Title: Robust Tensor Recovery with Fiber Outliers for Traffic Events
Event detection is gaining increasing attention in smart cities research. Large-scale mobility data serves as an important tool to uncover the dynamics of urban transportation systems, and more often than not the dataset is incomplete. In this article, we develop a method to detect extreme events in large traffic datasets, and to impute missing data during regular conditions. Specifically, we propose a robust tensor recovery problem to recover low-rank tensors under fiber-sparse corruptions with partial observations, and use it to identify events, and impute missing data under typical conditions. Our approach is scalable to large urban areas, taking full advantage of the spatio-temporal correlations in traffic patterns. We develop an efficient algorithm to solve the tensor recovery problem based on the alternating direction method of multipliers (ADMM) framework. Compared with existing l 1 norm regularized tensor decomposition methods, our algorithm can exactly recover the values of uncorrupted fibers of a low-rank tensor and find the positions of corrupted fibers under mild conditions. Numerical experiments illustrate that our algorithm can achieve exact recovery and outlier detection even with missing data rates as high as 40% under 5% gross corruption, depending on the tensor size and the Tucker rank of the low rank tensor. Finally, we apply our method on a real traffic dataset corresponding to downtown Nashville, TN and successfully detect the events like severe car crashes, construction lane closures, and other large events that cause significant traffic disruptions. more »« less
Chao, Zehan; Huang, Longxiu; Needell, Deanna(
, Journal of Imaging)
null
(Ed.)
Matrix completion, the problem of completing missing entries in a data matrix with low-dimensional structure (such as rank), has seen many fruitful approaches and analyses. Tensor completion is the tensor analog that attempts to impute missing tensor entries from similar low-rank type assumptions. In this paper, we study the tensor completion problem when the sampling pattern is deterministic and possibly non-uniform. We first propose an efficient weighted Higher Order Singular Value Decomposition (HOSVD) algorithm for the recovery of the underlying low-rank tensor from noisy observations and then derive the error bounds under a properly weighted metric. Additionally, the efficiency and accuracy of our algorithm are both tested using synthetic and real datasets in numerical simulations.
We propose an algorithm to impute and forecast a time series by transforming the observed time series into a matrix, utilizing matrix estimation to recover missing values and de-noise observed entries, and performing linear regression to make predictions. At the core of our analysis is a representation result, which states that for a large class of models, the transformed time series matrix is (approximately) low-rank. In effect, this generalizes the widely used Singular Spectrum Analysis (SSA) in the time series literature, and allows us to establish a rigorous link between time series analysis and matrix estimation. The key to establishing this link is constructing a Page matrix with non-overlapping entries rather than a Hankel matrix as is commonly done in the literature (e.g., SSA). This particular matrix structure allows us to provide finite sample analysis for imputation and prediction, and prove the asymptotic consistency of our method. Another salient feature of our algorithm is that it is model agnostic with respect to both the underlying time dynamics and the noise distribution in the observations. The noise agnostic property of our approach allows us to recover the latent states when only given access to noisy and partial observations a la a Hidden Markov Model; e.g., recovering the time-varying parameter of a Poisson process without knowing that the underlying process is Poisson. Furthermore, since our forecasting algorithm requires regression with noisy features, our approach suggests a matrix estimation based method-coupled with a novel, non-standard matrix estimation error metric-to solve the error-in-variable regression problem, which could be of interest in its own right. Through synthetic and real-world datasets, we demonstrate that our algorithm outperforms standard software packages (including R libraries) in the presence of missing data as well as high levels of noise.
We propose an algorithm to impute and forecast a time series by transforming the observed time series into a
matrix, utilizing matrix estimation to recover missing values and de-noise observed entries, and performing
linear regression to make predictions. At the core of our analysis is a representation result, which states that
for a large class of models, the transformed time series matrix is (approximately) low-rank. In effect, this
generalizes the widely used Singular Spectrum Analysis (SSA) in the time series literature, and allows us to
establish a rigorous link between time series analysis and matrix estimation. The key to establishing this link
is constructing a Page matrix with non-overlapping entries rather than a Hankel matrix as is commonly done
in the literature (e.g., SSA). This particular matrix structure allows us to provide finite sample analysis for
imputation and prediction, and prove the asymptotic consistency of our method. Another salient feature of
our algorithm is that it is model agnostic with respect to both the underlying time dynamics and the noise
distribution in the observations. The noise agnostic property of our approach allows us to recover the latent
states when only given access to noisy and partial observations a la a Hidden Markov Model; e.g., recovering
the time-varying parameter of a Poisson process without knowing that the underlying process is Poisson.
Furthermore, since our forecasting algorithm requires regression with noisy features, our approach suggests a
matrix estimation based method—coupled with a novel, non-standard matrix estimation error metric—to solve
the error-in-variable regression problem, which could be of interest in its own right. Through synthetic and
real-world datasets, we demonstrate that our algorithm outperforms standard software packages (including R
libraries) in the presence of missing data as well as high levels of noise.
Popa, Jonathan; Minkoff, Susan E.; Lou, Yifei(
, GEOPHYSICS)
null
(Ed.)
Seismic data are often incomplete due to equipment malfunction, limited source and receiver placement at near and far offsets, and missing crossline data. Seismic data contain redundancies because they are repeatedly recorded over the same or adjacent subsurface regions, causing the data to have a low-rank structure. To recover missing data, one can organize the data into a multidimensional array or tensor and apply a tensor completion method. We can increase the effectiveness and efficiency of low-rank data reconstruction based on tensor singular value decomposition (tSVD) by analyzing the effect of tensor orientation and exploiting the conjugate symmetry of the multidimensional Fourier transform. In fact, these results can be generalized to any order tensor. Relating the singular values of the tSVD to those of a matrix leads to a simplified analysis, revealing that the most square orientation gives the best data structure for low-rank reconstruction. After the first step of the tSVD, a multidimensional Fourier transform, frontal slices of the tensor form conjugate pairs. For each pair, a singular value decomposition can be replaced with a much cheaper conjugate calculation, allowing for faster computation of the tSVD. Using conjugate symmetry in our improved tSVD algorithm reduces the runtime of the inner loop by 35%–50%. We consider synthetic and real seismic data sets from the Viking Graben Region and the Northwest Shelf of Australia arranged as high-dimensional tensors. We compare the tSVD-based reconstruction with traditional methods, projection onto convex sets and multichannel singular spectrum analysis, and we see that the tSVD-based method gives similar or better accuracy and is more efficient, converging with runtimes that are an order of magnitude faster than the traditional methods. In addition, we verify that the most square orientation improves recovery for these examples by 10%–20% compared with the other orientations.
Repeatedly recording seismic data over a period of months or years is one way to identify trapped oil and gas and to monitor CO2 injection in underground storage reservoirs and saline aquifers. This process of recording data over time and then differencing the images assumes the recording of the data over a particular subsurface region is repeatable. In other words, the hope is that one can recover changes in the Earth when the survey parameters are held fixed between data collection times. Unfortunately, perfect experimental repeatability almost never occurs. Acquisition inconsistencies such as changes in weather (currents, wind) for marine seismic data are inevitable, resulting in source and receiver location differences between surveys at the very least. Thus, data processing aimed at improving repeatability between baseline and monitor surveys is extremely useful. One such processing tool is regularization (or binning) that aligns multiple surveys with different source or receiver configurations onto a common grid. Data binned onto a regular grid can be stored in a high-dimensional data structure called a tensor with, for example, x and y receiver coordinates and time as indices of the tensor. Such a higher-order data structure describing a subsection of the Earth often exhibits redundancies which one can exploit to fill in gaps caused by sampling the surveys onto the common grid. In fact, since data gaps and noise increase the rank of the tensor, seeking to recover the original data by reducing the rank (low-rank tensor-based completion) successfully fills in gaps caused by binning. The tensor nuclear norm (TNN) is defined by the tensor singular value decomposition (tSVD) which generalizes the matrix SVD. In this work we complete missing time-lapse data caused by binning using the alternating direction method of multipliers (or ADMM) to minimize the TNN. For a synthetic experiment with three parabolic events in which the time-lapse difference involves an amplitude increase in one of these events between baseline and monitor data sets, the binning and reconstruction algorithm (TNN-ADMM) correctly recovers this time-lapse change. We also apply this workflow of binning and TNN-ADMM reconstruction to a real marine survey from offshore Western Australia in which the binning onto a regular grid results in significant data gaps. The data after reconstruction varies continuously without the large gaps caused by the binning process.
Hu, Yue, and Work, Daniel B. Robust Tensor Recovery with Fiber Outliers for Traffic Events. Retrieved from https://par.nsf.gov/biblio/10310429. ACM Transactions on Knowledge Discovery from Data 15.1 Web. doi:10.1145/3417337.
Hu, Yue, & Work, Daniel B. Robust Tensor Recovery with Fiber Outliers for Traffic Events. ACM Transactions on Knowledge Discovery from Data, 15 (1). Retrieved from https://par.nsf.gov/biblio/10310429. https://doi.org/10.1145/3417337
@article{osti_10310429,
place = {Country unknown/Code not available},
title = {Robust Tensor Recovery with Fiber Outliers for Traffic Events},
url = {https://par.nsf.gov/biblio/10310429},
DOI = {10.1145/3417337},
abstractNote = {Event detection is gaining increasing attention in smart cities research. Large-scale mobility data serves as an important tool to uncover the dynamics of urban transportation systems, and more often than not the dataset is incomplete. In this article, we develop a method to detect extreme events in large traffic datasets, and to impute missing data during regular conditions. Specifically, we propose a robust tensor recovery problem to recover low-rank tensors under fiber-sparse corruptions with partial observations, and use it to identify events, and impute missing data under typical conditions. Our approach is scalable to large urban areas, taking full advantage of the spatio-temporal correlations in traffic patterns. We develop an efficient algorithm to solve the tensor recovery problem based on the alternating direction method of multipliers (ADMM) framework. Compared with existing l 1 norm regularized tensor decomposition methods, our algorithm can exactly recover the values of uncorrupted fibers of a low-rank tensor and find the positions of corrupted fibers under mild conditions. Numerical experiments illustrate that our algorithm can achieve exact recovery and outlier detection even with missing data rates as high as 40% under 5% gross corruption, depending on the tensor size and the Tucker rank of the low rank tensor. Finally, we apply our method on a real traffic dataset corresponding to downtown Nashville, TN and successfully detect the events like severe car crashes, construction lane closures, and other large events that cause significant traffic disruptions.},
journal = {ACM Transactions on Knowledge Discovery from Data},
volume = {15},
number = {1},
author = {Hu, Yue and Work, Daniel B.},
}
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