Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userWeighted Laplacian and Its Theoretical Applications
86
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper, we develop a novel weighted Laplacian method, which is partially inspired by the theory of graph Laplacian, to study recent popular graph problems, such as multilevel graph partitioning and balanced minimum cut problem, in a more convenient manner. Since the weighted Laplacian strategy inherits the virtues of spectral methods, graph algorithms designed using weighted Laplacian will necessarily possess more robust theoretical guarantees for algorithmic performances, comparing with those existing algorithms that are heuristically proposed. In order to illustrate its powerful utility both in theory and in practice, we also present two effective applications of our weighted Laplacian method to multilevel graph partitioning and balanced minimum cut problem, respectively. By means of variational methods and theory of partial differential equations (PDEs), we have established the equivalence relations among the weighted cut problem, balanced minimum cut problem and the initial clustering problem that arises in the middle stage of graph partitioning algorithms under a multilevel structure. These equivalence relations can indeed provide solid theoretical support for algorithms based on our proposed weighted Laplacian strategy. Moreover, from the perspective of the application to the balanced minimum cut problem, weighted Laplacian can make it possible for research of numerical solutions of PDEs to be a powerful tool for the algorithmic study of graph problems. Experimental results also indicate that the algorithm embedded with our strategy indeed outperforms other existing graph algorithms, especially in terms of accuracy, thus verifying the efficacy of the proposed weighted Laplacian.
rate research
Read More
In this paper, we generalize the combinatorial Laplace operator of Horak and Jost by introducing the $phi$-weighted coboundary operator induced by a weight function $phi$. Our weight function $phi$ is a generalization of Dawsons weighted boundary map. We show that our above-mentioned generalizations include new cases that are not covered by previous literature. Our definition of weighted Laplacian for weighted simplicial complexes is also applicable to weighted/unweighted graphs and digraphs.
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 interpretations and (iii) associated algorithms. Specifically, graph learning problems are posed as estimation of graph Laplacian matrices from some observed data under given structural constraints (e.g., graph connectivity and sparsity level). From a probabilistic perspective, the problems of interest correspond to maximum a posteriori (MAP) parameter estimation of Gaussian-Markov random field (GMRF) models, whose precision (inverse covariance) is a graph Laplacian matrix. For the proposed graph learning problems, specialized algorithms are developed by incorporating the graph Laplacian and structural constraints. The experimental results demonstrate that the proposed algorithms outperform the current state-of-the-art methods in terms of accuracy and computational efficiency.
Recently, Deutsch and Elizalde studied the largest and the smallest fixed points of permutations. Motivated by their work, we consider the analogous problems in weighted set partitions. Let $A_{n,k}(mathbf{t})$ denote the total weight of partitions on $[n+1]$ with the largest singleton ${k+1}$. In this paper, explicit formulas for $A_{n,k}(mathbf{t})$ and many combinatorial identities involving $A_{n,k}(mathbf{t})$ are obtained by umbral operators and combinatorial methods. As applications, we investigate three special cases such as permutations, involutions and labeled forests. Particularly in the permutation case, we derive a surprising identity analogous to the Riordan identity related to tree enumerations, namely, begin{eqnarray*} sum_{k=0}^{n}binom{n}{k}D_{k+1}(n+1)^{n-k} &=& n^{n+1}, end{eqnarray*} where $D_{k}$ is the $k$-th derangement number or the number of permutations of ${1,2,dots, k}$ with no fixed points.
Federated learning aims to protect data privacy by collaboratively learning a model without sharing private data among users. However, an adversary may still be able to infer the private training data by attacking the released model. Differential privacy provides a statistical protection against such attacks at the price of significantly degrading the accuracy or utility of the trained models. In this paper, we investigate a utility enhancement scheme based on Laplacian smoothing for differentially private federated learning (DP-Fed-LS), where the parameter aggregation with injected Gaussian noise is improved in statistical precision without losing privacy budget. Our key observation is that the aggregated gradients in federated learning often enjoy a type of smoothness, i.e. sparsity in the graph Fourier basis with polynomial decays of Fourier coefficients as frequency grows, which can be exploited by the Laplacian smoothing efficiently. Under a prescribed differential privacy budget, convergence error bounds with tight rates are provided for DP-Fed-LS with uniform subsampling of heterogeneous Non-IID data, revealing possible utility improvement of Laplacian smoothing in effective dimensionality and variance reduction, among others. Experiments over MNIST, SVHN, and Shakespeare datasets show that the proposed method can improve model accuracy with DP-guarantee and membership privacy under both uniform and Poisson subsampling mechanisms.
We present a novel spectral embedding of graphs that incorporates weights assigned to the nodes, quantifying their relative importance. This spectral embedding is based on the first eigenvectors of some properly normalized version of the Laplacian. We prove that these eigenvectors correspond to the configurations of lowest energy of an equivalent physical system, either mechanical or electrical, in which the weight of each node can be interpreted as its mass or its capacitance, respectively. Experiments on a real dataset illustrate the impact of weighting on the embedding.
suggested questions
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
