Most of real-world graphs are dynamic , i.e., they change over time. However, while the regression problem has been studied for static graphs, it has not been investigated for dynamic graphs, yet. In the current paper, first we present the notion of update-efficient matrix embedding that defines the conditions sufficient for a matrix embedding to be used for the dynamic graph regression problem, efficiently. We also show that some of the standard matrix embeddings, e.g., the (weighted) adjacency matrix, satisfy these conditions. Then, we prove that given an @math update-efficient matrix embedding, after an update operation in the graph, the optimal solution of the graph regression problem for the revised graph can be computed in @math time. In particular, using the (weighted) adjacency matrix as the matrix embedding of @math , it takes @math time to update the optimal solution, where @math is the number of nodes of the revised graph. To the best of our knowledge, this is the first result on updating the solution of the graph regression problem, in a time considerably less than the time of computing the solution from the scratch. Finally, we study a generalization of the dynamic graph regression problem and show that it can be solved in @math space.
@cite_0 presented representative multi-dimensional view smoothers on graphs, that are based on graph-based transductive learning @cite_11 . @cite_4 proposed a family of learning algorithms based on a new form of regularization so that some of transductive graph learning algorithms can be obtained as special cases. @cite_13 extended a model for nonparametric regression of nodes of an static graph, where distance between estimate and observation is measured at nodes by @math norm, and roughness is penalized on edges in the @math norm. In the literature, there also exist iterative methods, such as gradient descent and its improvements, that try to solve some of (static graph) learning problems without explicitly computing the Moore-Penrose pseudoinverse. However, the number of iterations they need is usually proportional to the condition number of the matrix embedding of the graph (e.g., the Laplacian matrix), which might be a (linear) function of @math .
@cite_0 @cite_4 @cite_13 @cite_11
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The ‘Signal plus Noise’ model for nonparametric regression can be extended to the case of observations taken at the vertices of a graph. This model includes many familiar regression problems. This article discusses the use of the edges of a graph to measure roughness in penalized regression. Distance between estimate and observation is measured at every vertex in the L2 norm, and roughness is penalized on every edge in the L1 norm. Thus the ideas of total variation penalization can be extended to a graph. The resulting minimization problem presents special computational challenges, so we describe a new and fast algorithm and demonstrate its use with examples. The examples include image analysis, a simulation applicable to discrete spatial variation, and classification. In our examples, penalized regression improves upon kernel smoothing in terms of identifying local extreme values on planar graphs. In all examples we use fully automatic procedures for setting the smoothing parameters. Supplemental materia... We present a new multi-class graph-based transduction algorithm. Examples are associated with vertices in an undirected weighted graph and edge weights correspond to a similarity measure between examples. Typical algorithms in such a setting perform label propagation between neighbours, ignoring the quality, or estimated quality, in the labeling of various nodes. We introduce an additional quantity of confidence in label assignments, and learn them jointly with the weights, while using them to dynamically tune the influence of each vertex on its neighbours. We cast learning as a convex optimization problem, and derive an efficient iterative algorithm for solving it. Empirical evaluations on seven NLP data sets demonstrate our algorithm improves over other state-of-the-art graph-based transduction algorithms. We consider estimation of worker skills from worker-task interaction data (with unknown labels) for the single-coin crowd-sourcing binary classification model in symmetric noise. We define the (worker) interaction graph whose nodes are workers and an edge between two nodes indicates whether or not the two workers participated in a common task. We show that skills are asymptotically identifiable if and only if an appropriate limiting version of the interaction graph is irreducible and has odd-cycles. We then formulate a weighted rank-one optimization problem to estimate skills based on observations on an irreducible, aperiodic interaction graph. We propose a gradient descent scheme and show that for such interaction graphs estimates converge asymptotically to the global minimum. We characterize noise robustness of the gradient scheme in terms of spectral properties of signless Laplacians of the interaction graph. We then demonstrate that a plug-in estimator based on the estimated skills achieves state-of-art performance on a number of real-world datasets. Our results have implications for rank-one matrix completion problem in that gradient descent can provably recover @math rank-one matrices based on @math off-diagonal observations of a connected graph with a single odd-cycle. We propose a new distributed algorithm based on alternating direction method of multipliers (ADMM) to minimize sum of locally known convex functions using communication over a network. This optimization problem emerges in many applications in distributed machine learning and statistical estimation. Our algorithm allows for a general choice of the communication weight matrix, which is used to combine the iterates at different nodes. We show that when functions are convex, both the objective function values and the feasibility violation converge with rate @math , where @math is the number of iterations. We then show that when functions are strongly convex and have Lipschitz continuous gradients, the sequence generated by our algorithm converges linearly to the optimal solution. In particular, an @math -optimal solution can be computed with @math iterations, where @math is the condition number of the problem. Our analysis highlights the effect of network and communication weights on the convergence rate through degrees of the nodes, the smallest nonzero eigenvalue, and operator norm of the communication matrix.