Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: An invariance principle of Random projection   Johnson-Lindenstrauss lemma states random projections can be used as a
topology preserving embedding technique for fixed vectors. In this paper, we
try to understand how random projections affect probabilistic properties of
random vectors. In particular we prove the distribution of inner product of two
independent random vectors $X, Z \in {R}^n$ is preserved by random projection
$S:{R}^n \to {R}^m$. More precisely,
  \[
  \sup_t \left| \text{P}(\frac{1}{C_{m,n}} X^TS^TSZ <t) -
\text{P}(\frac{1}{\sqrt{n}} X^TZ<t) \right| \le O\left(\frac{1}{\sqrt{n}}+
\frac{1}{\sqrt{m}} \right)
  \]
  As a by-product, we obtain product central limit theorem (product-CLT) for
$\sum_{k=1}^{n} X_k Y_k$, where $\{X_k\}$ is a martingale difference sequence,
and $\{Y_k\}$ has dependency within the sequence. We also obtain the rate of
convergence in the spirit of Berry-Esseen theorem.

28
The categories of this paper are: alg-geom
36
The categories of this paper are: cmp-lg
37
The categories of this paper are: supr-con
21
The categories of this paper are: adap-org
22
The categories of this paper are: chao-dyn
17
The categories of this paper are: q-fin
29
The categories of this paper are: funct-an
35
The categories of this paper are: mtrl-th
23
The categories of this paper are: solv-int
26
The categories of this paper are: bayes-an
33
The categories of this paper are: plasm-ph
5
The categories of this paper are: gr-qc
19
The categories of this paper are: eess
18
The categories of this paper are: econ
3
The categories of this paper are: physics
7
The categories of this paper are: hep-th