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.
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. 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.
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.
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.
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.
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.
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.
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.
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.
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 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 …
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 . 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 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.
In this paper, we generalize the dualistic structures on warped product manifolds to the dualistic structures on generalized warped product manifolds. we develop an expression of curvature for the connection of the generalized warped product in relation to those corresponding analogues of its base and fiber and warping…
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.
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.
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.
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.
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…
Recurrent Neural Networks (RNNs) are powerful models that achieve exceptional performance on several pattern recognition problems. However, the training of RNNs is a computationally difficult task owing to the well-known "vanishing/exploding" gradient problem. Algorithms proposed for training RNNs either exploit no (or…
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…
We propose an efficient method for approximating natural gradient descent in neural networks which we call Kronecker-Factored Approximate Curvature (K-FAC). K-FAC is based on an efficiently invertible approximation of a neural network's Fisher information matrix which is neither diagonal nor low-rank, and in some cases…
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
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.
This work analyzes Adam's preconditioning effect on quadratic functions and quantifies its impact on condition number.
problem Understanding and quantifying the preconditioning effect of Adam to alleviate ill-conditioning in gradient descent.
method Detailed analysis of Adam's preconditioning effect for quadratic functions, including empirical evidence.
result Adam can mitigate the condition number but at a dimension-dependent cost, with specific bounds for different types of Hessians.
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.
The paper constructs non-Riemannian Einstein solutions on S 2 i m e s T 2 S^2 imes T^2 S 2 im es T 2 using cohomologically calibrated affine connections.
problem Constructing non-Riemannian Einstein manifolds on S 2 i m e s T 2 S^2 imes T^2 S 2 im es T 2 . method Using cohomologically calibrated affine connections and analyzing the torsion tensor within the family T ω \mathcal{T}_ω T ω . result Explicit non-Riemannian Einstein solutions are constructed using a torsion tensor associated with the purelly harmonic 3-form.
Study homogeneous Einstein metrics on specific non-Kähler C-spaces.
problem Classify and analyze homogeneous Einstein metrics on non-Kähler C-spaces.
method Use painted Dynkin diagrams and mapping degree theory to classify and find Einstein metrics.
result Existence and classification of invariant Einstein metrics on specific spaces.
We extend the model of rational bubbles of Blanchard and of Blanchard and Watson to arbitrary dimensions d: a number d of market time series are made linearly interdependent via d times d stochastic coupling coefficients. We first show that the no-arbitrage condition imposes that the non-diagonal impacts of any asset i…
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.
A new method for unfolding histograms without matrix inversion.
problem Matrix inversion in experimental physics, especially in high-energy particle physics.
method Sampling many distributions, folding them through the response matrix, and choosing the closest one to the data.
result Performs as well as traditional methods in well-defined inverse problems and outperforms them in ill-defined ones.