Metric Non-Degenerate Source Condition (MNDSC)

• MNDSC is a quantitative refinement of the non-degenerate source condition that ensures exact sparse recovery in convex regularization problems using precise metric inequalities.
• It enables continuous and stable recovery by enforcing strict metric-based gaps and curvature conditions around the true support or extreme points.
• MNDSC has been formalized for measures, BV functions, and Wasserstein-regularized problems, offering actionable insights for super-resolution and imaging applications.

The Metric Non-Degenerate Source Condition (MNDSC) is a quantitative refinement of the Non-Degenerate Source Condition (NDSC) underpinning exact sparse recovery in convex-regularized inverse problems, especially those regularized by total variation or similar 1-homogeneous convex functionals. The MNDSC introduces a metric-based, quantitatively nondegenerate gap and curvature around the true support (or set of extreme points) ensuring not only identifiability in noiseless limits but also continuous, stable and exact recovery of locating and weighing atoms under small perturbations. This condition has been formalized and characterized for measures, bounded-variation functions, and Wasserstein-regularized multi-measure problems, and generalizes to broad Banach or metric settings (Duval, 2017, Carioni et al., 2023).

1. Mathematical Framework and Preliminaries

MNDSC applies in infinite-dimensional settings with Banach space $X$, often the dual of a separable predual $X_*$, and a Hilbert space $Y$. The primary problem is

$$(P_\lambda(y_0 + w))\; \min_{u \in X} \; \frac{1}{2}\|Ku - y_0 - w\|_Y^2 + \lambda G(u)$$

where $K: X \rightarrow Y$ is weak*-to-weak continuous, $G: X \rightarrow [0,\infty]$ is proper, convex, lower semi-continuous and positively 1-homogeneous. The unit ball $B = \{u \in X : G(u) \leq 1\}$, and its set of extreme points $\mathrm{Ext}(B)$, equipped with a compatible metric $d$, structure the support representation of optimal solutions. Every solution $u$ to either the exact or penalized problems can be represented as a finite (1-)positive combination of extreme points of $X_*$0.

The classical source condition produces a minimal-norm dual certificate $X_*$1, which acts on $X_*$2 via $X_*$3. The optimal measure is supported where $X_*$4; at other points, $X_*$5.

2. Formal Definition and Criteria of MNDSC

The MNDSC supplements the source condition by imposing strict, metric-based quantitative restrictions on the dual certificate in the neighborhoods of each true atom:

Let $X_*$6 with $X_*$7 and $X_*$8, and $X_*$9 linearly independent. Then $Y$0 satisfies the MNDSC if:

1. Source condition: $Y$1; a certificate $Y$2 exists with $Y$3.
2. Extreme-point uniqueness: The set $Y$4.
3. Second-order nondegeneracy: There exist $Y$5 such that for all $Y$6 and any distinct $Y$7, there is a $Y$8-admissible curve $Y$9 with $$(P_\lambda(y_0 + w))\; \min_{u \in X} \; \frac{1}{2}\|Ku - y_0 - w\|_Y^2 + \lambda G(u)$$0, $$(P_\lambda(y_0 + w))\; \min_{u \in X} \; \frac{1}{2}\|Ku - y_0 - w\|_Y^2 + \lambda G(u)$$1, $$(P_\lambda(y_0 + w))\; \min_{u \in X} \; \frac{1}{2}\|Ku - y_0 - w\|_Y^2 + \lambda G(u)$$2, and

$$(P_\lambda(y_0 + w))\; \min_{u \in X} \; \frac{1}{2}\|Ku - y_0 - w\|_Y^2 + \lambda G(u)$$3

For measures, the metric is induced by weak-* topology; for functions of bounded variation or measure pairs (with Wasserstein coupling), appropriate metrics are used on the extreme-point sets (Carioni et al., 2023, Duval, 2017).

3. The Determinantal and Metric Characterizations

The MNDSC is directly related to a determinantal criterion arising from the dual certificate construction. In the total variation regularized case (BLASSO), vanishing-derivative certificates $$(P_\lambda(y_0 + w))\; \min_{u \in X} \; \frac{1}{2}\|Ku - y_0 - w\|_Y^2 + \lambda G(u)$$4 are constructed so that at support points $$(P_\lambda(y_0 + w))\; \min_{u \in X} \; \frac{1}{2}\|Ku - y_0 - w\|_Y^2 + \lambda G(u)$$5, $$(P_\lambda(y_0 + w))\; \min_{u \in X} \; \frac{1}{2}\|Ku - y_0 - w\|_Y^2 + \lambda G(u)$$6 while $$(P_\lambda(y_0 + w))\; \min_{u \in X} \; \frac{1}{2}\|Ku - y_0 - w\|_Y^2 + \lambda G(u)$$7 away from $$(P_\lambda(y_0 + w))\; \min_{u \in X} \; \frac{1}{2}\|Ku - y_0 - w\|_Y^2 + \lambda G(u)$$8. The “infinitesimal T-system” determinant

$$(P_\lambda(y_0 + w))\; \min_{u \in X} \; \frac{1}{2}\|Ku - y_0 - w\|_Y^2 + \lambda G(u)$$9

is positive for all $K: X \rightarrow Y$0 if and only if the NDSC holds. The MNDSC is a quantitative version, requiring a nontrivial lower bound on $K: X \rightarrow Y$1 for $K: X \rightarrow Y$2 in terms of the metric $K: X \rightarrow Y$3 and constants $K: X \rightarrow Y$4 (Duval, 2017).

4. Implications for Exact and Stable Recovery

Under the MNDSC and linear independence of $K: X \rightarrow Y$5, it can be shown, using localization and subgradient calculus, that for sufficiently small regularization parameter $K: X \rightarrow Y$6 and noise norm $K: X \rightarrow Y$7, the penalized solution admits a unique expansion:

$K: X \rightarrow Y$8

and $K: X \rightarrow Y$9 as $G: X \rightarrow [0,\infty]$0, uniformly in $G: X \rightarrow [0,\infty]$1 (Carioni et al., 2023).

For measures, this reproduces the Duval–Peyré theorem: small-$G: X \rightarrow [0,\infty]$2 solutions are supported on exactly $G: X \rightarrow [0,\infty]$3 Diracs with weights $G: X \rightarrow [0,\infty]$4-close to the originals and locations likewise varying smoothly with the perturbation (Duval, 2017).

5. Concrete Instances and Specialized Criteria

The MNDSC has been characterized explicitly for several canonical problems:

Setting	$G: X \rightarrow [0,\infty]$5	$G: X \rightarrow [0,\infty]$6	Extreme points Ext$G: X \rightarrow [0,\infty]$7
BLASSO (TV on measures)	$G: X \rightarrow [0,\infty]$8	$G: X \rightarrow [0,\infty]$9	$B = \{u \in X : G(u) \leq 1\}$0
1D BV-seminorm	$B = \{u \in X : G(u) \leq 1\}$1	$B = \{u \in X : G(u) \leq 1\}$2	$B = \{u \in X : G(u) \leq 1\}$3
Wasserstein-regularized pairs	$B = \{u \in X : G(u) \leq 1\}$4	$B = \{u \in X : G(u) \leq 1\}$5	$B = \{u \in X : G(u) \leq 1\}$6

In these cases, the second-order nondegeneracy reduces to negative-definiteness of the second derivative of the dual certificate at each support, e.g., requiring $B = \{u \in X : G(u) \leq 1\}$7 in TV, or negative-definite Hessian in the Wasserstein setting (Carioni et al., 2023).

6. Sampling Regimes and Design Implications

For super-resolution, MNDSC enables support stability without geometric separation in some regimes. For Laplace kernels, NDSC and MNDSC hold unconditionally with $B = \{u \in X : G(u) \leq 1\}$8 sampling points anywhere in $B = \{u \in X : G(u) \leq 1\}$9 (no separation required); Gaussian convolution achieves MNDSC under both dense uniform sampling and confined, tightly clustered measurement locations (Duval, 2017). This underpins new design paradigms in applications such as microscopy or spectroscopy, enabling either spread-out or highly concentrated sensor placements with guarantees of exact, stable recovery in the presence of small noise.

7. Theoretical and Methodological Significance

MNDSC unifies the understanding of support stability and exact sparse recovery across different regularization functionals. The crucial insight is that algebraic positivity of rescaled determinant criteria can be metricized to guarantee a nonvanishing quantitative gap and curvature, ensuring continuous support recovery and robustness to noise. This generalization from classical NDSC to MNDSC facilitates the extension to new imaging modalities, complex sparse recovery problems, and broadens the scope for rigorous guarantees of uniqueness and stability beyond measures—covering general Banach-space settings and diverse convex regularizers (Carioni et al., 2023, Duval, 2017).