On identity testing of tensors, low-rank recovery and compressed sensing
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
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In this paper we study the problem of explicitly constructing a{\em dimension expander} raised by \cite{BISW}: Let $\mathbb{F}^n$ be the $n$ dimensional linear space over the field $\mathbb{F}$. Find a small (ideally constant) set of linear transformations from $\F^n$ to itself $\{A_i\}_{i \in I}$ such that for every linear subspace $V \subset \F^n$ of dimension $\dim(V) 0$ is some constant. In other words, the dimension of the subspace spanned by $\{ A_i(V) \}_{i\in I}$ should be at least $(1+\alpha) \cdot \dim(V)$. For fields of characteristic zero Lubotzky and Zelmanov \cite{LubotzkyZelmanov} completely solved the problem by exhibiting a set of matrices, of size independent of $n$, having the dimension expansion property. In this paper we consider the finite field version of the problem and obtain the following results. \begin{enumerate}\item We give a constant number of matrices that expand the dimension of every subspace of dimension $d 0$. \end{enumerate}Our constructions are algebraic in nature and rely on expanding Cayley graphs for the group $\mathbb{Z}/\mathbb{Z}n$ and small-diameter Cayley graphs for the group $\mathrm{SL}_2(p)$.