Gram matrix

About Gram matrix

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n Any... In a dimensional Euclidean space k A matrix formed by the inner product of two vectors , It's called this k A vector Gram matrix .

Vector inner product formula ：$$\vec{a}\cdot \vec{b}=\sum _{i=1}^n{a}_i{b}_i$$ therefore Gram The formula of a matrix is ：$$Gram({\vec{a}}_1\cdots {\vec{a}}_k)=\left(\begin{array}{ccc}({\vec{a}}_1\cdot {\vec{a}}_1)& \cdots & ({\vec{a}}_1\cdot {\vec{a}}_k)\\ ⋮& \ddots & ⋮\\ ({\vec{a}}_k\cdot {\vec{a}}_1)& \cdots & ({\vec{a}}_k\cdot {\vec{a}}_k)\end{array}\right)$$
It can be seen that ,Gram A matrix is a symmetric matrix .

Gram The matrix can be regarded as the eccentric covariance matrix between the data （ That is, the covariance matrix without subtracting the mean , The covariance matrix can be seen in This article ）

Gram Application of matrix

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For a set of features , We calculate its Gram matrix , It can reflect the correlation between two features . The elements on the diagonal can be understood as the information of the feature , The rest of the elements provide information about the relevance of different features , What are the characteristics of such a matrix , It can also reflect the closeness of different characteristics .

Gram The most direct application of matrix is in the field of image style transfer . For a picture , We want to extract its style , We need to use the network to extract local texture features 、 Image outline and other information , And then calculate Gram matrix , We can find out the correlation between features , This calculated Gram The matrix reflects the style of the image . At this point, if you want to compare the style similarity between the two pictures , Just compare their Gram matrix , if Gram The difference of matrix is small , It can be considered that the two images are similar in style .