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.
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.
@cite_35 @cite_8 @cite_55 @cite_32 @cite_44 @cite_63
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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. We present a novel deep neural network architecture for unsupervised subspace clustering. This architecture is built upon deep auto-encoders, which non-linearly map the input data into a latent space. Our key idea is to introduce a novel self-expressive layer between the encoder and the decoder to mimic the "self-expressiveness" property that has proven effective in traditional subspace clustering. Being differentiable, our new self-expressive layer provides a simple but effective way to learn pairwise affinities between all data points through a standard back-propagation procedure. Being nonlinear, our neural-network based method is able to cluster data points having complex (often nonlinear) structures. We further propose pre-training and fine-tuning strategies that let us effectively learn the parameters of our subspace clustering networks. Our experiments show that the proposed method significantly outperforms the state-of-the-art unsupervised subspace clustering methods. 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. 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. Subspace clustering algorithms are notorious for their scalability issues because building and processing large affinity matrices are demanding. In this paper, we introduce a method that simultaneously learns an embedding space along subspaces within it to minimize a notion of reconstruction error, thus addressing the problem of subspace clustering in an end-to-end learning paradigm. To achieve our goal, we propose a scheme to update subspaces within a deep neural network. This in turn frees us from the need of having an affinity matrix to perform clustering. Unlike previous attempts, our method can easily scale up to large datasets, making it unique in the context of unsupervised learning with deep architectures. Our experiments show that our method significantly improves the clustering accuracy while enjoying cheaper memory footprints. Adversarial perturbations are noise-like patterns that can subtly change the data, while failing an otherwise accurate classifier. In this paper, we propose to use such perturbations for improving the robustness of video representations. To this end, given a well-trained deep model for per-frame video recognition, we first generate adversarial noise adapted to this model. Using the original data features from the full video sequence and their perturbed counterparts, as two separate bags, we develop a binary classification problem that learns a set of discriminative hyperplanes – as a subspace – that will separate the two bags from each other. This subspace is then used as a descriptor for the video, dubbed discriminative subspace pooling. As the perturbed features belong to data classes that are likely to be confused with the original features, the discriminative subspace will characterize parts of the feature space that are more representative of the original data, and thus may provide robust video representations. To learn such descriptors, we formulate a subspace learning objective on the Stiefel manifold and resort to Riemannian optimization methods for solving it efficiently. We provide experiments on several video datasets and demonstrate state-of-the-art results. Image representation plays an important role in medical image analysis. The key to the success of different medical image analysis algorithms is heavily dependent on how we represent the input data, namely features used to characterize the input image. In the literature, feature engineering remains as an active research topic, and many novel hand-crafted features are designed such as Haar wavelet, histogram of oriented gradient, and local binary patterns. However, such features are not designed with the guidance of the underlying dataset at hand. To this end, we argue that the most effective features should be designed in a learning based manner, namely representation learning, which can be adapted to different patient datasets at hand. In this paper, we introduce a deep learning framework to achieve this goal. Specifically, a stacked independent subspace analysis (ISA) network is adopted to learn the most effective features in a hierarchical and unsupervised manner. The learnt features are adapted to the dataset at hand and encode high level semantic anatomical information. The proposed method is evaluated on the application of automatic prostate MR segmentation. Experimental results show that significant segmentation accuracy improvement can be achieved by the proposed deep learning method compared to other state-of-the-art segmentation approaches. Previous work on action recognition has focused on adapting hand-designed local features, such as SIFT or HOG, from static images to the video domain. In this paper, we propose using unsupervised feature learning as a way to learn features directly from video data. More specifically, we present an extension of the Independent Subspace Analysis algorithm to learn invariant spatio-temporal features from unlabeled video data. We discovered that, despite its simplicity, this method performs surprisingly well when combined with deep learning techniques such as stacking and convolution to learn hierarchical representations. By replacing hand-designed features with our learned features, we achieve classification results superior to all previous published results on the Hollywood2, UCF, KTH and YouTube action recognition datasets. On the challenging Hollywood2 and YouTube action datasets we obtain 53.3 and 75.8 respectively, which are approximately 5 better than the current best published results. Further benefits of this method, such as the ease of training and the efficiency of training and prediction, will also be discussed. You can download our code and learned spatio-temporal features here: http: ai.stanford.edu ∼wzou Recent success in training deep neural networks have prompted active investigation into the features learned on their intermediate layers. Such research is difficult because it requires making sense of non-linear computations performed by millions of parameters, but valuable because it increases our ability to understand current models and create improved versions of them. In this paper we investigate the extent to which neural networks exhibit what we call convergent learning, which is when the representations learned by multiple nets converge to a set of features which are either individually similar between networks or where subsets of features span similar low-dimensional spaces. We propose a specific method of probing representations: training multiple networks and then comparing and contrasting their individual, learned representations at the level of neurons or groups of neurons. We begin research into this question using three techniques to approximately align different neural networks on a feature level: a bipartite matching approach that makes one-to-one assignments between neurons, a sparse prediction approach that finds one-to-many mappings, and a spectral clustering approach that finds many-to-many mappings. This initial investigation reveals a few previously unknown properties of neural networks, and we argue that future research into the question of convergent learning will yield many more. The insights described here include (1) that some features are learned reliably in multiple networks, yet other features are not consistently learned; (2) that units learn to span low-dimensional subspaces and, while these subspaces are common to multiple networks, the specific basis vectors learned are not; (3) that the representation codes show evidence of being a mix between a local code and slightly, but not fully, distributed codes across multiple units. Many clustering algorithms are not applicable to high-dimensional feature spaces, because the clusters often exist only in specific subspaces of the original feature space. Those clusters are also called subspace clusters. In this paper, we propose the algorithm HiSC (Hierarchical Subspace Clustering) that can detect hierarchies of nested subspace clusters, i.e. the relationships of lower-dimensional subspace clusters that are embedded within higher-dimensional subspace clusters. Several comparative experiments using synthetic and real data sets show the performance and the effectivity of HiSC. While the research on convolutional neural networks (CNNs) is progressing quickly, the real-world deployment of these models is often limited by computing resources and memory constraints. In this paper, we address this issue by proposing a novel filter pruning method to compress and accelerate CNNs. Our work is based on the linear relationship identified in different feature map subspaces via visualization of feature maps. Such linear relationship implies that the information in CNNs is redundant. Our method eliminates the redundancy in convolutional filters by applying subspace clustering to feature maps. In this way, most of the representative information in the network can be retained in each cluster. Therefore, our method provides an effective solution to filter pruning for which most existing methods directly remove filters based on simple heuristics. The proposed method is independent of the network structure, thus it can be adopted by any off-the-shelf deep learning libraries. Experiments on different networks and tasks show that our method outperforms existing techniques before fine-tuning, and achieves the state-of-the-art results after fine-tuning. We combine linear discriminant analysis (LDA) and K-means clustering into a coherent framework to adaptively select the most discriminative subspace. We use K-means clustering to generate class labels and use LDA to do subspace selection. The clustering process is thus integrated with the subspace selection process and the data are then simultaneously clustered while the feature subspaces are selected. We show the rich structure of the general LDA-Km framework by examining its variants and their relationships to earlier approaches. Relations among PCA, LDA, K-means are clarified. Extensive experimental results on real-world datasets show the effectiveness of our approach. Many clustering algorithms tend to break down in high-dimensional feature spaces, because the clusters often exist only in specific subspaces (attribute subsets) of the original feature space. Therefore, the task of projected clustering (or subspace clustering) has been defined recently. As a solution to tackle this problem, we propose the concept of local subspace preferences, which captures the main directions of high point density. Using this concept, we adopt density-based clustering to cope with high-dimensional data. In particular, we achieve the following advantages over existing approaches: Our proposed method has a determinate result, does not depend on the order of processing, is robust against noise, performs only one single scan over the database, and is linear in the number of dimensions. A broad experimental evaluation shows that our approach yields results of significantly better quality than recent work on clustering high-dimensional data. Subspace clustering is an extension of traditional clustering that seeks to find clusters in different subspaces within a dataset. Often in high dimensional data, many dimensions are irrelevant and can mask existing clusters in noisy data. Feature selection removes irrelevant and redundant dimensions by analyzing the entire dataset. Subspace clustering algorithms localize the search for relevant dimensions allowing them to find clusters that exist in multiple, possibly overlapping subspaces. There are two major branches of subspace clustering based on their search strategy. Top-down algorithms find an initial clustering in the full set of dimensions and evaluate the subspaces of each cluster, iteratively improving the results. Bottom-up approaches find dense regions in low dimensional spaces and combine them to form clusters. This paper presents a survey of the various subspace clustering algorithms along with a hierarchy organizing the algorithms by their defining characteristics. We then compare the two main approaches to subspace clustering using empirical scalability and accuracy tests and discuss some potential applications where subspace clustering could be particularly useful. Since the emergence of extensive multimedia data, feature fusion has been more and more important for image and video retrieval, indexing and annotation. Existing feature fusion techniques simply concatenate a pair of different features or use canonical correlation analysis based methods for joint dimensionality reduction in the feature space. However, how to fuse multiple features in a generalized way is still an open problem. In this paper, we reformulate the multiple feature fusion as a general subspace learning problem. The objective of the framework is to find a general linear subspace in which the cumulative pairwise canonical correlation between every pair of feature sets is maximized after the dimension normalization and subspace projection. The learned subspace couples dimensionality reduction and feature fusion together, which can be applied to both unsupervised and supervised learning cases. In the supervised case, the pairwise canonical correlations of feature sets within the same classes are also counted in the objective function for maximization. To better model the high-order feature structure and overcome the computational difficulty, the features extracted from the same pattern source are represented by a single 2D tensor. The tensor-based dimensionality reduction methods are used to further extract low-dimensional discriminative features from the fused feature ensemble. Extensive experiments on visual data classification demonstrate the effectiveness and robustness of the proposed methods. 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. To uncover an appropriate latent subspace for data representation, in this paper we propose a novel Robust Structured Subspace Learning (RSSL) algorithm by integrating image understanding and feature learning into a joint learning framework. The learned subspace is adopted as an intermediate space to reduce the semantic gap between the low-level visual features and the high-level semantics. To guarantee the subspace to be compact and discriminative, the intrinsic geometric structure of data, and the local and global structural consistencies over labels are exploited simultaneously in the proposed algorithm. Besides, we adopt the @math -norm for the formulations of loss function and regularization respectively to make our algorithm robust to the outliers and noise. An efficient algorithm is designed to solve the proposed optimization problem. It is noted that the proposed framework is a general one which can leverage several well-known algorithms as special cases and elucidate their intrinsic relationships. To validate the effectiveness of the proposed method, extensive experiments are conducted on diversity datasets for different image understanding tasks, i.e., image tagging, clustering, and classification, and the more encouraging results are achieved compared with some state-of-the-art approaches. Person re-identification (ReID) aims at matching persons across different views scenes. In addition to accuracy, the matching efficiency has received more and more attention because of demanding applications using large-scale data. Several binary coding based methods have been proposed for efficient ReID, which either learn projections to map high-dimensional features to compact binary codes, or directly adopt deep neural networks by simply inserting an additional fully-connected layer with tanh-like activations. However, the former approach requires time-consuming hand-crafted feature extraction and complicated (discrete) optimizations; the latter lacks the necessary discriminative information greatly due to the straightforward activation functions. In this paper, we propose a simple yet effective framework for efficient ReID inspired by the recent advances in adversarial learning. Specifically, instead of learning explicit projections or adding fully-connected mapping layers, the proposed Adversarial Binary Coding (ABC) framework guides the extraction of binary codes implicitly and effectively. The discriminability of the extracted codes is further enhanced by equipping the ABC with a deep triplet network for the ReID task. More importantly, the ABC and triplet network are simultaneously optimized in an end-to-end manner. Extensive experiments on three large-scale ReID benchmarks demonstrate the superiority of our approach over the state-of-the-art methods. A clustering framework within the sparse modeling and dictionary learning setting is introduced in this work. Instead of searching for the set of centroid that best fit the data, as in k-means type of approaches that model the data as distributions around discrete points, we optimize for a set of dictionaries, one for each cluster, for which the signals are best reconstructed in a sparse coding manner. Thereby, we are modeling the data as a union of learned low dimensional subspaces, and data points associated to subspaces spanned by just a few atoms of the same learned dictionary are clustered together. An incoherence promoting term encourages dictionaries associated to different classes to be as independent as possible, while still allowing for different classes to share features. This term directly acts on the dictionaries, thereby being applicable both in the supervised and unsupervised settings. Using learned dictionaries for classification and clustering makes this method robust and well suited to handle large datasets. The proposed framework uses a novel measurement for the quality of the sparse representation, inspired by the robustness of the l 1 regularization term in sparse coding. In the case of unsupervised classification and or clustering, a new initialization based on combining sparse coding with spectral clustering is proposed. This initialization clusters the dictionary atoms, and therefore is based on solving a low dimensional eigen-decomposition problem, being applicable to large datasets. We first illustrate the proposed framework with examples on standard image and speech datasets in the supervised classification setting, obtaining results comparable to the state-of-the-art with this simple approach. We then present experiments for fully unsupervised clustering on extended standard datasets and texture images, obtaining excellent performance. A new form of document coupling called co-citation is defined as the frequency with which two documents are cited together. The co-citation frequency of two scientific papers can be determined by comparing lists of citing documents in the Science Citation Index and counting identical entries. Networks of co-cited papers can be generated for specific scientific specialties, and an example is drawn from the literature of particle physics. Co-citation patterns are found to differ significantly from bibliographic coupling patterns, but to agree generally with patterns of direct citation. Clusters of co-cited papers provide a new way to study the specialty structure of science. They may provide a new approach to indexing and to the creation of SDI profiles.