Pattern Recognition Framework Based on the Best Rank Tensor Decomposition

can be seen as a K-dimensional cube of data, in which each dimension correspond to a different factor of the input data space. With this definition, the j-mode vector of the K-th order tensor is a vector obtained from elements of $$\mathcal{T}$$ by varying only one its index n j while keeping all other indices fixed. Further, if from the tensor $$\mathcal{T}$$ the matrix T (j) is created, where





$${{\mathbf{T}}_{\left( j \right)}}\in {{\Re }^{{{N}_{j}}\times \left( {{N}_{1}}{{N}_{2}}\ldots {{N}_{j-1}}{{N}_{j+1}}\ldots {{N}_{K}} \right)}},$$

(1)

then columns of T (j) are j-mode vectors of $$\mathcal{T}$$ . Also, T (j) is a matrix representation of the tensor $$\mathcal{T}$$ , called a j-mode tensor flattening. The j-th index becomes a row index of T (j), whereas its column index is a product of all the rest K-1 indices. An analysis of sufficient computer representations of (1) are discussed in many publications, for instance in [5].

Further important concept is a p-mode product of a tensor $$\mathcal{T}\in {{\Re }^{{{N}_{1}}\times {{N}_{2}}\times \ldots {{N}_{K}}}}$$ with a matrix $$\mathbf{M}\in {{\Re }^{Q\times {{N}_{p}}}}$$ . A result of this operation is the tensor $$\mathcal{S}\in {{\Re }^{{{N}_{1}}\times {{N}_{2}}\times \ldots {{N}_{p-1}}\times Q\times {{N}_{p+1}}\times \ldots {{N}_{K}}}}$$ whose elements are obtained based on the following scheme:




$${{\mathcal{S}}_{{{n}_{1}}{{n}_{2}}\ldots {{n}_{p-1}}q{{n}_{p+1}}\ldots {{n}_{K}}}}={{\left( \mathcal{T}{{\times }_{p}}\mathbf{M} \right)}_{{{n}_{1}}{{n}_{2}}\ldots {{n}_{p-1}}q{{n}_{p+1}}\ldots {{n}_{K}}}}=\sum\limits_{{{n}_{p}}=1}^{{{N}_{p}}}{{{t}_{{{n}_{1}}{{n}_{2}}\ldots {{n}_{p-1}}{{n}_{p}}{{n}_{p+1}}\ldots {{n}_{K}}}}{{m}_{q{{n}_{p}}}}.}$$

(2)

As was shown, the p-mode product can be equivalently represented in terms of the flattened versions of the tensors T (p) and S (p). That is, if the following holds




$$\mathcal{S}=\mathcal{T}{{\times }_{p}}\mathbf{M},$$

(3)

then we have the following




$${{\mathbf{S}}_{\left( p \right)}}=\mathbf{M}\,{{\mathbf{T}}_{\left( p \right)}}$$

(4)

In some computations, it is more efficient to represent the tensor and matrix product given in (2) in an equivalent representation based on the p-mode tensor flattening and the Kronecker product. That is,




$$\mathcal{T}=\mathcal{Z}{{\times }_{1}}{{\mathbf{S}}_{1}}{{\times }_{2}}{{\mathbf{S}}_{2}}\ldots {{\times }_{K}}{{\mathbf{S}}_{K}},$$

(5)

can be equivalently represented as follows




$${{\mathbf{T}}_{\left( n \right)}}={{\mathbf{S}}_{n}}{{\mathbf{Z}}_{\left( n \right)}}{{\left[ {{\mathbf{S}}_{n+1}}\otimes {{\mathbf{S}}_{n+2}}\otimes \ldots \otimes {{\mathbf{S}}_{K}}\otimes {{\mathbf{S}}_{1}}\otimes {{\mathbf{S}}_{2}}\otimes \ldots \otimes {{\mathbf{S}}_{n-1}} \right]}^{T}},$$

(6)

where ⊗ denotes the Kronecker product between the matrices.

Equipped with the above concepts, the best rank-(R 1 , R 2 , …, R K ) decomposition of a tensor $$\mathcal{T}\in {{\Re }^{{{N}_{1}}\times {{N}_{2}}\times \ldots {{N}_{K}}}}$$ can be defined as a problem of computying a tensor $$\tilde {\cal T}$$ , which is characteristic of the ranks $$ran{k_1}\left( {\tilde {\cal T}} \right) = {R_1}$$ , $$ ran{k_2}\left( {\tilde {\cal T}} \right) = {R_2}$$ , …, $$ran{{k}_{K}}\left( {\tilde{\mathcal{T}}} \right)={{R}_{K}}$$ , and which as close as possible approximates to the input tensor $$\mathcal{T}$$ [11, 12], that is the following functional should be minimized:




$$E\left( {\tilde{\mathcal{T}}} \right)=\left\| \tilde{\mathcal{T}}-\mathcal{T} \right\|_{F}^{2},$$

(7)

where ||.|| F denotes the Frobenius norm . It can be shown that the approximated tensor $$\tilde {\cal T}$$ conveys as much of the “energy”, in the sense of the squared entries of a tensor, as the original tensor $${\cal T}$$ , under the requested rank constraints. A value of E is called the reconstruction error. Figure 1 depicts the best rank-(R 1 , R 2 , R 3 ) decomposition of a 3D tensor $$\mathcal{T}\in {{\Re }^{{{N}_{1}}\times {{N}_{2}}\times {{N}_{3}}}}$$ . However, contrary to the rank definition of the matrices, there are different rank definitions for tensors. For more discussion see [5, 11].



A329170_1_En_6_Fig1_HTML.gif


Fig. 1
Schematic representation of the best rank-(R 1 , R 2 , R 3 ) decomposition of a 3D tensor

It can be also easily observed that the assumed rank conditions mean that the approximation tensor $$\tilde {\cal T}$$ can be decomposed as follows




$$\tilde{\mathcal{T}} =\mathcal{Z}{{\times }_{1}}{{\mathbf{S}}_{1}}{{\times }_{2}}{{\mathbf{S}}_{2}}\ldots {{\times }_{K}}{{\mathbf{S}}_{K}},$$

(8)

Each of the matrices $${{\mathbf{S}}_{1}}\in {{\Re }^{{{N}_{1}}\times {{R}_{1}}}}$$ , $${{\mathbf{S}}_{2}}\in {{\Re }^{{{N}_{2}}\times {{R}_{2}}}}$$ , …, and $${{\mathbf{S}}_{K}}\in {{\Re }^{{{N}_{K}}\times {{R}_{K}}}}$$ in (8) has orthonormal columns. The number of columns for S i is given by R i .

The core tensor $$\mathcal{Z}\in {{\Re }^{{{R}_{1}}\times {{R}_{2}}\times \ldots \times {{R}_{K}}}}$$ is of dimensions R 1 , R 2 , …, R K . It can be computed from the original tensor $${\cal T}$$ as follows




$$\mathcal{Z} =\mathcal{T}{{\times }_{1}}\mathbf{S}_{1}^{T}{{\times }_{2}}\mathbf{S}_{2}^{T}\ldots {{\times }_{K}}\mathbf{S}_{K}^{T}$$

(9)

Summarizing, to find the best rank-(R 1 , R 2 , …, R K ) approximation of $${\cal T}$$ it is sufficient to determine only a set of S i in (8), and then $${\cal Z}$$ is computed from Eq. (9).

Further analysis is constrained exclusively to 3D tensors, such as the one shown in Fig. 1. It can be seen that this decomposition leads to a significant data reduction. The compression C ratio can be expressed as follows:




$$C={\left( {{R}_{1}}{{R}_{2}}{{R}_{3}}+{{N}_{1}}{{R}_{1}}+{{N}_{2}}{{R}_{2}}+{{N}_{3}}{{R}_{3}} \right)}/{\left( {{N}_{1}}{{N}_{2}}{{N}_{3}} \right)}\;.$$

(10)

As alluded to previously, the only control parameters of the method are the ranks R 1 , R 2 , and R 3 . A trade-off can be achieved between the compression ratio C in (10) with respect to the approximation error expressed in Eq. (7). This influences also pattern recognition accuracy, as will be discussed.



2.2 Pattern Classification with the Best Rank Tensor Decomposition


The already described, the subspace obtained after the best rank decomposition can be used to generate specific features of an image X, which can be then used for pattern recognition [16]. The features are obtained by projecting the image X of dimensions N 1 ×N 2 into the space spanned by the two matrices S 1 and S 2 in accordance with (9). However, at first the pattern X needs to be represented in an equivalent tensor form $$\mathcal{X}$$ which is of dimensions N 1 × N 2 × 1. Then, the feature tensor $$\mathcal{F}$$ of dimensions R 1 × R 2 × 1 is obtained by projecting $$\mathcal{X}$$ onto the space spanned by S 1 and S 2, as follows

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Jun 14, 2017 | Posted by in GENERAL SURGERY | Comments Off on Pattern Recognition Framework Based on the Best Rank Tensor Decomposition

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