3-manifold triangulation can be reconstructed from its intersection matrix.
problem Reconstructing the triangulation of 3-manifolds from their intersection matrix.
method Using the intersection matrix of a simplicial complex to determine the triangulation of a 3-manifold up to isomorphism.
result The intersection matrix is sufficient to determine the triangulation of a 3-manifold up to isomorphism.
Paper proves inequality for Green function on Kähler manifolds.
problem Estimating Green function on Kähler manifolds.
method Matrix Li-Yau-Hamilton inequality for Green function.
result Elliptic analogue of heat equation estimate for Kähler manifolds.
We relax indicator matrices to form a manifold for faster optimization.
problem Optimizing indicator matrices is NP-hard.
method Developed a Riemannian manifold (RIM) and Riemannian optimization methods.
result RIM manifold optimization is significantly faster and yields better results.
New method improves robust low-rank matrix completion for computer vision.
problem Robust low-rank matrix completion for partially observed data.
method Formulated as a nonsmooth Riemannian optimization problem over Grassmann manifold, solved with an alternating manifold proximal gradient continuation method.
result Demonstrated advantages over existing approaches in background extraction from surveillance videos.
Paper builds neural networks on matrix manifolds using gyrovector spaces.
problem Lack of concepts in gyrovector spaces for matrix manifolds.
method Generalized gyrovector space concepts for SPD and Grassmann manifolds, proposing new neural network models.
result Demonstrated effectiveness in human action recognition and knowledge graph completion.
Study connects Lie groups to specific Riemannian manifolds.
problem Understanding Lie groups through Riemannian manifold properties.
method Investigates Lie groups as 3D almost paracontact almost paracomplex Riemannian manifolds.
result Established correspondence between Lie algebra and matrix representation.
Homotopy commutativity in quasitoric manifolds depends on polytope structure and characteristic matrix type.
problem Conditions for homotopy commutativity in quasitoric manifolds.
method Analyzing characteristic matrices and polytope structures.
result Homotopy commutativity is determined by specific polytope and matrix conditions.
We derive an interpolation version of constrained matrix Li-Yau-Hamilton estimate on Kähler manifolds. As a result, we first get a constrained matrix Li-Yau-Hamilton estimate for heat equation on a Kähler manifold with fixed Kähler metric. Secondly, we get a corresponding estimate for forward conjugate heat equation on…
Researchers developed a new Riemannian manifold for SPD matrix-valued optimal transport problems.
problem Optimal transport between SPD matrix-valued measures.
method Formulated as a generalized optimal transport problem with block SPD matrices, endowed with a novel Riemannian manifold structure.
result The novel Riemannian manifold allows solving SPD matrix-valued optimal transport problems using Riemannian optimization.
New method reduces computational cost for nonnegative low rank matrix approximation.
problem Efficiently compute nonnegative low rank matrix approximation for nonnegative matrices.
method Alternating projections onto tangent spaces of fixed rank matrices manifold and nonnegative matrix manifold.
result Sequence converges linearly to optimal solutions, showing better performance in terms of computational time and accuracy.
We develop matrix models for Grassmann, flag, and Stiefel manifolds.
problem Creating efficient models for Grassmann, flag, and Stiefel manifolds.
method Orthogonally-equivariant matrix submanifold models derived for each manifold.
result Exhaustive list of orthogonally-equivariant submanifold models for the lowest dimensions.
Study non-convex matrix factorization using Riemannian geometry.
problem Matrix completion via non-convex optimization.
method Optimization over a Grassmannian manifold, analyzing principal angles.
result Geodesically convex region in matrix completion cost function.
Nonnegative sectional curvature linked to matrix displacement convexity.
problem Nonnegative sectional curvature in Riemannian manifolds.
method Matrix displacement convexity as a criterion for nonnegative sectional curvature.
result Entropy functional matrix displacement convexity implies nonnegative sectional curvature.
In this paper, we formulate the Canonical Correlation Analysis (CCA) problem on matrix manifolds. This framework provides a natural way for dealing with matrix constraints and tools for building efficient algorithms even in an adaptive setting. Finally, an adaptive CCA algorithm is proposed and applied to a change dete…
Researchers develop a new quantum invariant using a matrix dilogarithm for 3-manifolds.
problem Developing quantum invariants for 3-manifolds.
method Using a sl3 matrix dilogarithm and quantum groups. result The sl3 matrix dilogarithm can be considered as a 6j-symbol. We study low-dimensional representations of matrix groups over general rings, by considering group actions on CAT(0) spaces, spheres and acyclic manifolds.
New algorithm reduces complexity for SPD manifold optimization.
problem Efficiently minimize functions over SPD manifold.
method Low-complexity Riemannian subspace descent with sparse updates.
result Innovative updates avoid costly matrix operations.
Efficient CD algorithms on matrix manifolds for optimization problems.
problem Optimization on Riemannian manifolds with computational efficiency.
method Developed coordinate descent algorithms for various matrix manifolds, updating only a few variables at each iteration.
result Proposed algorithms achieve low cost per iteration and a more efficient variant via first-order approximation.
This paper proposes robust matrix variate regression models with rank constraints and vector regularization.
problem High dimensional and noisy matrix-valued predictors in regression models.
method Rank constraint, vector regularization, alternating projected gradient descent algorithm.
result The proposed method achieves the minimax rate of estimation errors.
The paper studies transformations of Frobenius manifolds and their properties.
problem Analyzing transformations of Frobenius manifolds and their properties.
method Analytic theory of Legendre-type transformations for Frobenius manifolds.
result Monodromy data, Stokes matrix, and central connection matrix are shared among Legendre-type transformations.
Transforms uniquely determine Higgs fields on real-analytic manifolds.
problem Determining Higgs fields from transforms on manifolds.
method Matrix-weighted real-analytic double fibration transforms.
result Higgs fields can be uniquely determined from transforms.
Random matrix ensembles yield uniform distributions on manifolds.
problem Understanding distributions of vectors in random matrix ensembles.
method Analyzing eigenvalues, singular values, and Autonne-Takagi vectors of various random matrix ensembles.
result Uniform distributions on specific manifolds for different types of random matrix ensembles.
Equivalent formulations for low-rank matrix optimization are proven.
problem Low-rank matrix optimization with rank constraints.
method Established geometric landscape connections between manifold and factorization formulations.
result Equivalence between manifold and factorization formulations at FOSPs, SOSPs, and strict saddles.
New deep learning model for matrix completion combining linear and nonlinear relationships.
problem Matrix completion considering only linear or nonlinear relations, ignoring latent relationships.
method Combines linear and nonlinear models in a latent variables framework, using a deep neural network with two branches for columns and rows, and manifold learning as an auxiliary task.
result Experimental results show the proposed method outperforms state-of-the-art matrix completion methods.
Solves weakly supervised regression using low-rank approximations and manifold regularization.
problem Weakly supervised regression with known, unknown, and uncertain labels.
method Combines manifold regularization and low-rank matrix decomposition for optimization.
result Improves solution quality and stability for large datasets.
Paper maps Hamiltonians and line elements in manifolds.
problem Mapping among generalized Hamiltonians and line elements.
method Constructing Calabi's Riemannian Line Elements and solving matrix Riccati equations.
result Analytical and exact solutions of mapping between manifolds.
A new optimization algorithm for Gaussian Variational Inference on precision matrices.
problem Complex models with positive definite constraints on covariance matrices.
method Manifold Gaussian Variational Bayes (MGVBP) with natural gradient updates.
result Empirically validated as a feasible and efficient solution for VI in complex models.
Derives matrix Harnack inequalities for semilinear heat equations on manifolds.
problem Bounding solutions of semilinear heat equations on manifolds with geometric constraints.
method Applies Li-Yau estimates to derive Harnack inequalities for positive solutions.
result Derives matrix Harnack inequalities for positive solutions of semilinear heat equations.
New Bayesian matrix completion method using Stiefel manifolds.
problem Efficient Bayesian matrix completion with uncertainty quantification.
method Geodesic Hamiltonian Monte Carlo on Stiefel manifolds.
result Improved sampling performance and accuracy on real-world problems.
The matrix completion problem consists of finding or approximating a low-rank matrix based on a few samples of this matrix. We propose a new algorithm for matrix completion that minimizes the least-square distance on the sampling set over the Riemannian manifold of fixed-rank matrices. The algorithm is an adaptation of…
Riemannian gradient descent escapes some spurious critical points on low-rank matrix manifold.
problem Spurious critical points on the boundary of low-rank matrix manifold.
method Riemannian gradient descent with dynamical low-rank approximation and rescaled gradient flow.
result Riemannian gradient descent escapes some spurious critical points on the boundary of the manifold.
The object of investigation are Lie groups considered as almost contact B-metric manifolds of the lowest dimension three. It is established a correspondence of all basic-class-manifolds of the Ganchev-Mihova-Gribachev classification of the studied manifolds and the explicit matrix representation of Lie groups. Some kno…
Since Li and Yau obtained the gradient estimate for the heat equation, related estimates have been extensively studied. With additional curvature assumptions, matrix estimates that generalize such estimates have been discovered for various time-dependent settings, including the heat equation on a Kähler manifold, Ricci…
Autoencoders are popular among neural-network-based matrix completion models due to their ability to retrieve potential latent factors from the partially observed matrices. Nevertheless, when training data is scarce their performance is significantly degraded due to overfitting. In this paper, we mit- igate overfitting…
We reframe linear dimensionality reduction as a problem of Bayesian inference on matrix manifolds. This natural paradigm extends the Bayesian framework to dimensionality reduction tasks in higher dimensions with simpler models at greater speeds. Here an orthogonal basis is treated as a single point on a manifold and is…
We proved a matrix Li-Yau-Hamilton type gradient estimates for the positive solutin of the heat equation on complete Kaehler manifolds with nonnegative bisectional curvature. As a consequence we obtain a comparison theorem for the distance function under this curvature assumption.
The contribution of reducible connections to the U(N) Chern-Simons invariant of a Seifert manifold M can be expressed in some cases in terms of matrix integrals. We show that the U(N) evaluation of the LMO invariant of any rational homology sphere admits a matrix model representation which agrees with the Chern-Simon…
Study Brownian motion on Grassmann manifold using matrix stochastic calculus.
problem Understanding Brownian motion on non-compact Grassmann manifold.
method Realize Brownian motion as matrix diffusion process, use matrix stochastic calculus, and hyperbolic Stiefel fibration.
result Connection to generalized Maass Laplacian of complex hyperbolic space.
Designs neural networks on matrix manifolds for improved performance in tasks like human action recognition.
problem Designing neural networks for tasks on non-Euclidean manifolds.
method Develops fully-connected and convolutional layers for SPD manifolds, and MLR on SPSD manifolds.
result Demonstrates improved performance in human action recognition and node classification tasks.
Knots and 4-manifolds linked via matrix kinking.
problem Understanding equivalence of symmetric matrices and their implications.
method Isotopy and kinking moves on Goeritz matrices.
result Every nonsingular symmetric integer matrix is kink-equivalent to positive or negative-definite matrices.
I-BBS identifies latent sub-manifolds from distance matrices, robust to noise.
problem Identifying latent sub-manifolds from distance matrices in high-dimensional spaces.
method Coordinate-free inference using random distance matrix theory and generative noise models.
result Recovering latent geometry from integer-stable signatures of eigenvalues.
In this paper we are concerned with the matrix Li-Yau-Hamilton estimates for nonlinear heat equations. Firstly, we derive such estimate on a Kähler manifold with a fixed Kähler metric. Then we consider the estimate on Kähler manifolds with Kähler metrics evolving under the rescaled Kähler-Ricci flow. Both of the estima…
Equivalence proven between equivariant K-theory and K-homology for certain matrix group actions.
problem Establishing Poincaré duality for proper cocompact matrix group actions.
method Using equivariant K-theory and K-homology, with geometric models of Baum and Douglas.
result Poincaré duality holds between equivariant K-theory and K-homology for G-spinc manifolds with compact quotient. Proposes CC-NMDF for analyzing manifold-valued data.
problem Nonlinear structure in manifold-valued data requires new analysis methods.
method Curvature-corrected nonnegative manifold data factorization (CC-NMDF) with an iterative algorithm.
result Demonstrates CC-NMDF on real-world diffusion tensor MRI data.
Improved heat equation estimates without gradient curvature assumption.
problem Improving Hamilton's matrix Harnack estimate for heat equation without gradient curvature assumption.
method New ingredients include a sharp Li-Yau estimate, a suitable vector field construction, and integral arguments.
result Removed the gradient curvature assumption in Hamilton's estimate for heat equation.
Develops Riemannian geometry for optimization on manifolds with detailed derivations.
problem Abstract high-level optimization on nonlinear spaces like matrix manifolds.
method Systematic derivation of geometric structures and constructions in coordinates and matrix form.
result Unified treatment of Riemannian geometry for optimization on manifolds.
Robust PCA is a widely used statistical procedure to recover a underlying low-rank matrix with grossly corrupted observations. This work considers the problem of robust PCA as a nonconvex optimization problem on the manifold of low-rank matrices, and proposes two algorithms (for two versions of retractions) based on ma…
Two algorithms estimate Wasserstein distance matrices from few entries for manifold learning.
problem Estimating Wasserstein distance matrices from limited data for manifold learning.
method Proposes two algorithms: matrix completion and Nyström completion for square Wasserstein matrices.
result Nyström completion can outperform matrix completion with a fixed sample budget and improve classification stability.