## The fastest mixing Markov process on a graph and a connection to a maximum variance unfolding problem (2006)

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Venue: | SIAM REVIEW |

Citations: | 47 - 5 self |

### BibTeX

@ARTICLE{Sun06thefastest,

author = {Jun Sun and Stephen Boyd and Lin Xiao and Persi Diaconis},

title = {The fastest mixing Markov process on a graph and a connection to a maximum variance unfolding problem},

journal = {SIAM REVIEW},

year = {2006},

volume = {48},

number = {4},

pages = {2006}

}

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### Abstract

We consider a Markov process on a connected graph, with edges labeled with transition rates between the adjacent vertices. The distribution of the Markov process converges to the uniform distribution at a rate determined by the second smallest eigenvalue λ2 of the Laplacian of the weighted graph. In this paper we consider the problem of assigning transition rates to the edges so as to maximize λ2 subject to a linear constraint on the rates. This is the problem of finding the fastest mixing Markov process (FMMP) on the graph. We show that the FMMP problem is a convex optimization problem, which can in turn be expressed as a semidefinite program, and therefore effectively solved numerically. We formulate a dual of the FMMP problem and show that it has a natural geometric interpretation as a maximum variance unfolding (MVU) problem, i.e., the problem of choosing a set of points to be as far apart as possible, measured by their variance, while respecting local distance constraints. This MVU problem is closely related to a problem recently proposed by Weinberger and Saul as a method for “unfolding ” high-dimensional data that lies on a low-dimensional manifold. The duality between the FMMP and MVU problems sheds light on both problems, and allows us to characterize and, in some cases, find optimal solutions.

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Citation Context ...vity constraints, and so is an SDP. We refer to (3) as the primal SDP. While problem (3) is not in one of the so-called standard forms for SDP, it is easily transformed to a standard form; see, e.g., =-=[5, 14]-=-. One immediate consequence of the SDP formulation (3) is that we can numerically solve the FMMP problem efficiently, using standard algorithms for SDP (e.g., [12, 2, 1, 9, 7, 13]; see [8] for a compr... |

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Citation Context ...vity constraints, and so is an SDP. We refer to (3) as the primal SDP. While problem (3) is not in one of the so-called standard forms for SDP, it is easily transformed to a standard form; see, e.g., =-=[5, 14]-=-. One immediate consequence of the SDP formulation (3) is that we can numerically solve the FMMP problem efficiently, using standard algorithms for SDP (e.g., [12, 2, 1, 9, 7, 13]; see [8] for a compr... |

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Citation Context ...ed to a standard form; see, e.g., [5, 14]. One immediate consequence of the SDP formulation (3) is that we can numerically solve the FMMP problem efficiently, using standard algorithms for SDP (e.g., =-=[12, 2, 1, 9, 7, 13]-=-; see [8] for a comprehensive list of current SDP software). For m no more than 1000 or so, interior-point methods can be used to solve the FMMP problem in a few minutes (or much less) on a small desk... |

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Citation Context ...ere c∗ is the optimal value of the FMMP (and MVU) problem. There seem to be some interesting connections between the results in this section and some well-known functions of graphs, the Lovász number =-=[18]-=-and the Colin de Verdière parameter [25]. In connection with the Lovász number, suppose we label the vertices of the graph with unit vectors in a d-dimensional Euclidean space, such that vectors assoc... |

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Citation Context ...ed to a standard form; see, e.g., [5, 14]. One immediate consequence of the SDP formulation (3) is that we can numerically solve the FMMP problem efficiently, using standard algorithms for SDP (e.g., =-=[12, 2, 1, 9, 7, 13]-=-; see [8] for a comprehensive list of current SDP software). For m no more than 1000 or so, interior-point methods can be used to solve the FMMP problem in a few minutes (or much less) on a small desk... |

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Citation Context ...osely related problems have been explored by the authors, including the discrete-time counterpart [4, 3], distributed algorithms for resource allocation [32], and distributed algorithms for averaging =-=[31]-=-. We will also see, in section 4.7, a somewhat surprising connection to recent work in machine learning in the area of manifold unfolding [28, 29]. 2. Interpretations. In this section we give some sim... |

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Citation Context ...thms for resource allocation [19], and distributed algorithms for averaging [20]. We will also see, in §4.7, a somewhat surprising connection to recent work in machine learning, on manifold unfolding=-= [16, 17]-=-. 2 Interpretations In this section we given some simple physical interpretations of the fastest mixing Markov process problem (1). None of this material is used in the sequel (except in interpretatio... |

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Citation Context ...ed to a standard form; see, e.g., [5, 14]. One immediate consequence of the SDP formulation (3) is that we can numerically solve the FMMP problem efficiently, using standard algorithms for SDP (e.g., =-=[12, 2, 1, 9, 7, 13]-=-; see [8] for a comprehensive list of current SDP software). For m no more than 1000 or so, interior-point methods can be used to solve the FMMP problem in a few minutes (or much less) on a small desk... |

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Citation Context ...istribution π(t) to uniform is determined by λ2; for example, we have sup �π(t) − (1/n)1�TV ≤ π(0) 1 2 n1/2e −λ2t , where � · �TV is the total variation distance between two distri=-=butions (see, e.g., [6])-=-. (The total variation distance is the maximum difference in probability assigned by the two distributions, over any subset of vertices.) Thus, the larger the second Laplacian eigenvalue λ2 is, the f... |

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Citation Context ...bject classifications. 60J25, 65F15, 68Q32, 90C22, 90C46 DOI. 10.1137/S0036144504443821 1. The Problem. The fastest mixing Markov chain problem was proposed and studied by Boyd, Diaconis, and Xiao in =-=[4]-=-. In that problem the mixing rate of a discrete-time Markov chain on a graph was optimized over the set of transition probabilities on the edges of a given graph. In this paper, we discuss its continu... |

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Citation Context ... FMMP problem are linear, we see that it is a convex optimization problem (since the objective is concave and is to be maximized). General background on eigenvalue optimization can be found in, e.g., =-=[21, 17]-=-.s686 JUN SUN,STEPHEN BOYD,LIN XIAO, AND PERSI DIACONIS 3.2. An Alternate Formulation. Here we give an alternate formulation of the FMMP problem which will be more convenient in what follows. Since th... |

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Citation Context ...lems can be solved using subgradient-type methods and by exploiting the problem structure; see, e.g., [5, 4]. More sophisticated methods for solving large scale problems of this type are discussed in =-=[22, 14, 20]-=-. However, we won’t pursue numerical methods for the FMMP problem in this paper. (4) 3.4. The Dual Problem. The dual of the SDP (3) is (the SDP) maximize Tr(I − (1/n)11 T )X subject to Xii + Xjj − Xij... |

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Citation Context ...lems can be solved using subgradient-type methods and by exploiting the problem structure; see, e.g., [5, 4]. More sophisticated methods for solving large scale problems of this type are discussed in =-=[22, 14, 20]-=-. However, we won’t pursue numerical methods for the FMMP problem in this paper. (4) 3.4. The Dual Problem. The dual of the SDP (3) is (the SDP) maximize Tr(I − (1/n)11 T )X subject to Xii + Xjj − Xij... |

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Citation Context ... FMMP problem are linear, we see that it is a convex optimization problem (since the objective is concave and is to be maximized). General background on eigenvalue optimization can be found in, e.g., =-=[21, 17]-=-.s686 JUN SUN,STEPHEN BOYD,LIN XIAO, AND PERSI DIACONIS 3.2. An Alternate Formulation. Here we give an alternate formulation of the FMMP problem which will be more convenient in what follows. Since th... |

43 | Convex analysis on the Hermitian matrices
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Citation Context ...xpressed as � wij(ui − uj) =ui, i =1,...,n. j∼i We note that Result 2 corresponds exactly with the case in which the objective λ2 in the FMMP problem is differentiable at an optimal point (see, e.g., =-=[16]-=-).sFASTEST MIXING MARKOV PROCESS ON A GRAPH 689 4. A Geometric Dual. 4.1. A Maximum Variance Unfolding Problem. In this section we transform the dual SDP (5) to an equivalent problem that has a simple... |

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Citation Context ... to both the FMMP problem (2) and its dual MVU problem (7) are readily found numerically (and simultaneously) using an SDP solver to solve the SDP (3) (and its dual). For example, SDPSOL (Wu and Boyd =-=[18]) gives us the F-=-MMP optimal edge weights w ∗ 12 = 1.776, w ∗ 23 = 3.276, w ∗ 13 = 0.362, w ∗ 34 = 4.500, w ∗ 45 = 1.500, w ∗ 46 = 1.250, with optimal objective value � d 2 ijw ∗ ij = 17.5. Along with ... |

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Citation Context ...(and MVU) problem. There seem to be some interesting connections between the results in this section and some well-known functions of graphs, the Lovász number [18]and the Colin de Verdière parameter =-=[25]-=-. In connection with the Lovász number, suppose we label the vertices of the graph with unit vectors in a d-dimensional Euclidean space, such that vectors associated with nonadjacent vertices are orth... |

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Citation Context ...tional areas, subject to a unit total volume, so as to make the circuit equilibrate charge as quickly as possible. For a related application of semidefinite programming to optimizing RC circuits, see =-=[15]-=-. 2.2 Isolated thermal system We can give a similar interpretation for a thermal system. We consider a thermal system consisting of n unit thermal masses, connected by some thermal conductances, but o... |

13 | Fastest mixing Markov chain on a path
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Citation Context ...oblem also gives it several physical interpretations (discussed in the next section). Several other closely related problems have been explored by the authors, including the discrete-time counterpart =-=[4, 3]-=-, distributed algorithms for resource allocation [32], and distributed algorithms for averaging [31]. We will also see, in section 4.7, a somewhat surprising connection to recent work in machine learn... |

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Citation Context ... main results can skip this section. We note that similar extremal eigenvalue problems have been studied for continuous domains, with the Laplacian matrix replaced by a Laplacian operator; see, e.g., =-=[6, 7]-=-. 2.1. Grounded Capacitor RC Circuit. Consider a connected grounded capacitor resistor-capacitor (RC) circuit, as shown in Figure 1. Each node has a grounded unit value capacitor, and nodes i and j ar... |

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Citation Context ...λn. If the subgraph consisting of edges with positive rates is connected, then λ2 > 0 (i.e., λ1 is isolated). Conversely, if λ2 > 0, the subgraph of edges with positive rates is connected. See, e.=-=g., [11]-=- for a survey of the properties of the Laplacian. In the context of the Laplacian, we refer to the rates wij as weights, since they can be thought of as weights on the edges of the graph. From L1 = 0 ... |

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Citation Context ...iscussed in the next section). Several other closely related problems have been explored by the authors, including the discrete-time counterpart [4, 3], distributed algorithms for resource allocation =-=[32]-=-, and distributed algorithms for averaging [31]. We will also see, in section 4.7, a somewhat surprising connection to recent work in machine learning in the area of manifold unfolding [28, 29]. 2. In... |

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Citation Context ...0 (since the graph is connected). Finally, we note that if w ∗ is optimal, the subgraph associated with w ∗ ij > 0 must be connected. The absolute algebraic connectivity problem, described by Fiedler =-=[9, 10]-=-, is a special case of the FMMP problem, with dij =1/ √ m for all {i, j} ∈E. The absolute algebraic connectivity problem is a very interesting topic in algebraic graph theory; the fact that it is a sp... |

6 | 2003): Where best to hold a drum fast
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Citation Context ... main results can skip this section. We note that similar extremal eigenvalue problems have been studied for continuous domains, with the Laplacian matrix replaced by a Laplacian operator; see, e.g., =-=[6, 7]-=-. 2.1. Grounded Capacitor RC Circuit. Consider a connected grounded capacitor resistor-capacitor (RC) circuit, as shown in Figure 1. Each node has a grounded unit value capacitor, and nodes i and j ar... |

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Citation Context ... be connected. The FMMP problem is closely related to several problems recently explored by the authors, including the discrete-time counterpart [4, 3], distributed algorithms for resource allocation =-=[19],-=- and distributed algorithms for averaging [20]. We will also see, in §4.7, a somewhat surprising connection to recent work in machine learning, on manifold unfolding [16, 17]. 2 Interpretations In th... |

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Citation Context ...0 (since the graph is connected). Finally, we note that if w ∗ is optimal, the subgraph associated with w ∗ ij > 0 must be connected. The absolute algebraic connectivity problem, described by Fiedler =-=[9, 10]-=-, is a special case of the FMMP problem, with dij =1/ √ m for all {i, j} ∈E. The absolute algebraic connectivity problem is a very interesting topic in algebraic graph theory; the fact that it is a sp... |

4 |
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Citation Context ..., e.g., [5, 14]. One immediate consequence of the SDP formulation (3) is that we can numerically solve the FMMP problem efficiently, using standard algorithms for SDP (e.g., [12, 2, 1, 9, 7, 13]; see =-=[8]-=- for a comprehensive list of current SDP software). For m no more than 1000 or so, interior-point methods can be used to solve the FMMP problem in a few minutes (or much less) on a small desktop compu... |

4 |
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Citation Context ...to a standard form; see, e.g., [5, 26]. One immediate consequence of the SDP formulation (3) is that we can numerically solve the FMMP problem efficiently, using standard algorithms for an SDP (e.g., =-=[23, 2, 1, 11, 24]-=-; see [13] for a comprehensive list of current SDP software). For m no more than 1000 or so, interior-point methods can be used to solve the FMMP problem in a minute or so (or much less) on a small pe... |

2 |
Fastest mixing Markov chain on a path,” The American Mathematical Monthly
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Citation Context ...s optimal, the subgraph associated with w ∗ ij > 0 must be connected. The FMMP problem is closely related to several problems recently explored by the authors, including the discrete-time counterpar=-=t [4, 3],-=- distributed algorithms for resource allocation [19], and distributed algorithms for averaging [20]. We will also see, in §4.7, a somewhat surprising connection to recent work in machine learning, on... |

2 |
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Citation Context ...on both problems, and allows us characterize, and, in some cases, find optimal solutions. 1 The problem The fastest mixing Markov chain problem was proposed and studied by Boyd, Diaconis, and Xiao in =-=[4]-=-. In that problem the mixing rate of a discrete-time Markov chain on a graph was optimized over the set of transition probabilities on the edges of a given graph. In this paper, we discuss its continu... |

2 |
Fast linear iterations for distributed averaging. Accepted for publication
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Citation Context ...ted to several problems recently explored by the authors, including the discrete-time counterpart [4, 3], distributed algorithms for resource allocation [19], and distributed algorithms for averaging =-=[20].-=- We will also see, in §4.7, a somewhat surprising connection to recent work in machine learning, on manifold unfolding [16, 17]. 2 Interpretations In this section we given some simple physical interp... |

1 |
SBmethod—a C++ Implementation of the Spectral Bundle Method, Tech. Report ZIB-Report 00-35, Konrad-Zuse-Zentrum für Informationstechnik
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Citation Context ...lems can be solved using subgradient-type methods and by exploiting the problem structure; see, e.g., [5, 4]. More sophisticated methods for solving large scale problems of this type are discussed in =-=[22, 14, 20]-=-. However, we won’t pursue numerical methods for the FMMP problem in this paper. (4) 3.4. The Dual Problem. The dual of the SDP (3) is (the SDP) maximize Tr(I − (1/n)11 T )X subject to Xii + Xjj − Xij... |