Invariance principle of random matrix
Duan, Jun Tao
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Random matrix has been found useful in many real-world applications. The celebrated Johnson-Lindenstrauss lemma states that certain geometric structure of deterministic vectors is preserved when projecting high dimensional space to a lower dimensional space. However, when random vectors are concerned, it is still unclear how the distribution of the geometry is affected by random matrices. Since random projection or embedding introduces dependence to independent random vectors, does it imply random matrices are inferior for transforming random vectors? In this work, we try to answer some of the questions. We will start with establishing a new central limit theorem for random variables with certain product dependence structure. At the same time, we obtained its Berry-Esseen type rate of convergence. Then we apply this general central limit theorem to random projection and embedding of two independent random vectors. In particular, we show the distribution of inner product structure is preserved by random matrices. Roughly speaking, two orthogonal random vectors remain orthogonal in the randomly projected lower dimensional space or randomly embedded high dimensional space. More importantly, we also quantitatively characterize the distortion of distribution introduced by random matrices. Then we also establish the fact that random matrices have low distortion on the norm of a random vector. It is first justified by establishing concentration of the projected or embedded norm under sub-Gaussian assumptions. A central limit theorem for the randomly projected norm is established as well similar to the CLT for inner product.