Abstract
We extend our recent work on scalable spectral clustering with cosine similarity (ICPR’18) to other kinds of similarity functions, in particular, the Gaussian RBF. In the previous work, we showed that for sparse or low-dimensional data, spectral clustering with the cosine similarity can be implemented directly through efficient operations on the data matrix such as elementwise manipulation, matrix-vector multiplication and low-rank SVD, thus completely avoiding the weight matrix. For other similarity functions, we present an embedding-based approach that uses a small set of landmark points to convert the given data into sparse feature vectors and then applies the scalable computing framework for the cosine similarity. Our algorithm is simple to implement, has clear interpretations, and naturally incorporates an outliers removal procedure. Preliminary results show that our proposed algorithm yields higher accuracy than existing scalable algorithms while running fast.
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Notes
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Available at http://yann.lecun.com/exdb/mnist/.
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Code available at http://www.cad.zju.edu.cn/home/dengcai/Data/Clustering.html.
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This empirical rule is derived as \(\ell =\frac{1}{2}\cdot \sqrt{\frac{n}{k}} \cdot k = \frac{1}{2}\sqrt{nk}\), with the intuition that the value of \(\ell \) should be proportional to both the (average) cluster size and number of clusters. For the data sets in Table 1, such an \(\ell \) is always a few hundred.
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Acknowledgments
We thank the anonymous reviewers for their helpful feedback. This work was motivated by a project sponsored by Verizon Wireless, which had the goal of grouping customers based on similar profile characteristics. G. Chen was supported by the Simons Foundation Collaboration Grant for Mathematicians.
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Chen, G. (2018). A Scalable Spectral Clustering Algorithm Based on Landmark-Embedding and Cosine Similarity. In: Bai, X., Hancock, E., Ho, T., Wilson, R., Biggio, B., Robles-Kelly, A. (eds) Structural, Syntactic, and Statistical Pattern Recognition. S+SSPR 2018. Lecture Notes in Computer Science(), vol 11004. Springer, Cham. https://doi.org/10.1007/978-3-319-97785-0_6
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