Upper bounds on algebraic connectivity via convex optimization

Extrapolation is a well-known technique for solving convex optimization and variational inequalities and recently attracts some attention for non-convex optimization. Several recent works have empirically shown its success in some machine learning tasks. However, it has not been analyzed for non-convex minimization and there still remains a gap between the theory and the practice. In this paper, we analyze gradient descent and stochastic gradient descent with extrapolation for finding an approximate first-order stationary point in smooth non-convex optimization problems. Our convergence upper bounds show that the algorithms with extrapolation can be accelerated than without extrapolation.

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LAPLACIAN EIGENVALUES OF GRAPHS WITH GIVEN DOMINATION NUMBER

Asian-European Journal of Mathematics ◽

10.1142/s1793557109000066 ◽

2009 ◽

Vol 02 (01) ◽

pp. 71-76 ◽

Cited By ~ 2

Author(s):

Lihua Feng ◽

Guihai Yu ◽

Xiqin Lin

Keyword(s):

Spectral Radius ◽

Domination Number ◽

Upper Bounds ◽

Algebraic Connectivity ◽

Laplacian Spectral Radius ◽

Laplacian Eigenvalues ◽

Eigenvalues Of Graphs

In this paper, we study the Laplacian eigenvalues of graphs on n vertices with domination number γ and present upper bounds for the Laplacian spectral radius and algebraic connectivity as well, which improve old results apparently.

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Computing tight upper bounds on the algebraic connectivity of certain graphs

Linear Algebra and its Applications ◽

10.1016/j.laa.2008.08.018 ◽

2009 ◽

Vol 430 (1) ◽

pp. 532-543 ◽

Cited By ~ 1

Author(s):

Oscar Rojo

Keyword(s):

Upper Bounds ◽

Algebraic Connectivity

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Defining quantum divergences via convex optimization

Quantum ◽

10.22331/q-2021-01-26-387 ◽

2021 ◽

Vol 5 ◽

pp. 387

Author(s):

Hamza Fawzi ◽

Omar Fawzi

Keyword(s):

Convex Optimization ◽

Semidefinite Programming ◽

Quantum Channel ◽

Upper Bounds ◽

Chain Rule ◽

Quantum Channels ◽

Optimization Program ◽

Rényi Divergence ◽

Strong Converse ◽

A Chain

We introduce a new quantum Rényi divergence Dα# for α∈(1,∞) defined in terms of a convex optimization program. This divergence has several desirable computational and operational properties such as an efficient semidefinite programming representation for states and channels, and a chain rule property. An important property of this new divergence is that its regularization is equal to the sandwiched (also known as the minimal) quantum Rényi divergence. This allows us to prove several results. First, we use it to get a converging hierarchy of upper bounds on the regularized sandwiched α-Rényi divergence between quantum channels for α>1. Second it allows us to prove a chain rule property for the sandwiched α-Rényi divergence for α>1 which we use to characterize the strong converse exponent for channel discrimination. Finally it allows us to get improved bounds on quantum channel capacities.

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Algebraic Connectivity and Disjoint Vertex Subsets of Graphs

Mathematical Problems in Engineering ◽

10.1155/2020/5763218 ◽

2020 ◽

Vol 2020 ◽

pp. 1-6

Author(s):

Yan Sun ◽

Faxu Li

Keyword(s):

Laplacian Matrix ◽

Upper Bounds ◽

Small Eigenvalue ◽

Algebraic Connectivity

It is well known that the algebraic connectivity of a graph is the second small eigenvalue of its Laplacian matrix. In this paper, we mainly research the relationships between the algebraic connectivity and the disjoint vertex subsets of graphs, which are presented through some upper bounds on algebraic connectivity.

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Bounds for Kirchhoff index and Laplacian-energy-like invariant of some derived graphs of a regular graph

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830920500226 ◽

2020 ◽

Vol 12 (02) ◽

pp. 2050022

Author(s):

Ruhul Amin ◽

Sk. Md. Abu Nayeem

Keyword(s):

Regular Graph ◽

Complete Bipartite Graph ◽

Double Cover ◽

Upper Bounds ◽

Algebraic Connectivity ◽

Kirchhoff Index ◽

Laplacian Eigenvalues ◽

Square Roots ◽

Laplacian Energy ◽

Better Than

The Kirchhoff index and Laplacian-energy-like invariant of a connected graph [Formula: see text], denoted by [Formula: see text] and [Formula: see text], are given by the number of vertex times the sum of the reciprocals of all nonzero Laplacian eigenvalues of [Formula: see text] and the sum of the square roots of all Laplacian eigenvalues of [Formula: see text], respectively. In this paper, we have obtained the Laplacian eigenvalues of some derived graphs, such as double graph, extended double cover and Mycielskian of an [Formula: see text]-regular graph [Formula: see text], in terms of the adjacency eigenvalues of [Formula: see text] and hence, we obtain some upper bounds of Kirchhoff index and Laplacian-energy-like (LEL) invariant of those derived graphs in terms of [Formula: see text], number of vertices and algebraic connectivity of [Formula: see text]. We have shown that the bounds obtained here are better than some existing bounds. We have also obtained the exact formulae for Kirchhoff index and LEL invariant of those derived graph when [Formula: see text] is a complete graph or a complete bipartite graph.

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