Announcements

(05/27/18) Phase retrieval via polytope optimization: Geometry, phase transitions, and new algorithms

May 27, 2018
In our recent paper, we study algorithms for solving quadratic systems of equations based on optimization methods over polytopes. Our work is inspired by a recently proposed convex formulation of the phase retrieval problem, which estimates the unknown signal by solving a simple linear program over a polytope constructed from the measurements. We present a sharp characterization of the high-dimensional geometry of the aforementioned polytope under Gaussian measurements. This... Read more about (05/27/18) Phase retrieval via polytope optimization: Geometry, phase transitions, and new algorithms

(05/08/18) Subspace estimation from incomplete observations: a high dimensional analysis

May 7, 2018
In our recent paper, we present a high-dimensional analysis of three popular algorithms, namely, Oja's method, GROUSE and PETRELS, for subspace estimation from streaming and highly incomplete observations.  We show that, with proper time scaling, the time-varying principal angles between the true subspace and its estimates... Read more about (05/08/18) Subspace estimation from incomplete observations: a high dimensional analysis

(04/16/18) ICASSP tutorial on nonconvex methods for high-dimensional statistical estimation (slides available online)

April 16, 2018

Together with Yuxin Chen (Princeton) and Yuejie Chi (CMU), I gave a tutorial at this year's ICASSP on recent advances on nonconvex statistical estimation. We will be covering topics include the landscapes of nonconvex estimation, analyzing gradient descent and stochastic gradient descent methods, spectral methods for initialization, and example applications to phase retrieval, low-rank matrix recovery and blind deconvolution.

Here are...

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(12/11/17) Best Student Paper Award (First Prize) at IEEE CAMSAP

December 11, 2017

Best Student Paper Award (first prize) at the 2017 IEEE CAMSAP Workshop:

O. Dhifallah and Y. M. Lu, Fundamental Limits of PhaseMax for Phase Retrieval: A Replica Analysis, 2017.

Congratulations to Oussama!

Note: The replica predictions derived in this paper has been rigorously proved in our more recent work, based on a convex version...

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(12/06/17) Understanding the Dynamics of Online Learning Algorithms via Scaling and Mean-Field Limits

December 6, 2017
In our recent paper, we present a tractable and asymptotically exact framework for analyzing the dynamics of online learning algorithms in the high-dimensional scaling limit. Our results are applied to two concrete examples: online regularized linear regression and principal component analysis. As the ambient dimension tends to infinity, and with proper time scaling, we show that the time-varying joint empirical measures of the target feature vector and its estimates provided by the... Read more about (12/06/17) Understanding the Dynamics of Online Learning Algorithms via Scaling and Mean-Field Limits

(08/15/17) Fundamental Limits of PhaseMax for Phase Retrieval: A Replica Analysis

August 16, 2017
In our recent paper, we establish the exact asymptotic performance and phase transition of an efficient convex relaxation algorithm, named PhaseMax, for solving the phase retrieval problem. Our analysis uses the replica method from statistical mechanics. (I have known about the replica method for a long while, and have even taught about this method in my classes. It's nice to finally apply it in our research.) Read more about (08/15/17) Fundamental Limits of PhaseMax for Phase Retrieval: A Replica Analysis

(02/21/17) Phase transitions of spectral initialization for nonconvex estimation

February 21, 2017

We study in our recent paper a widely used spectral method that serves as a key ingredient in a growing line of work using efficient iterative algorithms for estimating signals in nonconvex settings. Unlike previous work, which focuses on the phase retrieval setting and provides only bounds on the performance, we...

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