arXiv research

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

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…

This paper presents GRASTA (Grassmannian Robust Adaptive Subspace Tracking Algorithm), an efficient and robust online algorithm for tracking subspaces from highly incomplete information. The algorithm uses a robust $l^1$-norm cost function in order to estimate and track non-stationary subspaces when the streaming data …

In this work, we study the robust subspace tracking (RST) problem and obtain one of the first two provable guarantees for it. The goal of RST is to track sequentially arriving data vectors that lie in a slowly changing low-dimensional subspace, while being robust to corruption by additive sparse outliers. It can also b…

We study the problem of subspace tracking in the presence of missing data (ST-miss). In recent work, we studied a related problem called robust ST. In this work, we show that a simple modification of our robust ST solution also provably solves ST-miss and robust ST-miss. To our knowledge, our result is the first `compl…

A robust visual tracking system requires an object appearance model that is able to handle occlusion, pose, and illumination variations in the video stream. This can be difficult to accomplish when the model is trained using only a single image. In this paper, we first propose a tracking approach based on affine subspa…

Dynamic robust PCA refers to the dynamic (time-varying) extension of robust PCA (RPCA). It assumes that the true (uncorrupted) data lies in a low-dimensional subspace that can change with time, albeit slowly. The goal is to track this changing subspace over time in the presence of sparse outliers. We develop and study …

Principal Components Analysis (PCA) is one of the most widely used dimension reduction techniques. Robust PCA (RPCA) refers to the problem of PCA when the data may be corrupted by outliers. Recent work by Cand{è}s, Wright, Li, and Ma defined RPCA as a problem of decomposing a given data matrix into the sum of a low-ran…

Robust PCA methods are typically batch algorithms which requires loading all observations into memory before processing. This makes them inefficient to process big data. In this paper, we develop an efficient online robust principal component methods, namely online moving window robust principal component analysis (OMW…

In this work we study the problem of Subspace Tracking with missing data (ST-miss) and outliers (Robust ST-miss). We propose a novel algorithm, and provide a guarantee for both these problems. Unlike past work on this topic, the current work does not impose the piecewise constant subspace change assumption. Additionall…

We present a framework for supervised subspace tracking, when there are two time series $x_t$ and $y_t$, one being the high-dimensional predictors and the other being the response variables and the subspace tracking needs to take into consideration of both sequences. It extends the classic online subspace tracking work…

This work presents GROUSE (Grassmanian Rank-One Update Subspace Estimation), an efficient online algorithm for tracking subspaces from highly incomplete observations. GROUSE requires only basic linear algebraic manipulations at each iteration, and each subspace update can be performed in linear time in the dimension of…

We consider the problem of online subspace tracking of a partially observed high-dimensional data stream corrupted by noise, where we assume that the data lie in a low-dimensional linear subspace. This problem is cast as an online low-rank tensor completion problem. We propose a novel online tensor subspace tracking al…

Many applications in data analysis rely on the decomposition of a data matrix into a low-rank and a sparse component. Existing methods that tackle this task use the nuclear norm and L1-cost functions as convex relaxations of the rank constraint and the sparsity measure, respectively, or employ thresholding techniques. …

This work takes the first steps towards solving the "phaseless subspace tracking" (PST) problem. PST involves recovering a time sequence of signals (or images) from phaseless linear projections of each signal under the following structural assumption: the signal sequence is generated from a much lower dimensional subsp…

In an era of ubiquitous large-scale streaming data, the availability of data far exceeds the capacity of expert human analysts. In many settings, such data is either discarded or stored unprocessed in datacenters. This paper proposes a method of online data thinning, in which large-scale streaming datasets are winnowed…

Bicycle paths form geodesics in 3D subspaces, related to Kirchhoff rods.

Robust high-dimensional data processing has witnessed an exciting development in recent years, as theoretical results have shown that it is possible using convex programming to optimize data fit to a low-rank component plus a sparse outlier component. This problem is also known as Robust PCA, and it has found applicati…

New framework tracks communities in dynamic networks.

LASER compresses recursive model activations by exploiting their low-dimensional structure.

For many modern applications in science and engineering, data are collected in a streaming fashion carrying time-varying information, and practitioners need to process them with a limited amount of memory and computational resources in a timely manner for decision making. This often is coupled with the missing data pro…

Memory-efficient optimizers fail to track a subspace, leading to unpredictable model performance.

Index tracking is a popular form of asset management. Typically, a quadratic function is used to define the tracking error of a portfolio and the look back approach is applied to solve the index tracking problem. We argue that a forward looking approach is more suitable, whereby the tracking error is expressed as expec…

Recent work in distance metric learning has focused on learning transformations of data that best align with specified pairwise similarity and dissimilarity constraints, often supplied by a human observer. The learned transformations lead to improved retrieval, classification, and clustering algorithms due to the bette…

SOOTT framework optimizes target tracking with robust and learning-augmented algorithms.

The tracking method based on the extreme learning machine (ELM) is efficient and effective. ELM randomly generates input weights and biases in the hidden layer, and then calculates and computes the output weights by reducing the iterative solution to the problem of linear equations. Therefore, ELM offers the satisfying…

This paper will serve as an introduction to the body of work on robust subspace recovery. Robust subspace recovery involves finding an underlying low-dimensional subspace in a dataset that is possibly corrupted with outliers. While this problem is easy to state, it has been difficult to develop optimal algorithms due t…

WPCA improves subspace recovery robustness to outliers.

Hybrid approach combines transformer and Bayesian filtering for robust multiple particle tracking.

This paper addresses network anomography, that is, the problem of inferring network-level anomalies from indirect link measurements. This problem is cast as a low-rank subspace tracking problem for normal flows under incomplete observations, and an outlier detection problem for abnormal flows. Since traffic data is lar…

Paper proves IRLS converges to subspace from any start, with practical benefits.

This paper considers the problem of robust subspace recovery: given a set of $N$ points in $\mathbb{R}^D$, if many lie in a $d$-dimensional subspace, then can we recover the underlying subspace? We show that Tyler's M-estimator can be used to recover the underlying subspace, if the percentage of the inliers is larger t…

Theoretical guarantees for STE, a robust subspace recovery method.

We study the basic problem of robust subspace recovery. That is, we assume a data set that some of its points are sampled around a fixed subspace and the rest of them are spread in the whole ambient space, and we aim to recover the fixed underlying subspace. We first estimate "robust inverse sample covariance" by solvi…

Algorithm finds a subspace minimizing distances to inliers with outliers.

We present a simple and fast geometric method for modeling data by a union of affine subspaces. The method begins by forming a collection of local best-fit affine subspaces, i.e., subspaces approximating the data in local neighborhoods. The correct sizes of the local neighborhoods are determined automatically by the Jo…

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…

Paper extends multivariate rank tests for robust subspace detection.

This paper describes a novel approach to change-point detection when the observed high-dimensional data may have missing elements. The performance of classical methods for change-point detection typically scales poorly with the dimensionality of the data, so that a large number of observations are collected after the t…

Paper proposes $S^2$ConvSCN for robust subspace clustering and classification.

A new method tracks index using topological data analysis for sparse portfolios.

Two-layer networks trained on low-dimensional subspaces are vulnerable to adversarial examples.

We propose a neural network for unsupervised anomaly detection with a novel robust subspace recovery layer (RSR layer). This layer seeks to extract the underlying subspace from a latent representation of the given data and removes outliers that lie away from this subspace. It is used within an autoencoder. The encoder …

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…

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…

New method fuses audio and magnetic data to identify underlying subspaces.

A new model tracks indices without rebalancing, solving NP-hard problems.

Motivated by vision tasks such as robust face and object recognition, we consider the following general problem: given a collection of low-dimensional linear subspaces in a high-dimensional ambient (image) space, and a query point (image), efficiently determine the nearest subspace to the query in $\ell^1$ distance. In…