T product Tensors Part I: Inequalities

The T product operation between two three order tensors was invented around 2011 and it arises from many applications, such as signal processing, image feature extraction, machine learning, computer vision, and the multiview clustering problem. Although there are many pioneer works about T product tensors, there are no works dedicated to inequalities associated with T product tensors. In this work, we first attempt to build inequalities at the following aspects: (1) trace function nondecreasing and convexity; (2) Golden Thompson inequality for T product tensors; (3) Jensen T product inequality; (4) Klein T product inequality. All these inequalities are related to generalize celebrated Lieb concavity theorem from matrices to T product tensors. This new version of Lieb concavity theorem under T product tensor will be used to determine the tail bound for the maximum eigenvalue induced by independent sums of random Hermitian T product, which is the key tool to derive various new tail bounds for random T product tensors. Besides, Qi et. al introduces a new concept, named eigentuple, about T product tensors and they apply this concept to study nonnegative (positive) definite properties of T product tensors. The final main contribution of this work is to develop the Courant Fischer Theorem with respect to eigentuples, and this theorem helps us to understand the relationship between the minimum eigentuple and the maximum eigentuple. The main content of this paper is Part I of a serious task about T product tensors. The Part II of this work will utilize these new inequalities and Courant Fischer Theorem under T product tensors to derive tail bounds of the extreme eigenvalue and the maximum eigentuple for sums of random T product tensors, e.g., T product tensor Chernoff and T product tensor Bernstein bounds.


T-product Tensors
The T-product operation between two three order tensors was introduced by Kilmer and her collaborators in [2,3]. It has been shown as a powerful tool in many fields: signal processing [4,5], machine learning [6], computer vision [7,8], image processing [9], low-rank tensor approximation [10][11][12] etc. Due to wide applications of T-product, T-SVD and tubal ranks, Qi et. al [1]. extend eigentuple concept first defined from [13] to study properties for symmetry of T-product tensors and positive (nonnegative) semidefiniteness of T-product tensors by defining a T-quadratic form, whose variable is an m × p matrix, and whose value is a p-dimensional vector. They further show that a T-quadratic form is positive semidefinite (definite) if and only if the smallest eigentuple of the corresponding T-symmetric tensor is nonnegative (positive). Besides T-quadratic form, general functions for T-product tensors and their properties are also studied based on T-SVD, see [14][15][16][17]. However, none of these works discussed further issues about inequalities associated with T-product tensors. The first part of this work about T-product tensors is to develop several new T-product tensors inequalities, which are the main topics discussed by this paper. In the matrix setting, there are many useful applications about these similar inequalities under the traditional matrix product, e.g., quantum information processing [18]. The second part of this work is to apply these new inequalities about T-product tensors to tail bounds estimation of the maximum eigenvalue and the maximum eigentuple for sums of random T-product tensors, e.g., Chernoff and Bernstein bounds. We will introduce these new inequalities about T-product tensors obtained at this Part I work at the next subsection.

New Inequalities about T-product Tensors
In this work, we define trace, denoted by Tr, as the summation of f-diagonal entries of a given symmetric T-product tensor C ∈ C m×m×p and study properties of trace for T-product tensors. Our first main inequality about trace is following theorem: Theorem 1.1 (Monotonicity and Convexity of T-product Trace Function) Let f : R → R be a continuous function with non-decreasing / convex / strictly convex properties, then so is the mapping C → Tr (f (C)).
From trace definition, we will prove Golden-Thompson inequality for two Hermitian T-product tensors which will be utilized to prove T-product tensors martingale inequalities. A Hermitian T-product tensor is a tensor equal to its Hermiitan transpose, which is defined by Eq. (11). Theorem 1.2 (Golden-Thompson inequality for T-product Tensors) Given two Hermitian T-product tensors C, D ∈ C m×m×p , we have where ⋆ is the product operation between two T-product tensors defined by Eq. (15).
The next inequality we will show is Jensen's operator inequality (positive semidefinite relation between two T-product tensors). A tensor with T-positive definite (or T-positive semi-definite) will be abbreviated as TPD (or TPSD), see Section 2.3 for its definition. If we have a TPSD relation between two T-product tensors C and D represented as C D, then the difference T-product tensor (D − C) is a T-positive semi-definite tensor. Let I mmp ∈ C m×m×p be the identity tensor defined by Eq. (13).
where X i ∈ C m×m×p are bounded, Hermitian T-product tensors with all eigenvalues in the interval I and The immediate application of Theorem 1.1 is to prove Klein's inequality for T-product tensor.
In both situations, if f is strictly convex, equality holds if and only if C = D.
Previous theorems will help us to establish the following main theorem of this paper about Lieb's concavity for T-product tensors since tail bounds for sums of random T-product tensors will be derived based on such concavity relation.

Theorem 1.5 (Lieb's concavity theorem for T-product tensors) Let H be a Hermitian T-product tensor. Following map
is concave on the positive-definite cone.
We are ready to present the theorem for the tail bound of the maximum eigenvalue induced by independent sums of random Hermitian T-product tensors and this theorem will play a key role to establish various new tail bounds of the maximum eigenvalue generated by independent sums of random T-product tensors. Theorem 1.6 (Master Tail Bound for Independent Sum of Random T-product Tensors for Eigenvalue) Given a finite sequence of independent Hermitian T-product tensors {X i }, we have Similarly, we can generalize master tail bound for independent sum of random Hermitian T-product tensors with respect to eigenvalue from Theorem 1.6 to eigentuple version by the following theorem 1.7. We begin with operation defined in Proposition 2.1 from work [1]. Let a = (a 1 , a 2 , · · · , a p ) T ∈ C p , then operator circ to the vector a can be defined as and circ −1 (circ(a)) def = a. Suppose that a, b ∈ C p , we define where · is the standard matrix and vector multiplication. Then, we are ready to present the following theorem. Tail Bound for Independent Sum of Random T-product Tensors for Eigentuple)  Given a finite sequence of independent random Hermitian T-product

Theorem 1.7 (Master
where e tb ∈ C p is the exponential for the vector tb with respect to operation.
The last important theorem is the Courant-Fischer theorem for T-product tensors. This theorem will be used to figure out the relationship between the maximum eigentuple and the minimum eigentule of a T-product tensor.
where is the division (inverse operation) under .
All these inequalities and maximum/minimum eigentuples relation about T-product tensors will be utilized to derive a serious of new tail bounds for the extreme eigenvalue and eigentuple for sums of random T-product tensors. These new inequalities different from author Chang's previous works about bounds for sums of random tensors based on Einstein product [19,20].

Paper Organization
The rest of this paper is organized as follows. In Section 2, basic notions of T-product tensors are introduced. Lieb's concavity theorem under T-product will be studied in Section 3. General tail bounds for random Tproduct tensors are provided in Section 4. Courant-Fischer Theorem under T-product is given in Section 5. Finally, conclusion will be drawn in Section 6. Nomenclature: The sets of complex and real numbers are denoted by C and R, respectively. The symbol def = denotes mathematical definition.

T-product Tensors
In this section, we will review T-product operations briefly and discuss related properties in Sec. 2.1. The T-SVD decomposition of T-product tensors and T-Symmetric tensors will be presented in Sec. 2.2

What are T-product Tensors
For a third order tensor C ∈ C m×n×p , we define bcirc operation to the tensor C as: where C (1) , · · · , C (p) ∈ C m×n are frontal slices of tensor C. The inverse operation of bcirc is denoted as bcirc −1 with relation bcirc −1 (bcirc(C)) def = C. For a third order tensor C ∈ C m×m×p , we define Hermitian transpose of C, denoted by C H , as And a tensor D ∈ C m×m×p is called a Hermitian T-product tensor if D H = D. Similarly, we also define ranspose of C, denoted by C T , as And a tensor D ∈ C m×m×p is called a Symmetric T-product tensor if D T = D.
The identity tensor I mmp ∈ C m×m×p can be defined as: where I mp is the identity matrix in R mp×mp . A zero tensor, denoted as O mnp ∈ C m×n×p , is a tensor that all elements inside the tensor as 0.
In order to define the T-product operation, we need to define another kind of operation over a third order tensor. For a third order tensor C ∈ C m×n×p , we define unfold operation to the tensor C as: (2) . . .
where unfold(C) ∈ C mp×n , and the inverse operation of unfold is fold with the relation fold(unfold(C)) def = C. Given C ∈ C m×n×p and D ∈ C n×k×p , we define the T-product between C and D as where C ⋆ D ∈ C m×k×p .
where ≥ is the elementwise comparison between two vectors.
We define the T-product tensor trace for a tensor C = (c ijk ) ∈ C m×m×p , denoted by Tr(C), as following which is the summation of all entries in f-diagonal components. Then, we have following lemma about trace properties.

Lemma 1
For any tensors C, D ∈ C m×m×p , we have where c, d are two contants. And, the transpose operation will keep the same trace value, i.e., Finally, we have Proof: Eqs. (17) and (18) are true from trace definiton directly. From Eq. (15), the i-th frontal slice matrix of D ⋆ C is similarly, the i-th frontal slice matrix of C ⋆ D is Because the matrix trace of Eq. (20) and the matrix trace of Eq. (21) are same for each slice i due to linearity and invariant under cyclic permutations of matrix trace, we have Eq. (19) by summing over all frontal matrix slices.
Below, we will define the determinant of a T-product tensor C ∈ C m×m×p and its asscoiate properties. The determinant of a m × m × p tensor C is the m-linear alternating form defined as where V i ∈ C m×p is the i-th lateral matrix of the tensor C. Moreover, we require that det(I mmp ) = 1.
Given two tensors C, D ∈ C m×m×p , the determinant of C ⋆ D is det(C ⋆ D) = λdet(D) for some value λ.
If we set D as I mmp , we have Then, we have

T-SVD Decomposition
Given a tensor C ∈ C m×n×p , Theorem 4.1 in [2] proposed a T-singular value decomposition (T-SVD) for C as: where U ∈ C m×m×p and V ∈ C n×n×p are orthogonal tensors, and S ∈ C m×n×p is a f-diagonal tensor. We also have U T ⋆ U = I mmp and V T ⋆ V = I nnp . We define σ(C) be the spectrum of C, i.e., the set of s ∈ C, where s are nonzero entries from tensor S. We use · for the spectral norm, which is the largest singular value of a T-product tensor. Given any integer k and B ∈ C m×m×p , we define B k as where B k ∈ C m×m×p . Then, we have following corollary from T-SVD in Eq. (25).
Corollary 1 Suppose B ∈ C m×m×p is a Hermitian T-product tensor, and S −1 exists, where S is f-diagonal tensor obtained from the T-SVD of the tensor C. Then, we have Then, we can define the T-product tensor exponential function and the T-product tensor logarithm function under T-product as below with tensor power.
Definition 2 Given a tensor X ∈ C m×m×p , the tensor exponential of the tensor X is defined as where X 0 is defined as the identity tensor I mmp . Given a tensor Y, the tensor X is said to be a tensor logarithm of Y if e X = Y.
From T-SVD in Eq. (25), we can express a Hermitian T-product tensor C ∈ C m×m×p as where s iik are eigenvalues of the tensor C, and U i is obtained from the i-th lateral slice (matrix) of the tensor U with column vectors as u Note that we have U From Corollary 1 and Eq. (29), we can have following spectral mapping lemma.

Lemma 2
For any continous function f : R → R and any Hermitian T-product tensor C, we have (31)

Positive Semidefinite T-product Tensors
Given a Hermitian T-product tensor C ∈ C m×m×p , and a tensor X ∈ C m×1×p obtained from treating the matrix X ∈ C m×p as a tensor with dimensions R m×1×p . We define following quadratic form with respect to the matrix X as and we say that a tensor C is T-positive definite (TPD) (or T-positive semi-definite (TPSD)) if F C (X) > 0 (or F C (X) ≥ 0 ) for any X ∈ C m×p , where 0 is a zero vector with size p.
We now define eigentuples and eigenmatrices of a Hermitian T-product tensor which will be used to characterize TPD or TPSD for a given tensor. For a matrix X ∈ C m×p = (x (1) , · · · , x (p) ) , we define unfolding opeartion with respect to the matrix X columns, denoted by cunfold(X), as (1) x (2) . . .
where cunfold(X) ∈ C mp . Then, suppose that C ∈ C m×m×p is a Hermitian T-product tensor, we define C ⋆ X as where C ⋆ X ∈ C m×p . We also define a new product operation between a vector d = (d 1 , d 2 , · · · , d p ) T and a matrix X ∈ C m×p , denoted by •, as where · is the standard matrix multiplication. Suppose that X ∈ C m×p and X = O, and d ∈ C p , if we have we call d as an eigentuple of C, and X as an eigenmatrix of C corresponding to the eigentuple d. From Theorem 4.1 in [1], a T-square tensor C ∈ C m×m×p with eigentuples arranged as f-diagonal tensor S according to the standard form provided by Definition 1, i.e., s 1 ≥ s 2 ≥ · · · ≥ s m . Then C is TPD (or TPSD) if and only if the smallest eigentuple s m > (or ≥ )0. We use C vec to represent the spectral norm of eigentuple of the tensor C, which is defined as

T-product Tensors Analysis
We will begin with monotonicity and convexity discussions of the trace function.

Lemma 3 For a given continous and non-decreasing function f : R → R, the associated trace function on a Hermitian T-product tensor C is given by
Then we have Proof: We first assume that the function f is differentiable, then the first derivative of f is greater or equal than zero (monotonicity). We further define a trace function g(t) def = Tr (f (D + t(C − D))). Then, we have where we apply Lemma 1 at the last equality, and the last inequality comes from the nonnegative of f ′ . By applying the standard continuity argument, we can relax the requirement that f is continuously differentiable to the requirement that f is continuous. The next lemma will be used to show the convexity of trace function on a Hermitian T-product tensor C.
where V i is an eigenmatrix of C and it's the only case if f is strictly convex.
Proof: From spectral representation by Eq. (29), we have where the only inequality comes from the convexity of the function f . Since for each i, k, we have for some i ′ , k ′ , and is 0 otherwise.
From Lemma 3 and Lemma 4, we have the following theorem about convexity and monotonicity of a trace function. We recall theorem 1.1.

Theorem 1.1 (Monotonicity and Convexity of T-product Trace Function)
Let f : R → R be a continuous function with non-decreasing / convex / strictly convex properties, then so is the mapping C → Tr (f (C)).
Proof: Given two C, D ∈ C m×m×p Hermitian T-product tensors, f as a convex function, and V

[k]
i for 1 ≤ i ≤ m and 0 ≤ k ≤ p − 1 be an orthonormal basis of C m×p consisting of eigenmatrices of C+D 2 . Then, from Lemma 4, we have where inequalities come from Lemma 4. This demonstrates that the map C → Tr (f (C)) is midpoint convex.
For the strict convexity of f and Tr f ( i . From Lemma 4, the equality will be true when V [k] i are eigenmatrices for both tensors C and D. Then, we have which indicates that C = D. An obvious continuity argument now shows that if f continuous as well as convex, C → Tr (f (C)) is convex, and strictly convex so if f is strictly convex. Therefore, this Theorem is proved from Lemma 3 and above arguments.
From T-SVD, we have following relation for Hermitian T-product tensor: Above Eq. (46) is named as transfer rule. We have defined tensor exponential under Definition 2, and the exponential of an Hermitian T-product tensor is always TPD due to the spectral mapping Lemma 2. From transfer rule Eq. (46), the tensor exponential satisfies following relations for a Hermitian T-product tensor C ∈ C m×m×p that we will use at later theory development: and From Theorem 1.1 and the monotonicity of the exponential function, we have Below, we want to prove the monotonicity and the concavity of the logarithm function. We will begin definitions about T-product tensor monotonicity and convexity first and present several lemmas used to establish the monotonicity and the concavity of the logarithm function. Given two Hermitian T-product tensors C, D ∈ C m×m×p , a function f : R → R is said to be T-product tensor monotone if the following relation holds: A function f is said as a T-product tensor convex function if we have: where 0 ≤ t ≤ 1. Also, a function f is said as a T-product tensor concave function if −f is a T-product tensor convex function. The following derivation about the monotonicity and the concavity of the logarithm function is extended from matrices according to works in [21] and [22] to T-product tensors.
Proof: Since eigenvalues are roots of the characteristic polynomial, it is enough to show that det(λI mmp − C ⋆ D) = det(λI mmp − D ⋆ C). We first assume that C has inverse, then from Eq. (24), we have This shows that σ(C ⋆ D) = σ(D ⋆ C).
If C is not invertible, we choose a sequence {ǫ n } in C\σ(C) with ǫ n → 0, with property that all new tensors C n def = C − ǫ n I mmp are invertibale for each n. Then, Lemma 6 For every tensor C ∈ C m×m×p and every function f on σ(C H ⋆ C), we have Proof: Since σ(C H ⋆ C) = σ(C ⋆ C H ) from Lemma 5 and C ⋆ C H ⋆ C n = C ⋆ C H n ⋆ C, for n ∈ N, this lemma is hold for f is a polynomial. If the function f is an arbitary function on σ(C H ⋆ C) = [s 1 , · · · s n ], we define the Lagrance interpolation polynomial as where we have p(s i ) = f (s i ) for 1 ≤ i ≤ n. Then, we also have and this Lemma is proved. Following Lemma is adopted from Corollary 12 from [22].
is T-product tensor monotone function; 2.
x/f (x) is T-product tensor monotone function; is T-product tensor concave function;

1/f (x) is T-product tensor convex function.
Proof: The proof is based in Corollary 12 from [22]. But those facts about using Theorem 2.5.2 and Theorem 2.5.3 from [21] should be modified from matrices settings to T-product tensors settings. With help from Lemma 6 and transfer rules provided by Eq. (46), the proof about Theorem 2.5.2 and Theorem 2.5.3 from [21] for T-product tensors is straightforward by replacing matrix multiplication opertions to T-product operations.
We are ready to prove that the logarithmic function is T-product tensor monotone and concave function on (0, ∞).
In general, it is not practical to always working with Hermitian T-product tensor, we will apply dilations techqnique to expand any Ttensor into a Hermitian T-product tensor. For any tensor C ∈ C m×n×p , a dilation for the tensor C, denoted as D(C), will be where D(C) ∈ C (m+n)×(m+n)×p and we have (D(C)) H = D(C) (Hermitian T-product tensor after dilation). Also, we have Since Eq. (59) is zero trace, the largest eigenvalue of D(C) will be the same with the largest singular of C.
Since the expectation of a random T-product tensor can be considered as a convex combination, expectation preserves the TPSD order as: Also from Lemma 7, we know that the quadratic function f (x) = x 2 is T-product tensor convex, thus, we have We will present one more theorem in this section about Golden-Thompson inequality for T-product tensors. We recall theorem 1.2 Theorem 1.2 (Golden-Thompson inequality for T-product Tensors) Given two Hermitian T-product tensors C, D ∈ C m×m×p , we have where ⋆ is the product operation between two T-product tensors defined by Eq. (15).
Proof: From T-SVD decomposition and Eq. (29), we can express the tensor C as where λ are eigenvalues and P λ are corresponding projectors (T-product tensors) which are mutually orthgonal. Given X O, we define following mapping with respect to the tensor C as : Then, we have following properties about mapping P C (X ) 1. P C (X ) commutes with C; 2. Tr (P C (X ) ⋆ C) = Tr (X ⋆ C); The third property of P C (X ) is true due to the following relation: where Let he equality = 1 comes from the fact that the trace is multiplicative under the Kronecker product. The inequality ≤ 2 follows from inequality from the third property of P A ⊗m 2 (A ⊗m 1 ), the monotone of log and Tr exp ( ) functions, and the number of eigenvalues of A ⊗n 2 growing polynomially with n due to the fact that the number of distinct eigenvalues of A ⊗n 2 is bounded by the number of different types of sequences of mp symbols of length n, see Lemma II.1 in [23]. The equality = 3 utilizes the commutativity property for tensors P A ⊗m 2 (A ⊗m 1 ) and A ⊗m 2 based on the first property. Finally, the equality = 4 applies trace properties from the second property of the mapping P A ⊗m 2 (A ⊗m 1 ). If n → ∞, the result of this theorem is established.

Lieb's Concavity Under T-product
In this section, we will extend several trace inequalities to T-product tensors: Jensen's T-product tensor inequality in Section 3.1 and Klein's T-product tensor inequality in Section 3.2. These new T-product tensor inequalities will play important roles in establishing a new version of Lieb's concavity theorem under Tproduct in Section 3.3.

Jensen's T-product Inequality
In this subsection, we will derive Jensen's T-product tensor inequality in Theorem 1.3. We begin with a lemma which will be used in later proof in Thereom 1.3. Given two natural numbers m, n, we define a T-product tensor θ ∈ C m×m×p as exp(2π √ −1/n)×I mmp . Then, we can have tensor D ∈ C mn×mn×p obtained by where diag θ, θ 2 , · · · , θ n−1 , I mmp will be a matrix with entries as T-product tensors, and the diagonal part of this matrix is compsoded by tensors (θ, θ 2 , · · · , θ n−1 , I mmp ). Let D ∈ C mn×mn×p be another matrix of T-product tensor, i.e., entries d i,j as T-product tensors. We define a new operation * between two T-product tensors, A, B with dimensions belong to C mn×mn×p as: where both a i,k and c k,j are T-product tensors. Therefore, given any tensor C ∈ C mn×mn×p , the i, j-th entry (a T-product tensor) of C * D becoms exp(2π √ −1j/n) × c i,j , where c i,j ∈ C m×m×p is a T-product tensor.

Lemma 9
Given any tensor C ∈ C mn×mn×p and the tensor D defined by Eq. 67, we have where D k is the self-product of the tensor D by * operation k-times.
Proof: By direct computation with * , we have following: where this summation is zero for i = j, otherwise, it is c i,i . We are ready to prove theorem 1.3. Eq. (51). we have the following TPSD relation for each natural number n:

Theorem 1.3 (Jensen's T-product Inequality) For a continuous T-product tensor convex function f defined on an interval I. The definition of T-product tensor convex is given by
where X i ∈ C m×m×p are bounded, Hermitian T-product tensors with all eigenvalues in the interval I and Proof: Let us define a unitary tensor U = (u i,j ) ∈ C mn×mn×p for 1 ≤ i, j ≤ n as u i,j = C i , D = diag θ, θ 2 , · · · , θ n−1 , I mmp defined by Eq. (67), and define the tensor X ∈ C mn×mn×p as diag (X 1 , · · · , X n ). From Lemma 9, we have where the inequality comes from that the function f is a T-product tensor convex function.

Klein's T-product Inequality
The immediate application of Theorem 1.1 is to prove Klein's inequality for T-product tensor. We recall theorem 1.4.

Theorem 1.4 (Klein's T-product Inequality) For all C, D Hermitian T-product tensors and a differen-
tiable convex function f : R → R or for all C, D Hermitian T-product tensors and a differentiable convex function f : (0, ∞) → R, we have

In both situations, if f is strictly convex, equality holds if and only if C = D.
Proof: We define function F (t) as where t ∈ (0, 1). From Theorem 1.1, F (t) is a convex function. Then, we have then we obatin Klein's ineqaulity under T-product by rearrangement and substitution with Eq. (72).

Lieb's Concavity Theorem Under T-product
In this section, we will extend Lieb's concavity theorem to T-product tensors and we begin with the definition about the relative entropy between two T-product tensors.

Definition 3 Given two TPD tensors A ∈ C m×m×p and tensor B ∈ C m×m×p . The relative entropy between two T-product tensors A and B is defined as
We apply perspective function concept for T-product tensor convex and introduce the following lemma about the convexity of a T-product tensor convex function [24].
Lemma 10 Given f as a convex function, two commuting tensors X , Y ∈ C m×m×p , i.e., X ⋆ Y = Y ⋆ X , and the existence of the Y −1 , then the following map h is jointly convex in the sense that, given t

then we have
where ≤ 1 is based on the condition provided by Eq. (78) and Theorem 1.3.
Following lemma is given to establish the joint convexity property of relative entropy for T-product tensors.

Lemma 11 (Joint Convexity of Relative Entropy for T-product Tensors) The relative entropy function of two TPD tensors is a jointly convex function. That is
where t ∈ [0, 1] and all the following four tensors A 1 , B 1 , A 2 and B 2 , are TPD tensors.
Proof: From the definition 3, we wish to show the joint convexity of the function D(A B) with respect to the tensors A, B ∈ C m×m×p . Let us define tensor operators F(X ) def = A ⋆ X and G(X ) def = X ⋆ B for the variable tensor X ∈ C m×m×p . Then, we have F(X ) and G(X ) commuting on the inner product operation F(X ), G(X ) defined as: is jointly convex with respect to tensors A and B.
Lieb's concavity theorem is recalled below by theorem 1.5.

Theorem 1.5 (Lieb's concavity theorem for T-product tensors) Let H be a Hermitian T-product tensor. Following map
is concave on the positive-definite cone.
Proof: From Klein's inequality obtain from Theorem 1.4, the convexity of map t → t log t (which is strictly concave for t > 0) and Hermitian T-tensors X , Y, we have If we replace Y by e H+log A , we then have where D(X A) is the quantum relative entropy between two tensor operators. For real number t ∈ [0, 1] and two positive-definite tensors A 1 , A 2 , we have TrX H − D(X tA 1 ) + TrX where the first and last equalities are obtained based on the variational formula provided by Eq. (84), and the inequality is due to the joint convexity property of the relative entropy from Leamm 11. Based on Lieb's concavity theorem for T-product tensors, we have the following corollary.

Corollary 2 Let A be a fixed Hermitian T-product tensor, and let X be a random Hermitian T-product tensor, then we have
Proof 2 Define the random tensor Y = e X , we have where the inequality is based on Lieb's concavity theorem for T-product tensors obtained by Theorem 1.5 and Jensen's T-product tensor inequality by Theorem 1.3.

T-product Tensor Moments and Cumulants
Since the expectation of a random T-product tensor can be treated as convex combination, expectation will preserve the semidefinite order as From Jensen's T-product tensor inequality by Theorem 1.3, we also have Suppose a random Hermitian T-product tensor X having tensor moments of all orders, i.e., E(X n ) existing for all n, we can define the tensor moment-generating function, denoted as M X (t), and the tensor cumulant-generating function, denoted as K X (t), for the tensor X as where t ∈ R. Both the tensor moment-generating function and the tensor cumulant-generating function can be expressed as power series expansions: where ψ n is called tensor cumulant. The tensor cumulant ψ n can be expressed as a polynomial in terms of tensor moments up to the order n, for example, the first cumulant is the mean and the second cumulant is the variance: Finally, in this work, we also assume that all random variables are sufficiently regular for us to compute their expectations, interchange limits, etc.

Tail Bounds By Concatenation of Lieb's Concavity
The goal of this section is to develop several important tools which will be applied intensively in the proof of probability inequalities for the extreme eigentule (or eigenvalue) of a sum of independent random T-product tensors. The first tool is the Laplace transform method for T-product tensors discussed in Section 4.1, and the second tool is the tail bound for independent sums of random Hermitian T-product tensors presented by Section 4.2.

Laplace Transform Method for T-product Tensors
We extend the Laplace transform bound from matrices to T-product tensors based on [25]. Following lemma is given to establish the Laplace transform bound for the maximum eigenvalue of a T-product tensor.
Lemma 12 (Laplace Transform Method for T-product Tensors: Eigenvalue Version) Let X be a random Hermitian T-product tensor. For θ ∈ R, we have Proof 3 Given a fix value t, we have The first equality uses the homogeneity of the maximum eigenvalue map, the second equality comes from the monotonicity of the scalar exponential function, and the last relation is Markov's inequality. Because we have where the first equality used the spectral mapping theorem from Lemma 2, and the inequality holds because the exponential of an Hermitian T-product tensor is TPD and the maximum eigenvalue of a TPD tensor is dominated by the trace from Eq. (16). From Eqs (94) and (95), this lemma is established.
The Lemma 12 helps us to control the tail probabilities for the maximum eigenvalue of a random Hermitian T-product tensor by utilizing a bound for the trace of the tensor moment-generating function introduced in Section 3.4.
Since T-product tensors also have notions about eigentuples, we then extend Lemma 12 from eigenvalues version to eigentuples version. We begin with the derivation of Markov's inequality for random vectors.
Lemma 13 (Markov's inequality for Random Vector) If X ∈ R p is a nonnegative random vector and a > 0, then the probability that X is at least a = [a i ] can be bounded as: Proof: Because we have therefore, we have the desired inequality shown by Eq. (96).
We are ready to present following lemma about Laplace transform method for T-product tensors with eigentuples.

Lemma 14 (Laplace Transform Method for T-product Tensors: Eigentuple Version)
Let X ∈ C m×m×p be a random T-positive definite (TPD) tensor and an all one vector 1 p = [1, 1, · · · , 1] T ∈ C p . Suppose we have where t > 0 1 . Then, for b ∈ R p , we obtain where d max is the maximum eigentuple of the TPD tensor X .
Proof: Given a fix value t, we have The first equality uses the homogeneity of the maximum eigenvalue map, the second equality comes from the monotonicity of the exponential function with operation defined in Proposition 2.1 from work [1], and the last relation is Markov's inequality for random vector obtained from Lemma 13 since both E e dmax(tX ) and e tb are vectors with p entries. Then, we have where the first inequality comes from the relation that d max (tX ) ≤ λ max (tX )1 p , and the second inequality holds because e λmax(tX ) = λ max (e tX ) and the relation 1 p λ p max (e tX ) + 1 − 1 p ≤ Tr(e tX ). From Eqs (100) and (101), this lemma is established.

Tail Bounds for Independent Sums of Random T-product Tensors
This section will present the tail bound for the sum of independent random T-product tensors and several corollaries according to this tail bound for independent sums. We begin with the subadditivity lemma of tensor cumulant-generating functions.

Lemma 15 Given a finite sequence of independent random Hermitian T-product tensors
Proof: We begin with the following definition for the tensor cumulant-generating function for X i as: Then, we define the Hermitian T-product tensor H k as By applying Eq. (104) to Theorem 1.5 repeatedly for k = 1, 2, · · · , n, we have .
where the equality = 1 is based on the law of total expectation by defining E i as the conditional expectation given X 1 , · · · , X i . We are ready to present the theorem for the tail bound of independent sums of random Hermitian Tproduct tensors with respect to the maximum eigenvalue. We recell theorem 1.6 Theorem 1.6 (Master Tail Bound for Independent Sum of Random T-product Tensors for Eigenvalue) Given a finite sequence of independent Hermitian T-product tensors {X i }, we have Proof: By substituting the Lemma 15 into the Laplace transform bound provided by the Lemma 12, this theorem is established.
Several useful corollaries will be provided based on Theorem 1.6.

Corollary 3 Given a finite sequence of independent Hermitian random tensors
and a sequence of non-random Hermitian T-product tensors {A i } with following condition: Then, for all θ ∈ R, we have Proof: From the condition provided by Eq. (106) and Theorem 1.6, we have where the second inequality holds since we bound the trace of a TPD T-product tensor by the dimension size m × p multiplied by the maximum eigenvalue; the last equality is based on the spectral mapping theorem since the function f is nonnegative.

Corollary 4 Given a finite sequence of independent Hermitian random tensors
Proof: From T-tensor logarithm concavity property provided by Lemma 8, we have and from the trace exponential monotone property provided by Lemma 3, we have where the last inequality holds since we bound the trace of a positive-definite tensor by the dimension size m × p multiplied by the maximum eigenvalue and apply spectral mapping theorem twice. Similarly, we can generalize master tail bound for independent sum of random Hermitian T-product tensors for eigenvalue version from Theorem 1.6 to master tail bound for independent sum of random Hermitian T-product tensors for eigentuple version by the following Theorem 1.7.

Theorem 1.7 (Master Tail Bound for Independent Sum of Random T-product Tensors for Eigentuple) Given a finite sequence of independent random Hermitian T-product tensors
where e tb ∈ C p is the exponential for the vector tb with respect to operation.
Proof: By substituting the Lemma 15 into the Laplace transform bound provided by the Lemma 14, this theorem is established.
Some useful corollaries will be provided based on Theorem 1.7.

Corollary 5 Given a finite sequence of independent random Hermitian T-product tensors
Then, for all b ∈ R p and n i=1 tX i satisfing Eq. (98), we have Proof: From the condition provided by Eq. (112) and Theorem 1.7, we have where the second inequality holds since we bound the trace of a TPD T-tensor by the eigenvalue size with m × p multiplied by the maximum eigenvalue; the last equality is based on the spectral mapping theorem since the function f is nonnegative. This corollary is proved. Ee tX i ,

Corollary 6 Given a finite sequence of independent random Hermitian T-product tensors
and from the trace exponential monotone property provided by Lemma 3, we also have where the last inequality holds since we bound the trace of a TPD tensor by the eigenvalue size with m × p multiplied by the maximum eigenvalue; and spectral mapping theorem for log and exp functions.

Courant-Fischer Theorem under T-product Tensors and Minimum Eigenvalue/Eigentuple
In this section, Courant-Fischer theorem for T-product tensors will be proved and this theorem will be used to show the relationship between the maximum eigentuple and the minimum eigentule of TPD T-product tensors. Let us recall theorem 1.8. j } ∈ C m×p be orthnomal matrices for 1 ≤ j ≤ m and 0 ≤ l ≤ p − 1, S k be the space spanned by {U j } for k ≤ j ≤ m and 0 ≤ l ≤ p − 1. Then, we have where is the division (inverse operation) under .
Proof: We will just prove the first characterization of d k . The other can be proved similarly. First, we wish to show that d k is achievable. As S k is the space spanned by {U j } for 1 ≤ j ≤ k and 0 ≤ l ≤ p − 1. For every X ∈ S k , we can express X as where e = (1, 0, · · · , 0) T ∈ C p . To verify that this is the maximum eigentuple, as T k is the space spanned by {U j } for k ≤ j ≤ m and 0 ≤ l ≤ p − 1, for any S k with dimension k × p the intersection of S k with T k is non-empty. Then, we also have Any such X can be expressed as Therefore, all subspace of S k with dimension k × p, we have This theorem is proved since d k is achievable and is the maximum eigentuple. By applying Theorem 1.8, we have following relations: d min (X ) = −d max (−X ) and λ min (X ) = −λ max (−X ) (124)

Conclusion
In this Part I work, we try to establish following inequalities about T-product tensors: (1) trace function nondecreasing/convexity; (2) Golden-Thompson inequality for T-product tensors; (3) Jensen's T-product inequality; (4) Klein's T-product inequality. All these inequalities are used to generalize celebrated Lieb's concavity theorem from matrices to T-product tensors. Then, this new version of Lieb's concavity theorem under T-product tensor is utilized to build master tail bounds for the maximum eigenvalue and the maximum eigentuple induced by independent sums of random Hermitian T-product. In order to find the relationship between the minimum eigentuple and the maximum eigentuple, we also extended the Courant-Fischer Theorem from matrices to T-product tensors. How these new inequalities and Courant-Fischer Theorem under T-product are used to derive new tail bounds of the extreme eigenvalue and eigentuple for sums of random T-product tensors is the main goal of our Part II paper.