Any criterion to numerically check the orthogonality between two noisy vectors? If $\boldsymbol{x}=(x_1,\dots,x_n)^\top$ and $\boldsymbol{y}=(y_1,\dots,y_n)^\top$ are two standardized vectors, i.e., $\sum_{i=1}^nx_i=0$ and $\sqrt{\sum_i x_i^2}=1$, is there any method to numerically judge $\boldsymbol{x}\perp \boldsymbol{y}$?
Actually $\boldsymbol{x}$ and $\boldsymbol{y}$ are two observed noisy data vectors, so I think it is not appropriate to use $\boldsymbol{x}^\top \boldsymbol{y}=0$ as the criterion. 
 A: We measure, or compute, noisy vectors. Consider the case of $m$ vectors. Hoping for
$$
x = 
\left[ \begin{array}{c}
  x_{1} \\ \vdots \\x_{m}
\end{array} \right], \quad
%
y = 
\left[ \begin{array}{c}
  y_{1} \\ \vdots \\ y_{m}
\end{array} \right]
$$
we instead measure or compute
$$
x = 
\left[ \begin{array}{c}
  x_{1} + \epsilon_{1} \\ \vdots \\ x_{m} + \epsilon_{m}
\end{array} \right], \quad
%
y = 
\left[ \begin{array}{c}
  y_{1} + \delta_{1} \\ \vdots \\ y_{m} + \delta_{m}
\end{array} \right]
$$
To make progress, consider the $\epsilon$ and $\delta$ terms to come from distributions with $0$ mean. Further assume the distribution of errors is random.
Noise is inevitable. How big are $\epsilon$ and $\delta$? How much noise is in the data? Did we spend a lot of money and build a measurement apparatus with great fidelity? Are the computations to produce these data based on ill-conditioned systems or ill-posed differential equations?
The object of study is not
$$ 
  x^{*} y = x \cdot y = \sum_{k=1}^{m} x_{k} \bar{y}_{k} = 0.
$$
The object of study is instead how 
$$ 
  \sum_{k=1}^{m} x_{k} \bar{y}_{k} - 
  \sum_{k=1}^{m} 
  \left( x_{k} + \epsilon_{k} \right) 
  \left( \bar{y}_{k} + \delta_{k} \right),
\tag{1}
$$
a number which is $0$ on the blackboard and nonzero in computation.

To first order (that is, ignoring terms like $\epsilon \delta$), the error is 
$$ 
  \sum_{k=1}^{m} x_{k} \bar{y}_{k} - 
  \sum_{k=1}^{m} 
  \left( x_{k} + \epsilon_{k} \right) 
  \left( \bar{y}_{k} + \delta_{k} \right)
%
=
\sum_{k=1}^{m} x_{k} \delta_{k} + \bar{y}_{k} \epsilon_{k}
$$
In practice, you will set a threshold value for numeric $0$. This is the standard problem that arises with real numbers in finite precision representation: don't check $x-y=0$, check $\lvert x - y \rvert < \Delta$, where $\Delta$ is set by a linear combination of insight and hope.
