FOCS 2011
TechTalks from event: FOCS 2011
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5A

On the Power of Adaptivity in Sparse RecoveryThe goal of (stable) sparse recovery is to recover a $k$sparse approximation $x^*$ of a vector $x$ from linear measurements of $x$. Specifically, the goal is to recover $x^*$ such that \[ \norm{p}{xx^*} \le C \min_{k\text{sparse } x'} \norm{q}{xx'} \] for some constant $C$ and norm parameters $p$ and $q$. It is known that, for $p=q=1$ or $p=q=2$, this task can be accomplished using $m=O(k \log (n/k)$ {\em nonadaptive} measurements~\cite{CRT06:StableSignal} and that this bound is tight~\cite{DIPW,FPRU}. In this paper we show that if one is allowed to perform measurements that are {\em adaptive} , then the number of measurements can be considerably reduced. Specifically, for $C=1+\epsilon$ and $p=q=2$ we show:* A scheme with $m=O(\frac{1}{\eps}k \log \log (n\eps/k))$ measurements that uses $O(\sqrt{\log k} \cdot \log \log (n\eps/k))$ rounds. This is a significant improvement over the {\em best possible} nonadaptive bound. * A scheme with $m=O(\frac{1}{\eps}k \log (k/\eps) + k \log (n/k))$ measurements that uses {\em two} rounds. This improves over the {\em best known} nonadaptive bound. To the best of our knowledge, these are the first results of this type.

(1+eps)Approximate Sparse RecoveryThe problem central to sparse recovery and compressive sensing is that of \emph{stable sparse recovery}: we want a distribution $\mathcal{A}$ of matrices $A \in \R^{m \times n}$ such that, for any $x \in \R^n$ and with probability $1  \delta > 2/3$ over $A \in \mathcal{A}$, there is an algorithm to recover $\hat{x}$ from $Ax$ with \begin{align} \norm{p}{\hat{x}  x} \leq C \min_{k\text{sparse } x'} \norm{p}{x  x'} \end{align} for some constant $C > 1$ and norm $p$. The measurement complexity of this problem is well understood for constant $C > 1$. However, in a variety of applications it is important to obtain $C = 1+\eps$ for a small $\eps > 0$, and this complexity is not well understood. We resolve the dependence on $\eps$ in the number of measurements required of a $k$sparse recovery algorithm, up to polylogarithmic factors for the central cases of $p=1$ and $p=2$. Namely, we give new algorithms and lower bounds that show the number of measurements required is $k/\eps^{p/2} \textrm{polylog}(1/\eps)$. We also give matching bounds when the output is required to be $k$sparse, in which case we achieve $k/\eps^p \textrm{polylog}(1/\eps)$. This shows the distinction between the complexity of sparse and nonsparse outputs is fundamental.

NearOptimal ColumnBased Matrix ReconstructionWe consider lowrank reconstruction of a matrix using its columns and we present asymptotically optimal algorithms for both spectral norm and Frobenius norm reconstruction. The main tools we introduce to obtain our results are: (i) the use of fast approximate SVDlike decompositions for column reconstruction, and (ii) two deterministic algorithms for selecting rows from matrices with orthonormal columns, building upon the sparse representation theorem for decompositions of the identity that appeared in~\cite{BSS09}.

Near Linear Lower Bound for Dimension Reduction in L1Given a set of $n$ points in $\ell_{1}$, how many dimensions are needed to represent all pairwise distances within a specific distortion ? This dimensiondistortion tradeoff question is well understood for the $\ell_{2}$ norm, where $O((\log n)/\epsilon^{2})$ dimensions suffice to achieve $1+\epsilon$ distortion. In sharp contrast, there is a significant gap between upper and lower bounds for dimension reduction in $\ell_{1}$. A recent result shows that distortion $1+\epsilon$ can be achieved with $n/\epsilon^{2}$ dimensions. On the other hand, the only lower bounds known are that distortion $\delta$ requires $n^{\Omega(1/\delta^2)}$ dimension and that distortion $1+\epsilon$ requires $n^{1/2O(\epsilon \log(1/\epsilon))}$ dimensions. In this work, we show the first near linear lower bounds for dimension reduction in $\ell_{1}$. In particular, we show that $1+\epsilon$ distortion requires at least $n^{1O(1/\log(1/\epsilon))}$ dimensions. Our proofs are combinatorial, but inspired by linear programming. In fact, our techniques lead to a simple combinatorial argument that is equivalent to the LP based proof of BrinkmanCharikar for lower bounds on dimension reduction in $\ell_{1}$.