Low-rank tensor recover problem is the generalization of sparse vector recovery and low-rank matrix recover to tensor data[1, 2, 3]
. It has drawn lots of attention from researchers in different fields in the past several years. They have wide applications in data-mining, computer vision, signal/image processing, machine learning, etc.. The fundamental problem of low-rank tensor recovery is to find a tensor of (nearly) lowest rank from an underdetermined, where is a linear operator and is a -way tensor.
and is the mode- unfolding of , is the trace norm of the matrix
, i.e. the sum of the singular values of. For vector with noise or generated by an approximately low-rank tensor, a variant of (1) is 
Despite empirical success, the recovery guarantees of tensor recovery algorithms has not been fully elucidated. Recently, several authors [4, 5, 6] have got excellent results in the guarantees of sparse vector recovery and low-rank matrix recover. In this paper, we try to generalize these results to low-rank tensor recovery. To the best of our knowledge, this is the first paper that studies the guarantees of low-rank tensor recovery algorithm.
The order of a tensor is the number of dimensions, also known as ways or modes. Matrices (tensor of order two) are denoted by upper case letters, e.g. , and lower case letters for the elements, e.g. . Higher-order tensors (order three or higher) are denoted by Euler script letters, e.g. , and element of a -order tensor is denoted by . Fibers are the higher-order analogue of matrix rows and columns. A fiber is defined by fixing every index but one. The mode- fibers are all vectors that obtained by fixing the values of . The mode- unfolding, also knows as matricization, of a tensor is denoted by and arranges the model- fibers to be the columns of the resulting matrix. The unfolding operator is denoted as . The opposite operation is , denotes the refolding of the matrix into a tensor. The tensor element is mapped to the matrix element , where
Therefore, . The -rank of a -dimensional tensor , denoted as is the column rank of , i.e. the dimension of the vector space spanned by the mode- fibers. We say a tensor is rank when for , and denoted as . We introduce an ordering among tensors by the rank:. The inner product of two same-size tensors is defined as , where vec is a vectorization. The corresponding norm is , which is often called the Frobenius norm.
The -th mode product of a tensor with a matrix is denoted by and is of size . Elementwise, we have
Every tensor can be written as the product 
is a unitary matrix,
is a -tensor of which the subtensors , obtained by fixing the -th index to , have the properties of:
1) all-orthogonality: two subtensors and are orthogonal for all possible values of , and subject to : ,
2) ordering: for all possible values of , one has .
The Frobenius norms , symbolized by , are mode- singular values of , that means the singular values of .
This is called the higher-order singular value decomposition (HOSVD) of a tensorin . Some properties of this HOSVD which will be used in this paper are list below as lemmas:
Iii Motivations and contributions
To explain why model (4) is interesting, we conducted following tensor completion simulations
to compare it with model (1) based tensor completion
The relative errors are depicted as functions of the number of iterations in Figure 1(a).
Motivated by the above example, we show in this paper that any guarantees that problem (4) either recovers exactly or returns an approximate of it nearly as good as the solution of problem (1). Specifically, we show that several properties of , such as the null-space property (a simple condition used in, e.g., [3, 10, 11, 12, 13]), the restricted isometry principle , and the spherical section property , which have been used in the recovery guarantees for vectors and matrices, can also guarantee the tensor recovery by model (1) and (4).
Even though not known when is set,
is often easy to estimate. Whenis not available, using inequalities , one get the more conservative formulae . Furthermore, when satisfies the RIP, one has for some ; hence, one has the option to use the even more conservative formula .
Iv Tensor recovery guarantees
This section establishes recovery guarantees for the original and augmented trace norm models (1) and (4). The results are given based on the properties of including the null-space property (NSP) in Theorem 5 and 6, the restricted isometry principle (RIP)  in Theorem 8 and 9, the spherical section property (SSP)  in Theorem 10. These results adapt and generalize of the work in .
Iv-a Null space property
The wide use of NSP for recovering sparse vector and low-rank matrices can be found in e.g. [10, 11, 12, 13]. In this subsection, we extend the NSP conditions on for tensor recovery. Throughout this subsection, we let denote the -th largest singular value of matrix of rank or less, and denote the diagonal matrix of singular values and . denotes the spectral norms of .
We will need the following two technical lemmas for the introduction of the tensor NSP conditions.
( Theorem 7.4.51). Let and be two matrices of the same size. Then we have
( Equation (19)). Let and be two vectors of the same size, and . Then we have
Now we give a NSP type sufficient condition for problem (1).
Pick any tensor of rank or less and let . For any , we have . By using (11), we have
We can extend this result to problem (4) as follows.
Pick any tensor of rank or less and let . For any nonzero , we have . Thus
where the first inequality follows from (11) and (12), and the second inequality follows from (13) and (14). For any nonzero , Hence, from (18) and (17), it follows that leads to a strictly worse objective than . That is, is the unique solution to problem (4).
For any finite , (17) is stronger than (15) due to the extra term . Since various uniform recovery results establish conditions that guarantee (15), one can tighten these conditions so that they guarantee (17) and thus the uniform recovery by problem (4). How much tighter these conditions have to be depends on the value .
Iv-B Tensor restricted isometry principle
In this subsection, we generalize the RIP-based guarantees to tensor case and show that any guarantees exact recovery by (4).
(Tensor RIP). Let }. The RIP constant of linear operator is the smallest value such that
holds for all .
The following recovery theorems will characterize the power of the tensor restricted isometry constants. The first theorem generalizes Lemma 1.3 in  and Theorem 3.2 in to low-rank tensor recovery.
Suppose for some . Then is the only tensor of rank at most satisfying .
Assume, on the contrary, that there exists a tensor with rank or less satisfying and . Then is a nonzero tensor of rank at most , and . But then we would have which is a contradiction.
The proof of the preceding theorem is identical to the argument given by Candes and Tao and is an immediate consequence of our definition of the constant . No adjustment is necessary in the transition from sparse vectors and low-rank matrices to low-rank tensors. The key property used is the sub-additivity of the rank. Adapting results in [4, 6], we give the uniform recovery conditions for (1) below.
For any nonzero , , let be the HOSVD of . From Proposition 3.7 of  we have , thus we have . We decompose , where
and is the tensor obtained by fixing the n-th index to the index set , others to zero. Similarly we have . Due to the unitarily invariant of the Frobenius norm, we have , , …. Let and , , ,…, where is the -th largest mode- singular value. From the definition of HOSVD and Lemma 2, we have
Due to the mean-inequation, one has . Assume that with some . Then we have .
From (23) we have
So we have
Since , we have , by the above equations we have
We have a quadratic polynomial of with in the right-hand side of the above inequality. Hence, by calculus, this quadratic polynomial achieves its maximal value at . Therefore we obtain , where
, then , we get , which is
Next we carry out a similar study for the augmented model (4).
The proof of Theorem 8 establishes that any nonzero satisfies . Hence, if , notice , we have
For , we obtain , which proves the theorem.
Different values of