The Convex Geometry of Linear Inverse Problems (1012.0621v3)

Abstract: In applications throughout science and engineering one is often faced with the challenge of solving an ill-posed inverse problem, where the number of available measurements is smaller than the dimension of the model to be estimated. However in many practical situations of interest, models are constrained structurally so that they only have a few degrees of freedom relative to their ambient dimension. This paper provides a general framework to convert notions of simplicity into convex penalty functions, resulting in convex optimization solutions to linear, underdetermined inverse problems. The class of simple models considered are those formed as the sum of a few atoms from some (possibly infinite) elementary atomic set; examples include well-studied cases such as sparse vectors and low-rank matrices, as well as several others including sums of a few permutations matrices, low-rank tensors, orthogonal matrices, and atomic measures. The convex programming formulation is based on minimizing the norm induced by the convex hull of the atomic set; this norm is referred to as the atomic norm. The facial structure of the atomic norm ball carries a number of favorable properties that are useful for recovering simple models, and an analysis of the underlying convex geometry provides sharp estimates of the number of generic measurements required for exact and robust recovery of models from partial information. These estimates are based on computing the Gaussian widths of tangent cones to the atomic norm ball. When the atomic set has algebraic structure the resulting optimization problems can be solved or approximated via semidefinite programming. The quality of these approximations affects the number of measurements required for recovery. Thus this work extends the catalog of simple models that can be recovered from limited linear information via tractable convex programming.

• The paper develops a comprehensive framework for linear inverse problems using convex optimization and atomic norms to convert model simplicity into tractable penalties.
• It introduces atomic norms, derived from the convex hull of atomic sets, as the basis for computationally feasible recovery of various structured models, including sparse vectors and low-rank matrices.
• The work provides a geometric analysis using Gaussian widths to derive conditions and bounds for exact and robust model recovery, specifying the number of measurements required.

Overview of "The Convex Geometry of Linear Inverse Problems"

The paper "The Convex Geometry of Linear Inverse Problems" by Venkat Chandrasekaran, Benjamin Recht, Pablo A. Parrilo, and Alan S. Willsky provides a comprehensive framework for addressing ill-posed linear inverse problems through the development of convex optimization techniques. The core premise of the paper is to convert diverse notions of model simplicity into convex penalty functions, known as atomic norms, allowing for robust solutions to inverse problems with potentially fewer measurements than the dimensionality of the underlying models. This work extends beyond traditional sparse models and low-rank matrix recovery, encompassing a wide array of structured models within its framework.

The atomic norm approach is central to this work, where models are formed as a sum of a few components from an atomic set. The effective deployment of convex optimization methods for model recovery relies on exploiting the atomic norm, which is computationally defined utilizing the convex hull of these atomic sets. The research offers significant insights into how these norms can be employed in diverse problem domains, including sparse signal recovery, low-rank matrix completion, as well as more complex structures like low-rank tensors and permutation matrices.

Key Themes and Numerical Insights

1. Atomic Norms Formation: The paper introduces a method to design atomic norms for various structured models. These norms are non-trivial convex relaxations, constructed to yield tractable optimization formulations for model recovery given partial linear information.
2. Geometric Analysis: The analysis derives conditions under which atomic norm-based convex optimization succeeds in exact and robust recovery. The work provides estimates for the number of measurements needed, using the concept of Gaussian widths of tangent cones to atomic norm balls.
3. Structured Recovery Results: Propositions and theorems outlined in the paper provide recovery guarantees for several cases, emphasizing sparse vectors and low-rank matrices. The bounds derived are not only simplistic but often improve on existing results by presenting tighter dimensional constants and incorporating robust recovery under noise.
4. Semidefinite Programming and Algebraic Varieties: The paper details semidefinite programming methods as avenues for optimally or approximately solving the atomic norm minimization problems when the atomic sets exhibit algebraic structure. This area of research cross-links with real algebraic geometry, utilizing algebraic varieties to systematically derive relaxations of convex hulls.
5. Tradeoff Analysis: Particularly compelling is the exploration of tradeoffs between computational tractability and measurement needs when employing relaxations of the atomic norm through semidefinite approximations, exemplified by the tensor nuclear norm and cut polytope approximations.
6. Computational Experiments: While several key results are theoretical, the paper also conducts numerical experiments to observe empirical phase transitions in recovery performance from Gaussian measurements, aligning well with the theoretical constructions.

Implications and Future Directions

The implications of this research extend to a multitude of practical applications and theoretical investigations in signal processing, machine learning, and data science. One immediate application area is in compressed sensing, where the manipulation of large data sets and dimensionality reduction relies heavily on efficient recovery algorithms.

Future research could aim to explore more precise Gaussian width calculations for other atomic norm constructions and structured measurement schemes, potentially involving Fourier coefficients or structured random matrices. Another critical avenue is the development of highly efficient large-scale optimization algorithms, leveraging the proximal operators of atomic norms.

Furthermore, a rigorous examination of the loss incurred through norm relaxation and approximation, as well as determining robust methods for reconstructing original data models from approximate solutions, remains an open challenge. The paper of constructing atomic norms from domain-specific knowledge could also lead to customized optimization solutions, thereby enhancing the practical utility of atomic norms in handling high-dimensional and complex data structures.