Similarity Learning for High-Dimensional Sparse Data

A good measure of similarity between data points is crucial to many tasks in machine learning. Similarity and metric learning methods learn such measures automatically from data, but they do not scale well respect to the dimensionality of the data. In this paper, we propose a method that can learn efficiently similarity measure from high-dimensional sparse data. The core idea is to parameterize the similarity measure as a convex combination of rank-one matrices with specific sparsity structures. The parameters are then optimized with an approximate Frank-Wolfe procedure to maximally satisfy relative similarity constraints on the training data. Our algorithm greedily incorporates one pair of features at a time into the similarity measure, providing an efficient way to control the number of active features and thus reduce overfitting. It enjoys very appealing convergence guarantees and its time and memory complexity depends on the sparsity of the data instead of the dimension of the feature space. Our experiments on real-world high-dimensional datasets demonstrate its potential for classification, dimensionality reduction and data exploration.Comment: 14 pages. Proceedings of the 18th International Conference on Artificial Intelligence and Statistics (AISTATS 2015). Matlab code: https://github.com/bellet/HDS

Robust Spectral Clustering via Sparse Representation

Clustering high-dimensional data has been a challenging problem in data mining and machining learning. Spectral clustering via sparse representation has been proposed for clustering high-dimensional data. A critical step in spectral clustering is to effectively construct a weight matrix by assessing the proximity between each pair of objects. While sparse representation proves its effectiveness for compressing high-dimensional signals, existing spectral clustering algorithms based on sparse representation use those sparse coefficients directly. We believe that the similarity measure exploiting more global information from the coefficient vectors will provide more truthful similarity among data objects. The intuition is that the sparse coefficient vectors corresponding to two similar objects are similar and those of two dissimilar objects are also dissimilar. In particular, we propose two approaches of weight matrix construction according to the similarity of the sparse coefficient vectors. Experimental results on several real-world high-dimensional data sets demonstrate that spectral clustering based on the proposed similarity matrices outperforms existing spectral clustering algorithms via sparse representation

Similarity Learning via Kernel Preserving Embedding

Data similarity is a key concept in many data-driven applications. Many algorithms are sensitive to similarity measures. To tackle this fundamental problem, automatically learning of similarity information from data via self-expression has been developed and successfully applied in various models, such as low-rank representation, sparse subspace learning, semi-supervised learning. However, it just tries to reconstruct the original data and some valuable information, e.g., the manifold structure, is largely ignored. In this paper, we argue that it is beneficial to preserve the overall relations when we extract similarity information. Specifically, we propose a novel similarity learning framework by minimizing the reconstruction error of kernel matrices, rather than the reconstruction error of original data adopted by existing work. Taking the clustering task as an example to evaluate our method, we observe considerable improvements compared to other state-of-the-art methods. More importantly, our proposed framework is very general and provides a novel and fundamental building block for many other similarity-based tasks. Besides, our proposed kernel preserving opens up a large number of possibilities to embed high-dimensional data into low-dimensional space.Comment: Published in AAAI 201

Similarity modeling for machine learning

Similarity is the extent to which two objects resemble each other. Modeling similarity is an important topic for both machine learning and computer vision. In this dissertation, we first propose a discriminative similarity learning method, then introduce two novel sparse similarity modeling methods for high dimensional data from the perspective of manifold learning and subspace learning. Our sparse similarity modeling methods learn sparse similarity and consequently generate a sparse graph over the data. The generated sparse graph leads to superior performance in clustering and semi-supervised learning, compared to existing sparse graph based methods such as $\ell^{1}$-graph and Sparse Subspace Clustering (SSC). More concretely, our discriminative similarity learning method adopts a novel pairwise clustering framework by bridging the gap between clustering and multi-class classification. This pairwise clustering framework learns an unsupervised nonparametric classifier from each data partition, and searches for the optimal partition of the data by minimizing the generalization error of the learned classifiers associated with the data partitions. Regarding to our sparse similarity modeling methods, we propose a novel $\ell^{0}$ regularized $\ell^{1}$-graph ($\ell^{0}$-$\ell^{1}$-graph) to improve $\ell^{1}$-graph from the perspective of manifold learning. Our $\ell^{0}$-$\ell^{1}$-graph generates a sparse graph that is aligned to the manifold structure of the data for better clustering performance. From the perspective of learning the subspace structures of the high dimensional data, we propose $\ell^{0}$-graph that generates a subspace-consistent sparse graph for clustering and semi-supervised learning. Subspace-consistent sparse graph is a sparse graph where a data point is only connected to other data that lie in the same subspace, and the representative method Sparse Subspace Clustering (SSC) proves to generate subspace-consistent sparse graph under certain assumptions on the subspaces and the data, e.g. independent/disjoint subspaces and subspace incoherence/affinity. In contrast, our $\ell^{0}$-graph can generate subspace-consistent sparse graph for arbitrary distinct underlying subspaces under far less restrictive assumptions, i.e. only i.i.d. random data generation according to arbitrary continuous distribution. Extensive experimental results on various data sets demonstrate the superiority of $\ell^{0}$-graph compared to other methods including SSC for both clustering and semi-supervised learning. The proposed sparse similarity modeling methods require sparse coding using the entire data as the dictionary, which can be inefficient especially in case of large-scale data. In order to overcome this challenge, we propose Support Regularized Sparse Coding (SRSC) where a compact dictionary is learned. The data similarity induced by the support regularized sparse codes leads to compelling clustering performance. Moreover, a feed-forward neural network, termed Deep-SRSC, is designed as a fast encoder to approximate the codes generated by SRSC, further improving the efficiency of SRSC

Joint Projection Learning and Tensor Decomposition Based Incomplete Multi-view Clustering

Incomplete multi-view clustering (IMVC) has received increasing attention since it is often that some views of samples are incomplete in reality. Most existing methods learn similarity subgraphs from original incomplete multi-view data and seek complete graphs by exploring the incomplete subgraphs of each view for spectral clustering. However, the graphs constructed on the original high-dimensional data may be suboptimal due to feature redundancy and noise. Besides, previous methods generally ignored the graph noise caused by the inter-class and intra-class structure variation during the transformation of incomplete graphs and complete graphs. To address these problems, we propose a novel Joint Projection Learning and Tensor Decomposition Based method (JPLTD) for IMVC. Specifically, to alleviate the influence of redundant features and noise in high-dimensional data, JPLTD introduces an orthogonal projection matrix to project the high-dimensional features into a lower-dimensional space for compact feature learning.Meanwhile, based on the lower-dimensional space, the similarity graphs corresponding to instances of different views are learned, and JPLTD stacks these graphs into a third-order low-rank tensor to explore the high-order correlations across different views. We further consider the graph noise of projected data caused by missing samples and use a tensor-decomposition based graph filter for robust clustering.JPLTD decomposes the original tensor into an intrinsic tensor and a sparse tensor. The intrinsic tensor models the true data similarities. An effective optimization algorithm is adopted to solve the JPLTD model. Comprehensive experiments on several benchmark datasets demonstrate that JPLTD outperforms the state-of-the-art methods. The code of JPLTD is available at https://github.com/weilvNJU/JPLTD.Comment: IEEE Transactions on Neural Networks and Learning Systems, 202