论文标题
随机张量的广义汉森 - 赖特不等式
Generalized Hanson-Wright Inequality for Random Tensors
论文作者
论文摘要
Hanson-Wright不平等是独立随机变量中实际二次形式的尾部的上限。在这项工作中，我们扩展了Hanson-Wright不平等的Ky Fan k-norm，以在爱因斯坦产品下随机张量的二次张量总和的多项式函数。我们将二次张量分解为对角线部分和耦合部分。对于对角线部分，我们可以直接应用广义张量Chernoff绑定。但是，对于耦合部分，我们必须首先应用解耦方法，即，将不等式的不等式与带有独立随机随机张量的依赖随机张量进行绑定表达式，然后再应用ky fan $ k $ k $ k $ k $ k $ k $ norm的尾部的尾巴可能性。最后，可以通过从对角线总和和耦合总和部分结合的结合来获得随机张量二次张量的多项式函数的ky fan k-norm的普遍不等式。
The Hanson-Wright inequality is an upper bound for tails of real quadratic forms in independent random variables. In this work, we extend the Hanson-Wright inequality for the Ky Fan k-norm for the polynomial function of the quadratic sum of random tensors under Einstein product. We decompose the quadratic tensors sum into the diagonal part and the coupling part. For the diagonal part, we can apply the generalized tensor Chernoff bound directly. But, for the coupling part, we have to apply decoupling method first, i.e., decoupling inequality to bound expressions with dependent random tensors with independent random tensors, before applying generalized tensor Chernoff bound again to get the the tail probability of the Ky Fan $k$-norm of the coupling part sum of independent random tensors. At the end, the generalized Hanson-Wright inequality for the Ky Fan k-norm for the polynomial function of the quadratic sum of random tensors can be obtained by the combination of the bound from the diagonal sum part and the bound from the coupling sum part.
