Introduction
Main Contributions
Specifically, multiplicities in the spectrum of the matrix [AΓ −γIk]k→n ∈ Sn can be increased by suitable choices of γ, which in turn gives the normal cone NSH([AΓ−γIk]k→n)([AΓ− ) makes γIk]k→n) larger. Thus, a notable feature of the relaxation R+2(Λ,E) is that it is of the same size as R2(Λ,E), despite being a more strictly convex outer approximation in general to VR(Siep ) provided.
Finding Planted Subgraphs with Few Eigenvalues using the
Introduction
In section 2.2 we study the geometric properties of the Schur-Horn orthotope as they relate to the optimal conditions of the Schur-Horn relaxation. Specifically, the success of the Schur-Horn relaxation (P) depends on the existence of a suitable eigenspace E ⊂RkofAΓ.
Geometric Properties of the Schur-Horn Orbitope
Based on this characterization of the optimality conditions, the success of the Schur–Horn relaxation depends on the existence of a suitable double variable M ∈ Sn that satisfies two conditions. Furthermore, A is strictly spectrally comonotonic with B if and only if A and B are simultaneously diagonalizable and each of the inequalities (2.4) strictly holds.
Recovering Subgraphs Planted in Erdős-Rényi Random Graphs 18
In the numerical experiments presented next, we use the description (2.19) of the Schur-Horn orbit. We investigate the performance of the Schur-Horn relaxation (P) in planted subgraph problems with the four planted subgraphs listed in Figure 2.6.
Discussion
This leads to useful lower bounds on the editing distance if one of the graphs is "properly structured". In Section 3.2, we theoretically investigate the effectiveness of the Schur-Horn orbitope as an invariant convex set in providing lower bounds on the ordering distance of a graph over (P). We have shown the utility of the Schur-Horn orbitope as a set of constraints in (P) in obtaining bounds on the graph ordering distance between graphs G and G0.
We report the average ratio of the calculated lower bound on the graph modification distance to the number of modification operations in Figure 3.7. In general, the best known limits are on the size of certificates of infeasibility – that is, the degrees of the.
Convex Graph Invariant Relaxations For Graph Edit Distance 40
Theoretical Guarantees for the Schur-Horn Orbitope Constraint 48
This observation, together with several properties of the eigenspaces of G, plays a prominent role in the analysis in this section. We present here the statements of our main theoretical results regarding the performance of the Schur-Horn orthotope as a constraint captured in (P). The reason is that the quantity ξ(α, d,G) is a graph parameter (for any fixed α, d) and does not depend on a specific labeling of the vertices of G.
Pm ∈Sn indicates the projection matrices on the associated eigenspaces indexed in descending order of the corresponding eigenvalues. Let G be a vertex-transitive graph of n vertices consisting of m distinct eigenvalues, and let κ denote the multiplicity of the eigenvalue with the next highest multiplicity.
Numerical Illustrations with Invariants based on Stable Sets and
The function f(A) described in section 3.1 is an efficiently computable lower bound on the inverse of the stability number for a graph, and it is also a concave graph invariant. We thus investigate the utility of the CIS(G) constraint in settings where the edits consist mainly of edge deletions. As expected, the relaxation based on the constraint CIS(G) gives the best lower bounds of the three approaches.
Consequently, we should only expect the constraintCMC(G) to yield potentially useful lower bounds on graph edit distance in settings where most operations on a graph G correspond to edge additions. Since the operations mainly consist of edge additions, the constraint based on the Motzkin-Straus relaxation of the inverse of the stability number performs poorly.
Experiments with Real Data
Because of this larger size, calculating the exact average pairwise graph editing distance for the PAH dataset is prohibitively expensive. Indeed, to our knowledge, the exact average pairwise graph editing distance for the PAH dataset is unknown to this date [17]. Therefore, obtaining guaranteed lower bounds on the average graph editing distance for the PAH data set is particularly useful as a way to compare with known average upper bounds.
Specifically, for the Alkane dataset, the average lower bound is 10.72 obtained using our convex programming framework, and the exact value of the average graph editing distance is 15.3 (which is obtained using combinatorial approaches). In particular, the best-known upper limit of the average PAH graph editing distance is 29.8 [38].
Discussion
This note describes promising experimental results of the performance of semidefinite relaxations for affine IEPs. However, a strong alternative to the system −1 ∈ I2+ + Σ leads to a convex outer approximation R+2(Λ,E) of VR(Siep), which is generally tighter than R2(Λ,E); in addition to all constraints defining R2(Λ,E) in (4.8), the set R+2(Λ,E) consists of additional constraints Pn. In each of the two settings, we maximized 1000 random linear functionals over R2(Λ,E) and obtained the element VR(Siep) in all cases.
Here, u(x) and p(x) are functions, and λ is a parameter that is an eigenvalue of the system. The optimization problem (I2) is feasible on a potentially larger set than the optimization problem (I1), but its unique optimal value is still reached by the same matrix – the adjacency matrix of the planted clique.
Sum of Squares Based Convex Relaxations for Inverse Eigen-
Introduction
At one end of the spectrum, there have been several attempts aimed at providing necessary and sufficient conditions for the existence of a solution to a given IEP. First, our framework is applicable to general affine IEPs, while some of the previous convex approaches are only useful for some structured problem instances; see Section 4.3 for the wide range of examples in which we apply our methods. From a dual perspective, these relaxations can also be seen as providing a sequence of outer convex approximations R1(Λ,E) ⊇ R2(Λ,E conv (VR(Siep)), leading to a natural heuristic for trying to get solutions of the Siep system.
Therefore, although we describe the general mechanism by which semidefinite relaxations of increasing size can be generated, we limit our attention in numerical experiments to the performance of the relaxations R1(Λ,E) and R2(Λ,E). Our system (4.1) for the affine IEP can be seen as a matrix analog of those arising in the literature on combinatorial problems, since the idempotency constraints Zi2−Zi = 0 represent a generalization of the scalar Boolean constraints x2i −xi = 0.
Semidefinite Relaxations for Affine IEPs
In the search for impossibility certificates of the form −1 = p+q, p ∈ Σ, q ∈ I, it can be checked that without losing generality the search for p can also be limited to the sum of squares of polynomials of limited degree; . Thus, the search on a limited family of impossibility certificates through the degree restriction of the coefficients of the elements of hf1,. The elements of the truncated ideal I1 can be related to the above problem through the relations1 =−A, h(i)2 =−di, h(i)3 =−Bii, h(k)4 =−ξk, and then by observed that the constraints in (4.4) are equivalent to checking this.
From the results in [4, 31], we have that if η is the expected value of the square of the Euclidean distance from a Gaussian random matrix to the normal cone at X. The resulting set I2 consists of polynomials of at most two , and therefore we can restrict our attention to elements of Σ of at most two in the search for infeasibility certificates of the form −1∈ I2+ Σ.
Numerical Illustrations
In the first set of results, we compare the relative strength of the two relaxations described in Section 4.2 in certifying impossibility, or from a dual point of view, in approximating conv(VR(Siep)). Here, the dimension of the solution set VR(Siep) is at most three, and the feasibility regions corresponding to R1(Λ,E) and R2(Λ,E) in Figures 4.1c and 4.1d represent two-dimensional projections (on the (X11, X22) plane of S3) of these sets. With this approach, we prove that the octahedral graph with 6 nodes and 12 edges (shown in Figure 4.2a) is not contained as an induced subgraph in any of the larger graphs shown in Figure 4.2b (of 20 nodes with 44 edges ) and Figure 4.2c (on 15 nodes with 38 edges).
1 ∈ I1 + Σ, and there is a form −1 ∈ I2+ + Σ, which confirms that the octahedral graph is again not an induced subgraph. Our first convex relaxation confirms that the octahedral graph is not an induced subgraph of the graph shown in Figure 4.2b.
Conclusions
We demonstrate that our framework can be used to find solutions or prove the infeasibility of the fundamental inverse eigenvalue problem. We determine the utility of our framework by performing numerical experiments on various instances of the inverse affine eigenvalue problem, such as the discrete inverse Sturm-Liouville problem. It may be possible to adapt some of the methods described in this thesis to answer such questions.
This is due to the additional restriction on the nuclear norm of the variable, and the fact that the nuclear norm of the k-clique (with 1's on the diagonal) is equal to k. Because the adjacency matrix of the k-clique (including the 1's on the diagonal) satisfies this constraint, it is still the unique optimal solution of the resulting problem.
Conclusions
Summary of Contributions
Convex graph invariant relaxations for the graph ordering distance The graph ordering distance is a dominant similarity metric between two graphs. Calculating the editing distance of a graph is generally NP-hard due to the basic combinatorial setting. Based on our ideas from Chapter 2, we introduce a family of tractable convex relaxations to accurately compute or provide lower bounds on the graph ordering distance between two graphs.
We present the conditions in terms of certain graph parameters under which our graph spectrum-based relaxation accurately computes the ordering distance between two graphs. We confirm the applicability of our method through numerical experiments on real and synthetic graph editing distance problems.
Future Directions
We express this problem in terms of polynomial equations and investigate convex relaxations for it that arise from previous levels of the corresponding sum-of-squares hierarchy. For example, to solve a certain "singular value inverse problem" on nonsymmetric n × n matrices, one might consider applying majorization inequalities to the eigenvalues of a 2n × 2n matrix whose eigenvalues correspond to plus and minus the desired singular value. ues. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM (JACM.
The solvability conditions for the inverse eigenvalue problems of centrosymmetric matrices. Linear algebra and its applications. In the appendix, we further investigate the relationship between our method and Ames and Vavasis's kernel norm minimization approach [5], for the special case of the plant deck clique problem.
Proof of Lemma 29
