Element-wise uniqueness, prior knowledge, and data-dependent resolution

Abstract

Techniques for finding regularized solutions to underdetermined linear systems can be viewed as imposing prior knowledge on the unknown vector. The success of modern techniques, which can impose priors such as sparsity and non-negativity, is the result of advances in optimization algorithms to solve problems which lack closed-form solutions. Techniques for characterization and analysis of the system to determine when information is recoverable, however, still typically rely on closed-form solution techniques such as singular value decomposition or a filter cutoff estimate. In this letter we propose optimization approaches to broaden the approach to system characterization. We start by deriving conditions for when each unknown element of a system admits a unique solution, subject to a broad class of types of prior knowledge. With this approach we can pose a convex optimization problem to find “how unique” each element of the solution is, which may be viewed as a generalization of resolution to incorporate prior knowledge. We find that the result varies with the unknown vector itself, i.e., it is data-dependent, such as when the sparsity of the solution improves the chance it can be uniquely reconstructed. The approach can be used to analyze systems on a case-by-case basis, estimate the amount of important information present in the data, and quantitatively understand the degree to which the regularized solution may be trusted.

Appendix

In this appendix we will describe how a selection of variations on prior knowledge can be formulated as linear inequality constraints. Again the classical case with no prior knowledge is based on the solution set $F_{EC}$, with $\mathbf D=\mathbf 0$ and $\mathbf d=\mathbf 0$.

$$\begin{aligned} F_\mathrm{{EC}} = \{\mathbf x \in \mathbb {R}^n | \mathbf A \mathbf x = \mathbf b \}. \end{aligned}$$

(10)

Application of our bounds testing problem with the feasible set $\mathbf x \in F_{EC}$ forms an equality-constrained linear program [16], for which optimality conditions give the row space condition $\mathbf A^T \mathbf y = \mathbf e_k$.

Non-negativity results in the solution set

$$\begin{aligned} F_{NN} = \{\mathbf x \in \mathbb {R}^n | \mathbf A \mathbf x = \mathbf b , \mathbf x \ge \mathbf 0 \}. \end{aligned}$$

(11)

This can be implemented in our system with the simple definitions, $\mathbf D = \mathbf I$, $\mathbf d = \mathbf 0$, using the identity matrix and a vector of zeros.

$\ell _1-regularization$ can be formulated as a case of non-negativity, which can be used to determine whether we have a unique optimal solution to the Basis Pursuit problem [8],

$$\begin{aligned}&\alpha = \underset{\mathbf x}{\min } \; \Vert \mathbf x \Vert _1 \nonumber \\&\mathbf A \mathbf x = \mathbf b. \end{aligned}$$

(12)

This can be tested by analyzing in the uniqueness of the solutions in the following set,

$$\begin{aligned} F_{BP}&= \{\mathbf x \in \mathbb {R}^n | \mathbf A \mathbf x = \mathbf b , \, \Vert \mathbf x \Vert _1 = \alpha \} \nonumber \\&= \{\mathbf x \in \mathbb {R}^n | \mathbf A \mathbf x = \mathbf b , \, \Vert \mathbf x \Vert _1 \le \alpha \}. \end{aligned}$$

(13)

This is equivalent to the following non-negative system,

$$\begin{aligned} F_{NN} = \{{\hat{\mathbf {x}}} \in \mathbb {R}^{2n} | {\hat{\mathbf {A}}} {\hat{\mathbf {x}}} = {\hat{\mathbf {b}}} , {\hat{\mathbf {x}}} \ge \mathbf 0 \}. \end{aligned}$$

(14)

With the definitions

$$\begin{aligned} {\hat{\mathbf {A}}} = \begin{pmatrix} \mathbf A, -\mathbf A \\ \mathbf 1^T \end{pmatrix}, \;\; {\hat{\mathbf {b}}} = \begin{pmatrix} \mathbf b \\ \alpha \end{pmatrix}. \end{aligned}$$

(15)

This can be seen by defining $\mathbf x = {\hat{\mathbf {x}}}_{(1)} - {\hat{\mathbf {x}}}_{(2)}$, where ${\hat{\mathbf {x}}}^T =\begin{pmatrix}{\hat{\mathbf {x}}}_{(1)}^T, {\hat{\mathbf {x}}}_{(2)}^T\end{pmatrix}$ and ${\hat{\mathbf {x}}}_{(1)} \ge \mathbf 0$, ${\hat{\mathbf {x}}}_{(2)}\ge \mathbf 0$. We relate bounds found using the feasible set of Eq. (14) to the bounds for the set of Eq. (13) by noting that at the minimum, where we get $\alpha$ as the optimal for Eq. (12), ${\hat{\mathbf {x}}}_{(1)}$ and ${\hat{\mathbf {x}}}_{(1)}$ are complementary. If they were not, we could take advantage of this fact to reduce the minimum of $\Vert \mathbf x \Vert _1 = {\hat{\mathbf {x}}}_{(1)} + {\hat{\mathbf {x}}}_{(2)}$ further.

Box constraints define the following set,

$$\begin{aligned} F_{BOX} = \{\mathbf x \in \mathbb {R}^n \; | \; \mathbf A \mathbf x = \mathbf b , \; \mathbf d_{min} \le \mathbf x \le \mathbf d_{max} \}. \end{aligned}$$

(16)

Here $\mathbf d_{min}$ and $\mathbf d_{max}$ are vectors defining the box. We can formulate this as Eq. (2) with the definitions

$$\begin{aligned} \mathbf D = \begin{pmatrix} +\mathbf I \\ -\mathbf I \end{pmatrix}, \;\; \mathbf d = \begin{pmatrix} +\mathbf d_{min} \\ -\mathbf d_{max} \end{pmatrix}. \end{aligned}$$

(17)

We can view this as a more general version of regularization with the infinity norm, e.g.,

$$\begin{aligned} F_{BOX} = \{\mathbf x \in \mathbb {R}^n | \mathbf A \mathbf x = \mathbf b , \, \Vert \mathbf x \Vert _\infty \le d \}. \end{aligned}$$

(18)

Denoising can be viewed as a dual to regularization, where rather than requiring the solution set be regular, we require the error in the linear system to be regular, as in the following,

$$\begin{aligned} F_{DN} = \{\mathbf x \in \mathbb {R}^n \; | \; \Vert \mathbf A \mathbf x - \mathbf b \Vert \le \sigma , \}. \end{aligned}$$

(19)

We can also form a denoised version of the non-negativity case using the infinity norm as follows,

$$\begin{aligned} F_{NND} = \{\mathbf x \in \mathbb {R}^n \; | \; \Vert \mathbf A \mathbf x - \mathbf b \Vert _\infty \le \sigma , \; \mathbf x \ge \mathbf 0 \}. \end{aligned}$$

(20)

This can be formulated as mixed constraints with no linear constraint term (i.e., “$\mathbf A$” and “$\mathbf b$” in the original linear system are all zeros), and with

$$\begin{aligned} \mathbf D =\begin{pmatrix} -\mathbf A \\ \mathbf A \\ \mathbf I \end{pmatrix}, \;\; \mathbf d = \begin{pmatrix} -\mathbf b - \sigma \mathbf 1 \\ \mathbf b - \sigma \mathbf 1 \\ \mathbf 0 \end{pmatrix}. \end{aligned}$$

(21)