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Register a new userKernel Cuts: MRF meets Kernel & Spectral Clustering
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Added by
Yuri Boykov
Publication date
2015
fields
Informatics Engineering
and research's language is
English
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We propose a new segmentation model combining common regularization energies, e.g. Markov Random Field (MRF) potentials, and standard pairwise clustering criteria like Normalized Cut (NC), average association (AA), etc. These clustering and regularization models are widely used in machine learning and computer vision, but they were not combined before due to significant differences in the corresponding optimization, e.g. spectral relaxation and combinatorial max-flow techniques. On the one hand, we show that many common applications using MRF segmentation energies can benefit from a high-order NC term, e.g. enforcing balanced clustering of arbitrary high-dimensional image features combining color, texture, location, depth, motion, etc. On the other hand, standard clustering applications can benefit from an inclusion of common pairwise or higher-order MRF constraints, e.g. edge alignment, bin-consistency, label cost, etc. To address joint energies like NC+MRF, we propose efficient Kernel Cut algorithms based on bound optimization. While focusing on graph cut and move-making techniques, our new unary (linear) kernel and spectral bound formulations for common pairwise clustering criteria allow to integrate them with any regularization functionals with existing discrete or continuous solvers.
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We propose a deep learning approach for discovering kernels tailored to identifying clusters over sample data. Our neural network produces sample embeddings that are motivated by--and are at least as expressive as--spectral clustering. Our training objective, based on the Hilbert Schmidt Information Criterion, can be optimized via gradient adaptations on the Stiefel manifold, leading to significant acceleration over spectral methods relying on eigendecompositions. Finally, our trained embedding can be directly applied to out-of-sample data. We show experimentally that our approach outperforms several state-of-the-art deep clustering methods, as well as traditional approaches such as $k$-means and spectral clustering over a broad array of real-life and synthetic datasets.
Kernel methods are popular in clustering due to their generality and discriminating power. However, we show that many kernel clustering criteria have density biases theoretically explaining some practically significant artifacts empirically observed in the past. For example, we provide conditions and formally prove the density mode isolation bias in kernel K-means for a common class of kernels. We call it Breimans bias due to its similarity to the histogram mode isolation previously discovered by Breiman in decision tree learning with Gini impurity. We also extend our analysis to other popular kernel clustering methods, e.g. average/normalized cut or dominant sets, where density biases can take different forms. For example, splitting isolated points by cut-based criteria is essentially the sparsest subset bias, which is the opposite of the density mode bias. Our findings suggest that a principled solution for density biases in kernel clustering should directly address data inhomogeneity. We show that density equalization can be implicitly achieved using either locally adaptive weights or locally adaptive kernels. Moreover, density equalization makes many popular kernel clustering objectives equivalent. Our synthetic and real data experiments illustrate density biases and proposed solutions. We anticipate that theoretical understanding of kernel clustering limitations and their principled solutions will be important for a broad spectrum of data analysis applications across the disciplines.
An important goal in visual recognition is to devise image representations that are invariant to particular transformations. In this paper, we address this goal with a new type of convolutional neural network (CNN) whose invariance is encoded by a reproducing kernel. Unlike traditional approaches where neural networks are learned either to represent data or for solving a classification task, our network learns to approximate the kernel feature map on training data. Such an approach enjoys several benefits over classical ones. First, by teaching CNNs to be invariant, we obtain simple network architectures that achieve a similar accuracy to more complex ones, while being easy to train and robust to overfitting. Second, we bridge a gap between the neural network literature and kernels, which are natural tools to model invariance. We evaluate our methodology on visual recognition tasks where CNNs have proven to perform well, e.g., digit recognition with the MNIST dataset, and the more challenging CIFAR-10 and STL-10 datasets, where our accuracy is competitive with the state of the art.
These are notes on the method of normalized graph cuts and its applications to graph clustering. I provide a fairly thorough treatment of this deeply original method due to Shi and Malik, including complete proofs. I include the necessary background on graphs and graph Laplacians. I then explain in detail how the eigenvectors of the graph Laplacian can be used to draw a graph. This is an attractive application of graph Laplacians. The main thrust of this paper is the method of normalized cuts. I give a detailed account for K = 2 clusters, and also for K > 2 clusters, based on the work of Yu and Shi. Three points that do not appear to have been clearly articulated before are elaborated: 1. The solutions of the main optimization problem should be viewed as tuples in the K-fold cartesian product of projective space RP^{N-1}. 2. When K > 2, the solutions of the relaxed problem should be viewed as elements of the Grassmannian G(K,N). 3. Two possible Riemannian distances are available to compare the closeness of solutions: (a) The distance on (RP^{N-1})^K. (b) The distance on the Grassmannian. I also clarify what should be the necessary and sufficient conditions for a matrix to represent a partition of the vertices of a graph to be clustered.
Multiple datasets containing different types of features may be available for a given task. For instance, users profiles can be used to group users for recommendation systems. In addition, a model can also use users historical behaviors and credit history to group users. Each dataset contains different information and suffices for learning. A number of clustering algorithms on multiple datasets were proposed during the past few years. These algorithms assume that at least one dataset is complete. So far as we know, all the previous methods will not be applicable if there is no complete dataset available. However, in reality, there are many situations where no dataset is complete. As in building a recommendation system, some new users may not have a profile or historical behaviors, while some may not have a credit history. Hence, no available dataset is complete. In order to solve this problem, we propose an approach called Collective Kernel Learning to infer hidden sample similarity from multiple incomplete datasets. The idea is to collectively completes the kernel matrices of incomplete datasets by optimizing the alignment of the shared instances of the datasets. Furthermore, a clustering algorithm is proposed based on the kernel matrix. The experiments on both synthetic and real datasets demonstrate the effectiveness of the proposed approach. The proposed clustering algorithm outperforms the comparison algorithms by as much as two times in normalized mutual information.
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