Spectral sparsification improves Laplacian-constrained graph learning.
problem Improving accuracy of Laplacian-constrained graph learning.
method Spectral graph sparsification as a post-estimation operation.
result Improved accuracy of Laplacian-constrained graph learning.
Paper proves conditions for estimating precision matrices with Laplacian constraints.
problem Estimating high-dimensional precision matrices with Laplacian constraints.
method Minimizing Stein's loss with conditions on graph connectivity and Laplacian constraints.
result High-dimensional consistency achieved with Laplacian constraints, independent of graph structure.
The ℓ1-norm fails to produce sparse solutions in Laplacian constrained graphical models, leading to a complete graph.
problem Learning a sparse graph under Laplacian constrained Gaussian graphical models.
method Introduced a nonconvex sparsity penalty and proposed a new estimator using a sequence of weighted ℓ1-norm penalized sub-problems. Developed a projected gradient descent algorithm with linear convergence rate. result The proposed estimator can recover the edges correctly with high probability and is effective on both synthetic and real-world data sets.
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.
The paper derives Cramer-Rao bounds for Laplacian matrix estimation under various constraints.
problem Estimating Laplacian matrices with structural constraints and sparsity.
method Linear reparametrization and closed-form expressions for Cramer-Rao bounds tailored to Laplacian matrix estimation.
result The derived CRBs provide performance limits for Laplacian matrix estimation and are validated in various applications.
The symmetry operators for the Laplacian in flat space were recently described and here we consider the same question for the square of the Laplacian. Again, there is a close connection with conformal geometry. There are three main steps in our construction. The first is to show that the symbol of a symmetry is constra…
LA-VDM accelerates VDM using landmarks to improve data analysis.
problem Efficiently analyzing complex datasets with nonuniform sampling densities.
method Landmark-constrained two-stage normalization to accelerate VDM.
result LA-VDM accurately recovers parallel transport and converges to the connection Laplacian.
Method constrains spectral gaps of hyperbolic spin surfaces using identities and semidefinite programming.
problem Bounding Laplacian and Dirac spectra of hyperbolic spin manifolds and orbifolds.
method Infinite family of spectral identities, semidefinite programming, and Selberg trace formula.
result Upper bounds on spectral gaps nearly saturated by specific orbifolds.
Paper proves convergence of bi-stochastically normalized graph Laplacian to manifold Laplacian and robustness to outlier noise.
problem Convergence of bi-stochastically normalized graph Laplacian to manifold Laplacian and robustness to outlier noise.
method Proves convergence of bi-stochastically normalized graph Laplacian to manifold Laplacian with rates, and proposes an approximate and constrained matrix scaling problem to achieve the same consistency rate.
result Graph Laplacian consistency rate matches the rate for clean manifold data plus an additional term proportional to the boundedness of the inner-products of the noise vectors.
Paper proposes a method to improve graph clustering by integrating node textual metadata with node signals in GGMs.
problem Graph learning in Gaussian Graphical Models with auxiliary node metadata.
method Laplacian-constrained Gaussian Graphical Models with majorization-minimization algorithm.
result The proposed method outperforms state-of-the-art approaches that use either signals or metadata alone.
The paper analyzes spectral properties of connection Laplacian on tori, proving convergence to real torus.
problem Spectral analysis of connection Laplacian on tori.
method Employing parallel orthonormal basis in pullback bundle, examining eigenvalues of connection Laplacian on real and discrete tori.
result Eigenvalues of connection Laplacian on discrete tori converge to those on real torus, with unique twist in torsion matrix.
ANN clusters multi-view data by agglomerating subviews and avoiding postprocessing.
problem Challenges in multi-view clustering, especially subview agglomeration.
method Agglomerative Neural Network (ANN) using Constrained Laplacian Rank.
result ANN outperforms state-of-the-art methods in multi-view clustering.
Finslerian graph neural networks recover nonlinear diffusion geometry
problem Graph neural networks on point clouds
method Estimates of the Finsler Laplacian
result Recovery of Finsler geometry
Constrained clustering has been well-studied for algorithms such as K-means and hierarchical clustering. However, how to satisfy many constraints in these algorithmic settings has been shown to be intractable. One alternative to encode many constraints is to use spectral clustering, which remains a developing area. I…
The potential of recovering the topology of a grid using solely publicly available market data is explored here. In contemporary whole-sale electricity markets, real-time prices are typically determined by solving the network-constrained economic dispatch problem. Under a linear DC model, locational marginal prices (LM…
We consider the problem of estimating a function defined over n locations on a d-dimensional grid (having all side lengths equal to n1/d). When the function is constrained to have discrete total variation bounded by Cn, we derive the minimax optimal (squared) ℓ2 estimation error rate, parametrized by …
Grid security and open markets are two major smart grid goals. Transparency of market data facilitates a competitive and efficient energy environment, yet it may also reveal critical physical system information. Recovering the grid topology based solely on publicly available market data is explored here. Real-time ener…
We propose a graph spectral representation of time series data that 1) is parsimoniously encoded to user-demanded resolution; 2) is unsupervised and performant in data-constrained scenarios; 3) captures event and event-transition structure within the time series; and 4) has near-linear computational complexity in both …
Bounds on spectral gaps of hyperbolic 3-manifolds and orbifolds.
problem Constraining the spectra of Laplace operators on hyperbolic manifolds and orbifolds.
method Linear programming and spectral identities derived from the conformal bootstrap and Selberg trace formula.
result Upper bounds on the first and second Laplacian eigenvalues, and spectral gaps of hyperbolic 3-manifolds and orbifolds.
AEGCN uses autoencoder constraints to improve graph node classification.
problem Node classification on graph domains with reduced information loss.
method Autoencoder-constrained graph convolutional network (AEGCN).
result Adding autoencoder constraints significantly improves graph convolutional network performance.
Defines vector Laplacian on statistical manifolds.
problem No specific problem stated; focuses on mathematical definition.
method Defines and derives vector Laplacian formula.
result Derives formula for vector Laplacian.
BIG Laplacians bridge combinatorial and Hodge Laplacians for discrete data.
problem Comparing combinatorial and Hodge Laplacians for discrete data.
method Introducing Boundary-Induced Graph (BIG) Laplacians using DEC.
result BIG Laplacian eigenvalues converge to Hodge Laplacian for simple shapes.
Paper introduces magnetic Hodge Laplacian for differential forms.
problem No specific problem stated; general spectral analysis of differential forms.
method Introduced magnetic Hodge Laplacian, discussed spectral results.
result Similarities and differences with magnetic Laplacian on functions.
Proves Laplacian and Lichnerowicz Laplacian are sectorial in weighted Hölder spaces.
problem Analyzing sectoriality of Laplacian and Lichnerowicz Laplacian on asymptotically hyperbolic spaces.
method Proves sectoriality in weighted Hölder spaces using asymptotically hyperbolic metrics.
result Analytic semigroups apply, yielding well-posedness results for parabolic evolution equations.
The paper extends Laplacian spectra approximations to vector bundles.
problem Approximating the spectrum of the connection Laplacian.
method Extending the graph connection Laplacian to vector bundles and proving spectrum approximation.
result The spectrum of the extended operator approximates the spectrum of the connection Laplacian.
Survey of Laplacian-based methods for data dimensionality reduction and embedding.
problem Efficiently reducing high-dimensional data to lower dimensions while preserving important features and structures.
method Laplacian-based methods including spectral clustering, Laplacian eigenmap, locality preserving projection, graph embedding, and diffusion map.
result Comprehensive overview of various optimization variants and applications of Laplacian-based techniques.
The graph Laplacian plays key roles in information processing of relational data, and has analogies with the Laplacian in differential geometry. In this paper, we generalize the analogy between graph Laplacian and differential geometry to the hypergraph setting, and propose a novel hypergraph p-Laplacian. Unlike the …
The paper sets up eigenvalue comparison theorems for specific Laplacians on manifolds.
problem Eigenvalue comparison theorems for Witten-Laplacian and weighted p-Laplacian on manifolds with modified Ricci curvature. method Established Cheng-type eigenvalue comparison theorems for the first Dirichlet eigenvalues of the Witten-Laplacian and weighted p-Laplacian on geodesic balls. result Successfully set up eigenvalue comparison theorems for the Witten-Laplacian and weighted p-Laplacian. Extended bounds on small eigenvalues for pseudo-Laplacians on hyperbolic surfaces.
problem Bounding small eigenvalues of pseudo-Laplacians on hyperbolic surfaces.
method Extended Otal-Rosas bound and Colin de Verdière's spectral theory to hyperbolic surfaces with multiple cusps.
result Extended bounds on small eigenvalues for pseudo-Laplacians.
Paper investigates rigidity phenomena for weighted Ricci curvature bounds with Laplacian comparison theorem.
problem Investigating rigidity phenomena for weighted Ricci curvature bounds.
method Derived comparison geometric estimates and generalized for non-symmetric Laplacian.
result Obtained rigidity results for Laplacian comparison theorem, diameter comparisons, and volume comparisons.
Study Hodge Laplacians for manifold data, improving error bounds.
problem Approximating Laplace-Beltrami operator on differential forms.
method Higher-order graph Laplacians (Hodge Laplacians) as approximations.
result High-probability error bound for Dirichlet forms.
Root Laplacian Eigenmaps help in spectral embedding of graphs.
problem Efficient spectral embedding of graphs.
method Square root of graph-Laplacian operator.
result Improved spectral embedding techniques.
New perspective on G2-structures flow from DeTurck Laplacian.
problem Understanding G2-structures and their flows.
method Introducing a new flow (DeTurck Laplacian flow) for G2-structures.
result DeTurck Laplacian flow is a flow of G2-structures.
Explains BV Laplacian on half-densities in simple terms.
problem None explicitly stated; focuses on explanation.
method Didactical review of BV Laplacian on half-densities.
result Explains BV Laplacian concept in plain language.
Study solves sub-Laplacian equivalence on a specific Heisenberg group.
problem Contact equivalence problem for sub-Laplacians on the second Heisenberg group.
method Solves the contact equivalence problem for generalised sub-Laplacians on $\He^2$.
result Parameterises sub-Laplacians on $\He^2$ by R+. A kinematic method selects the deformation Laplacian for fluid dynamics on Riemannian manifolds.
problem Ambiguity in viscous operator choice for Navier-Stokes equations on Riemannian manifolds.
method Kinematic construction of strain rate from Lie-dragged vectors, excluding Hodge Laplacian due to antisymmetric part.
result Kinematic selection uniquely identifies the deformation Laplacian, resolving analytical obstructions.
Study shows rates for Laplacian-eigenmap methods in nonparametric regression.
problem Minimizing error in nonparametric regression using Laplacian-eigenmap.
method Adaptive and non-adaptive minimax rates using Sobolev space constraints.
result Extends minimax rates to various weighted Laplacian matrices.
Study eigenvalues of p-Laplacian on quaternionic Kähler manifolds.
problem Finding lower bounds for eigenvalues of p-Laplacian on quaternionic Kähler manifolds.
method Analytical proofs for both Neumann and Dirichlet boundary conditions.
result Established lower bounds for eigenvalues on compact quaternionic Kähler manifolds.
The paper finds new inequalities for Laplacian and biharmonic eigenvalues on manifolds.
problem Eigenvalue inequalities for Laplacian and biharmonic operators on submanifolds.
method Using Sobolev inequalities to establish new eigenvalue inequalities.
result Established new inequalities for Laplacian and biharmonic eigenvalues.
Researchers find second-order estimates for p-Laplacian in RCD spaces.
problem Estimating functions with p-Laplacian in RCD spaces. method Establishing quantitative second-order Sobolev regularity.
result Second-order estimates for p-Laplacian functions in RCD spaces. Study on G2-structures using Laplacian coflow and solitons.
problem Characterizing and understanding G2-structures and their solitons. method Using the irreducible G2-decomposition of the Hodge Laplacian and Lie derivative, characterizing infinitesimal symmetries and soliton conditions. result Proof of the absence of compact shrinking solitons for the Laplacian coflow.
Study of Yang-Mills fields on 4-manifolds using modified Lévy Laplacians.
problem Connection between Yang-Mills fields and modified Lévy Laplacians on 4-manifolds.
method Analysis of modified Lévy Laplacians and their relation to Yang-Mills equations under nontrivial holonomy groups.
result Existence of a modified Lévy Laplacian related to Yang-Mills self-duality equations.
For a bounded domain Ω with a piecewise smooth boundary in an n-dimensional Euclidean space Rn, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. First we give a general inequality for eigenvalues of the Laplacian. As an application, we study lower order eigenvalues of the Lap…
Fractional Laplacian inverse problem solved for connection Laplacians.
problem Determining structures from metric, bundle, and map knowledge.
method Local knowledge of metric, bundle, and map determines global structures.
result Global structures determined from local knowledge of metric, bundle, and map.
Universal inequalities found for Laplacian eigenvalues on convex domains.
problem Finding bounds for Laplacian eigenvalues on convex domains.
method Established two universal inequalities.
result Found new bounds for Laplacian eigenvalues.
Universal inequalities for Laplacian eigenvalues on convex domains.
problem Eigenvalue distribution of the Laplacian on convex domains.
method Established two universal inequalities.
result Two new inequalities for Laplacian eigenvalues.
New upper bound for Neumann Laplacian eigenvalues on convex domains.
problem Bounding Neumann eigenvalues on convex domains.
method Deriving a new upper bound for eigenvalues.
result Universal inequalities for Neumann eigenvalues derived from the upper bound.
New inequalities for planar convex domains' Laplacian eigenvalues.
problem Neumann eigenvalues of the Laplacian on planar convex domains.
method Established two new universal inequalities.
result New inequalities for Laplacian eigenvalues on convex domains.