Paper reduces nonholonomic systems with symmetries.
problem Nonholonomic systems with symmetries and conserved quantities.
method Two-step reduction procedure: first results in Chaplygin system, second in almost symplectic structure.
result Almost symplectic manifolds coincide with reduced nonholonomic brackets.
Paper shows spectra can't distinguish naturally reductive manifolds.
problem Cannot distinguish naturally reductive manifolds using Laplace-Beltrami spectrum.
method Characterized naturally reductive 2-step nilpotent Lie groups via Ambrose-Singer's structures; constructed isospectral pairs of 9-dimensional nilmanifolds.
result Spectra of Laplace-Beltrami operator can't distinguish naturally reductive manifolds from non-naturally reductive ones.
New SVRG and SARAH schemes reduce tuning effort for variance reduction.
problem Optimal performance of SVRG and SARAH requires tuning of parameters.
method Introduces Barzilai-Borwein step sizes, averaging, and adaptive inner loop length.
result Improves SVRG, SARAH, and BB variants' convergence rates and performance.
A deep learning method for probabilistic weather forecasting.
problem Probabilistic forecasting of weather.
method Two chained machine-learning steps: dimension reduction and density estimation using normalizing flows.
result The method produces accurate conditional forecast distributions for weather.
Paper proves naturally reductive property is inaudible for certain manifolds.
problem Proving naturally reductive property is inaudible for specific manifolds.
method Using a pair of non-compact 11-dimensional generalized Heisenberg groups, the paper proves the naturally reductive property is inaudible.
result The naturally reductive property is inaudible for certain manifolds.
Study on special Lie groups with Lorentzian metrics.
problem Characterize structure of 2 2 2 -step nilpotent Lorentzian naturally reductive Lie groups. method Develop framework for naturally reductive Lie groups, extend to Lorentzian context, analyze degenerate and non-degenerate cases.
result Complete structural description of naturally reductive 2 2 2 -step Lorentzian nilpotent Lie groups. This paper deals with naturally reductive pseudo-Riemannian 2-step nilpotent Lie groups $(N, \la \,,\,\ra_N)$ , such that $\la \,,\,\ra_N$ is invariant under a left action. The case of nondegenerate center is completely characterized. In fact, whenever $\la \,,\, \ra_N$ restricts to a metric in the center it is proved h…
Develops a new SPP algorithm with variance reduction for weakly convex optimization.
problem Weakly convex, composite optimization problems.
method Inexact semismooth Newton framework with variance reduction for stochastic proximal point updates.
result Establishes convergence results for the proposed algorithm.
SiD distills pretrained diffusion models into a fast one-step generator.
problem Efficiently distilling pretrained diffusion models into a fast generator.
method Reformulates forward diffusion processes as semi-implicit distributions and uses three score-related identities to create a loss mechanism.
result Achieves high FID performance and significantly reduces generation time.
We provide examples of naturally reductive pseudo-Riemannian spaces, in particular an example of a naturally reductive pseudo-Riemannian 2-step nilpotent Lie group ( N , < , > N ) (N, < \,,\,>_N) ( N , < , > N ) , such that < , > N < \,,\,>_N < , > N is invariant under a left action and for which the center is degenerate. The metric does not correspond to a bi-in…
Reduces S L ( 3 ) SL(3) S L ( 3 ) pre-buildings from spectral curves.
problem Constructing pre-buildings from spectral curves.
method Reduction steps from spectral curves to pre-buildings.
result Core of pre-building constructed from spectral curve differential.
Boosting random forests reduces bias and improves predictive performance.
problem Reducing bias in random forest predictions.
method Extract residuals from random forest, fit another random forest to residuals, sum predictions.
result One-step boosted forest has reduced bias and improved predictive performance.
Adaptive step sizes improve optimization for convex and nonconvex problems.
problem Optimizing functions that are not strongly convex.
method Bridge nonconvex and strongly convex problems via regularization, then apply Barzilai-Borwein step sizes with SARAH.
result Regularized SARAH methods achieve better complexity in nonconvex problems.
Study G2-instantons on specific Lie groups, finding conditions and structures.
problem Characterize G2-instantons on 2-step nilpotent Lie groups.
method Analyze connections arising from characteristic connections, use Lie group structure and torsion.
result Establish necessary and sufficient conditions for G2-instantons, define naturally reductive structures.
This work examines consistency issues in Gaussian Mixture Model reduction algorithms.
problem Consistency issues in Gaussian Mixture Model reduction algorithms.
method Discussion of the importance of dissimilarity measure choice and consistency of GMR algorithms.
result Most existing GMR algorithms are not consistent with a unique measure, leading to suboptimal reduced GMs.
Geodesic orbit property studied for Lorentz manifolds.
problem Geodesic orbit property for Lorentz manifolds.
method Defined naturally reductive for pseudo-Riemannian manifolds and proved theorems for Lorentz nilmanifolds.
result Geodesic orbit property holds for Lorentz nilmanifolds under specific conditions.
New method reduces PDE surrogate model training costs by selectively acquiring time steps.
problem High computational cost of generating training data for PDE surrogate models.
method STAP (Selective Time-Step Acquisition for PDEs) framework that acquires only important time steps.
result Demonstrated effectiveness on several benchmark PDEs, reducing training costs.
Study of symmetry distributions in Lorentzian naturally reductive nilmanifolds.
problem Understanding symmetry in Lorentzian naturally reductive nilmanifolds.
method Analysis of 2-step nilpotent Lorentzian Lie groups with transitive isometry subgroups.
result Fixed points of isotropy representation indicate the distribution of symmetry.
The paper shows how to recover true node positions from a graph or similarity matrix.
problem Recovering true distances and positions from a graph or similarity matrix.
method Two steps: matrix factorisation followed by nonlinear dimension reduction.
result Nonlinear dimension reduction can recover latent positions close to a manifold where geodesic distance is encoded.
Study of symplectic and Poisson reduction, proposing Poisson implosion.
problem Understanding and generalizing symplectic reduction to Poisson manifolds.
method Recalled and reviewed symplectic and Poisson reduction, proved cross-section theorem for Poisson manifolds.
result Generalized Guillemin-Sternberg theorem for Poisson manifolds, identified Poisson transversals.
Injectivity and support theorem for X-ray transform on specific Lie groups.
problem Injectivity and support theorem for X-ray transform on 2-step nilpotent Lie groups.
method General reduction principle for manifolds with uniformly escaping geodesics.
result Injectivity and support theorem for X-ray transform on 2-step nilpotent Lie groups.
In this work we introduce a category of discrete Lagrange--Poincare systems LP_d and study some of its properties. In particular, we show that the discrete mechanical systems and the discrete mechanical systems obtained by the Lagrangian reduction of symmetric discrete mechanical systems are objects in LP_d. We introdu…
Regression aims at estimating the conditional mean of output given input. However, regression is not informative enough if the conditional density is multimodal, heteroscedastic, and asymmetric. In such a case, estimating the conditional density itself is preferable, but conditional density estimation (CDE) is challeng…
A new deep neural network tackles nonlinear functional regression with improved dimensionality reduction.
problem Nonlinear functional regression in infinite-dimensional functional data analysis.
method Functional deep neural network with adaptive kernel embedding and projection steps.
result Explicit rates of approximating nonlinear smooth functionals are derived, and the network is shown to be effective in both simulated and real datasets.
In Carnot groups of step 3, all subriemannian geodesics are proved to be normal. The proof is based on a reduction argument and the Goh condition for minimality of singular curves. The Goh condition is deduced from a reformulation and a calculus of the end-point mapping which boils down to the graded structures of Carn…
This work connects LLE, factor analysis, and probabilistic PCA through a stochastic perspective.
problem Exploring the theoretical connection between LLE, factor analysis, and probabilistic PCA.
method Solving the stochastic linear reconstruction of LLE using expectation maximization.
result LLE, factor analysis, and probabilistic PCA are shown to be connected through a stochastic perspective.
New method improves scalability of SGD for large datasets.
problem High variance in stochastic gradient descent.
method Adaptive measure reduction with Carathéodory's theorem.
result Improved scalability to high-dimensional spaces.
A 2-step nilpotent Lie algebra n is called nonsingular if ad(X): n --> [n,n] is onto for any X not in [n,n]. We explore nonsingular algebras in several directions, including the classification problem (isomorphism invariants), the existence of canonical inner products (nilsolitons) and their automorphism groups (maxima…
We provide a reduction in the classification problem for non-compact, homogeneous, Einstein manifolds. Using this work, we verify the (Generalized) Alekseevskii Conjecture for a large class of homogeneous spaces.
Predicts entity-wise topical behavior from network logs.
problem Predicting entity-wise topical behavior from network logs.
method Combines RNN and CNN learning architectures with reduction steps to form homogeneous topical metrics.
result Improves prediction accuracy with both temporal and spatial gains compared to MLP.
Repeated self-distillation improves model performance significantly.
problem How much gain is possible by applying multiple steps of self-distillation?
method Investigated linear regression tasks, applied multiple steps of self-distillation, analyzed excess risk reduction.
result Multi-step self-distillation reduces excess risk by a factor as large as d d d , where d d d is the input dimension. A method for high-dimensional Bayesian optimization reduces dimensionality using EDR and Gaussian process.
problem Extending Bayesian optimization to high-dimensional settings.
method Two-step framework: EDR subspace identification followed by Gaussian process optimization.
result Algorithm converges in high-dimensional contexts, validated by numerical experiments.
Study geodesic orbit Lorentz nilmanifolds, proving structural properties.
problem Understanding the structure of geodesic orbit Lorentz nilmanifolds.
method Analyzing geodesic orbit Lorentz nilmanifolds with reductive decompositions.
result Proves properties of nilpotent subgroups and their nilpotency steps.
GAMs combine autoregressive and log-linear components for data-efficient sequence learning.
problem Poor performance of standard autoregressive models under small-data conditions.
method Introduce Global Autoregressive Models (GAMs) combining autoregressive and log-linear components, trained in two steps.
result GAMs show a strong perplexity reduction over standard models in language modelling.
Decentralized optimization on dynamic manifolds with improved regret bound.
problem Optimizing on nonstationary Riemannian manifolds in decentralized systems.
method Decentralized projected Riemannian gradient descent with weighted Frechet mean consensus.
result Achieved dynamic regret bound of O ( T ( 1 + P T ) / ( 1 − σ 2 ( W ) ) ) {\cal O}(\sqrt{T(1+P_T)}/\sqrt{(1-σ_2(W))}) O ( T ( 1 + P T ) / ( 1 − σ 2 ( W )) ) . A novel Riemannian extension of stochastic variance reduction for manifold optimization.
problem Optimization on the Grassmann manifold for large-scale problems.
method Riemannian stochastic variance reduced gradient (R-SVRG) on the Grassmann manifold.
result The proposed algorithm outperforms standard Riemannian SGD on various problems.
New method improves convergence and reduces variance in noisy optimization problems.
problem Computing exact minimizers with noisy gradient information.
method Stochastic mirror descent with interacting particles.
result Interaction helps improve convergence and reduce variance.
New algorithm balances spatial data approximation and prediction accuracy.
problem Lack of methods considering spatial correlation and downstream modeling in dimension reduction.
method Formalizes approximation and modeling utility as metrics, proposes a balanced algorithm.
result Optimal trade-off between approximation accuracy and downstream modeling utility.
New algorithm reduces optimization complexity in adaptive mirror descent.
problem Optimizing complex, non-smooth, non-convex functions efficiently.
method SVRAMD: Variance Reduced Adaptive Mirror Descent.
result Variance reduction accelerates convergence in adaptive mirror descent.
Novel unsupervised feature selection method using multi-step Markov transition probability.
problem Neglected relationships between non-adjacent data points in feature selection.
method MMFS (Multi-step Markov transition probability for Feature Selection) approach, employing positive and negative viewpoints.
result MMFS effectively maintains data structure in unsupervised feature selection.
Improved variance reduction for Riemannian non-convex optimization with adaptive batch size.
problem Optimizing non-convex functions on Riemannian manifolds.
method Batch size adaptation in R-SVRG, R-SRG, and R-SPIDER.
result Achieves lower total complexities for various non-convex functions.
The paper tackles manifold overfitting in deep generative models.
problem Manifold overfitting occurs when generative models learn the manifold itself instead of the distribution on it.
method The authors propose a two-step procedure: dimensionality reduction followed by maximum-likelihood density estimation.
result The two-step procedure avoids manifold overfitting and enables density estimation on learned manifolds.
Introduces generalized moment maps for almost Hermitian settings.
problem Extending classical moment map theory to almost Hermitian settings.
method Introduces momentumly closed forms and proves a variant of the Darboux-Weinstein theorem.
result Establishes convexity property and constructs reduction space for generalized moment maps.
Improved multi-step TD learning with control variates reduces variance and improves performance.
problem Variance in multi-step TD learning causes divergence in off-policy settings.
method Per-decision control variates for multi-step TD algorithms.
result Control variates significantly improve performance in both on and off-policy tasks.
Transformers interpreted as probabilistic Laplacian Eigenmaps steps.
problem Improving transformer performance through probabilistic interpretation.
method Probabilistic Laplacian Eigenmaps model derivation and graph diffusion step.
result Subtracting identity from attention matrix improves transformer performance.
PLS-Lasso integrates dimension reduction into regression for financial index tracking.
problem Dimension reduction and regression are traditionally treated separately in multivariate data analysis.
method PLS-Lasso integrates dimension reduction directly into the regression process, presenting two formulations: PLS-Lasso-v1 and PLS-Lasso-v2.
result PLS-Lasso-v1 and PLS-Lasso-v2 outperform Lasso in financial index tracking.
Study on reducing dimensionality in high-dimensional regression with kernel methods and stability analysis.
problem Analyzing errors in high-dimensional regression with dimensionality reduction and kernel regression.
method Derive a stability result for kernel regression with Wasserstein distance and apply it to PCA to deduce convergence rates.
result Two-step procedure yields useful convergence rates in semi-supervised settings.
KineticSim: A lightweight, high-performance execution engine for real-time market simulators
problem Simulating financial markets at scale with multi-agent models
method Reusable parallel design pattern: persistent, state-carrying clearing for iterative multi-agent reductions
result Reduces per-step critical-path depth from Theta(L+A) to Theta(log L + ceil(A/L))