- Can someone explain these two-property, I am not able to understand this.
- I can’t find from where these are taken in the paper.
Thank You!
Property 1: given a vector n \in \mathbb{R}^d, the set \{z \in \mathbb{R}^d : n^Tz = 0\} just means the set of vectors that are perpendicular to n (remember that the dot product of perpendicular vectors is zero). We could write it as \{z \in \mathbb{R}^d : n \perp z\}. This set of vectors will form a hyperplane perpendicular to n, so n will be the normal vector to that hyperplane.
It also says that there are two sides to the hyperplane. A vector’s side depends on the sign of its dot product with n, meaning all vectors z \in \mathbb{R}^d satisfying n^Tz \gt 0 lie on one side, and all those satisfying n^Tz \lt 0 lie on the other side of the hyperplane.
Property 2: we have a randomly drawn vector z \in \mathbb{R}^d of which each entry is independent with mean 0 and variance 1 – that’s what z \sim \mathcal{N}(0,\,I_d) means.
Remember from property 1 that a normal vector n defines a hyperplane. The probability of this z being “close” to that hyperplane is at least \left(1-3e^{-cd}\right) \left(1-\frac{2}{\alpha}e^{-\alpha^2/2}\right).
Being “close” here means the dot product of n and z is smaller than the constant 2\alpha\sqrt{\frac{d}{d-2}}. Remember that the dot product being zero means the vector is part of the hyperplane, so the dot product being small means that the vector is “close” to being in the hyperplane.
To put it in numerical terms, see observation 3:
When d = 512, we have P(|n^T z| > 5.0) < 1e^{−6}.
Proof for property 2 is given in Appendix C, if you want to check it out.
Section 2.1 explains how these properties are used in the paper, let me know if you would like me to try to break it down for you.
Hope that helps you understand it better.
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