Graph-regularized semi-supervised learning has been used effectively for classification when (i) instances are connected through a graph, and (ii) labeled data is scarce. If available, using multiple relations (or graphs) between the instances can improve the prediction performance. On the other hand, when these relations have varying levels of veracity and exhibit varying relevance for the task, very noisy and or irrelevant relations may deteriorate the performance. As a result, an effective weighing scheme needs to be put in place. In this work, we propose a robust and scalable approach for multi-relational graph-regularized semi-supervised classification. Under a convex optimization scheme, we simultaneously infer weights for the multiple graphs as well as a solution. We provide a careful analysis of the inferred weights, based on which we devise an algorithm that filters out irrelevant and noisy graphs and produces weights proportional to the informativeness of the remaining graphs. Moreover, the proposed method is linearly scalable w.r.t. the number of edges in the union of the multiple graphs. Through extensive experiments we show that our method yields superior results under different noise models, and under increasing number of noisy graphs and intensity of noise, as compared to a list of baselines and state-of-the-art approaches.
Mostafavi al @cite_33 @cite_29 construct the optimal composite graph by solving a linear regression problem using labeled data, later also used by Luo al @cite_37 . Different from most semi-supervised transductive methods that leverage both labeled and unlabeled data, these use only the labeled data to estimate the graph weights.
@cite_37 @cite_29 @cite_33
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In the literature, most existing graph-based semi-supervised learning methods only use the label information of observed samples in the label propagation stage, while ignoring such valuable information when learning the graph. In this paper, we argue that it is beneficial to consider the label information in the graph learning stage. Specifically, by enforcing the weight of edges between labeled samples of different classes to be zero, we explicitly incorporate the label information into the state-of-the-art graph learning methods, such as the low-rank representation (LRR), and propose a novel semi-supervised graph learning method called semi-supervised low-rank representation. This results in a convex optimization problem with linear constraints, which can be solved by the linearized alternating direction method. Though we take LRR as an example, our proposed method is in fact very general and can be applied to any self-representation graph learning methods. Experiment results on both synthetic and real data sets demonstrate that the proposed graph learning method can better capture the global geometric structure of the data, and therefore is more effective for semi-supervised learning tasks. This paper presents a semi-supervised graph-based method for the classification of hyperspectral images. The method is designed to handle the special characteristics of hyperspectral images, namely, high-input dimension of pixels, low number of labeled samples, and spatial variability of the spectral signature. To alleviate these problems, the method incorporates three ingredients, respectively. First, being a kernel-based method, it combats the curse of dimensionality efficiently. Second, following a semi-supervised approach, it exploits the wealth of unlabeled samples in the image, and naturally gives relative importance to the labeled ones through a graph-based methodology. Finally, it incorporates contextual information through a full family of composite kernels. Noting that the graph method relies on inverting a huge kernel matrix formed by both labeled and unlabeled samples, we originally introduce the Nystro umlm method in the formulation to speed up the classification process. The presented semi-supervised-graph-based method is compared to state-of-the-art support vector machines in the classification of hyperspectral data. The proposed method produces better classification maps, which capture the intrinsic structure collectively revealed by labeled and unlabeled points. Good and stable accuracy is produced in ill-posed classification problems (high dimensional spaces and low number of labeled samples). In addition, the introduction of the composite-kernel framework drastically improves results, and the new fast formulation ranks almost linearly in the computational cost, rather than cubic as in the original method, thus allowing the use of this method in remote-sensing applications. Problems related to computing optimal paths have been abundant in computer science since its emergence as a field. Yet for a large number of such problems we still do not know whether the state-of-the-art algorithms are the best possible. A notable example of this phenomenon is the all pairs shortest paths problem in a directed graph with real edge weights. The best algorithm (modulo small polylogarithmic improvements) for this problem runs in cubic time, a running time known since the 1960s (by Floyd and Warshall). Our grasp of many such fundamental algorithmic questions is far from optimal, and the major goal of this thesis is to bring some new insights into efficiently solving path problems in graphs. We focus on several path problems optimizing different measures: shortest paths, maximum bottleneck paths, minimum nondecreasing paths, and various extensions. For the all-pairs versions of these path problems we use an algebraic approach. We obtain improved algorithms using reductions to fast matrix multiplication. For maximum bottleneck paths and minimum nondecreasing paths we are the first to break the cubic barrier, obtaining truly subcubic strongly polynomial algorithms. We also consider a nonalgebraic, combinatorial approach, which is considered more efficient in practice compared to methods based on fast matrix multiplication. We present a combinatorial data structure that maintains a matrix so that products with given sparse vectors can be computed efficiently. This allows us to obtain good running times for path problems in unweighted sparse graphs. This thesis also gives algorithms for some single source path problems. We obtain the first linear time algorithm for the single source minimum nondecreasing paths problem. We give some extensions to this, including an algorithm to find cheapest minimum nondecreasing paths. Besides finding optimal paths, we consider the related problem of finding optimal cycles. In particular, we focus on the problem of finding in a weighted graph a triangle of maximum weight sum. We obtain the first truly subcubic algorithm for finding a maximum weight triangle in a node-weighted graph. We also present algorithms for the edge-weighted case. These algorithms immediately imply good algorithms for finding maximum weight k-cliques, or arbitrary maximum weight pattern subgraphs of fixed size. Active and semi-supervised learning are important techniques when labeled data are scarce. We combine the two under a Gaussian random field model. Labeled and unlabeled data are represented as vertices in a weighted graph, with edge weights encoding the similarity between instances. The semi-supervised learning problem is then formulated in terms of a Gaussian random field on this graph, the mean of which is characterized in terms of harmonic functions. Active learning is performed on top of the semisupervised learning scheme by greedily selecting queries from the unlabeled data to minimize the estimated expected classification error (risk); in the case of Gaussian fields the risk is efficiently computed using matrix methods. We present experimental results on synthetic data, handwritten digit recognition, and text classification tasks. The active learning scheme requires a much smaller number of queries to achieve high accuracy compared with random query selection.