Bispectrum Inversion with Application to Multireference Alignment

We consider the problem of estimating a signal from noisy circularly-translated versions of itself, called multireference alignment (MRA). One natural approach to MRA could be to estimate the shifts of the observations first, and infer the signal by aligning and averaging the data. In contrast, we consider a method based on estimating the signal directly, using features of the signal that are invariant under translations. Specifically, we estimate the power spectrum and the bispectrum of the signal from the observations. Under mild assumptions, these invariant features contain enough information to infer the signal. In particular, the bispectrum can be used to estimate the Fourier phases. To this end, we propose and analyze a few algorithms. Our main methods consist of non-convex optimization over the smooth manifold of phases. Empirically, in the absence of noise, these non-convex algorithms appear to converge to the target signal with random initialization. The algorithms are also robust to noise. We then suggest three additional methods. These methods are based on frequency marching, semidefinite relaxation and integer programming. The first two methods provably recover the phases exactly in the absence of noise. In the high noise level regime, the invariant features approach for MRA results in stable estimation if the number of measurements scales like the cube of the noise variance, which is the information-theoretic rate. Additionally, it requires only one pass over the data which is important at low signal-to-noise ratio when the number of observations must be large

Temperature Dependence of a Sub-wavelength Compact Graphene Plasmon-Slot Modulator

We investigate a plasmonic electro-optic modulator with an extinction ratio exceeding 1 dB/um by engineering the optical mode to be in-plane with the graphene layer, and show how lowering the operating temperature enables steeper switching. We show how cooling Graphene enables steeping thus improving dynamic energy consumption. Further, we show that multi-layer Graphene integrated with a plasmonic slot waveguide allows for in-plane electric field components, and 3-dB device lengths as short as several hundred nanometers only. Compact modulators approaching electronic length-scales pave a way for ultra-dense photonic integrated circuits with smallest footprint