Laplacian Embedding (LE) is a powerful method to reveal the intrinsic geometry of high-dimensional data by using graphs. Imposing the orthogonal and nonnegative constraints onto the LE objective has proved to be effective to avoid degenerate and negative solutions, which, though, are challenging to achieve simultaneously because they are nonlinear and nonconvex. In addition, recent studies have shown that using the p-th order of the L2-norm distances in LE can find the best solution for clustering and promote the robustness of the embedding model against outliers, although this makes the optimization objective nonsmooth and difficult to efficiently solve in general. In this work, we study LE that uses the p-th order of the L2-norm distances and satisfies both orthogonal and nonnegative constraints. We introduce a novel smoothed iterative reweighted method to tackle this challenging optimization problem and rigorously analyze its convergence. We demonstrate the effectiveness and potential of our proposed method by extensive empirical studies on both synthetic and real data sets.
Different to traditional clustering methods that deal with one single type of data, High-Order Co- Clustering (HOCC) aims to cluster multiple types of data simultaneously by utilizing the inter- or/and intra-type relationships across different data types. In existing HOCC methods, data points routinely enter the objective functions with squared residual errors. As a result, outlying data samples can dominate the objective functions, which may lead to incorrect clustering results. Moreover, existing methods usually suffer from soft clustering, where the probabilities to different groups can be very close. In this paper, we propose an L1 -norm symmetric nonnegative matrix tri-factorization method to solve the HOCC problem. Due to the orthogonal constraints and the symmetric L1 -norm formulation in our new objective, conventional auxiliary function approach no longer works. Thus we derive the solution algorithm using the alternating direction method of multipliers. Extensive experiments have been conducted on a real world data set, in which promising empirical results, including less time consumption, strictly orthogonal membership matrix, lower local minima etc., have demonstrated the effectiveness of our proposed method.
more » « less- Award ID(s):
- 1652943
- PAR ID:
- 10084447
- Date Published:
- Journal Name:
- Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence
- Page Range / eLocation ID:
- 2454 to 2460
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Bringmann, Karl ; Grohe, Martin ; Puppis, Gabriele ; Svensson, Ola (Ed.)This paper considers correlation clustering on unweighted complete graphs. We give a combinatorial algorithm that returns a single clustering solution that is simultaneously O(1)-approximate for all 𝓁_p-norms of the disagreement vector; in other words, a combinatorial O(1)-approximation of the all-norms objective for correlation clustering. This is the first proof that minimal sacrifice is needed in order to optimize different norms of the disagreement vector. In addition, our algorithm is the first combinatorial approximation algorithm for the 𝓁₂-norm objective, and more generally the first combinatorial algorithm for the 𝓁_p-norm objective when 1 < p < ∞. It is also faster than all previous algorithms that minimize the 𝓁_p-norm of the disagreement vector, with run-time O(n^ω), where O(n^ω) is the time for matrix multiplication on n × n matrices. When the maximum positive degree in the graph is at most Δ, this can be improved to a run-time of O(nΔ² log n).more » « less
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.24963/ijcai.2018/340
