Haantjes algebras help in diagonalizing operators on manifolds.
problem Diagonalizing operators on differentiable manifolds.
method Introducing Haantjes algebra, a family of operator fields with vanishing Haantjes torsion and compatibility conditions.
result Simultaneous diagonalization of operators in local coordinates or block-diagonal form in general cases.
New theory allows simultaneous block-diagonalization of commuting operator fields.
problem Normal forms of operator fields.
method Generalized Nijenhuis torsions and generalized Haantjes algebra.
result Simultaneous block-diagonalization of commuting operator fields.
Paper proposes a new method for brain disease classification using connectome data.
problem Challenges in classifying brain diseases due to small sample size and high dimensionality.
method Simultaneous approximate diagonalization of adjacency matrices to compute stable eigenstructures.
result The method outperforms simple baselines and state-of-the-art approaches for Alzheimer's disease detection.
SEM-DNN learns reciprocal interactions from observational data without external instruments.
problem Estimating bidirectional interactions from endogenous data.
method Heteroscedastic neural simultaneous-equation estimator (SEM-DNN) that learns reciprocal structural interactions.
result SEM-DNN recovers structural effects more reliably than other methods under increasing information.
Researchers found a canonical form for pairs of Hermitian and antilinear operators.
problem Simultaneous normalization of pairs of Hermitian and antilinear operators in differential geometry.
method Finding a canonical form for pairs of Hermitian and antilinear operators.
result Generalized previous results on simultaneous normalization of such pairs.
Paper estimates GMMs with unknown covariances using sparse regularization.
problem Estimating GMMs with unknown diagonal covariances from samples.
method Employed Beurling-LASSO (BLASSO) for sparse estimation of component means, covariances, and weights.
result Established non-asymptotic recovery guarantees with nearly parametric convergence rates.
The paper analyzes heat kernel asymptotics for real powers of Laplacians on manifolds.
problem Analyzing the small-time behavior of heat kernels for real powers of Laplacians.
method Analyzes asymptotics on the diagonal and away from it, proving non-triviality and non-locality of coefficients.
result Logarithmic terms appear only if the manifold dimension is odd and the power is rational with even denominator.
This paper improves neural network generalization by dynamically learning kernel parameters.
problem Improving neural network generalization and adaptability.
method Diagonal adaptive kernel model that learns kernel eigenvalues and output coefficients during training.
result The diagonal adaptive kernel model significantly improves generalization over fixed-kernel methods.
The paper introduces a penalized matrix estimation procedure aiming at solutions which are sparse and low-rank at the same time. Such structures arise in the context of social networks or protein interactions where underlying graphs have adjacency matrices which are block-diagonal in the appropriate basis. We introduce…
Non-orthogonal joint diagonalization (NJD) free of prewhitening has been widely studied in the context of blind source separation (BSS) and array signal processing, etc. However, NJD is used to retrieve the jointly diagonalizable structure for a single set of target matrices which are mostly formulized with a single da…
Clustering analysis is one of the most widely used statistical tools in many emerging areas such as microarray data analysis. For microarray and other high-dimensional data, the presence of many noise variables may mask underlying clustering structures. Hence removing noise variables via variable selection is necessary…
Proves existence of equivariant maps avoiding diagonal in n-space.
problem Existence of equivariant maps between spaces.
method Different approach to vanishing all obstructions.
result Vanishing of all equivariant obstructions.
The paper proves a mapping from a space of holonomy varieties to Teichmüller spaces, with a non-empty discrete intersection.
problem Intersection of Poincaré holonomy varieties and their properties.
method Holomorphic mapping and branched covering proof.
result Intersection of arbitrary Poincaré holonomy varieties is a non-empty discrete set.
Using generalized Tanaka-Webster connection, we considered a real hypersurface M M M in a complex two-plane Grassmannian G 2 ( C m + 2 ) G_2({\mathbb C}^{m+2}) G 2 ( C m + 2 ) when the GTW Reeb Lie derivative of the structure Jacobi operator coincides with the Reeb Lie derivative. Next using the method of simultaneous diagonalization, we prove a comp…
This paper solves matrix blind joint block diagonalization with noise.
problem Identifying the diagonalizer and block diagonal structure of matrices under noise.
method Bi-block diagonalization method.
result The method can identify the exact solution under certain conditions.
Diagonalizes metrics of 3D Lorentzian manifolds.
problem Diagonalizing metrics of 3D Lorentzian manifolds.
method Applying the technique of moving frames.
result Every smooth Lorentzian 3-manifold admits an atlas with a diagonal metric.
In the first quarter of 2006 Chicago Board Options Exchange (CBOE) introduced, as one of the listed products, options on its implied volatility index (VIX). This created the challenge of developing a pricing framework that can simultaneously handle European options, forward-starts, options on the realized variance and …
Study Ricci vector fields on 2D space with diagonal metrics.
problem Understanding Ricci vector fields on 2D space with specific metrics.
method Examined Ricci vector fields on R 2 \mathbb R^2 R 2 with a diagonal metric. result Characterized Ricci vector fields on R 2 \mathbb R^2 R 2 with a diagonal metric. Develops a novel stochastic algorithm for diagonal estimation of large matrices.
problem Efficient diagonal estimation for large or implicit matrices.
method Adaptive parameter selection in a stochastic algorithm.
result Lower bound on random query vectors needed for estimation.
Octagon map accelerates diagonal changes algorithm.
problem Improving the efficiency of diagonal changes algorithm.
method Octagon Farey map as an acceleration.
result Octagon map accelerates diagonal changes algorithm.
New diagonal knots found with non-torus structure.
problem Identifying knots with diagonal grid diagrams.
method Analysis of knots represented by diagonal grid diagrams.
result All diagonal knots are positive, and a new non-torus example is found.
Study finds symmetries in a special 3D space with a diagonal metric.
problem Identifying symmetries in a specific 3D space.
method Determining Killing vector fields on a diagonal metric in R 3 \mathbb R^3 R 3 . result Killing vector fields on the space R 3 \mathbb R^3 R 3 with a diagonal metric have been identified. We use mathematical induction to prove that the horizontal composition in the class of coherently diagonal complexes is indeed a binary operation. That is to say, the embedding of two coherently diagonal complexes in an alternating planar diagram produces a coherently diagonal complex.
Most known four-dimensional cohomogeneity-one Einstein metrics are diagonal in the basis defined by the left-invariant one-forms, though some essentially non-diagonal ones are known. We consider the problem of explicitly seeking non-diagonal Einstein metrics, and we find solutions which in some cases exhaust the possib…
Classifies weakly Einstein curvature tensors in 4D Euclidean space.
problem Classifying algebraic curvature tensors in 4D Euclidean space.
method Algebraic formulation and geometric interpretation of weakly Einstein manifolds.
result Complete classification of non-Einstein weakly Einstein curvature tensors in dimension four.
New method identifies structural parameters without assuming uncorrelated errors.
problem Identifying structural parameters in simultaneous equation models.
method Exploits higher-order cumulant restrictions, not requiring uncorrelated errors.
result Simple diagonality condition on h h h th-order cumulants identifies structural parameter matrix. New method improves Pham's algorithm for joint diagonalization.
problem Optimizing joint diagonalization of matrices for statistical learning.
method Quasi-Newton method for Pham's diagonalization criterion.
result Proposed method outperforms Pham's algorithm in experiments.
Equal diagonal energies proven on Liouville surfaces.
problem Diagonal energies on Liouville surfaces.
method Analyzing parameter curves and rectangles on Liouville surfaces.
result Diagonal energies are equal in n-dimensional Liouville manifolds.
Proposes QDF to improve multi-step time-series forecasting.
problem Ignoring label autocorrelation and unequal task weights in training objectives.
method Quadratic-form weighted training objective and QDF learning algorithm.
result Improves performance of various forecast models, achieving state-of-the-art results.
Diagonal linear networks converge to lasso regularization path during training.
problem Understanding the regularization behavior of diagonal linear networks.
method Analyzing the training trajectory of diagonal linear networks and comparing it to the lasso regularization path.
result The training trajectory of diagonal linear networks is closely related to the lasso regularization path.
We show that a basis of a semisimple Lie algebra of compact type, for which any diagonal left-invariant metric has a diagonal Ricci tensor, is characterized by the Lie algebraic condition of being "nice". Namely, the bracket of any two basis elements is a multiple of another basis element. This extends the work of Laur…
New diagonal move simplifies knots and links efficiently.
problem Efficiently unknotting knots and links.
method Introduces diagonal move, proving its effectiveness for classical and welded knots.
result Diagonal move reduces any knot or link to the unknot or unlink with fewer operations.
Tensor factorization arises in many machine learning applications, such knowledge base modeling and parameter estimation in latent variable models. However, numerical methods for tensor factorization have not reached the level of maturity of matrix factorization methods. In this paper, we propose a new method for CP te…
We obtain the natural diagonal almost product and locally product structures on the total space of the cotangent bundle of a Riemannian manifold. We find the Riemannian almost product (locally product) and the (almost) para-Hermitian cotangent bundles of natural diagonal lift type. We prove the characterization theorem…
Hydro storage system optimization is becoming one of the most challenging tasks in Energy Finance. While currently the state-of-the-art of the commercial software in the industry implements mainly linear models, we would like to introduce risk aversion and a generic utility function. At the same time, we aim to develop…
Study grid homology of diagonal knots, finding key terms related to prime factors and decompositions.
problem Determine grid homology of diagonal knots and compare them to other knot types.
method Use grid diagrams and combinatorial knot Floer homology to analyze diagonal knots.
result Grid homology detects the number of prime factors and decompositions of the knot into non-integer tangles.
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. 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. 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.
The author connects Poincaré embeddings to Reidemeister traces and diagonal maps.
problem Existence of Poincaré embeddings for specific spaces.
method Relates total obstruction to Reidemeister trace and uses Poincaré duality.
result Diagonal maps admit Poincaré embeddings under certain conditions.
Study shows mean-field approximation fails to improve PAC-Bayes bounds for neural networks.
problem Understanding why overparametrized neural networks achieve low risk and zero empirical risk.
method Optimized PAC-Bayes bounds using variational inference (VI), investigating mean-field approximation.
result Mean-field approximation does not provide significant improvements in PAC-Bayes bounds for neural networks.
Conditions for flat 3-manifolds with diagonal metrics are identified.
problem Characterizing flat 3-manifolds with diagonal metrics.
method Provided necessary and sufficient conditions for flatness.
result Characterized flat manifolds of warped product-type.
Constructs coordinates to diagonalize Toda flow on matrices with simple spectrum.
problem Diagonalizing the Toda flow on matrices with simple spectrum.
method Lie theoretic methods applied to complex semisimple Lie algebras and their real forms.
result Decouples the Toda vector field into simpler components.
Paper proposes ABDR for convex subspace clustering with adaptive block diagonal representation.
problem Subspace clustering with block diagonal structure for noisy data.
method ABDR explicitly pursues block diagonality without sacrificing convexity, using a specially designed convex regularizer.
result Experimental results show ABDR outperforms state-of-the-arts.
Paper studies deep diagonal circulant neural networks and introduces training techniques.
problem Understanding and training deep neural networks with structured weight matrices.
method Theoretical analysis and practical training techniques including initialization and non-linearity use.
result Deep diagonal circulant networks outperform other structured models in accuracy and weight efficiency.
This paper optimizes diagonal preconditioning to improve matrix condition numbers.
problem Optimizing diagonal preconditioning to reduce matrix condition numbers.
method Reformulated as a quasi-convex problem, solved with bisection and Newton updates.
result Optimal diagonal preconditioners can significantly improve iterative methods.
Classify projective subvarieties in Bogomolov-Guan manifolds using quasi-diagonals.
problem Classify projective subvarieties in non-Kahler holomorphically symplectic manifolds.
method Use quasi-diagonals to classify projective subvarieties.
result Prove that any projective subvariety belongs to a fiber of the Lagrangian fibration.
New adaptive methods improve deep learning performance.
problem Training deep networks efficiently and effectively.
method Block-diagonal matrix adaptation for gradient updates.
result Block-diagonal methods outperform adaptive diagonal methods and vanilla SGD.