Learning Graphs With Monotone Topology Properties and Multiple Connected Components

Abstract

Recent papers have formulated the problem of learning graphs from data as an inverse covariance estimation with graph Laplacian constraints. While such problems are convex, existing methods cannot guarantee that solutions will have specific graph topology properties (e.g., being a tree or k-partite), which are desirable for some applications. In fact, the problem of learning a graph with given topology properties, e.g., finding the k-partite graph that best matches the data, is in general nonconvex. In this paper, we develop novel theoretical results that provide performance guarantees for an approach to solve these problems. Our solution decomposes this graph learning problem into two sub-problems, for which efficient solutions are known. Specifically, a graph topology inference (GTI) step is employed to select a feasible graph topology, i.e., one having the desired topology property. Then, a graph weight estimation (GWE) step is performed by solving a generalized graph Laplacian estimation problem, where edges are constrained by the topology found in the GTI step. Our main result is a bound on the error of the GWE step as a function of the error in the GTI step. This error bound indicates that the GTI step should be solved using an algorithm that approximates the similarity matrix (which in general corresponds to a complete weighted graph) by another matrix whose entries have been thresholded to zero to have the desired type of graph topology. The GTI stage can leverage existing methods (e.g., state of the art approaches for graph coloring) which are typically based on minimizing the total weight of removed edges. Since the GWE stage is formulated as an inverse covariance estimation problem with linear constraints, it can be solved using existing convex optimization methods. We demonstrate that our two step approach can achieve good results for both synthetic and texture image data.

Connecting the Dots: Identifying Network Structure via Graph Signal Processing

“…Interestingly, this problem can be tackled using a convexification technique known as semidefinite relaxation [31]. More precisely, (41) can be recast as a Boolean quadratic program and then equivalently expressed as a semidefinite program subject to a rank constraint. Dropping this latter constraint, one arrives at a convex relaxation with provable approximation bounds; see [56] for full algorithmic, complexity, and performance details.…”

“…Gaussian models are ubiquitous in machine learning and statistical analysis of real-valued network data, because of their widespread applicability and analytical tractability. Most recent advances to GMRF model selection have explored ways of incorporating Laplacian or otherwise graph topological constraints in the precision matrix estimation task [11], [21], [41], [43]. These approaches are well suited to settings when prior information dictates that e.g., feasible graphs should have a tree structure or edge weights should be positive given the physics of the problem.…”

“…The graph recovered by network deconvolution [13] is illustrated in the bottom left corner of Figure 5 (a), whereas the one recovered using the approach in (32) (with the sparsity-promoting x M and x H . At the macro or large timescale, we average the decomposed signals x L for all sample points within each scanning session with different sequence type, and evaluate the variance of the magnitudes of the averaged signals across all the scanning sessions and sequence types [40], [41]. For the 6 week experiment, session with different sequence types.…”

“…The evaluated variance then averaged across all participants of the experiment of intere Figure 8 displays the variance of the decomposed sign x L , x M and x H at two different temporal scales of the t experiments. For the 6 week dataset, 3 session-sequence co the macro or large timescale, we average the als x L for all sample points within each scanning ferent sequence type, and evaluate the variance s of the averaged signals across all the scanning ence types [40], [41]. For the 6 week experiment, ning sessions and 3 different sequence types, so session with different sequence types.…”

Learning Graphs From Data: A Signal Representation Perspective

“…Third, although the current lines of work reviewed in this survey mainly focus on the signal representation, it is also possible to put constraints directly on the learned graphs by enforcing certain graph properties that go beyond the common choice of sparsity, which has been adopted explicitly in the optimization problems in many existing methods such as the ones in [15], [25], [42], [45], [46], [55], [62]. One example is the work in [82], where the authors have proposed to infer graphs with monotone topology properties.…”