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 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.
Paper derives inequalities for eigenvalues of Witten-Laplacian under fixed volume constraint.
problem Eigenvalue inequalities of Witten-Laplacian on bounded domains.
method Rearrangement technique and trial functions under fixed weighted volume constraint.
result Several isoperimetric inequalities for eigenvalues of Witten-Laplacian.
Paper learns Cartesian product graphs with Laplacian constraints.
problem Learning Cartesian product graphs from Laplacian constraints.
method Penalized maximum likelihood estimation (MLE) and efficient algorithm.
result Statistical consistency for Cartesian product Laplacian estimation.
Study financial market graphs with Laplacian constraints.
problem Learning undirected graphs in financial markets.
method Proposes algorithms to estimate graphs accounting for financial data properties.
result Guidelines for estimating graphs in financial markets.
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.
Graphs are fundamental mathematical structures used in various fields to represent data, signals and processes. In this paper, we propose a novel framework for learning/estimating graphs from data. The proposed framework includes (i) formulation of various graph learning problems, (ii) their probabilistic interpretatio…
Paper finds inequalities for eigenvalues of buckling problems on special metric spaces.
problem Eigenvalue inequalities for buckling problems of drifting Laplacian.
method Investigated on bounded domains in complete smooth metric measure spaces (SMMSs) with special functions.
result General inequalities for eigenvalues derived under curvature constraints.
The paper finds extremum values for mixed Laplacian eigenvalues on triangles and trapezoids.
problem Finding extremum values for mixed eigenvalues of the Laplacian on triangles and trapezoids.
method Characterizations obtained under suitable geometric constraints.
result Characterizations of extremum values for mixed eigenvalues of the Laplacian on triangles and trapezoids.
The paper decomposes spacelike hypersurface properties for general relativistic vacuum equations.
problem Analyzing properties of spacelike hypersurfaces in general relativity.
method Used L2-orthogonal decomposition and Ahlfors Laplacian. result Decomposed the second fundamental form of spacelike hypersurfaces.
Constructs metrics with zero eigenvalues for Hodge-Laplacian.
problem Eigenvalues of Hodge-Laplacian under sectional curvature constraints.
method One-parameter family of metrics with bounded sectional curvature.
result k-th positive eigenvalue converges to zero.
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.
Learning a graph with a specific structure is essential for interpretability and identification of the relationships among data. It is well known that structured graph learning from observed samples is an NP-hard combinatorial problem. In this paper, we first show that for a set of important graph families it is possib…
The Ahlfors Laplacian is applied to solve geometric and relativistic problems.
problem Solving geometric and relativistic problems using the Ahlfors Laplacian.
method Orthogonal decompositions and expansions of tensor components are used to study the Ahlfors Laplacian's applications.
result The Ahlfors Laplacian is applied to construct solutions of general relativistic constraint equations in vacuum.
The paper examines geometric properties of domains for the p-Laplacian in Euclidean and hyperbolic spaces.
problem Exploring geometric properties of unbounded extremal domains for the p-Laplacian operator.
method Analyzing properties in Euclidean and hyperbolic spaces, proving constraints on domains and their asymptotic boundaries.
result Extremal domains in two dimensions must be balls, and in hyperbolic space, they have specific geometric constraints.
Paper proves a new isoperimetric inequality for Steklov eigenvalues.
problem Finding a new isoperimetric inequality for Steklov eigenvalues.
method Proving a Brock-type inequality under specific conditions.
result Extension of Brock's classical result to Witten-Laplacian.
Paper finds eigenvalue bounds for hyperbolic space domains.
problem Finding eigenvalue bounds for Robin Laplacian in hyperbolic space.
method Lower and upper bounds derived for eigenvalues.
result Geodesic ball maximizes eigenvalue in negative boundary parameter case.
The paper explores learning graphs in financial markets using Laplacian constraints.
problem Learning undirected graphical models for financial assets.
method Alternating Direction Method of Multipliers for graph learning.
result Laplacian matrix as a model for financial assets' precision matrix.
Multi-view spectral clustering, which aims at yielding an agreement or consensus data objects grouping across multi-views with their graph laplacian matrices, is a fundamental clustering problem. Among the existing methods, Low-Rank Representation (LRR) based method is quite superior in terms of its effectiveness, intu…
Local Linear embedding (LLE) is a popular dimension reduction method. In this paper, we first show LLE with nonnegative constraint is equivalent to the widely used Laplacian embedding. We further propose to iterate the two steps in LLE repeatedly to improve the results. Thirdly, we relax the kNN constraint of LLE and p…
Eigen-stratified models reduce model size and improve performance.
problem Large model size in Laplacian-regularized stratified models.
method Formulate eigen-stratified models with linear combinations of bottom eigenvectors of the graph Laplacian.
result Significant reduction in model size with eigen-stratified models.
A new method for few-shot learning using Laplacian regularization.
problem Few-shot learning with limited labeled data.
method Transductive Laplacian-regularized inference for feature embeddings.
result Our method outperforms state-of-the-art methods across various benchmarks.
In this paper, we successfully generalize the eigenvalue comparison theorem for the Dirichlet p-Laplacian (1<p<∞) obtained by Matei [A.-M. Matei, First eigenvalue for the p-Laplace operator, Nonlinear Anal. TMA 39 (8) (2000) 1051--1068] and Takeuchi [H. Takeuchi, On the first eigenvalue of the p-Laplacian …
In the classic sparsity-driven problems, the fundamental L-1 penalty method has been shown to have good performance in reconstructing signals for a wide range of problems. However this performance relies on a good choice of penalty weight which is often found from empirical experiments. We propose an algorithm called t…
Paper proves rigidity of weak solutions for anisotropic N-Laplacian equations with Neumann or Robin boundary conditions.
problem Rigidity of weak solutions for anisotropic N-Laplacian equations with boundary conditions.
method Established a key integral inequality involving anisotropic gradient and second fundamental form, proving rigidity under natural monotonicity assumptions.
result All weak solutions to Neumann boundary problems are constant without a priori boundedness assumption.
Paper bounds the lowest spectrum of manifolds with curvature constraints.
problem Estimating the lowest energy level of manifolds with curvature restrictions.
method Used a minimal positive Green's function and its properties.
result Proved an upper bound for the bottom of the spectrum.
Study graph-based algorithms for multi-manifold clustering with sufficient conditions.
problem Clustering data from a union of manifolds with different dimensions and intersections.
method Investigate sufficient conditions for similarity graphs to capture geometric information.
result High probability error bounds for spectral approximation of tensorized Laplacian.
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…
Unified analysis of multi-task functional linear regression with manifold and composite penalties.
problem Estimating slope functions from functional data with multi-task learning.
method Penalized splines with manifold constraint and composite quadratic penalty.
result Unified convergence upper bound and phase transition behaviors for estimators.
Study on compact manifolds for exact G2-Structures without additional constraints.
problem Whether compact 7-manifolds support exact G2-Structures. method Investigate exact G2-Structures on compact manifolds, considering relationships with other conditions. result Initiate a study on exact G2-Structures on compact manifolds without additional constraints. Graphs are naturally sparse objects that are used to study many problems involving networks, for example, distributed learning and graph signal processing. In some cases, the graph is not given, but must be learned from the problem and available data. Often it is desirable to learn sparse graphs. However, making a grap…
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.
In this paper, we propose a scalable algorithm for spectral embedding. The latter is a standard tool for graph clustering. However, its computational bottleneck is the eigendecomposition of the graph Laplacian matrix, which prevents its application to large-scale graphs. Our contribution consists of reformulating spect…
A new method for spectral barycentre of graph datasets.
problem Creating a summary graph from a set of graphs with community structure.
method Using multiscale spectral distance based on normalized graph Laplacian eigenvalues.
result The barycentre inherits the topological structure of the graphs in the sample dataset.
We define a new family of similarity and distance measures on graphs, and explore their theoretical properties in comparison to conventional distance metrics. These measures are defined by the solution(s) to an optimization problem which attempts find a map minimizing the discrepancy between two graph Laplacian exponen…
Post-processing corrects bias in ML systems without retraining.
problem Correcting bias in ML systems that are already in use.
method Proposes general post-processing algorithms for individual fairness based on graph Laplacian regularization.
result Empirically, post-processing algorithms correct individual biases in large-scale NLP models while preserving accuracy.
New metrics on 3D manifolds with large Steklov eigenvalues.
problem Finding metrics with large Steklov eigenvalues on compact manifolds.
method Expressed Steklov spectrum of warped products and applied to metrics with fixed volume.
result Examples of metrics on 3D manifolds with arbitrarily large first non-zero Steklov eigenvalue.
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.
The paper finds metrics for surfaces with boundaries that match specific eigenvalues and areas.
problem Finding metrics for surfaces with boundaries that match specific eigenvalues and areas.
method The approach involves constructing a metric on a compact surface with boundary that satisfies given eigenvalues and area constraints.
result A metric can be constructed on a compact surface with boundary that matches a given sequence of eigenvalues and area.
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.
We analyze the performance of a class of manifold-learning algorithms that find their output by minimizing a quadratic form under some normalization constraints. This class consists of Locally Linear Embedding (LLE), Laplacian Eigenmap, Local Tangent Space Alignment (LTSA), Hessian Eigenmaps (HLLE), and Diffusion maps.…
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.