Paper proves isoperimetric inequalities for non-reversible Finsler manifolds.
problem Proving isoperimetric inequalities for non-reversible Finsler manifolds.
method Constructing needle decompositions and using curvature-dimension condition CD(K,N).
result Established isoperimetric inequality for non-reversible Finsler manifolds.
The paper provides precise estimates for isoperimetric inequalities on weighted manifolds.
problem Quantitative isoperimetric inequalities on weighted Riemannian manifolds.
method Analyzes L1, Lp, and W2 estimates for the push-forward of measures. result Close approximation of the guiding function's push-forward to Gaussian measure.
Study improves isoperimetric inequality for weighted Riemannian manifolds.
problem Improving isoperimetric inequality for weighted Riemannian manifolds.
method Use of Klartag's needle decomposition and Bakry-Ledoux's Gaussian isoperimetric inequality.
result Established a quantitative upper bound for symmetric difference volumes.
Proves inequality in metric measure spaces with non-branching structure.
problem Proving Heintze-Karcher inequality in metric measure spaces.
method Used needle decomposition technique for metric measure spaces.
result Characterizes equality case in spaces with positive curvature.
Study on rigidity of logarithmic Sobolev inequality on manifolds.
problem Rigidity of logarithmic Sobolev inequality on weighted Riemannian manifolds.
method Needle decomposition method.
result Splitting off of 1-dimensional Gaussian space when equality holds.
Study rigidity of spectral gap on Finsler manifolds with specific curvature bounds.
problem Rigidity of spectral gap on Finsler manifolds with Ricci curvature bound.
method Analysis of spectral gap, splitting phenomena, and needle decomposition.
result Rigidity results for spectral gap, logarithmic Sobolev, and Bakry-Ledoux inequalities.
New curvature-dimension condition for Lagrangians on manifolds.
problem Establishing a curvature-dimension condition for autonomous Lagrangians.
method Generalizing Klartag's needle decomposition technique to Lagrangian setting.
result Equivalence of curvature-dimension condition to displacement convexity of entropy.
The localization technique from convex geometry is generalized to the setting of Riemannian manifolds whose Ricci curvature is bounded from below. In a nutshell, our method is based on the following observation: When the Ricci curvature is non-negative, log-concave measures are obtained when conditioning the Riemannian…
New findings show different cost functions yield equivalent curvature bounds.
problem Establishing equivalence of curvature bounds under various transport costs.
method Needle decomposition and localization technique for optimal transport.
result All CDp(K,N) conditions are equivalent for p>1. ANN learner finds sparse needles in nonlinear haystacks with high probability.
problem Finding sparse features in nonlinear data.
method Generalized LASSO penalty with stochastic gradient descent, warm-start algorithm.
result Phase transition in ANN learner's ability to find needles, better than other learners.
Develops semigroup approach for Finsler geometry, proving isoperimetric inequality.
problem Proving isoperimetric inequality for Finsler manifolds.
method Semigroup approach, Bochner-Weitzenböck formula, gradient estimate, lower weighted Ricci curvature bound.
result Proves Bakry-Ledoux's Gaussian isoperimetric inequality for non-reversible metrics.
In this paper, we study continuous Kakeya line and needle configurations, of both the oriented and unoriented varieties, in connected Lie groups and some associated homogenous spaces. These are the analogs of Kakeya line (needle) sets (subsets of Rn where it is possible to turn a line (respectively an inter…
Learning to approximate a separable function is hard, requiring many samples even with sparse networks.
problem Learning the separable function x↦∑i=1dxi2 with limited samples. method Sparse neural networks vs. dense neural networks, explicit regularization.
result The sample complexity for dense networks is O(d2.5) with explicit regularization, better than O(d4). Deep learning identifies space objects from uncorrelated observations.
problem Finding small groups of observations of the same space objects from a large set of uncorrelated data.
method Training a deep learning model on a large data set of uncorrelated observations to identify groups of observations likely of the same space objects.
result The model correctly identified 83.1% of observation pairs as belonging to the same space object.
Study uses Bayesian Optimization to analyze noise effects in materials research.
problem Optimizing materials with many variables and experimental noise.
method Batch Bayesian Optimization with synthetic data analysis.
result Noise sensitivity varies by problem landscape, impacting optimization outcomes.
CCA helps find hidden connections in complex biomedical data.
problem Analyzing large, multi-variable datasets in biology and medicine.
method Canonical correlation analysis (CCA) for exploring relationships between two sets of variables.
result CCA uncovers essential hidden associations between diverse data types.
Solves dual imbalance in detecting sparse anomalies in MIL.
problem Detecting scarce and sparse anomalous samples in MIL.
method Reformulates MIL as a fine-grained PU learning problem, addressing imbalance at both macro and micro levels.
result Demonstrates effectiveness of BFGPU framework on synthetic and real-world datasets.
SCORE technique reduces BO's high-dimensional search costs.
problem Bayesian optimization's high computational costs in high-dimensional spaces.
method 1D reparametrization trick to maintain linear time complexity.
result Successfully finds global minimum in high-dimensional optimization.
This paper tackles label-efficient evaluation in extreme class imbalance.
problem Challenges in obtaining a sufficient sample for accurate evaluation in tasks with extreme class imbalance.
method Develops a framework for online evaluation based on adaptive importance sampling.
result Establishes strong consistency and a central limit theorem for performance estimates.
Bayesian method suppresses low-frequency pulses in audio recordings.
problem Suppressing long pulses caused by mechanical defects in audio recordings.
method Bayesian approach using Gaussian Process for pulse location, signal interpolation, and tail estimation.
result Perceptual results similar to previous methods, performs well on naturally degraded signals.
Lasso-Zero improves support recovery in high-dimensional linear models.
problem Support recovery in high-dimensional linear models with sparse β0. method Lasso-Zero uses an 'overfit, then threshold' approach with noise dictionaries.
result Lasso-Zero outperforms competitors in support recovery and trade-off between true positives and false discoveries.
Network analysis detects insider trading by flagging coordinated trades.
problem Detecting insider trading due to limited labelled data.
method Data-driven network approach using SEC trade data.
result Algorithm identifies insider trading clusters with high accuracy.
Study develops a method to select penalty parameters for sparse neural networks without cross-validation.
problem Selecting optimal penalty parameters for sparse neural networks without cross-validation.
method Established theoretical foundation to bound the infinite norm of the gradient of the loss function at zero.
result Proposed method effectively selects penalty parameters for sparse neural networks.
Researchers describe and compare decompositions of Poincaré duality pairs.
problem Understanding and comparing different decompositions of Poincaré duality pairs.
method Developed and described edge splittings of decompositions based on group properties.
result Compared decompositions with two other related decompositions.
Novel method detects group differences in temporal data using scan statistics.
problem Detecting group differences in temporally evolving data with poor effect sizes.
method Parametric model for SPD matrix trends, generalized scan statistics for graph structures.
result Identifies scientifically interesting group differences not seen in full graph models.
The paper proposes and discusses semiorthogonal decompositions for moduli spaces of vector bundles.
problem Decompositions of moduli spaces of vector bundles with fixed determinant of odd degree.
method Semiorthogonal decompositions, Grothendieck ring of varieties, mirror symmetry, graph potentials, Fukaya category.
result Evidence for a conjectural semiorthogonal decomposition of moduli spaces of rank 2 bundles with odd determinant.
New complexity notion connects finite decomposition and asymptotic property C.
problem Understanding and connecting different properties in metric spaces.
method Introducing finite APC-decomposition complexity and proving its implications.
result Finite APC-decomposition complexity implies property A for metric spaces.
The paper classifies decompositions of 3-sphere and lens spaces with handlebodies.
problem Classifying decompositions of 3-manifolds with handlebodies.
method Studied decompositions of 3-sphere and lens spaces with three handlebodies, using stabilizations.
result Determined whether decompositions are stabilized.
This paper generalizes octahedral decomposition to links in thickened surfaces.
problem Understanding the geometry of links in thickened surfaces.
method Octahedral decomposition of links in thickened surfaces.
result Nonpositive curvature of the complement and essential-ness of edges proved.
A new tensor decomposition method using a dictionary for better interpretability.
problem Ensuring interpretability in tensor decomposition models.
method Dictionary-based tensor canonical polyadic decomposition with sparse coding.
result Improves parameter identifiability and estimation accuracy in tensor decomposition.
Paper studies polynomial decomposition in system identification and machine learning.
problem Identifying a polynomial decomposition model in system identification and machine learning.
method Introduces X-rank decomposition and proves results on generic/maximal rank and identifiability.
result Proves new results on identifiability of a polynomial decomposition model.
A new tensor decomposition method that minimizes KL divergence.
problem Tensor reconstruction accuracy.
method Legendre decomposition, based on information geometry.
result Minimizes KL divergence and improves tensor reconstruction accuracy.
Researchers compute Goeritz groups for all (1,1)-link decompositions.
problem Computing Goeritz groups for all (1,1)-link decompositions.
method Analyzing surface decompositions and isotopy classes of homeomorphisms.
result Computed Goeritz groups for all (1,1)-link decompositions.
Study concordance of decompositions from defining sequences in 3-sphere.
problem Understanding concordance and bordism of decompositions from defining sequences.
method Relate to invariants of toroidal decompositions and cobordism of homology manifolds.
result At least uncountably many concordance classes of decompositions in 3-sphere.
Characterizes conditions for quotient spaces of decompositions to be manifolds.
problem Conditions for quotient spaces of decompositions to be manifolds.
method Generalized characterizations of upper semi-continuity for decomposition into one for a class decomposition.
result Characterizations of necessary and sufficient conditions for quotient spaces of decompositions to be k-manifolds (k=1,2). The paper establishes new tools for studying decomposition complexity in metric spaces.
problem Understanding decomposition complexity in metric spaces.
method Three equivalent definitions for decomposition complexity are provided, and new methods are discussed to show metric spaces do not have finite decomposition complexity.
result Metric spaces with finite hyperbolic dimension have finite (weak) decomposition complexity.
Study shows OAT decomposition generates unexplained profit and loss, while SU decompositions depend on risk factor order.
problem Understanding profit and loss attribution in financial markets.
method Used financial market data from 2003 to 2022 to compare OAT, SU, and ASU decompositions.
result SU decompositions are sensitive to risk factor order and cannot identify all relevant risk factors.
Given a Delaunay decomposition of a compact hyperbolic surface, one may record the topological data of the decomposition, together with the intersection angles between the `empty disks' circumscribing the regions of the decomposition. The main result of this paper is a characterization of when a given topological decom…
New tensor network decompositions improve CNN performance.
problem Limited exploration of tensor network decompositions for CNNs.
method Characterized a new class of CNN modules and experimentally compared various decompositions.
result Some nonlinear decompositions outperform existing ones in terms of accuracy and efficiency.
A new algorithm speeds up CP decomposition for large tensors.
problem Efficiently processing large-scale tensors in real-time.
method Randomized online CP decomposition (ROCP) algorithm.
result ROCP reduces computing time and memory usage significantly.
New proof presented for double pants decompositions.
problem Transitivity of flip-twist groupoid on decompositions.
method New proof accessible without initial paper.
result Corrected proof of transitivity of flip-twist groupoid.
Paper characterizes optimization landscape of Tucker decomposition.
problem Finding exact Tucker decomposition is a nonconvex optimization problem.
method Characterized the optimization landscape and provided a local search algorithm.
result All local minima are globally optimal if tensor has an exact Tucker decomposition.
A double pants decomposition of a 2-dimensional surface is a collection of two pants decomposition of this surface introduced in arXiv:1005.0073v2. There are two natural operations acting on double pants decompositions: flips and handle twists. It is shown in arXiv:1005.0073v2 that the groupoid generated by flips and h…
Smooth 4-manifolds have simple horizontal decompositions.
problem Classifying smooth, closed, orientable 4-manifolds.
method Horizontal handlebody decomposition.
result Simplest horizontal decompositions classify closed 4-manifolds.
Profinite group analysis reveals manifold decompositions.
problem Decomposing 3-manifolds into simpler parts.
method Profinite completion of fundamental groups.
result Profinite completion determines manifold decompositions.
Let J1 be the real form of a complex simple Jordan algebra such that the automorphism group is F4(−20). By using some orbit types of F4(−20) on J1, for F4(−20), explicitly, we give the Iwasawa decomposition, the Oshima--Sekiguchi's Kε−Iwasawa decomp…
We study the topological types of pants decompositions of a surface by associating to any pants decomposition P, in a natural way its pants decomposition graph, Γ(P). This perspective provides a convenient way to analyze the maximum distance in the pants complex of any pants decomposition to a pants decomposition c…
New method uses random decompositions for high-dimensional Bayesian optimization.
problem Learning accurate decompositions for high-dimensional black-box functions.
method Data-independent random tree-based decomposition sampling.
result Random decomposition upper-confidence bound algorithm (RDUCB) yields significant empirical gains.