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Kernel Cuts: Kernel and Spectral Clustering Meet Regularization

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Abstract

This work bridges the gap between two popular methodologies for data partitioning: kernel clustering and regularization-based segmentation. While addressing closely related practical problems, these general methodologies may seem very different based on how they are covered in the literature. The differences may show up in motivation, formulation, and optimization, e.g. spectral relaxation versus max-flow. We explain how regularization and kernel clustering can work together and why this is useful. Our joint energy combines standard regularization, e.g. MRF potentials, and kernel clustering criteria like normalized cut. Complementarity of such terms is demonstrated in many applications using our bound optimization Kernel Cut algorithm for the joint energy (code is publicly available). While detailing combinatorial move-making, our main focus are new linear kernel and spectral bounds for kernel clustering criteria allowing their integration with any regularization objectives with existing discrete or continuous solvers.

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Notes

  1. Iverson brackets \([\cdot ]\) enclosing a logical proposition, e.g. \([S_p=k]\), return 1 or 0 depending on true or false value of this proposition.

  2. Here spectral bound means spectral auxiliary function in the context of optimization, not to be confused with bounds on eigenvalues.

  3. The name probabilistic K-means in the general clustering context was coined by Kearns et al. (1997). They formulated (14) after representing distortion energy (13) as ML fitting of Gibbs models \(\frac{1}{Z_d}e^{-\Vert x-m\Vert _d}\) for arbitrary integrable metrics.

  4. These can be isometrically embedded into a Hilbert space Hein et al. (2004).

  5. This could be a name for some clustering techniques constructing explicit embeddings (Belkin and Niyogi 2003; Yu et al. 2015) instead of working with pairwise affinities/kernels.

  6. Mercer’s theorem is a similar eigen decomposition for continuous p.d. kernels k(xy) giving infinite-dimensional Hilbert embedding \(\phi (x)\). Discrete kernel embedding \(\phi _p\equiv \phi (I_p)\) in Sect. 4.1 (60) has finite dimension \(|\varOmega |\), which is still much higher than the dimension of points \(I_p\), e.g. \({\mathcal {R}}^3\) for colors. Section 4.1 also shows lower dimensional embeddings \(\tilde{\phi }_p\) approximating isometry (20).

  7. Similar to fuzzy K-means in MacKay (2003), Duda et al. (2001a), Rose (1998) if extended to \(k\)KM.

  8. Function \({{\mathop {e}\limits ^{\frown }}}\) and gradient \({\nabla {\mathop {e}\limits ^{\frown }}}\) are defined only at non-zero indicators \(X_t\) where \(w' X_t >0\). We can formally extend \({{\mathop {e}\limits ^{\frown }}}\) to \(X=\mathbf{0}\) and make the bound \(T_t\) work for \({{\mathop {e}\limits ^{\frown }}}\) at \(X_t=\mathbf{0}\) with some supergradient. However, \(X_t=0\) is not a problem in practice since it corresponds to an empty segment.

  9. If k is given as a continuous kernel \(k(x,y):{\mathcal {R}}^N\times {\mathcal {R}}^N \rightarrow {\mathcal {R}}\) matrix \(\mathcal{K}\) is its restriction to finite data set \(\{I_p|p\in \varOmega \} \subset {\mathcal {R}}^N \).

  10. Without \(K\!N\!N\) or other special kernel accelerations.

  11. KM procedure (23) (weighted version) is not practical for objective (32) for points \(\phi _p\) in \({\mathcal R}^{|\varOmega |}\). Instead, Dhillon et al. (2004) later suggested pairwiseKM procedure (24) (weighted version) using kernel \(\mathcal{K}_{pq}\equiv \langle \phi _p,\phi _q\rangle \).

  12. The smoothness weights for different energies are not directly comparable; Fig. 19 shows all the curves for better visualization.

  13. We found that for the GrabCut database adding texture features to RGB does not improve the results.

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Acknowledgements

We greatly thank Carl Olsson (Lund University, Sweden) for hours of stimulating discussions, as well as for detailed feedback and valuable recommendations at different stages of our work. We appreciate his tremendous patience when our thoughts were much more confusing than they might be now. Ivan Stelmakh (PhysTech, Russia) also gave helpful feedback on our draft and caught several errors. Anders Eriksson (Lund University, Sweden) helped with related work on NC with constraints. We also thank Jianbo Shi (UPenn, USA) for his excellent spectral-relaxation optimization code for NC.

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Tang, M., Marin, D., Ben Ayed, I. et al. Kernel Cuts: Kernel and Spectral Clustering Meet Regularization. Int J Comput Vis 127, 477–511 (2019). https://doi.org/10.1007/s11263-018-1115-1

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