| Evan Chou | |
| Courant Institute of Mathematical Sciences | |
| Email: chou (at cims) | |
| Office: 608 Warren Weaver Hall |
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.
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 by the psuedoinverse.
Otherwise, much like in the theory for bandlimited functions, since \(D^{r}\) gives rise to a 'high-pass' sequence, we can do better with a 'low-pass' recovery operator. In this case, of all the possible left-inverses, choose the one \(F\) that minimizes the operator norm of \(FD^{r}\), so that \[ \|x - \hat{x}\| = \|Fy - Fq\| = \|FD^{r}u\| \leq \|FD^{r}\| \|u\| \]
Already proven that if \(\Phi\) is taken from the Gaussian ensemble, we get a recovery rate behaving like \(C(m/k)^{\alpha r}\) 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 \(\{z: \|D^{-r}\Phi z - Q(x)\|_\infty < \delta\}\) 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.