Quadratic Killing tensors on Lie groups are always decomposable.
problem Characterize Killing tensors on Lie groups.
method Analyzing the algebraic structure of Killing tensors on Lie groups.
result Quadratic Killing tensors on compact Lie groups are decomposable.
Quadratic Killing tensors are not always decomposable on certain symmetric spaces.
problem Characterizing when quadratic Killing tensors are decomposable on symmetric spaces.
method Analyzing Killing tensor fields on spaces of constant curvature and higher rank.
result Not all quadratic Killing tensors are decomposable on quaternionic projective spaces and Cayley projective plane.
This research shows how quadratic models can recover tensors with fewer samples than traditional methods.
problem Predicting missing entries in tensors with limited observations.
method Examined non-convex methods for learning quadratic models and their sample complexity.
result All local minima of the mean squared error objective are global minima, recovering the original tensor with linear samples.
The paper classifies Killing tensor fields on Riemannian symmetric spaces.
problem Understanding Killing tensor fields on Riemannian symmetric spaces.
method Reduced study to compact irreducible spaces, introduced top slot Killing tensor fields, and classified quadratic fields.
result Quadratic Killing tensor fields on Riemannian symmetric spaces of rank one are spanned by top-slot and decomposable fields.
Researchers found non-Killing tensor fields on certain symmetric spaces.
problem Understanding Killing tensors on all Riemannian symmetric spaces.
method Constructed explicit examples of quadratic Killing tensors on quaternionic and Cayley projective spaces.
result Quadratic Killing tensors can be non-Killing on some symmetric spaces.
Classifies extended Abelian Chern-Simons theories using quadratic modules.
problem Classifying extended Abelian Chern-Simons theories.
method Using quadratic modules to classify theories.
result Finite quadratic modules classify extended Abelian Chern-Simons theories.
Researchers found indecomposable Killing tensor fields on quaternionic projective spaces.
problem Characterizing Killing tensor fields on projective spaces.
method Analyzing Killing tensor fields on quaternionic and complex projective spaces, proving algebraic properties.
result Generated algebras of Killing tensor fields on quaternionic and complex projective spaces.
The study classifies critical metrics on manifolds with specific curvature conditions.
problem Characterizing critical metrics for quadratic curvature functionals.
method Analyzing closed n-dimensional manifolds with Ricci, scalar curvature, and Riemannian curvature tensor.
result Critical metrics are Einstein under certain curvature conditions.
Paper develops RGN method for estimating low-rank tensors from noisy measurements.
problem Estimating low-rank tensors from noisy linear measurements.
method Riemannian Gauss-Newton (RGN) method for efficient low-rank tensor estimation.
result First local quadratic convergence guarantee of RGN for low-rank tensor estimation in noisy settings.
New tensor recovery method uses Riemannian optimization on Segre manifold.
problem Recovering low-rank tensors from noisy measurements.
method Riemannian Gradient Descent (RGD) and Riemannian Gauss-Newton (RGN) algorithms over the Segre manifold.
result Proven convergence rates for RGD and RGN under mild noise assumptions.
It is shown that the variational derivative of the integral of Branson's Q-curvature is the ambient obstruction tensor of Fefferman-Graham. A classification of irreducible conformally invariant tensors modulo quadratic and higher degree terms in curvature is established.
We call a metric m-quasi-Einstein if RicXm (a modification of the m-Bakry-Emery Ricci tensor in terms of a suitable vector field X) is a constant multiple of the metric tensor. It is a generalization of Einstein metrics which contains Ricci solitons. In this paper, we focus on left-invariant vector fields and…
Systematic prolongation for Killing two-tensors in symmetric spaces.
problem Understanding Killing two-tensors in symmetric spaces.
method Systematic prolongation procedure for Killing two-tensors, focusing on locally symmetric spaces.
result Natural quadratic mapping from Killing fields to Killing two-tensors on irreducible locally symmetric spaces of compact type.
The paper constructs compatible Poisson brackets on gl(N).
problem Constructing compatible Poisson brackets on gl(N).
method Using constant tensors and Schouten brackets, the paper explicitly constructs quadratic Poisson brackets compatible with the standard Lie-Poisson bracket.
result Explicit construction of quadratic Poisson brackets compatible with the standard Lie-Poisson bracket on gl(N).
New tensor completion method reduces impact of outliers.
problem Recover tensors from incomplete data with outliers.
method Proposes a new correntropy-based objective function and half-quadratic minimization.
result Demonstrates robust performance with real and synthetic data.
New findings on shrinking Ricci solitons with vanishing Bach-like tensors.
problem Characterizing gradient shrinking Ricci solitons with vanishing Bach-like tensors.
method Defining and analyzing Bach-like tensors, proving rigidity results, and deriving variational formulas.
result Vanishing Bach-like tensors force solitons to be either Einstein or isometric to the Gaussian soliton.
The paper classifies Weyl tensors in Riemannian 4-manifolds via Lorentzian deformation.
problem Classifying Weyl tensors in Riemannian 4-manifolds.
method Deforming the metric into a Lorentzian one via a nonzero vector T. result Only Petrov Types I and D can occur, and each is determined by the number of critical points of the associated Lorentzian quadratic form.
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.
We formulate natural conformally invariant conditions on a 4-manifold for the existence of a metric whose Schouten tensor satisfies a quadratic inequality. This inequality implies that the eigenvalues of the Ricci tensor are positively pinched.
New findings on gradient expanding Ricci solitons with finite scalar curvature ratio.
problem Understanding the behavior of gradient expanding Ricci solitons with finite scalar curvature ratio.
method Analyzing complete gradient expanding Ricci solitons with nonnegative Ricci curvature.
result Riemann curvature tensor must have at least sub-quadratic decay for finite asymptotic scalar curvature ratio.
Deep tensor factorization benefits from implicit regularization with polynomial growth.
problem Tensor factorization's implicit regularization effect in deep networks is not well understood.
method Investigated the implicit regularization in deep tensor factorization, showing polynomial growth.
result Implicit regularization in deep tensor factorization grows polynomially with depth, improving estimation accuracy and convergence.
Weak harmonic Weyl metrics found on all 4D closed manifolds.
problem Finding canonical metrics on 4D closed manifolds.
method Critical points of a quadratic functional involving the divergence of the Weyl tensor.
result Every 4D closed manifold admits a unique weak harmonic Weyl metric.
Conformally recurrent pseudo-Riemannian manifolds of dimension n>4 are investigated. The Weyl tensor is represented as a Kulkarni-Nomizu product. If the square of the Weyl tensor is nonzero, a covariantly constant symmetric tensor is constructed, that is quadratic in the Weyl tensor. Then, by Grycak's theorem, the expl…
The paper extends results on Bach-flat solitons to new types.
problem Analyzing Bach-like tensors on complete gradient Ricci solitons.
method Extending previous results to new types of solitons.
result Results on Bach-flat solitons extended to new types.
In Lorentzian manifolds of any dimension the concept of causal tensors is introduced. Causal tensors have positivity properties analogous to the so-called ``dominant energy condition''. Further, it is shown how to build, from ANY given tensor A, a new tensor quadratic in A and ``positive'', in the sense that it is …
Solutions to a quadratic matrix equation are linked to strongly regular graphs and multiplicative characters.
problem Solving a specific quadratic matrix equation in Riemannian geometry.
method Constructing nonzero solutions using group rings and multiplicative characters of finite fields.
result Solutions relate to strongly regular graphs and multiplicative characters of finite fields.
Study of SO(3)-irreducible geometry in complex 5D and ternary Pauli exclusion principle.
problem Exploring SO(3)-irreducible geometry in complex 5D.
method Defined a ternary skew-symmetric tensor, split the 10D space into irreducible SO(3) subspaces, found invariants and defined geometric structures.
result Defined a SO(3)-irreducible geometric structure on a 5D complex Hermitian manifold.
This paper defines families of quadratic differentials for a surface.
problem Understanding spaces of quadratic differentials on surfaces.
method Defining families of quadratic differentials and their parametrization.
result A complete account of the theory needed for spaces of quadratic differentials.
We study the irreducible decomposition under Sp(2n, R) of the space of torsion tensors of almost symplectic connections. Then a description of all symplectic quadratic invariants of torsion-like tensors is given. When applied to a manifold M with an almost symplectic structure, these instruments give preliminary insigh…
The Eisenhart problem of finding parallel tensors is solved for the symmetric case in the regular f-Kenmotsu framework. On this way, the Olszack-Rosca example of Einstein manifolds provided by f-Kenmotsu manifolds via locally symmetric Ricci tensors is recovered as well as a case of Killing vector fields. Some othe…
Efficiently decomposes tensors with Boolean factors using BMP.
problem Tensor decomposition with Boolean factors is challenging due to non-convexity and combinatorial constraints.
method Binary Matching Pursuit (BMP) iteratively searches for atoms in a greedy fashion, solving the greedy atom search step via MAXCUT-like boolean quadratic program.
result BMP converges sublinearly to the optimal solution and recovers factors under mild conditions.
The study examines algebraic structures of specific tensor forms in four-dimensional spacetimes.
problem Investigating algebraic features of certain tensor forms in spacetimes.
method General treatment followed by specialization to four-dimensional spacetimes, focusing on invariant subspaces and generalizing relations.
result Generalized relations such as the Ruse-Lanczos identity, Bel-Matte decomposition, and Lovelock-like quadratic identities.
We solve tensor balancing, rescaling an Nth order nonnegative tensor by multiplying N tensors of order N - 1 so that every fiber sums to one. This generalizes a fundamental process of matrix balancing used to compare matrices in a wide range of applications from biology to economics. We present an efficient balancing a…
Nijenhuis tensors N on Courant algebroids compatible with the pairing are studied. This compatibility condition turns out to be of the form N+N∗=aI for irreducible Courant algebroids, in particular for the extended tangent bundles TM⊕T∗M. It is proved that compatible Nijenhuis tensors on irreducible Coura…
Study rigidity of Einstein metrics as critical points of curvature functionals.
problem Characterize Einstein metrics as critical points of quadratic curvature functionals.
method Analyze pointwise inequalities involving Weyl curvature and traceless Ricci curvature.
result Provide rigidity results for Einstein metrics and locally conformally flat critical metrics.
Unified framework for coupled tensor completion improves recovery accuracy.
problem Improving recovery accuracy in coupled tensor completion.
method Unified framework using tensor ring (TR) decomposition with shared latent factors and novel optimization model.
result The proposed method achieves superior recovery accuracy on real-world data compared to state-of-the-art methods.
Stationary and axially symmetric space-times play an important role in astrophysics, particularly in the theory of neutron stars and black holes. The static vacuum sub-class of these space-times is known as Weyl's class, and contains the Schwarzschild space-time as its most prominent example. This paper is going to stu…
A new method reduces the complexity of tensor products from cubic to quadratic, improving both speed and accuracy.
problem Efficiently computing high-dimensional tensor products for polynomial kernels.
method Complex-to-Real (CtR) modification of sketches using complex random projections.
result Achieves state-of-the-art performance in accuracy and speed.
We consider the two logarithmic strain measures\[ω_{\rm iso}=\|\mathrm{dev}_n\log U\|=\|\mathrm{dev}_n\log \sqrt{F^TF}\|\quad\text{ and }\quad ω_{\rm vol}=|\mathrm{tr}(\log U)|=|\mathrm{tr}(\log\sqrt{F^TF})|\,,\]which are isotropic invariants of the Hencky strain tensor logU, and show that they can be uniquely char…
New methods solve tensor-on-tensor regression with unknown rank, revealing benefits of over-parameterization.
problem Connecting tensor responses to tensor covariates with unknown intrinsic rank.
method Riemannian gradient descent and Riemannian Gauss-Newton methods for tensor-on-tensor regression.
result Riemannian optimization methods converge linearly and quadratically to a statistically optimal estimate in rank over-parameterized settings.
Given a space it is easy to obtain the system of geodesic equations on it. In this paper the inverse problem of reconstructing the space from the geodesic equations is addressed. A procedure is developed for obtaining the metric tensor from the Christoffel symbols. The procedure is extended for determining if a second …
Revisits CP tensor decomposition for noisy, non-orthogonal data.
problem Statistical optimality and convergence of ALS in noisy, non-orthogonal, higher-rank settings.
method Statistical analysis and TASD method for initialization.
result ALS with TASD achieves optimal error in rank-one setting within one or two iterations.
We present the theory of tensors with Young tableau symmetry as an efficient computational tool in dealing with the polynomial first integrals of a natural system in classical mechanics. We relate a special kind of such first integrals, already studied by Lundmark, to Beltrami's theorem about projectively flat Riemanni…
We propose the convex factorization machine (CFM), which is a convex variant of the widely used Factorization Machines (FMs). Specifically, we employ a linear+quadratic model and regularize the linear term with the ℓ2-regularizer and the quadratic term with the trace norm regularizer. Then, we formulate the CFM o…
Paper compares expressive power of GNNs, proving approximation guarantees for practical architectures.
problem Understanding the expressive power of Graph Neural Networks (GNNs).
method Theoretical framework comparing invariant and equivariant GNNs, proving approximation guarantees for practical architectures.
result Folklore Graph Neural Networks (FGNN) are the most expressive architectures for a given tensor order.
Information geometry provides a geometric approach to families of statistical models. The key geometric structures are the Fisher quadratic form and the Amari-Chentsov tensor. In statistics, the notion of sufficient statistic expresses the criterion for passing from one model to another without loss of information. Thi…
Based on a suggestion of Richard Hamilton, we give an alternate proof of his matrix Harnack inequality for solutions of the Ricci flow with positive curvature operator. This Harnack inequality says that a certain endomorphism, consisting of an expression in the curvature and its first two covariant derivatives, of the …
The study finds conditions for a third rank Killing tensor field on a 2D Riemannian torus.
problem Conditions for the existence of a third rank Killing tensor field on a 2D Riemannian torus.
method Analyzes the metric of the torus and uses Fourier coefficients to derive conditions for the function λ.
result Equations relating Fourier coefficients of the function λ determine the existence of a third rank Killing tensor field.