Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization

Abstract: The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and collaborative filtering. Although specific instances can often be solved with specialized algorithms, the general affine rank minimization problem is NP-hard. In this paper, we show that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum rank solution can be recovered by solving a convex optimization problem, namely the minimization of the nuclear norm over the given affine space. We present several random ensembles of equations where the restricted isometry property holds with overwhelming probability. The techniques used in our analysis have strong parallels in the compressed sensing framework. We discuss how affine rank minimization generalizes this pre-existing concept and outline a dictionary relating concepts from cardinality minimization to those of rank minimization.

• The paper proposes solving affine rank minimization problems using nuclear norm minimization under certain conditions, generalizing compressed sensing.
• Under a restricted isometry property (RIP) for the linear constraints, minimizing the nuclear norm guarantees finding the minimum-rank solution.
• This approach has practical implications in fields like data compression, signal processing, and statistical modeling, with numerical results showing effectiveness.

Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization

This paper presents an in-depth exploration of the affine rank minimization problem and proposes an approach to solve it using nuclear norm minimization. The affine rank minimization problem seeks the matrix with the minimum rank that satisfies a set of linear equality constraints. Such problems span a wide range of applications, including system identification, control, Euclidean embedding, and collaborative filtering. The paper highlights the NP-hard nature of the general problem due to its equivalence to vector cardinality minimization.

The authors introduce a key condition known as the restricted isometry property (RIP) for the linear transformations defining the constraints. Under this condition, the minimum-rank solution can be achieved by solving a convex optimization problem that minimizes the nuclear norm over the affine space. Notably, this approach generalizes the well-known $\ell_1$ minimization method used in compressed sensing to the rank minimization domain.

The paper discusses several random ensembles of linear equations where this RIP holds with a high degree of probability. Specifically, these conditions are satisfied when the subspace's codimension is sufficiently large, quantified as $\Omega(r(m+n)\log mn)$, where $m$ and $n$ represent the matrix dimensions, and $r$ is the matrix rank.

The relationship between affine rank minimization and compressed sensing is further elucidated through a comprehensive analogy, providing a dictionary of concepts linking cardinality minimization to rank minimization. This analogy extends the theoretical foundation to include techniques applicable in compressed sensing, allowing for the translation of guarantees from $\ell_1$ minimization to the field of matrix ranks.

Algorithmically, the paper explores several solutions to the norm minimization problem, such as interior point methods, gradient projection methods, and low-rank factorization techniques. These methods balance between computational efficiency and solution accuracy, indicating the applicability of each depending on problem size and computational resources available.

The authors support their theoretical findings with numerical demonstrations, showing that nuclear norm minimization effectively recovers low-rank solutions from significantly fewer constraints than theoretical guarantees might suggest.

From a practical standpoint, the implications of this research could profoundly impact fields requiring parsimonious models, especially as computing low-rank matrix solutions finds direct application in areas like data compression, signal processing, and statistical model identification.

In conclusion, while this study does not claim the nuclear norm heuristic approaches optimality in all scenarios, it establishes substantial ground where it can be reliably used with mathematical assurances, potentially guiding future developments in larger-scale affine rank minimization problems and alternative parsimony-based optimization models. Future work can extend these foundations to other non-Euclidean problems or leverage problem-specific structures to design more efficient algorithms.