The paper tackles sparse graph learning under Laplacian-related constraints, improving upon existing methods.
problem Learning a sparse undirected graph from multivariate data under Laplacian-related constraints.
method Modifications to penalized log-likelihood approaches to enforce total positivity and lasso/adaptive lasso penalties using ADMM.
result The proposed constrained adaptive lasso approach significantly outperforms existing Laplacian-based approaches.
New proofs given for space curves with totally positive torsion.
problem Description of convex hulls of space curves with totally positive torsion.
method New proofs of parametric representation, surface area, and volume formulas.
result Recovery of formulas for convex hull's surface area and volume.
Method estimates M-matrices in graphical models with improved accuracy.
problem Estimating M-matrices as precision matrices in Gaussian graphical models.
method Adaptive multiple-stage estimation method solving weighted ℓ1-regularized problems.
result Method outperforms state-of-the-art methods in precision matrix estimation and graph edge identification.
The study refines contingency matrices for complex stratification and braid group cohomology.
problem Combinatorics of contingency matrices and their applications.
method Refinement of complex stratification and study of braid group cohomology.
result Totally positive meta-matrix formed by contingency matrix sizes.
The paper proposes a new portfolio allocation method combining RMT and machine learning.
problem Optimal allocation instability in high-dimensional portfolios.
method Combines Random Matrix Theory covariance estimators with Nested Clustered Optimization.
result The modified NCO algorithm achieves stable allocations without risky short positions.
Introduces Θ-positivity in Lie groups, generalizing Lusztig's positivity.
problem Generalizing Lusztig's total positivity to a broader class of Lie groups.
method Introduces and studies Θ-positivity in real simple Lie groups. result Four families of Lie groups admit Θ-positive structures. Shows large unknotting number for simple knots.
problem Finding minimum crossing changes for unknotting.
method Positive-to-negative crossing changes without increasing genus.
result Genus non-increasing totally positive unknotting number can be large.
We prove in this paper that the weighted volume of the set of integral transportation matrices between two integral histograms r and c of equal sum is a positive definite kernel of r and c when the set of considered weights forms a positive definite matrix. The computation of this quantity, despite being the subject of…
Method estimates noise transition matrix from noisy labels without relying on unreliable class-posterior estimation.
problem Estimating noise transition matrix from noisy data.
method Total variation regularization to encourage distinguishable predicted probabilities.
result Consistent estimator of the noise transition matrix under mild assumptions.
Study on nonspin manifolds with spin boundary, showing nonconnectedness and nontrivial fundamental group.
problem Understanding spaces of positive scalar curvature metrics on totally nonspin manifolds.
method Analysis of positive scalar curvature metrics on manifolds with spin boundary, using propagation techniques.
result Spaces of positive scalar curvature metrics are not connected and have nontrivial fundamental groups for certain dimensions.
Proposes a new model for image restoration combining deep learning and total variation.
problem Restoring images from limited data with low-rank constraints insufficient.
method Regularized Deep Matrix Factorized (RDMF) model using deep neural network's low-rank bias and total variation.
result Outperforms state-of-the-art models in image restoration from few observations.
The paper proves uniformization for specific curvature types on manifolds.
problem Uniformizing fourth order conformal curvature on Riemannian manifolds.
method Proving existence of conformal deformations for specific curvature conditions.
result Existence of conformal deformations for positive Yamabe invariant and total Q-curvature.
The paper defines and studies the geometric mean for tensors and its associated Riemannian geometry.
problem Defining and studying the geometric mean for tensors.
method Generalized geometric mean for tensors using T-product, verified properties, and investigated Riemannian manifold.
result Geometric mean of T-positive definite tensors is a unique solution of algebraic Riccati tensor equations and a midpoint of geodesics.
Classifies totally geodesic submanifolds in Hopf-Berger spheres.
problem Classifying totally geodesic submanifolds in Hopf-Berger spheres.
method Investigates Hopf-Berger spheres, a special family of homogeneous spaces diffeomorphic to spheres constructed via Hopf fibrations.
result Discovered intriguing examples of totally geodesic submanifolds, including isometric submanifolds of real projective spaces and non-congruent submanifolds.
Positive simplicial volume found for certain non-positively curved manifolds with specific submanifolds.
problem Determining conditions for positive simplicial volume in non-positively curved manifolds.
method Analyzing isolated, closed totally geodesic submanifolds of codimension one and their impact on simplicial volume.
result Positive simplicial volume for certain non-positively curved manifolds with specific submanifolds.
Total torsion of 3D lines of curvature is an integer multiple of 2π.
problem Understanding the total torsion of 3D lines of curvature in Riemannian manifolds.
method Analyzing the properties of well-positioned lines of curvature and using the total torsion theorem for spherical curves.
result The total torsion of a well-positioned line of curvature is an integer multiple of 2π.
Paper proves curvature estimate for curved spaces.
problem Estimating total curvature of closed hypersurfaces.
method Used non-positively curved symmetric spaces.
result Proved total curvature estimate and isoperimetric inequality.
Efficiently solves large portfolio optimization problems by reducing and sparsifying covariance matrices.
problem Large and dense covariance matrices limit efficient portfolio optimization.
method Dimension reduction and increased sparsity based on machine learning predictions.
result Improved portfolio performance and reduced runtime compared to full dense covariance matrices.
Totally geodesic submanifolds in product spaces imply special curvature properties.
problem Characterizing manifolds based on the existence of totally geodesic submanifolds.
method Analyzing totally geodesic submanifolds in products of non-positively curved manifolds with transversality conditions.
result If infinitely or just a single dense such submanifolds exist, the manifold must be locally symmetric.
In this paper, we show that steady or shrinking complete gradient Yamabe solitons with finite total scalar curvature and non-positive Ricci curvature are Ricci flat. Moreover, under certain pinching condition for Ricci curvature, we show that steady or shrinking complete gradient Yamabe solitons with finite total scala…
Totally geodesically embeddings of infinitely many closed 7-manifolds into 13-dimensional positively curved closed Riemannian manifolds are constructed. The problems of computing pinching constants and existence of other totally geodesical embeddings are discussed.
In this note it is shown that every 7-dimensional Eschenburg space can be totally geodesically embedded into infinitely many topologically distinct 13-dimensional Bazaikin spaces. Furthermore, examples are given which show that, under the known construction, it is not always possible to totally geodesically embed a pos…
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.
The paper tackles feature cross search for linear models, providing approximation algorithms and structural results.
problem Maximizing AUC of a linear model trained on feature crosses.
method Submodular optimization, greedy algorithm, and connections to total variation and kernel matrices.
result Simple greedy (1−1/e)-approximation algorithm for maximizing AUC. This paper tackles fitting multilevel low rank matrices by addressing three problems.
problem Fitting a given matrix by an MLR matrix in the Frobenius norm.
method Factor fitting, rank allocation, and hierarchical partitioning.
result The proposed methods can fit a given matrix by an MLR matrix in the Frobenius norm.
Proves Gromov's conjecture on total mean curvature using surgery and positive mass theorems.
problem Proving Gromov's conjecture on total mean curvature of fill-ins.
method Surgery to reduce to fill-ins of spheres, positive mass theorems, and quantitative surgery process.
result Proves Gromov's conjecture on total mean curvature in various cases.
Workshop notes on positivity in Lie groups and its applications.
problem Understanding total and Θ-positivity in semisimple Lie groups. method Discussion and analysis of existing theories and recent developments.
result Progress in classifying higher Teichmüller spaces through Θ-positivity. We construct the first examples of complete, properly embedded minimal surfaces in H2×R with finite total curvature and positive genus. These are constructed by gluing copies of horizontal catenoids or other nondegenerate summands. We also establish that every horizontal catenoid is nondegen…
We consider an asymptotically flat Riemannian spin manifold of positive scalar curvature. An inequality is derived which bounds the Riemann tensor in terms of the total mass and quantifies in which sense curvature must become small when the total mass tends to zero.
This work formulates a novel song recommender system as a matrix completion problem that benefits from collaborative filtering through Non-negative Matrix Factorization (NMF) and content-based filtering via total variation (TV) on graphs. The graphs encode both playlist proximity information and song similarity, using …
We use the Cartan representations of SO(3) and SU(3), and an irreducible 14-dimensional representation of Sp(3) to construct certain totally geodesic submanifolds in "skew" position in the complex quadrics, the complex 2-Grassmannians and the quaternionic 2-Grassmannians.
This paper describes a simple framework for structured sparse recovery based on convex optimization. We show that many structured sparsity models can be naturally represented by linear matrix inequalities on the support of the unknown parameters, where the constraint matrix has a totally unimodular (TU) structure. For …
New measures quantify dependence between variables without distribution estimation.
problem Measuring dependence between variables in arbitrary dimensions.
method Proposed matrix-based normalized total correlation and dual total correlation measures.
result Measures are differentiable and statistically more powerful than existing methods.
This paper proposes a representational model for grid cells. In this model, the 2D self-position of the agent is represented by a high-dimensional vector, and the 2D self-motion or displacement of the agent is represented by a matrix that transforms the vector. Each component of the vector is a unit or a cell. The mode…
We prove that there exists a metric of positive curvature in a three-sphere which admits a given torus knot as a closed geodesic.We also sketch a construction of a metric in a four sphere, very likely of positive curvature, which admits a totally geodesic projective plane with Euler number four. Surpisingly, the techni…
Establishes geometric properties of elements in the positive semigroup of a general real semisimple Lie group.
problem Generalizing Lusztig's total positivity to the setting of general real semisimple Lie groups.
method Classifying Lie groups admitting a positive structure and establishing key properties of unipotent positive semigroups.
result Establishes key geometric properties of elements in the positive semigroup.
Total curvatures of certain hypersurfaces are continuous.
problem Continuity of curvatures in geometric settings.
method Hausdorff distance for hypersurfaces and convex bodies in Riemannian manifolds and Cartan-Hadamard spaces.
result Total generalized mean curvatures are continuous.
The study proves stable minimal immersions in positively curved manifolds are totally geodesic.
problem Proving stable minimal immersions in positively curved manifolds are totally geodesic.
method Formulating stable Bernstein type theorems in certain positively curved ambient manifolds.
result Proves stable minimal immersions in positively curved manifolds are totally geodesic.
Constructs infinitely many examples of large manifolds with circle bundles of positive scalar curvature.
problem Existence of circle bundles over large manifolds with positive scalar curvature metrics.
method Symplectic geometry techniques.
result Infinitely many examples of macroscopically large manifolds with circle bundles of positive scalar curvature.
Estimates Bartnik mass for metrics with nonnegative Gauss curvature.
problem Estimating Bartnik mass for specific metric configurations.
method Using area, total mean curvature, and a metric roundness measure.
result Estimate approaches sharp value for round spheres.
We use planar coordinates as well as hyperbolic coordinates to separate the de Sitter spacetime into two parts. These two ways of cutting the de Sitter give rise to two different spatial infinities. For spacetimes which are asymptotic to either half of the de Sitter spacetime, we are able to provide definitions of the …
The paper studies totally nonnegative parts of flag varieties and their topologies.
problem Understanding the topology of totally nonnegative flag varieties.
method Algebraic, geometric, and dynamical perspectives; orbit context; gradient flows; Riemannian metrics.
result Positivity is preserved in certain metrics on the totally nonnegative part of flag varieties.
Revisits conformal metrics with finite Q-curvature, providing necessary and sufficient conditions.
problem Understanding conformal metrics with finite total Q-curvature.
method Introduces conformal mass and provides necessary and sufficient conditions for normality.
result Derives volume comparison theorems and proves a positive mass type theorem related to Q-curvature.
Study variations of Riemannian submersions to maintain geodesic fibers and positive curvatures.
problem Maintain geodesic fibers and positive sectional curvatures in Riemannian submersions.
method Vary Riemannian metrics while keeping fibers totally geodesic and horizontal distribution fixed.
result Conditions for making sectional curvatures positive and existence of fat submersions.
We prove that some Riemannian manifolds with boundary under an explicit integral pinching are spherical space forms. Precisely, we show that 3-dimensional Riemannian manifolds with totally geodesic boundary, positive scalar curvature and an explicit integral pinching between the L2-norm of their scalar curvature and…
The study classifies certain types of incomplete surfaces with low curvature.
problem Classifying incomplete affine spheres with specific curvature constraints.
method Analyzing total curvature and asymptotic behavior of surfaces.
result New examples of incomplete affine spheres with positive genus found.
New approach to analyze matrix denoising using gradient flow and fixed point equations.
problem Positive semi-definite matrix denoising in extensive-rank and high-dimensional settings.
method Gradient flow and fixed point equations derived from linear pencil techniques of random matrix theory.
result Continuous phase transitions in the extensive-rank and high-dimensional regime.
We define the total energy-momenta for (4+1)-dimensional asymptotically anti-de Sitter spacetimes, and prove the positive energy theorem for such spacetimes.