The discrepancy method: randomness and complexity
The discrepancy method: randomness and complexity
Sparse representations and approximation theory
Journal of Approximation Theory
| Hi-index | 0.00 |
Let $d$ be a fixed integer, and let $W$ be any $d$-dimensional linear subspace of $\mathbb{R}^n$. There then exists a subset $I$ of the $n$ coordinates $\{1,2,\dots,n\}$ of $\mathbb{R}^n$ of cardinality at least $(\frac{1}{2}-o(1))n$ such that for every vector $w=(w_1,\dots,w_n)\in W$ we have $\sum_{i\in I}|w_i|\leq\sum_{i\notin I}|w_i|$. Equivalently, let $P$ be any multiset of $n$ arbitrary vectors in $\mathbb{R}^d$. Then there exists a subset $S$ of $P$ of size at least $(\frac{1}{2}-o(1))n$ such that for every vector $u\in\mathbb{R}^d$ we have $\sum_{x\in S}|\langle x,u\rangle|\leq\sum_{x\in P\setminus S}|\langle x,u\rangle|$. A continuous analogue of the former result is also considered.