M. A. Iwen

M. A. Iwen

A compressed sensing method consists of a rectangular measurement matrix, $M \in \mathbbm{R}^{m \times N}$ with $m \ll N$, together with an associated recovery algorithm, $\mathcal{A}: \mathbbm{R}^m \rightarrow \mathbbm{R}^N$. Compressed sensing methods aim to construct a high quality approximation to any given input vector ${\bf x} \in \mathbbm{R}^N$ using only $M {\bf x} \in \mathbbm{R}^m$ as input. In particular, we focus herein on instance optimal nonlinear approximation error bounds for $M$ and $\mathcal{A}$ of the form $ \| {\bf x} - \mathcal{A} (M {\bf x}) \|_p \leq \| {\bf x} - {\bf x}^{\rm opt}_k \|_p + C k^{1/p - 1/q} \| {\bf x} - {\bf x}^{\rm opt}_k \|_q$ for ${\bf x} \in \mathbbm{R}^N$, where ${\bf x}^{\rm opt}_k$ is the best possible $k$-term approximation to ${\bf x}$. In this paper we develop a compressed sensing method whose associated recovery algorithm, $\mathcal{A}$, runs in $O((k \log k) \log N)$-time, matching a lower bound up to a $O(\log k)$ factor. This runtime is obtained by using a new class of sparse binary compressed sensing matrices of near optimal size in combination with sublinear-time recovery techniques motivated by sketching algorithms for high-volume data streams. The new class of matrices is constructed by randomly subsampling rows from well-chosen incoherent matrix constructions which already have a sub-linear number of rows. As a consequence, fewer random bits than previously required are needed in order to select the rows utilized by the fast reconstruction algorithms considered herein.

M. A. Iwen

We study the problem of estimating the best B term Fourier representation for a given frequency-sparse signal (i.e., vector) $\textbf{A}$ of length $N \gg B$. More explicitly, we investigate how to deterministically identify B of the largest magnitude frequencies of $\hat{\textbf{A}}$, and estimate their coefficients, in polynomial$(B,\log N)$ time. Randomized sub-linear time algorithms which have a small (controllable) probability of failure for each processed signal exist for solving this problem. However, for failure intolerant applications such as those involving mission-critical hardware designed to process many signals over a long lifetime, deterministic algorithms with no probability of failure are highly desirable. In this paper we build on the deterministic Compressed Sensing results of Cormode and Muthukrishnan (CM) \cite{CMDetCS3,CMDetCS1,CMDetCS2} in order to develop the first known deterministic sub-linear time sparse Fourier Transform algorithm suitable for failure intolerant applications. Furthermore, in the process of developing our new Fourier algorithm, we present a simplified deterministic Compressed Sensing algorithm which improves on CM's algebraic compressibility results while simultaneously maintaining their results concerning exponential decay.

M. A. Iwen, D. Needell, E. Rebrova et al.

In this paper new general modewise Johnson-Lindenstrauss (JL) subspace embeddings are proposed that are both considerably faster to generate and easier to store than traditional JL embeddings when working with extremely large vectors and/or tensors. Corresponding embedding results are then proven for two different types of low-dimensional (tensor) subspaces. The first of these new subspace embedding results produces improved space complexity bounds for embeddings of rank-$r$ tensors whose CP decompositions are contained in the span of a fixed (but unknown) set of $r$ rank-one basis tensors. In the traditional vector setting this first result yields new and very general near-optimal oblivious subspace embedding constructions that require fewer random bits to generate than standard JL embeddings when embedding subspaces of $\mathbb{C}^N$ spanned by basis vectors with special Kronecker structure. The second result proven herein provides new fast JL embeddings of arbitrary $r$-dimensional subspaces $\mathcal{S} \subset \mathbb{C}^N$ which also require fewer random bits (and so are easier to store - i.e., require less space) than standard fast JL embedding methods in order to achieve small $ε$-distortions. These new oblivious subspace embedding results work by $(i)$ effectively folding any given vector in $\mathcal{S}$ into a (not necessarily low-rank) tensor, and then $(ii)$ embedding the resulting tensor into $\mathbb{C}^m$ for $m \leq C r \log^c(N) / ε^2$. Applications related to compression and fast compressed least squares solution methods are also considered, including those used for fitting low-rank CP decompositions, and the proposed JL embedding results are shown to work well numerically in both settings.