# Relationship between L2-norm of the noisy observations

by nOp   Last Updated June 12, 2019 06:19 AM - source

Assume $$\mathbf X_1^n$$ is a vector of size $$n$$ whose elements are either $$+1$$ or $$-1$$. Then, we define $$\mathbf Y^n=\mathbf X_1^n+\mathbf N^n$$ where $$\mathbf N^n$$ is Gaussian additive noise with distribution $$\mathbf N^n\sim \mathcal{N}(\mathbf 0,\sigma^2\mathbf I)$$, i.e., noise elements are identically independently distributed (i.i.d) with variance $$\sigma^2$$.

Now, consider $$\mathbf X_2^n$$ which is also a vector of $$+1$$'s and $$-1$$'s, and is the same as $$\mathbf X_1^n$$ except for $$k positions in which its values is opposite of the $$\mathbf X_1^n$$. For example for $$n=5$$ and $$k=2$$, we could have $$\mathbf X_1^5=[-1,-1,+1,+1,-1];\\ \mathbf X_2^5=[+1,-1,+1,+1,+1];\\$$ where two vectors differ in the first and the last positions. My question is can we obtain a relationship between $$||\mathbf Y^n-\mathbf X_1^n||^2_2$$ and $$||\mathbf Y^n-\mathbf X_2^n||^2_2$$. Numerically, I have seen that as $$k$$ gets large $$||\mathbf Y^n-\mathbf X_2^n||^2_2$$ gets larger than $$||\mathbf Y^n-\mathbf X_1^n||^2_2$$. I am looking for a mathematical expression to relate the two.

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