In this section, some preliminaries of t-SVD are first introduced. Then, the related works are presented.
For a given matrix
, define the nuclear norm and spectral norm of
respectively as:
where
, and
are the singular values of
in a non-ascending order. The
-norm,
-norm, Frobenius norm,
-norm of a tensor
is defined as:
where
is an indicator function whose value is 1 if the condition
C is true, and 0 otherwise.
Given two matrices
, we define their inner product as follows:
where
denotes conjugate transpose of matrix
and
denotes the conjugation of complex number
. Given two 3-way tensors
, we define their inner product as follows:
2.1. Tensor Singular Value Decomposition
We first define 3 operators based on block matrices which are introduced in [
33]. For a given 3-way tensor
, we define its block vectorization
and the inverse operation
in the following equation:
We further define the block circulant matrix
of any 3-way tensor
as follows:
Equipped with above defined operators, we are now in a position to define the t-product of 3-way tensors.
Definition 1 (t-product [
33]).
Given two tensors and , the t-product of and is a new 3-way tensor with size : A more intuitive interpretation of t-SVD is as follows [
33]. If we treat a 3-way tensor
as a matrix of size
whose entries are the tube fibers, then the tensor t-product can be analogously understood as the “matrix multiplication” where the standard scalar product is replaced with the vector circular convolution between the tubes (i.e., vectors):
where ⋆ represent the operation of circular convolution [
33] of two vectors
defined as
.
We also define the block diagonal matrix
of any 3-way tensor
and its inverse
as follows:
We also use
(or
) to denote the block diagonal matrix of tensor
(i.e., the Fourier version of
) i.e.,
Then the relationship between DFT and circular convolution further indicates that the conducting t-product in the original domain is equivalent to performing standard matrix product on the Fourier block diagonal matrices [
33]. Since matrix product on the Fourier block diagonal matrices can be parallel written as matrix product of all the frontal slices in the Fourier domain, we have the following relationships:
The relationship between the t-product and FFT also indicates that the inner product of two 3-way tensors
and the inner product of their corresponding Fourier block diagonal matrices
satisfy the following relationship:
When
, one has:
We further define the concepts of tensor transpose, identity tensor, f-diagonal tensor and orthogonal tensor as follows.
Definition 2 (tensor transpose [
33]).
Given a tensor , then define its transpose tensor of size which can be formed through first transposing all the frontal slices of and then exchanging each k-th transposed frontal slice with the -th transposed frontal slice for all . For example, consider 3-way tensor
with 4 frontal slices, the tensor transpose
of
is:
Definition 3 (identity tensor [
33]).
The identity tensor is a tensor whose first frontal slice is the n-by-n identity matrix with all other frontal slices are zero matrices. Definition 4 (f-diagonal tensor [
33]).
We call a 3-way tensor f-diagonal if all the frontal slices of it are diagonal matrices. Definition 5 (orthogonal tensor [
33]).
We call a tensor an orthogonal tensor if the following equations hold: Then, the tensor singular value decomposition (t-SVD) can be given as follows.
Definition 6 (Tensor singular value decomposition, and Tensor tubal rank [
38]).
Given any 3-way tensor , then it has the following factorization called tensor singular value decomposition (t-SVD):where the left and right factor tensors and are orthogonal, and the middle tensor is a rectangular f-diagonal tensor. A visual illustration for the t-SVD is shown in
Figure 1. It can be computed efficiently by FFT and IFFT in the Fourier domain according to Equation (
4). For more details, see [
2].
Definition 7 (Tensor tubal rank [
38]).
The tensor tubal rank of any 3-way tensor is defined as the number of non-zero tubes of in its t-SVD shown in Equation (5), i.e., Definition 8 (Tubal average rank [
38]).
The tubal average rank of any 3-way tensor is defined as the averaged rank of all frontal slices of as follows, Definition 9 (Tensor operator norm [
2,
38]).
The tensor operator norm of any 3-way tensor is defined as follows:
The relationship between t-product and FFT indicates that:
Definition 10 (Tensor spectral norm [
38]).
The tensor spectral norm of any 3-way tensor is defined as the matrix spectral norm of , i.e., We further define the tubal nuclear norm.
Definition 11 (Tubal nuclear norm [
2]).
For any tensor with t-SVD , the tubal nuclear norm (TNN) of is defined as:
where . To understand the tubal nuclear norm, first note that:
where (i) holds because of the definition of DFT [
2], (ii) holds by the property of
-norm, and (iii) is a result of DFT [
2]. Thus, the tubal rank of
is also the number of non-zero diagonal elements of
, i.e., the first frontal slice of tensor
in the t-SVD of
. Similar to the matrix singular values, the values
are also called the singular values of tensor
. As the matrix nuclear norm is the sum of matrix singular values, the tubal nuclear norm can be similarly understood as the sum of tensor singular values.
One can also verify by the property of DFT [
2] that:
which indicates that the TNN of
is also the averaged nuclear norm all frontal slices of
. Thus, TNN indeed models the low-rankness of Fourier domain.
Now, we will show that the low-tubal-rank model is ideal to some real-world tensor data, such as color images and videos.
First, we consider a natural image of size
, shown in
Figure 2a. In
Figure 2b, we plot the distribution of its singular values, i.e., the values of
along with the index
i. As can be seen from
Figure 2b, there are only a small number of singular values with large magnitude, and most of the singular values are close to 0. Then, we can say that some natural color images are approximately low tubal rank.
Then, consider a commonly used YUV sequence
Mother-daughter_qcif (These data can be download from the following link
https://sites.google.com/site/subudhibadri/fewhelpfuldownloads.) whose first frame is shown in
Figure 3a. We use the Y components of the first 30 frames, and get a tensor of size
and show the distribution of tensor singular values in
Figure 3b. We can see from
Figure 3b that similar to
Figure 2b, there are only a small number of singular values with large magnitude, and most of the singular values are close to 0. Then, we can say that some videos can be well approximately low tubal rank.
For TNN and tensor spectral norm, we highlight the following two lemmas.
Lemma 1. [2] TNN is the convex envelop of the tensor average rank in the unit ball of tensor spectral norm . Lemma 2. [2] The TNN and the tensor spectral norm are dual norms to each other. 2.2. Related Works
In this subsection, we briefly introduce some related works. The proposed STPCP is tightly related to the Tensor Robust Principal Component Analysis (TRPCA) which aims to recover a low-rank tensor
and a sparse tensor
from their sum
. This is a special case of our measurement Model (
1) where the noise tensor
is a zero tensor.
In [
39], the SNN-based TRPCA model is proposed by modeling the underlying tensor as a low Tucker rank one:
where SNN (Sum of Nuclear Norms) is defined as
where
and
is the mode-
k matricization of
[
40].
Model (
14) indeed assumes the underlying tensor to be low Tucker rank, which can be too strong for some real tensor data. The TNN-based TRPCA model uses TNN to impose low-rankness in the final solution
as follows:
As shown in [
2], when the underlying tensor
satisfy the tensor incoherent conditions, by solving Problem (
15), one can exactly recover the underlying tensor
and
with high probability with parameter
.
When the noise tensor
is not zero, the robust tensor decomposition based on SNN is proposed in [
36] as follows:
where
and
are positive regularization parameters. The estimation error on
and
is analyzed with an upper bound in [
36].
In [
37], the TNN-based RTD model is proposed as follows:
where
is an upper estimate of
-norm of the underlying tensor
. An upper bound on the estimation error is also established. However, in the analysis of Model (
17), the error does not vanish as the noise tensor
vanishes which means the analysis cannot guarantee exact recovery in the noiseless setting (which can be provided by the analysis of TNN-based TRPCA (
15) by Lu et al. [
2]).
The Bayesian approach is also used for robust tensor recovery. The CP decomposition under sparse corruption and small dense noise is considered [
41], and tensor rank estimation is achieved using Bayesian approach. In [
42], CP decomposition under missing value and small dense noise is considered with rank estimation similar to [
41]. A sparse Bayesian CP model is proposed in [
43] to recover a tensor with missing value, outliers and noises. In [
44], a fully Bayesian treatment is proposed to recover a low-tubal-rank tensor corrupted by both noises and outliers.