Geometric Regularization Framework

• Geometric regularization is a rigorous framework that imposes geometric constraints on inverse problems to encode prior knowledge and enhance solution stability.
• It characterizes solution sets as convex polyhedra with explicit face structures defined by active sign patterns and minimal faces.
• The approach ensures unique recovery and efficient algorithm design by linking kernel intersection properties to the geometric features of feasible solutions.

Geometric regularization is a rigorous methodological framework that imposes geometric constraints or penalties on the solution set of inverse problems, statistical estimation, or learning procedures. It is widely used to encode prior knowledge about the structure or smoothness of solutions, improve stability and conditioning, facilitate uniqueness, and—crucially—articulate the geometric nature of feasible solutions, particularly in high-dimensional, sparsity-promoting, or ill-posed regimes. In the context of sparse analysis regularization, geometric regularization elucidates the polyhedral nature of solution sets, their face structure, and criteria for extremality, enabling precise characterization of feasible solutions, uniqueness, and algorithmic exploration (Dupuis et al., 2019).

1. Formal Problem Setting and Notation

The archetypal geometric regularization setup for sparse inverse problems involves minimizing over $x\in\mathbb{R}^n$: $$L(x) = \frac{1}{2}\|y - \Phi x\|_2^2 + \lambda \|A x\|_1,$$ where:

• $y\in\mathbb{R}^p$ are observed measurements,
• $\Phi:\mathbb{R}^n\to\mathbb{R}^p$ is a linear sensing operator,
• $A:\mathbb{R}^n\to\mathbb{R}^m$ is an analysis operator (e.g., finite-difference, incidence matrix),
• $R(x) = \|A x\|_1$ is an $\ell_1$-analysis regularizer with parameter $\lambda>0$.

Key notation includes the sign pattern $\operatorname{sign}(v)\in\{-1,0,+1\}^m$, the support $\operatorname{supp}(v)$, and cosupport $\operatorname{cosupp}(v)$ (the complement of the support in $\{1,\dots,m\}$). For any subset $I$, $A_I$ denotes the submatrix of $A$ formed from columns indexed by $I$.

2. Polyhedral Geometry of Solution Sets

The fundamental geometric insight is that the solution set $$S(y) = \mathop{\mathrm{Argmin}}_x \left[\frac{1}{2}\|y-\Phi x\|^2 + \lambda \|A x\|_1\right]$$ forms a (possibly unbounded) convex polyhedron. The critical structure arises from the half-space representation of the $\ell_1$ sub-level set: $$\{x : \|A x\|_1 \leq t\} = \bigcap_{s\in\{-1,0,+1\}^m} \{x : \langle A s, x \rangle \leq t\}.$$ Explicitly, the solution set can be represented as: $$S(y) = (x^* + \ker \Phi) \cap \{x:\|A x\|_1 \leq r\},$$ where $x^*\in S(y)$ is a solution and $r = \|A x^*\|_1$. The affine dimension of $S(y)$ is given by $\dim(\ker \Phi \cap \ker A_{J}^T)$, where $J$ is the cosupport given by $J = \operatorname{cosupp}(A x^*)$.

3. Face Structure, Support Patterns, and Minimal Faces

Geometric regularization allows precise identification of faces and minimal face structure in the solution polyhedron. Any exposed face $F$ of the set $\{x : \|A x\|_1 \leq r\}$ corresponds to a unique maximal sign pattern $s$: $$F = \{x : \|A x\|_1 \leq r\,,\, \langle A s, x \rangle = r\} = \{x : \langle A s, x \rangle = r,\, \operatorname{sign}(A x) \preceq s\}.$$ For any solution $x^*$, its minimal face is characterized as: $$F_\text{min}(x^*) = (x^* + \ker \Phi) \cap \{x:\langle A s^*, x \rangle = r,\, \operatorname{sign}(A x) \preceq s^*\},$$ with $s^* = \operatorname{sign}(A x^*)$.

The dimension of the minimal face is exactly $\dim(\ker \Phi \cap \ker A_{J}^T)$, with $J = \operatorname{cosupp}(s^*)$, enabling explicit link between sign pattern and affine dimensionality.

4. Algebraic Characterization of Extremal Points

A rigorous algebraic test enables recovery of extreme points—those solutions that cannot be written as convex combinations of other members of $S(y)$. Formally, $x^*$ is extreme if and only if: $$\ker \Phi \cap (A s^*)^\perp \cap \ker A_{J}^T = \{0\},$$ where $s^*$ is the sign pattern at $x^*$ and $J = \operatorname{cosupp}(s^*)$. This vanishing direction space guarantees that $x^*$ is a vertex of the solution polyhedron.

5. Construction and Realization of Sub-Polyhedra

Geometric regularization theory ensures the realization of any specific exposed sub-polyhedron as the solution set of a regularized problem. For any affine $A$-subspace $\mathcal{H}$ such that $$\emptyset \neq \mathcal{H} \cap \{\|A x\|_1 \leq r\} \subset \{\|A x\|_1 = r\}$$, it is possible to construct $\Phi, y, \lambda$ so that: $S(y) = \mathcal{H} \cap \{\|A x\|_1 \leq r\}.$ This demonstrates the capacity of geometric regularization to encode arbitrary geometric constraints and sub-structure, connecting specialized cases such as total variation (TV), Fused Lasso, and Graph TV to generic polyhedral faces of the $\ell_1$ analysis ball.

6. Implications for Uniqueness, Stability, and Algorithms

The property $\ker \Phi \cap \ker A^T = \{0\}$ characterizes uniqueness: when satisfied, $S(y)$ is zero-dimensional and bounded. More refined uniqueness criteria arise with active-set analysis, i.e., $\ker \Phi \cap \ker A_{J}^T = \{0\}$ at the computed sign pattern $s^*$. These conditions facilitate the design of linear programs that exploit knowledge of active sign and cosupport, allowing rapid exploration of extremal solutions, optimality certification, and bounding of coordinate variations. In classical cases:

• Lasso corresponds to $A=I$ and solutions are intersections of affine subspaces with the $\ell_1$ ball;
• TV regularization with $A$ as finite-difference confines solutions to specific facets determined by jump patterns;
• Fused Lasso and Graph TV translate to tailored matrix $A$, embedding the same polyhedral face structures.

7. Geometric Interpretation and Broader Context

Each feasible sign pattern corresponds to an exposed face of the analysis $\ell_1$ ball (a zonotope), and the solution set is a slice of that face by the affine subspace $\Phi x = \text{const}$. This link between algebraic patterns in $A x$ and geometric facets in the regularizer clarifies the nature of extreme solutions as polyhedral vertices, detected algebraically by cosupport intersection tests.

The geometric regularization methodology provides a unified framework for:

• Characterizing solution sets as polyhedra with explicable affine and facial structure,
• Constructing arbitrary sub-polyhedra as solutions of regularized problems,
• Enabling uniqueness and stability via intersection properties of linear operator kernels,
• Designing efficient, face-aware algorithms for sparse inverse problems.

This paradigm is foundational for theoretical and algorithmic advances in analysis-sparse recovery, encompassing both classical and modern high-dimensional regression and denoising models (Dupuis et al., 2019).