Problem 8: Mixed Norms

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Suggested by Piotr Indyk
Source Kanpur 2006
Short link https://sublinear.info/8

For any vector $x$, let $\|x\|_0$ be a norm-like function computing the number of non-zero elements in $x$. Consider the following norm-like function $\|\cdot\|_{2,0}$ over $n \times n$ matrices $A = [a_1 \ldots a_n]$: \[ \|A\|_{2,0} = \left ( \sum_{i=1}^n (\|a_i\|_0)^2 \right )^{1/2}. \]

Assume we are given a stream of $m$ updates $(i,j,\delta)$ to $A$, interpreted as $A[i,j]:=A[i,j]+\delta$, starting from $A=0$. What is the smallest space needed by a streaming algorithm estimating $\|A\|_{2,0}$ up to a factor of $1 \pm \epsilon$? An upper bound of $O(\operatorname{poly}(\epsilon^{-1})\cdot\sqrt{n}\cdot\operatorname{polylog}(n))$ is known as long as $A \ge 0$ [CormodeM-05b]. There are no non-trivial lower bounds known.