João M. Pereira 
I'm a Postdoctoral Associate in
the Oden
Institute for Computational Engineering and
Sciences at the University of Texas at
Austin, under the supervision of Joe
Kileel and Rachel
Ward. Previously, I was a postdoc of Vahid
Tarokh, at Duke University, and a Ph.D. student of
Emmanuel Abbe and Amit Singer at Princeton
University.
I am an applied mathematician studying machine learning, multilinear algebra and information theory. My research centers on scalable methods and statistical fundamental limits of highdimensional inverse problems, that include cryoelectron microscopy, tensor decomposition and partial and stochastic differential equations. In addition, I have interests in machine learning, deep learning, statistics, information theory and optimization. 
Tensor
Decompositions
We proposed the Subspace Power Method (SPM) for decomposing lowrank symmetric tensors. Numerical experiments indicate that this method is faster by an order of magnitude than the state of the art. In a followup work we studied its optimization landscape and showed that, with a suitable initialization, it is provably efficient in certain regimes. Furthermore, the algorithm may be used to decompose moment tensors implicitly, enabling moment tensors which would otherwise occupy 100PB of memory to be decomposed in a matter of seconds. 
Illustration of symmetric tensor
decomposition
Credit: Joe Kileel 
Learning Laws
of Datasets
We use machine learning tools for discovering laws and equations, such as partial differential equations (PDEs) and stochastic differential equations (SDEs), that govern highdimensional datasets. We proposed a method that, given noisy data points of a function that is a solution of a PDE, it both learns this function and the underlying PDE. On SDE's, we developed a method for learning a latent unknown lowdimensional SDE that governs observed highdimensional data. As an example, given a video of a 2D ball moving in the plane according to an SDE, it learns the twodimensional underlying SDE that governs the ball coordinates. 
Performance of the method for
learning PDEs from data. Given true function
values (left plot), corrupted with noise (middle
plot), the method denoises the function (right
plot) and learns the Helmholtz equation.

Fundamental Limits of
multireference alignment and CryoEM
Cryoelectron microscopy (cryoEM) is a technique to determine the 3D structure of molecules. A crucial challenge in cryoEM consists of estimating the 3D electrostatic potential of the molecule from (very noisy) 2D projections of the molecule potential, over unknown viewing directions. This problem can be seen as an instance of MRA, where the observations result from a group action on the signal, which is then corrupted with noise. In the low signaltonoise ratio (SNR) regime, which is the prevalent regime in cryoEM. it has been showed that the fundamental limits in MRA are determined by the lower order moments of the data. This result follows from a Taylor expansion of the KullbackLeibler divergence of GMM models, valid in the low SNR regime, which I obtained and extended in my Ph.D. thesis for other random mixture models. 
Singleparticle reconstruction
problem in CryoEM
Credit: Amit Singer 
Pseudospectra of
TimeFrequency localization operators
We used inequalities arising from the trace and norm of timefrequency localization operators to study their pseudospectra and obtain other important results. Moreover, we used similar ideas to show that a class of determinantal point processes, associated with the Schrödinger representation of the Heinsenberg group, belong to a state of matter called hyperuniform. 
Left: Poisson process, not uniform;
Middle: hyperuniform;
Right: Lattice/Crystal, also hyperuniform. Credit: Wikipedia 