text: Subspace clustering methods based on data self-expression have become very popular for learning from data that lie in a union of low-dimensional linear subspaces. However, the applicability of subspace clustering has been limited because practical visual data in raw form do not necessarily lie in such linear subspaces. On the other hand, while Convolutional Neural Network (ConvNet) has been demonstrated to be a powerful tool for extracting discriminative features from visual data, training such a ConvNet usually requires a large amount of labeled data, which are unavailable in subspace clustering applications. To achieve simultaneous feature learning and subspace clustering, we propose an end-to-end trainable framework, called Self-Supervised Convolutional Subspace Clustering Network (S @math ConvSCN), that combines a ConvNet module (for feature learning), a self-expression module (for subspace clustering) and a spectral clustering module (for self-supervision) into a joint optimization framework. Particularly, we introduce a dual self-supervision that exploits the output of spectral clustering to supervise the training of the feature learning module (via a classification loss) and the self-expression module (via a spectral clustering loss). Our experiments on four benchmark datasets show the effectiveness of the dual self-supervision and demonstrate superior performance of our proposed approach.
  @cite_35: In this paper, we present a deep extension of Sparse Subspace Clustering, termed Deep Sparse Subspace Clustering (DSSC). Regularized by the unit sphere distribution assumption for the learned deep features, DSSC can infer a new data affinity matrix by simultaneously satisfying the sparsity principle of SSC and the nonlinearity given by neural networks. One of the appealing advantages brought by DSSC is: when original real-world data do not meet the class-specific linear subspace distribution assumption, DSSC can employ neural networks to make the assumption valid with its hierarchical nonlinear transformations. To the best of our knowledge, this is among the first deep learning based subspace clustering methods. Extensive experiments are conducted on four real-world datasets to show the proposed DSSC is significantly superior to 12 existing methods for subspace clustering.
  @cite_8: Subspace clustering refers to the problem of segmenting high dimensional data drawn from a union of subspaces into the respective subspaces. In some applications, partial side-information to indicate “must-link” or “cannot-link” in clustering is available. This leads to the task of subspace clustering with side-information. However, in prior work the supervision value of the side-information for subspace clustering has not been fully exploited. To this end, in this paper, we present an enhanced approach for constrained subspace clustering with side-information, termed Constrained Sparse Subspace Clustering plus (CSSC+), in which the side-information is used not only in the stage of learning an affinity matrix but also in the stage of spectral clustering. Moreover, we propose to estimate clustering accuracy based on the partial side-information and theoretically justify the connection to the ground-truth clustering accuracy in terms of the Rand index. We conduct experiments on three cancer gene expression datasets to validate the effectiveness of our proposals.
  @cite_55: Most current speech recognition systems use hidden Markov models (HMMs) to deal with the temporal variability of speech and Gaussian mixture models (GMMs) to determine how well each state of each HMM fits a frame or a short window of frames of coefficients that represents the acoustic input. An alternative way to evaluate the fit is to use a feed-forward neural network that takes several frames of coefficients as input and produces posterior probabilities over HMM states as output. Deep neural networks (DNNs) that have many hidden layers and are trained using new methods have been shown to outperform GMMs on a variety of speech recognition benchmarks, sometimes by a large margin. This article provides an overview of this progress and represents the shared views of four research groups that have had recent successes in using DNNs for acoustic modeling in speech recognition.
  @cite_32: Subspace clustering aims to cluster unlabeled samples into multiple groups by implicitly seeking a subspace to fit each group. Most of existing methods are based on a shallow linear model, which may fail in handling data with nonlinear structure. In this paper, we propose a novel subspace clustering method -- deeP subspAce clusteRing with sparsiTY prior (PARTY) -- based on a new deep learning architecture. PARTY explicitly learns to progressively transform input data into nonlinear latent space and to be adaptive to the local and global subspace structure simultaneously. In particular, considering local structure, PARTY learns representation for the input data with minimal reconstruction error. Moreover, PARTY incorporates a prior sparsity information into the hidden representation learning to preserve the sparse reconstruction relation over the whole data set. To the best of our knowledge, PARTY is the first deep learning based subspace clustering method. Extensive experiments verify the effectiveness of our method.
  @cite_44: 
  @cite_63: Most existing subspace clustering methods hinge on self-expression of handcrafted representations and are unaware of potential clustering errors. Thus they perform unsatisfactorily on real data with complex underlying subspaces. To solve this issue, we propose a novel deep adversarial subspace clustering (DASC) model, which learns more favorable sample representations by deep learning for subspace clustering, and more importantly introduces adversarial learning to supervise sample representation learning and subspace clustering. Specifically, DASC consists of a subspace clustering generator and a quality-verifying discriminator, which learn against each other. The generator produces subspace estimation and sample clustering. The discriminator evaluates current clustering performance by inspecting whether the re-sampled data from estimated subspaces have consistent subspace properties, and supervises the generator to progressively improve subspace clustering. Experimental results on the handwritten recognition, face and object clustering tasks demonstrate the advantages of DASC over shallow and few deep subspace clustering models. Moreover, to our best knowledge, this is the first successful application of GAN-alike model for unsupervised subspace clustering, which also paves the way for deep learning to solve other unsupervised learning problems.
summary: Explicit Feature Space Deep learning has gained a lot of research interests due to its powerful ability to learn hierarchical features in an end-to-end trainable way @cite_55 @cite_44 . Recently, there are a few works that use techniques in deep learning for feature extraction in subspace clustering. For example, in @cite_32 @cite_35 , a fully connected deep auto-encoder network with hand-crafted features ( , SIFT or HOG features) combined with a sparse self-expression model is developed; in @cite_8 , a stacked convolutional auto-encoder network with a plus-in self-expression model is proposed. While promising clustering accuracy has been reported, these methods are still suboptimal because the potentially useful supervision information from the clustering result has been taken into the feature learning step a joint optimization framework for fully combining feature learning and subspace clustering has been developed. More recently, in @cite_63 , a deep adversarial network with a subspace-specific generator and a subspace-specific discriminator is adopted in the framework of @cite_8 for subspace clustering. However, the discriminator need to use the dimension of each subspace, which is usually unknown.