Efficient Information Processing via Exploiting Geometry in Data Representations
Abstract
The tremendous amount of data collected by modern sensing, surveillance and imaging devices creates both opportunities and challenges for data-driven decision making: these data are often high-dimensional, meaning the number of unknowns is typically much larger than the number of collected samples, and therefore, the inference tasks of interest are often ill-posed and suffer from the curse of dimensionality. Fortunately, in many problems of interest to Navy, the data often possess intrinsic geometric structures in a low-dimensional space, induced by the physics of data generation or the underlying source of information, examples including sparsity, low-rank structure, positivity, (cyclo-)stationarity, graph structures, to name a few.The objective of the proposed research program is to develop computational and statistical efficient and robust procedures for recovering, detecting, and learning low-dimensional geometric representations from the acquired data that can be highly noisy and incomplete. Specifically, it aims to provide theoretical and algorithmic advances along the following thrusts.? Thrust 1: accelerating first-order algorithms for nonconvex statistical estimation. This thrust aims to develop new insights into the surprising ~implicit regularization~ phenomenon of gradient descent for solving nonconvex optimization problems arising from a statistical context. It will also develop variants of gradient descent for nonconvex statistical estimation that are robust to ill-conditioning, leveraging ideas such as momentum and pre-conditioning,with provable performance guarantees.? Thrust 2: provable unsupervised learning of latent representations. This thrust aims to shed insights on implementing unsupervised learning using neural networks, by developing performance guarantees for dictionary learning using auto-encoders in terms of optimization landscape, sample complexity and rate of convergence.?Thrust 3: exploitation of low-dimensional structures in second-order statistics. This thrust aims to develop efficient algorithms for exploiting low-dimensional geometric structures in the second-order statistics, i.e. the covariance matrices, for the collected data. Two approaches based on penalized maximum likelihood estimation and moment matching will be developed with provable performance guarantees.?Thrust 4: detection procedures of low-dimensional structures. This thrust aims to develop detection and hypothesis testing procedures for existence and classification of lowdimensional structures from highly incomplete data observations. Distributed hypothesis testing will be developed to accommodate the important scenarios when the data are distributed across a network topology.Techniques from high-dimensional statistics, statistical signal processing, machine learning, optimization and random matrix theory will be drawn and combined in an interdisciplinary manner to attack the proposed research questions. The proposed research program will generate deeper understandings of the synergies between statistics (data) and optimization (algorithms), with a focus on the role of geometry ~ how geometric structures in the data shape the optimization landscapes. The proposed algorithms will be validated over a wide range of applications of interest to the Navy, including but not limited to processing space-time data from sensing systems such assonar and radar, network inference, and inverse scattering.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Jun 13, 2019
- Source ID
- N000141912404
Entities
People
- Yuejie Chi
Organizations
- Massachusetts Institute of Technology
- Office of Naval Research
- United States Navy