Sparse PCA method for clustering Gaussian mixtures.
problem Clustering Gaussian mixture models.
method Sparse Principal Component Analysis (SPCA) for clustering.
result Comparison with IF-PCA method and discussion of non-diagonal covariance matrices.
Researchers find non-diagonal Einstein metrics in various signatures.
problem Finding non-diagonal four-dimensional cohomogeneity-one Einstein metrics in different signatures.
method Explicitly seeking and constructing new examples of non-diagonal Einstein metrics, particularly in neutral signature.
result Construct new examples of neutral signature non-diagonal Bianchi type VIII Einstein metrics with self-dual Weyl tensor.
Exact recovery method for community detection in Gaussian mixtures with dependent noise.
problem Community detection in Gaussian mixtures with dependent and heterogeneous noise.
method Maximum likelihood estimator (MLE) for constrained quadratic optimization problem, using Σ Σ Σ -whitened separation and local inequalities. result Sharp exact-recovery threshold and no-gap mechanism in the unknown-size setting.
Study on stability of non-diagonal Einstein metrics on specific homogeneous spaces.
problem Stability analysis of non-diagonal Einstein metrics on H i m e s H / Δ K H imes H/ΔK H im esH /Δ K . method Formula for scalar curvature, study of stability with Hilbert action.
result Non-diagonal Einstein metrics on M M M are unstable with different coindexes. Extends IBP for non-diagonal latent covariance structures, improving feature recovery and denoising.
problem Modeling latent features with smoothness characteristics.
method Extend Indian Buffet Process to include non-diagonal latent covariance structures.
result Smoothness prior improves feature recovery and denoising under appropriate conditions.
Study of Lorentz hypersurfaces with specific curvature properties.
problem Characterizing Lorentz hypersurfaces with complex eigenvalues and constant mean curvature.
method Analyzing hypersurfaces in E 1 n + 1 E_{1}^{n+1} E 1 n + 1 satisfying r i a n g l e H ⃗ = α H ⃗ riangle \vec {H}= α\vec {H} r ian g l e H = α H with non-diagonal shape operator. result Hypersurfaces with at most five distinct principal curvatures have constant mean curvature.
New integrable systems constructed for non-diagonal Killing tensors.
problem Constructing integrable Hamiltonian systems with quadratic momenta.
method Using Nijenhuis geometry and gl-regular Nijenhuis operators.
result Reproduces classical Stäckel construction and finds new systems for n≥3.
Study on invariant Einstein metrics on specific flag manifolds.
problem Existence of invariant Einstein metrics on real flag manifolds.
method Analysis of isotropy representations and Riemannian metrics.
result Existence of non-diagonal Einstein metrics on real flag manifolds.
Quaternionic Brownian motion on flag manifold linked to sphere diffusion.
problem Modeling quaternionic stochastic areas on quaternionic flag manifolds.
method Relating quaternionic Brownian motion to symplectic Brownian motion and using radial dynamics.
result Quaternionic stochastic areas follow a multivariate normal distribution.
Study shows non-positivity of Einstein-Hilbert action for certain metrics.
problem Analyzing the non-positivity of the Einstein-Hilbert action for specific metrics.
method Using spectral triples and modular operator computations.
result Recovery of earlier results on noncommutative tori and new Gauss-Bonnet theorem.
New method efficiently learns positive-definite curvature for neural nets.
problem Efficiently learn positive-definite curvature for neural net training.
method Spectral-factorized positive-definite curvature learning approach.
result Efficiently applies arbitrary matrix roots and generic curvature learning.
We study the anti-self-dual equation for non-diagonal SU(2)-invariant metrics and give an equivalent ninth-order system. This system reduce to a sixth-order system if the metric is in the conformal class of scalar-flat-Kaehler metric.
Study left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups.
problem Characterize left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups.
method Analyze left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups, classify Lie algebras and groups.
result New results on left-invariant Lorentzian metrics with harmonic curvature and non-parallel Ricci operator.
AlgoPerf competition evaluates neural network training speed-ups.
problem Improving neural network training speed using better algorithms.
method Compared 18 diverse submissions from 10 teams on multiple workloads.
result Schedule Free AdamW algorithm achieved best results in self-tuning ruleset.
Study uses instanton Floer theory to find definite lattices from certain 4-manifolds.
problem Determining definite lattices from smooth 4-manifolds bounded by homology 3-spheres.
method Extends Froyshov's methods using instanton Floer theory.
result Identifies specific definite lattices for +1 surgery on the (2,5) torus knot.
We provide an affirmative answer to a question posed by Tod \cite{Tod:1995b}, and construct all four-dimensional Kahler metrics with vanishing scalar curvature which are invariant under the conformal action of Bianchi V group. The construction is based on the combination of twistor theory and the isomonodromic problem …
VI struggles to fully quantify uncertainty when distributions don't factorize.
problem Uncertainty quantification in non-factorizable distributions.
method Analysis of variational inference trade-offs and divergence choices.
result Different divergences yield different measures of uncertainty in VI.
New Hopf surfaces found in LCK manifolds with potential.
problem Characterizing Hopf surfaces in LCK manifolds with potential.
method Analyzing quotient spaces and embedding properties.
result Non-Vaisman LCK manifolds with potential contain Hopf surfaces.
Study explores new actions on product manifolds with asymmetric factors.
problem Exploring effective circle actions on product manifolds with asymmetric factors.
method Proves existence of infinite families of distinct non-diagonal effective circle actions on products of asymmetric manifolds with S n S^n S n . result Infinite family of distinct non-diagonal effective circle actions on M i m e s S 2 M imes S^2 M im es S 2 . Study on determinant properties of elliptic operators with counterexamples and positive results.
problem Does the determinant of a matrix solution to a second order elliptic equation satisfy the unique continuation property?
method Analyzes counterexamples and positive results for various operators, including reductions to special cases.
result Partial answers and counterexamples provided, with positive results for specific cases.
Paper proposes efficient methods for clustering and signal recovery in high-dimensional data with block structures.
problem High-dimensional clustering and signal recovery under block signal structures.
method CFA-PCA and MA-PCA methods for sparse and dense block signals.
result Proposed methods achieve computational minimax optimality for clustering and signal recovery.
Efficient neural networks compute various differential operators cheaply.
problem Efficient computation of higher time complexity differential operators.
method Restricted neural network architectures with diagonal and hollow Jacobian matrices, allowing efficient extraction of dimension-wise derivatives.
result Demonstrated efficient computation of differential operators for various applications.
Novel duality theory for operator Frobenius algebras solves long-standing hydrodynamic integrable systems problem.
problem Long-standing Eisenhart-Stäckel problem for non-degenerate integrable systems.
method Introduce duality for operator Frobenius algebras and use mutual symmetry assumption.
result Construct new infinite-dimensional integrable systems of hydrodynamic type.
The paper extends statistical manifold structures to generalized warped product manifolds.
problem Generalizing statistical manifold structures to warped product manifolds.
method Developed expressions for curvature and dualistic structures on generalized warped products.
result Dualistic structures on base and fiber induce a dualistic structure on the generalized warped product.
Bayesian approach learns linear operators from noisy data.
problem Learning linear operators from noisy data in infinite-dimensional spaces.
method Bayesian approach with Gaussian priors.
result Establishes posterior contraction rates and generalization error guarantees.
In this paper, We construct the symmetric tensor field G f 1 f 2 G_{f_1f_2} G f 1 f 2 and h f 1 f 2 h_{f_1f_2} h f 1 f 2 on a product manifold and we give conditions under which G f 1 f 2 G_{f_1f_2} G f 1 f 2 becomes a metric tensor, theses tensors fields will be called the generalized warped product, and then we develop an expression of curvature for the connection of th…
We investigate the daily correlation present among market indices of stock exchanges located all over the world in the time period Jan 1996 - Jul 2009. We discover that the correlation among market indices presents both a fast and a slow dynamics. The slow dynamics reflects the development and consolidation of globaliz…
Algorithm samples composite logconcave densities efficiently.
problem Sampling from composite logconcave densities efficiently.
method Uses a restricted Gaussian oracle and gradient queries.
result Achieves strong total variation distance guarantees.
Algorithm learns both stochastic and adversarial MDPs with best-of-both-worlds guarantees.
problem Learning episodic MDPs with known transition and bandit feedback.
method Follow-the-Regularized-Leader method with a hybrid regularizer.
result Achieves O ( l o g T ) \mathcal{O}(log T) O ( l o g T ) regret for stochastic losses and i l d e O ( T ) ilde{\mathcal{O}}(\sqrt{T}) i l d e O ( T ) regret for adversarial losses. We use Chern-Weil theory for Hermitian holomorphic vector bundles with canonical connections for explicit computation of the Chern forms of trivial bundles with special non-diagonal Hermitian metrics. We prove that every del-dellbar exact real form of the type (k,k) on an n-dimensional complex manifold X arises as a di…
adaQN improves training RNNs with low cost and good performance.
problem Training RNNs is computationally difficult due to vanishing/exploding gradient issues.
method Stochastic quasi-Newton algorithm with L-BFGS updating, low per-iteration cost.
result adaQN is competitive with popular RNN training algorithms on language modeling tasks.
Unified representation for tree ensembles indexed by nodes
problem Unifying geometric object for tree ensembles indexed by nodes
method KPP indexes feature map by nodes, weighted by path metric
result Unified non-diagonal Gram for prediction, additive attribution, robust radius, and risk bounds
K-FAC approximates neural networks' Fisher info matrix for faster optimization.
problem Efficiently optimizing neural networks with natural gradient descent.
method Approximates Fisher information matrix using Kronecker-factored matrices.
result K-FAC produces updates that make more progress than stochastic gradient descent.
Paper proves spectral sequences of knot spaces are isomorphic over fields.
problem Proving isomorphism of spectral sequences related to knot spaces.
method Using embedding calculus and Thom space models.
result Spectral sequences of knot spaces are isomorphic over fields.
The paper analyzes how Gaussian kernel parameters affect posterior covariance in Gaussian processes.
problem Understanding the influence of Gaussian kernel parameters on posterior covariance in Gaussian processes.
method Geometric analysis and a posteriori error estimation techniques from adaptive finite element methods.
result The bandwidth parameter and spatial distribution of observations significantly influence posterior covariance and its matrix.
CovRegRF estimates covariance matrix from covariates using random forests.
problem Estimating conditional covariances or correlations among multivariate responses.
method Random forest trees with a custom splitting rule to maximize covariance difference.
result Accurate covariance matrix estimates and controlled Type-1 error.
The paper explores using historical data to improve clinical trial analysis by optimizing covariate weights.
problem Limited covariates in small clinical trials reduce the effectiveness of analysis.
method Leverage historical data to pre-specify covariate weights as a composite covariate.
result A composite covariate improves the cost/benefit ratio and reduces overfitting in small clinical trials.
New method needed for class prior estimation when covariates are reduced.
problem Class prior estimation fails under covariate shift when covariates are reduced.
method Propose a probing algorithm for class prior estimation.
result Provable transformations preserving covariate shift are necessary for class prior estimation.
Enhanced Transformer models predict ETF portfolio performance by optimizing covariance and semi-covariance matrices.
problem Static covariance estimates fail to capture dynamic market fluctuations and non-linear correlations.
method Transformer-based models for real-time covariance and semi-covariance predictions.
result Portfolios optimized with semi-covariance matrix outperform those with standard covariance matrix, especially in volatile conditions.
Paper introduces a novel method for dynamic covariance estimation with random forests.
problem Estimating high-dimensional dynamic covariance matrices with multiple covariates.
method Nonparametric approach using random forests.
result Uniform consistency theory and error rates established for high-dimensional scenarios.
NeurT-FDR controls FDR by incorporating auxiliary covariates in deep learning.
problem Controlling FDR in complex large-scale problems with indirect relations among covariates.
method NeurT-FDR uses a deep Black-Box framework that parametrizes test-level covariates as a neural network and adjusts auxiliary covariates through a regression framework.
result NeurT-FDR makes substantially more discoveries in real datasets compared to competitive baselines.
S-VNNs improve VNNs by sparsifying covariance matrices.
problem Spurious correlations in covariance matrices degrade VNNs' performance and efficiency.
method Apply sparsification techniques on sample covariance matrix and integrate into VNN architecture.
result S-VNNs achieve improved performance, stability, and reduced computational time.
We introduce and study covariance fields of distributions on a Riemannian manifold. At each point on the manifold, covariance is defined to be a symmetric and positive definite (2,0)-tensor. Its product with the metric tensor specifies a linear operator on the respected tangent space. Collectively, these operators form…
Estimates joint inverse covariances with shared structure across groups.
problem Joint estimation of structured inverse covariance matrices.
method Optimization algorithm exploiting shared linear structure.
result Improves estimation of inverse covariances.
Geodesic curves improve flexibility in covariance estimation.
problem Inflexible covariance families limit spatiotemporal modeling.
method Use geodesic curves to build more flexible covariance families.
result Natural projection minimizes geodesic distance to sample covariance.
CSTs improve stability in covariance spectrum analysis without training.
problem Stability and expressiveness in covariance spectrum analysis.
method Sequential application of covariance wavelet filters to input data.
result Stable and expressive hierarchical representations in low-data settings.
New method estimates sparse covariance matrices in logit mixtures.
problem Estimating correlations among random coefficients in logit models.
method Mixed-integer optimization (MIO) with Markov Chain Monte Carlo (MCMC) for posterior draws.
result Correctly recovers true covariance structure from synthetic data.
Deep model predicts shapes of curves with multiple covariates.
problem Predicting shapes of planar curves with various covariates.
method Deep learning model using complex-valued functions, conditional covariance smoother with modality-specific encoders.
result Model accurately predicts shapes of curves with multimodal covariates.