error bounds.
problem Modeling diffusion processes on Riemannian manifolds with geometric SDEs.
method Introduced a new construction of geometric SDEs and provided non-asymptotic error bounds.
result First non-asymptotic error bound for geometric Euler-Murayama discretization.
Algorithms compute geometric intersection numbers of curves efficiently.
problem Computing the minimal number of intersections of curves on surfaces.
method Simple algorithms for computing geometric intersection number, constructing curves, and deciding if intersections are zero.
result Efficient algorithms with polynomial time complexity for various curve intersection problems.
Geometric Dirichlet Means algorithm improves topic inference efficiency.
problem Improving topic inference in Latent Dirichlet Allocation models.
method Optimization of a geometric loss function for weighted clustering with geometric corrections.
result Achieves comparable accuracy to Gibbs sampling but with computational efficiency.
Establishes geometric convergence of iterative optimization algorithms.
problem Analyzes convergence of iterative optimization algorithms under general assumptions.
method General framework for iterative optimization algorithms, proving asymptotic geometric convergence and providing convergence rates.
result Asymptotic geometric convergence of iterative optimization algorithms with exact rate.
The paper proposes an efficient NMF algorithm using geometric assumptions and rank-one NMFs.
problem Nonnegative matrix factorization (NMF) for clustering and factorization.
method Geometric assumption on data matrices, rank-one NMF initialization, and clustering.
result The proposed algorithm provides faster speeds and comparable relative errors to classical NMF algorithms.
Geometric Block Model improves community detection in sparse graphs.
problem Improving community detection in sparse graphs.
method Proposes a new geometric block model and a triangle-counting algorithm.
result Triangle-counting algorithm performs near-optimal in sparse graphs.
Survey of Sinkhorn algorithm for optimal transport, emphasizing its geometric origins.
problem Solving optimal transport problems efficiently and accurately.
method Discretization of a non-linear integral equation.
result Geometric interpretation and discretization of the Sinkhorn algorithm.
Kernel-based algorithms improve integral estimation with near-geometric speed.
problem Estimating integrals with target measures that are nearly atomic.
method Weighted kernel herding and sequential Bayesian quadrature.
result Near-geometric rate of convergence for nearly atomic target measures.
Efficient algorithms learn geometric shapes privately with limited data.
problem Learning geometric shapes privately with minimal data.
method Differentially private algorithms for learning unions of polygons.
result Achieves ( α , β ) (α,β) ( α , β ) -PAC learning and ( ε , δ ) (ε,δ) ( ε , δ ) -differential privacy with a sample size of $ ilde{O}\left(\frac{1}{αε}k\log d
ight)$ . Enhanced spectral clustering for geometric graphs improves clustering accuracy.
problem Ineffective standard spectral clustering for geometric graphs.
method Higher-order spectral clustering using higher-order eigenvectors.
result Established weak and strong consistency for Soft Geometric Block Model.
Survey of de Casteljau's algorithm's applications in geometric data analysis.
problem No specific problem stated; focuses on algorithm applications.
method Constructive approach to generalize parametric smooth curves to manifolds.
result Algorithm provides principled way to analyze geometric data.
Geometric step decay schedules improve stochastic algorithms' convergence on sharp nonconvex problems.
problem Convergence of stochastic algorithms on sharp nonconvex problems.
method Geometric step decay schedule applied to stochastic algorithms.
result Geometric step decay schedules lead to local linear convergence rates for sharp nonconvex problems.
This work investigates a mixture of LMC and RMHMC with MMALA for geometric ergodicity.
problem Lack of geometric ergodicity study in Riemannian manifold and Lagrangian Monte Carlo methods.
method Investigates a mixture of LMC and RMHMC with MMALA to achieve geometric ergodicity.
result Demonstrates geometric ergodicity in the mixture of LMC and RMHMC with MMALA.
Adyan and Rabin showed that most properties of groups cannot be algorithmically recognized from a finite presentation alone. We prove that, if one is also given a solution to the word problem, then the class of fundamental groups of closed, geometric 3-manifolds is algorithmically recognizable. In our terminology, the …
Improved private geometric median estimation with nearly-linear time complexity.
problem Estimating the geometric median of a dataset while maintaining privacy.
method Improved algorithm using subsampling and geometric aggregation, achieving nearly-linear runtime.
result Achieves the same approximation quality as previous methods but with nearly-linear runtime.
Study of active learning in geometric block model for community detection.
problem Active learning for community detection in geometric block model.
method Proposed two active learning algorithms combining motif-counting with label query policies.
result Sampling labels of a vanishingly small fraction of nodes is sufficient for exact recovery.
Motivated by a question of Gordon and Wilton, we consider the question of which collections of words are "virtually geometric". In particular, we prove that some words (e.g. bbaaccabc) are not virtually geometric.
This paper introduces online algorithms to estimate robust geometric median in large data streams.
problem Detecting outliers in large data sets using robust statistical measures.
method Online stochastic Newton methods for estimating the geometric median.
result Rates of convergence for online estimation of the geometric median.
This thesis classifies pseudo-Anosov homeomorphisms using geometric Markov partitions.
problem Classifying pseudo-Anosov homeomorphisms up to topological conjugacy.
method Algorithmic approach using geometric Markov partitions.
result Geometric type is a complete invariant of conjugation.
Algorithm simplifies geometric intersection computation.
problem Computing geometric intersection between curves.
method Linear programming and triangulation simplification.
result Algorithm runs in polynomial time.
Geometric framework explains deep learning performance.
problem Understanding why deep learning works well across various tasks.
method Comparing deep learning to quantum computations and diffeomorphic template matching.
result Geometric structures of different deep learning systems.
We construct geometric shrinkage priors for Kählerian signal filters. Based on the characteristics of Kähler manifolds, an efficient and robust algorithm for finding superharmonic priors which outperform the Jeffreys prior is introduced. Several ansätze for the Bayesian predictive priors are also suggested. In particul…
Study reveals geometric properties of neural network activation spaces.
problem Understanding the geometric structure of neural network activation spaces.
method Efficient approximation algorithm to characterize convex hull of activation spaces.
result Four common geometric properties of activation spaces are concluded.
We analyze the generalization and robustness of the batched weighted average algorithm for V-geometrically ergodic Markov data. This algorithm is a good alternative to the empirical risk minimization algorithm when the latter suffers from overfitting or when optimizing the empirical risk is hard. For the generalization…
Algorithm checks if geometrically triangulated manifolds are isometric.
problem Determining if two geometric triangulations of manifolds are isometric.
method Sequence of Pachner moves and barycentric subdivisions with bounds on lengths.
result Bounding the length of transformations between triangulations.
Study on convergence of forward-backward algorithm with geometric conditions.
problem Convergence analysis of the forward-backward algorithm under geometric constraints.
method Revisit geometric notions over arbitrary sets, analyze convex minimization problems, derive Łojasiewicz inequalities, and connect geometry to inverse problems.
result Derive new linear rates for inverse problems with low-complexity priors.
Introduces geometric formulation of EM algorithm for robust inference and various applications.
problem Statistical inference with missing data or unobservables.
method Information geometric formulation of EM algorithm and its extensions.
result Outlier-robust inference algorithm and various applications in deep learning.
Unified framework for geometric computation of minimum-area homotopy.
problem Computing the minimum homotopy area of a closed curve.
method Unified combinatorial word approach combining geometric and algebraic methods.
result Unified geometric proof and constructive algorithm for minimum area homotopy.
New technique tracks geometric properties to correct base algorithms' poor performance.
problem Discrepancy between empirical performance and theoretical regret bounds in linear bandits.
method Data-driven technique that incorporates geometric information to formulate frequentist regret bound.
result Course-corrected algorithms achieve minimax optimal regret of i l d e O ( d T ) ilde{\mathcal{O}}(d\sqrt{T}) i l d e O ( d T ) . Equivalence found between algorithmic regularization and convex penalization for convex losses.
problem Understanding the relationship between algorithmic regularization and convex penalization.
method Introducing a geometric condition and showing equivalence through optimization paths.
result Optimization paths of iterative algorithms on unregularized problems match those of corresponding penalized problems under certain conditions.
We give a recipe to compute the geometric intersection number of an integral lamination with a particular type of integral lamination on an n-times punctured disk. This provides a way to find the geometric intersection number of two arbitrary integral laminations when combined with an algorithm of Dynnikov and Wiest.
Study of control problems on Carnot groups with SO(3) symmetry using geometric algebra.
problem Control problems on Carnot groups with SO(3) symmetry.
method Geometric algebra approach to understand geodesics and develop a control algorithm.
result New algorithm for local control developed.
Viewing Dehn's algorithm as a rewriting system, we generalise to allow an alphabet containing letters which do not necessarily represent group elements. This extends the class of groups for which the algorithm solves the word problem to include nilpotent groups, many relatively hyperbolic groups including geometrically…
In this paper we prove two results, one semi-historical and the other new. The semi-historical result, which goes back to Thurston and Riley, is that the geometrization theorem implies that there is an algorithm for the homeomorphism problem for closed, oriented, triangulated 3-manifolds. We give a self-contained proof…
We introduce a new geometric approach that constructs a transition kernel of Markov chain. Our method always minimizes the average rejection rate and even reduce it to zero in many relevant cases, which cannot be achieved by conventional methods, such as the Metropolis-Hastings algorithm or the heat bath algorithm (Gib…
New models discover new topics over time in topic modeling.
problem Discovering new topics over time in topic modeling.
method Nonparametric Bayesian models and Hungarian matching algorithm.
result Significantly faster than existing methods, discovering new topics in large datasets.
The paper improves Kaczmarz algorithm with momentum for linear least squares.
problem Improving convergence of the Kaczmarz algorithm for linear least squares.
method Integrates geometrically smoothed momentum into the randomized Kaczmarz algorithm.
result Proves expected error reduction in singular vector directions.
Algorithm morphs graphs on hyperbolic surfaces.
problem Morphing graphs on hyperbolic surfaces.
method Generalization of Tutte's spring embedding theorem.
result First algorithm for morphing graphs on hyperbolic surfaces.
New algorithm adapts to optimize non-convex problems efficiently.
problem Inefficient hyperparameter tuning in existing optimization methods.
method Combining geometrization and SARAH algorithms.
result Achieves adaptivity to both accuracy and PL constant.
Enhances quantum circuit synthesis using deep learning and geometric methods.
problem Optimizing quantum circuits for time efficiency.
method Combining deep learning with geometric control techniques.
result Improved time-optimal control in quantum circuit synthesis.
Braids can be represented geometrically as curve diagrams. The geometric complexity of a braid is the minimal complexity of a curve diagram representing it. We introduce and study the corresponding notion of geometric generating function. We compute explicitly the geometric generating function for the group of braids o…
A new geometrical setting for classical field theories is introduced. This description is strongly inspired in the one due to Skinner and Rusk for singular lagrangians systems. For a singular field theory a constraint algorithm is developed that gives a final constraint submanifold where a well-defined dynamics exists.…
GeoAdaLer enhances geometric understanding of Adam for stochastic optimization.
problem Understanding geometric principles behind Adam's success in stochastic optimization.
method Introduces GeoAdaLer, an adaptive learning method based on geometric properties.
result Extends interpretability and effectiveness in complex optimization scenarios.
We give a more geometric approach to an algorithm for deciding whether two hyperbolic 3-manifolds are homeomorphic. We also give a more algebraic approach to the homeomorphism problem for geometric, but non-hyperbolic, 3-manifolds.