Evan Chou Courant Institute of Mathematical Sciences Email: chou (at cims) Office: 608 Warren Weaver Hall

## Research

In general, I am interested in mathematical models of signal processing; specifically, quantization. My current project (with Sinan Güntürk): Investigating noise-shaping (Sigma-Delta) as a means to boost sparse recovery of compressed sensing measurements.

Measurements $$y=\Phi x$$ are encoded in a finite alphabet via Sigma-Delta, finding $$q$$ such that $$y-q=D^{r}u$$ for some bounded $$u$$. ($$D$$ is the finite-difference linear operator). The effect of this on compressed sensing: if we can recover the support of a sparse $$x$$ (say via original $$l^1$$ minimization methods), then we can boost the recovery further. Let $$T$$ be the support and $$\Phi_T$$ the corresponding submatrix.

Originally, if we have no information about the noise profile of $$y-q$$, (or say $$y-q$$ behaves like white-noise, which happens in the case of scalar quantization), then the best we can do is to recover using the psuedoinverse of $$\Phi_T$$.

Otherwise, we can tailor the reconstruction to take advantage of the structure: of all the possible left-inverses of $$\Phi_T$$, choose the one $$F$$ that minimizes the operator norm of $$FD^{r}$$, and use the bound $\|x - \hat{x}\| = \|Fy - Fq\| = \|FD^{r}u\| \leq \|FD^{r}\| \|u\|$

Already proven that if $$\Phi$$ is taken from the Gaussian ensemble, we can derive a recovery rate behaving like $$C_r(m/k)^{\alpha r}$$ where $$m$$ is the number of measurements, $$k$$ is the sparsity, $$0 < \alpha < 1$$ and $$r$$ is the order of Sigma-Delta. (This is all in the Sobolev Dual paper by Güntürk, et. al.)

An alternative is to examine the quantization cell of $$x$$, described entirely by the set $$\{z: \|D^{-r}(\Phi z - Q(x))\|_\infty \leq \delta\}$$ where $$\delta$$ is the alphabet resolution. Experiments of recovery methods based on the quantization cell directly seem to exhibit a better rate (for large $$m$$) than above.

## Notes

Below are links to notes I have typed up for various classes I have taken.