skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


Title: Probabilistic Sparse Subspace Clustering Using Delayed Association.
Discovering and clustering subspaces in high-dimensional data is a fundamental problem of machine learning with a wide range of applications in data mining, computer vision, and pattern recognition. Earlier methods divided the problem into two separate stages of finding the similarity matrix and finding clusters. Similar to some recent works, we integrate these two steps using a joint optimization approach. We make the following contributions: (i) we estimate the reliability of the cluster assignment for each point before assigning a point to a subspace. We group the data points into two groups of “certain” and “uncertain”, with the assignment of latter group delayed until their subspace association certainty improves. (ii) We demonstrate that delayed association is better suited for clustering subspaces that have ambiguities, i.e. when subspaces intersect or data are contaminated with outliers/noise. (iii) We demonstrate experimentally that such delayed probabilistic association leads to a more accurate self-representation and final clusters. The proposed method has higher accuracy both for points that exclusively lie in one subspace, and those that are on the intersection of subspaces. (iv) We show that delayed association leads to huge reduction of computational cost, since it allows for incremental spectral clustering  more » « less
Award ID(s):
1712977
PAR ID:
10094431
Author(s) / Creator(s):
Date Published:
Journal Name:
Proceedings of the 2018 24th International Conference on Pattern Recognition (ICPR)
Page Range / eLocation ID:
2087 - 2092
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ; ; ; (, Information and Inference: A Journal of the IMA)
    null (Ed.)
    Abstract Subspace clustering is the unsupervised grouping of points lying near a union of low-dimensional linear subspaces. Algorithms based directly on geometric properties of such data tend to either provide poor empirical performance, lack theoretical guarantees or depend heavily on their initialization. We present a novel geometric approach to the subspace clustering problem that leverages ensembles of the $$K$$-subspace (KSS) algorithm via the evidence accumulation clustering framework. Our algorithm, referred to as ensemble $$K$$-subspaces (EKSSs), forms a co-association matrix whose $(i,j)$th entry is the number of times points $$i$$ and $$j$$ are clustered together by several runs of KSS with random initializations. We prove general recovery guarantees for any algorithm that forms an affinity matrix with entries close to a monotonic transformation of pairwise absolute inner products. We then show that a specific instance of EKSS results in an affinity matrix with entries of this form, and hence our proposed algorithm can provably recover subspaces under similar conditions to state-of-the-art algorithms. The finding is, to the best of our knowledge, the first recovery guarantee for evidence accumulation clustering and for KSS variants. We show on synthetic data that our method performs well in the traditionally challenging settings of subspaces with large intersection, subspaces with small principal angles and noisy data. Finally, we evaluate our algorithm on six common benchmark datasets and show that unlike existing methods, EKSS achieves excellent empirical performance when there are both a small and large number of points per subspace. 
    more » « less
  2. ; ; ; (, Internationl Conference on Machine Learning)
    The K-subspaces (KSS) method is a generalization of the K-means method for subspace clustering. In this work, we present local convergence analysis and a recovery guarantee for KSS, assuming data are generated by the semi-random union of subspaces model, where N points are randomly sampled from K ≥ 2 overlapping subspaces. We show that if the initial assignment of the KSS method lies within a neighborhood of a true clustering, it converges at a superlinear rate and finds the correct clustering within (log logN) iterations with high probability. Moreover, we propose a thresholding inner-product based spectral method for initialization and prove that it produces a point in this neighborhood. We also present numerical results of the studied method to support our theoretical developments. 
    more » « less
  3. ; ; ; ; ; (, International Conference on Computer Vision)
    We consider the problem of simultaneously clustering and learning a linear representation of data lying close to a union of low-dimensional manifolds, a fundamental task in machine learning and computer vision. When the manifolds are assumed to be linear subspaces, this reduces to the classical problem of subspace clustering, which has been studied extensively over the past two decades. Unfortunately, many real-world datasets such as natural images can not be well approximated by linear subspaces. On the other hand, numerous works have attempted to learn an appropriate transformation of the data, such that data is mapped from a union of general non-linear manifolds to a union of linear subspaces (with points from the same manifold being mapped to the same subspace). However, many existing works have limitations such as assuming knowledge of the membership of samples to clusters, requiring high sampling density, or being shown theoretically to learn trivial representations. In this paper, we propose to optimize the Maximal Coding Rate Reduction metric with respect to both the data representation and a novel doubly stochastic cluster membership, inspired by state-of-the-art subspace clustering results. We give a parameterization of such a representation and membership, allowing efficient mini-batching and one-shot initialization. Experiments on CIFAR-10, -20, -100, and TinyImageNet-200 datasets show that the proposed method is much more accurate and scalable than state-of-the-art deep clustering methods, and further learns a latent linear representation of the data. 
    more » « less
  4. ; (, 2017 12th International Workshop on Self-Organizing Maps and Learning Vector Quantization, Clustering and Data Visualization (WSOM))
    We extend the self-organizing mapping algorithm to the problem of visualizing data on Grassmann manifolds. In this setting, a collection of k points in n-dimensions is represented by a k-dimensional subspace, e.g., via the singular value or QR-decompositions. Data assembled in this way is challenging to visualize given abstract points on the Grassmannian do not reside in Euclidean space. The extension of the SOM algorithm to this geometric setting only requires that distances between two points can be measured and that any given point can be moved towards a presented pattern. The similarity between two points on the Grassmannian is measured in terms of the principal angles between subspaces, e.g., the chordal distance. Further, we employ a formula for moving one subspace towards another along the shortest path, i.e., the geodesic between two points on the Grassmannian. This enables a faithful implementation of the SOM approach for visualizing data consisting of k-dimensional subspaces of n-dimensional Euclidean space. We illustrate the resulting algorithm on a hyperspectral imaging application. 
    more » « less
  5. ; (, IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP))
    An evolutionary self-expressive model for clustering a collection of evolving data points that lie on a union of low-dimensional evolving subspaces is proposed. A parsimonious representation of data points at each time step is learned via a non-convex optimization framework that exploits the self-expressiveness property of the evolving data while taking into account data representation from the preceding time step. The resulting scheme adaptively learns an innovation matrix that captures changes in self-representation of data in consecutive time steps as well as a smoothing parameter reflective of the rate of data evolution. Extensive experiments demonstrate superiority of the proposed framework overs state-of-the-art static subspace clustering algorithms and existing evolutionary clustering schemes. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email