Is there a function to tell if two probability vectors' max values are in the same dimension?

Question

Is there a method or function to tell two probability vectors' max values are in the same dimension?

Or

Is there a bound for the angle of two normalized probability vector which their max values are in the same dimension?

Thank you

Answer 1

For the second question: if the size of the vector is at most $N$, then something can be said. Let $m_1,m_2$ be the maximal values; they are at least $1/N$. If they are in the same dimension, the inner product is at least $m_1m_2$. The square of $L^2$ norm of the $i$th vector, $i=1,2$, is at most $\sum p_k^2 \le m_i \sum p_k = m_i$. Thus, the cosine of the angle is at least $$\frac{m_1m_2}{\sqrt{m_1}\sqrt{m_2}} = \sqrt{m_1m_2} \ge \frac{1}{N}$$

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Without control on the size, two such vectors can be arbitrarily close to being orthogonal. Take $$\left(\frac2n , 0, \frac1n, 0, \frac1n,\dots, 0, \frac1n\right) $$ and $$\left(\frac2n , \frac1n, 0, \frac1n, 0, \dots, \frac1n, 0\right) $$ The inner product of these vectors is $4/n^2$. The $L^2$ norm of each is of the order $1/\sqrt{n} $. Hence, the cosine of angle is of order $1/n$, which can be arbitrarily small.

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For the first question: I'm afraid there is no method other than to locate the maximal elements and note their positions. Aggregate quantities like $L^2$ nor and inner product cannot tell exactly where the maximum happens, since the position of maximal entry can change under a slight perturbation.