The problem of data recovery in multiway arrays (i.e., tensors) arises in many fields such as computer vision, image processing, and traffic data analysis. In this paper, we propose a scalable and fast algorithm for recovering a low-
A tensor is a multidimensional array. It is the higher-order generalization of vector and matrix, which has many applications in information sciences, computer vision, graph analysis [
In the two-dimensional case, that is, the matrix case, the “rank” and “sparsity” are the most useful tools for matrix-valued data analysis. Chandrasekaran et al. [
Although the low-rank matrix recovery problem has been well studied, there is not much work on tensors. Li et al. [
In this paper, a new algorithm for low-
For the usage of this algorithm, it is applied to real world data recovery including traffic data recovery, image restoration, and background modeling.
In traffic data analysis area, due to detector and communication malfunctions, traffic data often confronts with the noising data phenomenon, especially the outlier value noise, which has a great impact on the performance of Intelligent Transportation System (ITS). Therefore, it is essential to solve the issues caused by outlier data in order to fully explore the applicability of the data and realize the ITS applications. In the application part of this paper, we introduce the tensor form to model the traffic data, which can encode the multimode (e.g., week, day, record) correlations of the traffic data simultaneously and preserve the multiway nature of traffic data. For example, it is assumed that a loop detector collects traffic volume data every 15 minutes. Thus, it will have 96 records in a day. If we have 20 weeks traffic volume data, these data can be formed into a tensor of size
It is observed that the multichannel image can be seen as a tensor with multidimensions. For example, RGB image has three channels including Red channel, Green channel, and Black channel. Thus, it can be represented as
The rest of the paper is organized as follows. Section
In this paper, the nomenclatures and the notations in [
Therefore,
The
If the
The inner product of two same-size tensors
The corresponding Frobenius norm is
The
This section is separated into 2 parts. In Section
The derivation starts with the general version [
Problem (
In [
It is observed that (
Equation (
The core idea of solving the optimization problem in (
As shown in [
The operator can be extended to the matrix or tensor case by performing the shrinkage operator towards each element.
By the well-known
It is easy to show that the solution to (
It is easy to show that the solution to (
The pseudo-code of the proposed MALM-TR algorithm is summarized in Algorithm
( ( ( ( Where ( ( ( ( ( ( ( ( ( (
Under some rather general conditions, when
In this section, using both the numerical simulations and the real world data, we evaluate the performance of our proposed algorithm and then compare the results with RSTD on the low-
In all the experiments, the Lanczos bidiagonalization algorithm with partial reorthogonalization [
A low-
The entries of sparse tensor
The simulated tensor used in the experiments is of size
Tables
spr | Algorithm: MALM-TR | Algorithm: RSTD | ||||||
---|---|---|---|---|---|---|---|---|
|
|
# iter | Time (s) |
|
|
# iter | Time (s) | |
0.05 | 0.01 | 0.006 | 136 | 17.7 | 450 | 430 | 226 | 19.5 |
0.15 | 0.06 | 0.02 | 167 | 19.5 | 910 | 440 | 330 | 27.4 |
0.25 | 1.1 | 0.3 | 281 | 29.1 | 1510 | 490 | 714 | 44.4 |
0.35 | 2010 | 390 | 450 | 41.9 | 5620 | 1140 | 608 | 44.1 |
spr | Algorithm: MALM-TR | Algorithm: RSTD | ||||||
---|---|---|---|---|---|---|---|---|
|
|
# iter | Time (s) |
|
|
# iter | Time (s) | |
0.05 | 0.2 | 0.1 | 243 | 29.4 | 400 | 230 | 411 | 33.8 |
0.10 | 10.1 | 4.2 | 323 | 43.5 | 520 | 270 | 568 | 51.3 |
0.15 | 120 | 42 | 596 | 62.5 | 650 | 260 | 1103 | 68.5 |
0.20 | 1250 | 370 | 972 | 88.3 | 2650 | 780 | 1235 | 88.1 |
One straightforward application of our algorithm is the image restoration. Same as [
Comparisons in terms of visual effects. The rows (1), (2), and (3) correspond to the images before recovery, the obtained results by MALM-TR and RSTD [
Another application of our algorithm is to estimate a good model for the background variations in a scene (i.e., background modeling). In this situation, it is natural to model the background variation as approximately low rank. Foreground objects generally occupy only a fraction of the image pixels and hence can be treated as sparse part.
We test our algorithm using an example from [
Background modeling. Top: original video sequence of a scene. Middle: foreground object recovered by MALM-TR. Bottom: foreground object recovered by RSTD [
In our previous work [
In the fourth part of the experiment section, we will apply the proposed algorithm to traffic data recovery. The data used in the experiment are collected by a fixed loop in Sacramento County and downloaded from
Tensor model of size
In our previous work [
According to the above description, the traffic data are reasonably converted into a tensor form which can be approximated by a low-
Table
Comparison of RSE on traffic data.
SPR | MALM-TR(RSE) | RSTD(RSE) | Unrecovered(RSE) |
---|---|---|---|
0.05 | 0.0406 | 0.0525 | 0.2614 |
0.10 | 0.0432 | 0.0543 | 0.4117 |
0.15 | 0.0593 | 0.1370 | 0.5427 |
0.20 | 0.0784 | 0.2198 | 0.6859 |
0.25 | 0.1478 | 0.3959 | 0.8218 |
Comparison of RSE curves on traffic data.
In this paper, we extend the matrix recovery problem to low-
In the future, we would like to investigate how to automatically choose the parameters in our algorithm and develop more efficient method for tensor recovery problem. Also we will explore more applications of our method.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The research was supported by NSFC (Grant nos. 61271376, 51308115, and 91120010), the National Basic Research Program of China (973 Program: no. 2012CB725405), and Beijing Natural Science Foundation (4122067). The authors would like to thank Professor Bin Ran from the University of Wisconsin-Madison and Yong Li from the University of Notre Dame for the suggestive discussions.