Study connects Powell Conjecture to reducing sphere complex's connectivity.
problem Prove Powell Conjecture for genus g Heegaard surfaces. method Analyze reducing sphere complex R(Σg) and its connectedness. result Powell Conjecture true if and only if R(Σg) is connected. The paper shows how reducible complexes affect local indicability.
problem The local indicability of subcomplexes in reducible complexes.
method Characterization of diagrammatic reducibility and application to local indicability.
result Injective labeled oriented trees are locally indicable if reducible of degree 2.
Reduces connectivity problem for genus-4 Heegaard surface in 3-sphere.
problem Connectivity problem in reducing sphere complex for genus-4 Heegaard surface.
method Presented a sufficient condition for a non-separating weak reducing pair to be separated by a reducing sphere.
result Reduced connectivity problem to showing disjointness of representative reducing spheres from a fixed disk.
Fixed point sets of certain group actions are contractible.
problem Fixed point sets of group actions on specific types of complexes.
method Analyzing group actions on diagrammatically reducible complexes with fine 1-skeleton.
result Fixed point sets are contractible under certain conditions.
For a boundary-reducible 3-manifold M with ∂M a genus g surface, we show that if M admits a genus g+1 Heegaard surface S, then the disk complex of S is simply connected. Also we consider the connectedness of the complex of reducing spheres. We investigate the intersection of two reducing spheres…
New Q-learning method reduces variance and achieves optimal sample complexity.
problem Improving Q-learning to reduce variance and improve sample efficiency. method Introduces variance-reduced Q-learning and analyzes its sample complexity. result Achieves minimax optimal sample complexity for estimating optimal Q-function. The paper provides examples of keen weakly reducible bridge spheres for links in b-bridge position.
problem Characterizing and finding examples of keen weakly reducible bridge spheres.
method Analyzing bridge spheres and their properties in terms of compressing disks and width complex.
result Infinitely many examples of keen weakly reducible bridge spheres for links in b-bridge position.
Study reduces financial dynamics complexity using PCA for NASDAQ, oil, gold, and USD.
problem Understanding complex financial interactions among multiple assets.
method Time-delay embedding and PCA for dimensionality reduction, followed by linear regression.
result Limited number of principal components capture dominant dynamics of each asset.
The Jones polynomial can be expressed in terms of spanning trees of the graph obtained by checkerboard coloring a knot diagram. We show there exists a complex generated by these spanning trees whose homology is the reduced Khovanov homology. The spanning trees provide a filtration on the reduced Khovanov complex and a …
In this paper a novel modification of the multilevel Monte Carlo approach, allowing for further significant complexity reduction, is proposed. The idea of the modification is to use the method of control variates to reduce variance at level zero. We show that, under a proper choice of control variates, one can reduce t…
This paper proposes a method to reduce complexity in GLMs with categorical predictors.
problem Wasteful, hard-to-interpret, and prone to overfitting of traditional one-hot encoding for high-cardinality categorical predictors.
method Clustering categories of categorical predictors through a numerical method that preserves or improves accuracy while reducing the number of coefficients.
result Clustering categories of categorical predictors reduces complexity substantially without harming accuracy.
A new method reduces model complexity in DMD using LARS.
problem Building accurate reduced-order models from data.
method Least Angle Regression (LARS) for Dynamic Mode Decomposition (DMD).
result LARS4DMD produces comparable performance to DMDSP with less complexity.
We apply L0 regularization to reduce neural network complexity and interpretability.
problem Over-parameterization in neural networks leads to susceptibility to attacks, loss of interpretability, and increased SWaP-C.
method We use L0 regularization to reduce complexity and evaluate the trade-off between complexity and desired metrics.
result L0 regularization captures saliency in the input space and reduces complexity of neural networks.
CDEFs reduce model complexity and uncover time correlations.
problem Model complexity and data efficiency in probabilistic modeling.
method Builds on deep exponential families, ties weights for reduced parameters.
result CDEFs uncover time correlations with fewer parameters.
New framework reduces LLM complexity by directly finetuning in Boolean domain.
problem Reducing the complexity of large language models (LLMs) while maintaining performance.
method Proposes a novel framework using multi-kernel Boolean parameters for direct finetuning in the Boolean domain.
result Significantly reduces complexity during both finetuning and inference, outperforming recent techniques.
A new algorithm reduces time complexity for binary time series classification.
problem High time complexity of ensemble shapelet transform limits its application.
method Introduces short isometric shapelet transform with two strategies: fixed shapelet length and single linear classifier.
result Demonstrates superior performance and reduced time complexity.
Improved BLS by reducing pseudoinverse complexity for added inputs.
problem High computational complexity in pseudoinverse for incremental learning.
method Used inverse of a sum of matrices to reduce matrix inversion size.
result Significant reduction in computational complexity (1.24 - 1.30 speedups).
We show that the complex of weak reducing disks for the unknot in 3-bridge position is contractible.
Proposes a method to reduce parallel complexity of MLMC in SGD.
problem Poor scalability of MLMC in SGD on parallel platforms.
method Proposes a delayed MLMC gradient estimator to reduce parallel complexity.
result Proves reduction in average parallel complexity per iteration at the cost of slightly worse convergence rate.
This paper explores the non-convex composition optimization in the form including inner and outer finite-sum functions with a large number of component functions. This problem arises in some important applications such as nonlinear embedding and reinforcement learning. Although existing approaches such as stochastic gr…
The paper analyzes reducing model complexity for better generalization.
problem Improving model generalization with reduced complexity networks.
method Upper bound on Vapnik-Chervonenkis dimension, pruning, quantization, and a novel loss function.
result Quantization and the proposed loss function lead to sparser models with comparable accuracy.
The abstract describes a strategy to construct reduced Khovanov homology for links in lens spaces.
problem Constructing reduced Khovanov homology for links in lens spaces.
method Generalizing a symplectic interpretation of reduced Khovanov homology for links in S3 and constructing cochain complexes for links in S3 and S2imesS1. result The cohomology of the constructed cochain complex for links in S2imesS1 may be a link invariant. The paper describes the structure of injective LOT-complexes and proves they are aspherical.
problem The unresolved asphericity question for labeled oriented trees encoding spines of ribbon discs.
method Complete description of the link of a reduced injective LOT complex, proving asphericity.
result Reduced injective LOT complexes are aspherical, with specific conditions for non-boundary sub-LOTs.
Proposes DCADL for efficient image classification with reduced complexity.
problem Efficiency and discriminative capability in DL methods for image classification.
method Jointly learns a convolutional analysis dictionary and a universal classifier, reducing time complexity.
result Achieves competitive accuracy with reduced computational cost.
Confirming the Powell Conjecture for genus-3 Heegaard splittings of the 3-sphere.
problem Proving the finitely generated nature of the Goeritz group for genus-3 Heegaard splittings of the 3-sphere.
method Establishing the connectivity of reducing sphere complexes for the genus-3 case.
result Confirmation of the Powell Conjecture for genus-3 Heegaard splittings of the 3-sphere.
Reduced modeling of a computationally demanding dynamical system aims at approximating its trajectories, while optimizing the trade-off between accuracy and computational complexity. In this work, we propose to achieve such an approximation by first embedding the trajectories in a reproducing kernel Hilbert space (RKHS…
By the work of Harer, the reduced homology of the complex of curves is a fundamental cohomological object associated to all torsion free finite index subgroups of the mapping class group. We call this homology group the Steinberg module of the mapping class group. It was previously known that the curve complex has the …
VRL-SGD reduces communication complexity in non-identical data settings.
problem Training machine learning models with non-identical data distribution.
method VRL-SGD, which eliminates gradient variance dependency and achieves linear speedup with lower communication complexity.
result VRL-SGD reduces communication complexity from $O(T^{ rac{3}{4}} N^{ rac{3}{4}})$ to $O(T^{ rac{1}{2}} N^{ rac{3}{2}})$.
Neural networks have proven to be extremely powerful tools for modern artificial intelligence applications, but computational and storage complexity remain limiting factors. This paper presents two compatible contributions towards reducing the time, energy, computational, and storage complexities associated with multil…
Generalised contact structures are studied from the point of view of reduced generalised complex structures, naturally incorporating non-coorientable structures as non-trivial fibering. The infinitesimal symmetries are described in detail, with a geometric description given in terms of gerbes. As an application of the …
Paper applies FloatSD8 to LSTM networks, reducing complexity and power.
problem Training and inference complexity of LSTM networks.
method Applied FloatSD8 for weights, 8-bit quantization for gradients/activations, reduced arithmetic precision.
result Successfully trained LSTM models with reduced complexity and preserved accuracy.
Cubic regularization (CR) is an optimization method with emerging popularity due to its capability to escape saddle points and converge to second-order stationary solutions for nonconvex optimization. However, CR encounters a high sample complexity issue for finite-sum problems with a large data size. %Various inexact …
Genus 3 Heegaard groups of lens space connected sums are finitely generated.
problem Understanding the structure of mapping class groups of genus 3 Heegaard splittings.
method Proved finitely generated property through connected reducing sphere complexes.
result Mapping class groups are finitely generated and complexes are connected.
A second part of detailed elementary introduction into Khovanov homologies. This part is devoted to reduced Jones superpolynomials. The story is still about a hypercube of resolutions of a link diagram. Each resolution is a collection of non-intersecting cycles, and one associates a 2-dimensional vector space with each…
Computer experiments reveal complex knots that don't simplify.
problem Understanding the dynamics of complex knots under self-repulsion.
method Computer simulations of knot theory, focusing on rational knots and tangles.
result Discovered hard unknots and complexified knots that do not reduce to simpler forms under self-repulsion.
We show that the Spivak normal fibration of an orientable 4-dimensional Poincaré complex has a vector bundle reduction.
Backward exploration reduces sample complexity in policy evaluation.
problem Empirical policy evaluation in reinforcement learning.
method Backward exploration algorithms from high-cost states.
result Reduced average-case sample complexity to O(logS). A hybrid model reduces graph complexity for improved classification accuracy.
problem High computational complexity and large number of parameters in higher-order graph convolutional networks.
method Weight sharing mechanism and novel fusion pooling layer to reduce parameters and complexity.
result The proposed model achieves highest classification accuracy with fewer trainable parameters.
Sliced Inverse Regression reduces parameter space for estimating complex financial models.
problem High-dimensional parameter space in stochastic differential equations.
method Sliced Inverse Regression for dimension reduction.
result Reduced computational costs in estimating parameters.
A new method reduces complexity and uncertainty in neural networks.
problem Uncertainty quantification in complex neural networks.
method Condensed Stein Variational Gradient Descent (cSVGD) method.
result Condensed SVGD provides uncertainty quantification on parameters.
Active learning reduces spin network inference complexity by 10^6-fold.
problem Difficulty in inferring direct interactions in complex networks.
method Information geometry framework to quantify inference difficulty and information gain from perturbations.
result Designed perturbations reduce sampling complexity by 10^6-fold across various network architectures.
New algorithm reduces matrix multiplication time for sparse matrices.
problem Efficiently multiply large sparse matrices with limited space.
method Exploits sparsity to reduce QR decompositions and time complexity.
result Time complexity reduced to $\widetilde{O}\left((
nz(X)+
nz(Y))\ell+n\ell^2
ight)$ in expectation.
LeanResNet reduces CNN complexity without sacrificing accuracy.
problem High computational cost in CNNs, especially in ResNets.
method Introduces lean convolution operators that reduce parameters and complexity.
result LeanResNet achieves comparable results to other reduced architectures with fewer parameters.
VRCQ algorithm reduces variance in Q-learning for MDPs, achieving optimal sample complexity.
problem Estimating the optimal Q-function in MDPs with synchronous sampling.
method VRCQ combines direct variance reduction and Cascade Q-learning.
result VRCQ is minimax optimal and instance optimal for single-action problems.
Paper tackles uncertainties in reduced-order modeling of complex systems.
problem Model-form uncertainties in reduced-order modeling of complex systems.
method Combines Riemannian projection and retraction operators on a subset of the Stiefel manifold with an information-theoretic formulation.
result Identifies and quantifies the impact of model-form uncertainties on inferred operators.
Paper introduces probabilistic methods to approximate archetypal analysis, reducing complexity.
problem Inherent computational complexity of archetypal analysis limits its practical applicability.
method Two preprocessing techniques: dimensionality reduction and representation cardinality reduction, using probabilistic geometry.
result The method effectively reduces scaling and provides near-optimal solutions for prediction errors.
Aramayona and Leininger have provided a "finite rigid subset" X(Σ) of the curve complex C(Σ) of a surface Σ=Σgn, characterized by the fact that any simplicial injection X(Σ)→C(Σ) is induced by a unique element of the mapping class group Mod(Σ). In this…
BayPOD-AL learns reduced-order models from high-fidelity data efficiently.
problem Capturing dynamics of complex systems with large training datasets.
method Bayesian active learning based on uncertainty-aware POD.
result BayPOD-AL reduces computational cost and improves model accuracy.