arXiv research

Is it possible to find the sparsest vector (direction) in a generic subspace $\mathcal{S} \subseteq \mathbb{R}^p$ with $\mathrm{dim}(\mathcal{S})= n < p$? This problem can be considered a homogeneous variant of the sparse recovery problem, and finds connections to sparse dictionary learning, sparse PCA, and many other …

Unified framework for clustering with sparse convex combinations.

Unions of subspaces provide a powerful generalization to linear subspace models for collections of high-dimensional data. To learn a union of subspaces from a collection of data, sets of signals in the collection that belong to the same subspace must be identified in order to obtain accurate estimates of the subspace s…

Stochastic Sparse Subspace Clustering improves subspace clustering by reducing over-segmentation through dropout.

The problem of finding the sparsest vector (direction) in a low dimensional subspace can be considered as a homogeneous variant of the sparse recovery problem, which finds applications in robust subspace recovery, dictionary learning, sparse blind deconvolution, and many other problems in signal processing and machine …

Extracting the underlying low-dimensional space where high-dimensional signals often reside has long been at the center of numerous algorithms in the signal processing and machine learning literature during the past few decades. At the same time, working with incomplete (partly observed) large scale datasets has recent…

PCA is one of the most widely used dimension reduction techniques. A related easier problem is "subspace learning" or "subspace estimation". Given relatively clean data, both are easily solved via singular value decomposition (SVD). The problem of subspace learning or PCA in the presence of outliers is called robust su…

SSVI efficiently trains sparse Bayesian neural networks with minimal compression and performance loss.

Subspace clustering is the problem of clustering data points into a union of low-dimensional linear/affine subspaces. It is the mathematical abstraction of many important problems in computer vision, image processing and machine learning. A line of recent work (4, 19, 24, 20) provided strong theoretical guarantee for s…

Given an overcomplete dictionary $A$ and a signal $b$ that is a linear combination of a few linearly independent columns of $A$, classical sparse recovery theory deals with the problem of recovering the unique sparse representation $x$ such that $b = A x$. It is known that under certain conditions on $A$, $x$ can be re…

New method discovers concepts in hidden feature layers using sparse subspace clustering.

BO method identifies sparse subspaces for efficient high-dimensional optimization.

We study the Dictionary Learning (aka Sparse Coding) problem of obtaining a sparse representation of data points, by learning \emph{dictionary vectors} upon which the data points can be written as sparse linear combinations. We view this problem from a geometry perspective as the spanning set of a subspace arrangement,…

We study sparse principal components analysis in high dimensions, where $p$ (the number of variables) can be much larger than $n$ (the number of observations), and analyze the problem of estimating the subspace spanned by the principal eigenvectors of the population covariance matrix. We introduce two complementary not…

Simplicial learning improves classification by generating compact sparse representations.

Fast robust subspace tracking in sparse data-dependent noise with near-optimal delay.

Paper improves $\ell^{0}$-SSC for noisy data by proving SDP and proposing Noisy-DR-$\ell^{0}$-SSC.

In this paper we consider the problem of group invariant subspace clustering where the data is assumed to come from a union of group-invariant subspaces of a vector space, i.e. subspaces which are invariant with respect to action of a given group. Algebraically, such group-invariant subspaces are also referred to as su…

Subspace clustering refers to the problem of clustering high-dimensional data into a union of low-dimensional subspaces. Current subspace clustering approaches are usually based on a two-stage framework. In the first stage, an affinity matrix is generated from data. In the second one, spectral clustering is applied on …

Given an overcomplete dictionary $A$ and a signal $b = Ac^*$ for some sparse vector $c^*$ whose nonzero entries correspond to linearly independent columns of $A$, classical sparse signal recovery theory considers the problem of whether $c^*$ can be recovered as the unique sparsest solution to $b = A c$. It is now well-…

In many real-world problems, we are dealing with collections of high-dimensional data, such as images, videos, text and web documents, DNA microarray data, and more. Often, high-dimensional data lie close to low-dimensional structures corresponding to several classes or categories the data belongs to. In this paper, we…

New framework tackles high-dimensional reliability analysis using surrogate models and active subspaces.

Multivariate functional data from a complex system are naturally high-dimensional and have complex cross-correlation structure. The complexity of data structure can be observed as that (1) some functions are strongly correlated with similar features, while some others may have almost no cross-correlations with quite di…

New algorithm reduces dimensionality in federated learning.

Online detection of abrupt changes in high-dimensional data streams.

This paper considers the problem of subspace clustering under noise. Specifically, we study the behavior of Sparse Subspace Clustering (SSC) when either adversarial or random noise is added to the unlabelled input data points, which are assumed to be in a union of low-dimensional subspaces. We show that a modified vers…

High-dimensional data often lie in low-dimensional subspaces corresponding to different classes they belong to. Finding sparse representations of data points in a dictionary built using the collection of data helps to uncover low-dimensional subspaces and address problems such as clustering, classification, subset sele…

Unified framework for structured principal subspace estimation with bounds and rates.

We investigate the difficulties of training sparse neural networks and make new observations about optimization dynamics and the energy landscape within the sparse regime. Recent work of \citep{Gale2019, Liu2018} has shown that sparse ResNet-50 architectures trained on ImageNet-2012 dataset converge to solutions that a…

RaSE ensemble framework improves sparse classification accuracy.

Sparse subspace clustering (SSC) is an elegant approach for unsupervised segmentation if the data points of each cluster are located in linear subspaces. This model applies, for instance, in motion segmentation if some restrictions on the camera model hold. SSC requires that problems based on the $l_1$-norm are solved …

This paper proposes a subspace decomposition method based on an over-complete dictionary in sparse representation, called "Sparse Signal Subspace Decomposition" (or 3SD) method. This method makes use of a novel criterion based on the occurrence frequency of atoms of the dictionary over the data set. This criterion, wel…

State-of-the-art algorithms for sparse subspace clustering perform spectral clustering on a similarity matrix typically obtained by representing each data point as a sparse combination of other points using either basis pursuit (BP) or orthogonal matching pursuit (OMP). BP-based methods are often prohibitive in practic…

Orthogonal Matching Pursuit (OMP) plays an important role in data science and its applications such as sparse subspace clustering and image processing. However, the existing OMP-based approaches lack of data adaptiveness so that the data cannot be represented well enough and may lose the accuracy. This paper proposes a…

In multi-label learning, each sample is associated with several labels. Existing works indicate that exploring correlations between labels improve the prediction performance. However, embedding the label correlations into the training process significantly increases the problem size. Moreover, the mapping of the label …

Subspace clustering assumes that the data is sepa-rable into separate subspaces. Such a simple as-sumption, does not always hold. We assume that, even if the raw data is not separable into subspac-es, one can learn a representation (transform coef-ficients) such that the learnt representation is sep-arable into subspac…

Proposes an optimization framework for sparse robust subspace estimation.

Paper proposes estimators for sparse PCA with oracle property.

Nonnegative matrix factorization (NMF) with group sparsity constraints is formulated as a probabilistic graphical model and, assuming some observed data have been generated by the model, a feasible variational Bayesian algorithm is derived for learning model parameters. When used in a supervised learning scenario, NMF …

Subspace clustering is the problem of partitioning unlabeled data points into a number of clusters so that data points within one cluster lie approximately on a low-dimensional linear subspace. In many practical scenarios, the dimensionality of data points to be clustered are compressed due to constraints of measuremen…

This paper considers the problem of clustering a collection of unlabeled data points assumed to lie near a union of lower-dimensional planes. As is common in computer vision or unsupervised learning applications, we do not know in advance how many subspaces there are nor do we have any information about their dimension…

In this paper, we study the adversarial robustness of subspace learning problems. Different from the assumptions made in existing work on robust subspace learning where data samples are contaminated by gross sparse outliers or small dense noises, we consider a more powerful adversary who can first observe the data matr…

The sparse representation classifier (SRC) has been utilized in various classification problems, which makes use of L1 minimization and works well for image recognition satisfying a subspace assumption. In this paper we propose a new implementation of SRC via screening, establish its equivalence to the original SRC und…

Paper proposes distributed sparse multicategory discriminant analysis for classification.

Sparse representations using learned dictionaries are being increasingly used with success in several data processing and machine learning applications. The availability of abundant training data necessitates the development of efficient, robust and provably good dictionary learning algorithms. Algorithmic stability an…

Sparse Subspace Clustering (SSC) is one of the most popular methods for clustering data points into their underlying subspaces. However, SSC may suffer from heavy computational burden. Orthogonal Matching Pursuit applied on SSC accelerates the computation but the trade-off is the loss of clustering accuracy. In this pa…

A low-rank transformation learning framework for subspace clustering and classification is here proposed. Many high-dimensional data, such as face images and motion sequences, approximately lie in a union of low-dimensional subspaces. The corresponding subspace clustering problem has been extensively studied in the lit…

The implementation of conventional sparse principal component analysis (SPCA) on high-dimensional data sets has become a time consuming work. In this paper, a series of subspace projections are constructed efficiently by using Household QR factorization. With the aid of these subspace projections, a fast deflation meth…