NSP Constants in Sparse Recovery

• Null space property constants are key parameters that define the threshold for unique and stable sparse recovery from underdetermined linear systems.
• They underpin techniques in compressed sensing, high-dimensional statistics, and coding theory by setting sharp geometric and algorithmic thresholds for recovery.
• Recent advances extend NSP theory to weighted, dictionary-based, and robust models, enabling reliable reconstruction even in noisy and structured environments.

Null space property (NSP) constants are foundational parameters quantifying when unique and stable reconstruction of structured signals is possible from underdetermined linear systems. In compressed sensing, high-dimensional statistics, frame-based signal recovery, coding theory, and system identification, NSP constants define sharp geometric and algorithmic thresholds for successful sparse recovery. The evolution of NSP theory spans classical $\ell_1$ problems, weighted and structured formulations, frame and dictionary settings, extensions to nonconvex or nonseparable regularizations, and recent theoretical connections to isometry properties and robust recovery.

1. Definitions and Variants of Null Space Properties

The null space property characterizes the ability of a linear system $y = Ax$ to uniquely determine sparse or structured solutions via convex or nonconvex optimization. For a matrix $A \in \mathbb{R}^{m \times N}$, NSP of order $s$ with constant $\gamma$ requires: $$\forall v \in \ker(A) \setminus \{0\},~\forall T \subset [N],~|T| \leq s:\quad \|v_T\|_1 < \gamma \|v_{T^c}\|_1,~~ 0<\gamma<1$$ where $v_T$ denotes restriction to index set $T$, and $\gamma$ is the null space constant—the precise measure of how much "leakage" outside the support is required for a null space vector. When $\gamma < 1$, unique sparse recovery via $\ell_1$ minimization is guaranteed.

Extensions of the NSP include:

• Weighted NSP: Adapts to prior information or nonuniform penalty via weight vector $\omega$, leading to

$$\|\upsilon_S\|_{\omega,1} < \|\upsilon_{S^c}\|_{\omega,1}$$

for all $S$ with a weighted sparsity constraint (using a sparse function $\nu_\omega(S)$) (1412.1565, 2410.06794).

• Dictionary-based NSP (D-NSP): For frames or dictionaries $D$, $A$ is D-NSP of order $k$ if for all $T$ with $|T| \leq k$, and all $v \in D^{-1}(\ker A \setminus \{0\})$, there exists $u \in \ker D$ such that

$\|v_T + u\|_1 < \|v_{T^c}\|_1$

relating recovery in overcomplete representations (1302.7074).

• Robust/Strong NSP: Incorporates stability and resilience to noise:

$\|v_{T^c}\|_1 - \|v_T\|_1 \geq c \|v\|_2$

for $c>0$, controlling $\ell_2$ error in recovery (1302.7074, 1506.03040).

• Weak NSP: Requires the property to hold for most (not all) support sets, often justified via small coherence or random matrix models (1606.09193).

2. Null Space Property Constants: Computation and Quantitative Bounds

NSP constants admit explicit or computable probabilistic bounds in random matrix models. The threshold for NSP, and thus reliable sparse recovery, often coincides with critical phase transitions in the $(s, m, N)$ parameter regime.

• For Gaussian random matrices, the precise region where NSP holds with high probability is delineated by explicit inequalities involving normalized sparsity $\rho = s/n$, undersampling $\delta = n/p$, and the NSP constant $C$ (1405.6417). The probability that $\ker(A)$ fails $\mathrm{NSP}(s, C)$ decays exponentially outside this phase transition.
• For weighted or partially-informed cases, the sharp value of the NSP constant directly reflects both the weights and support estimate accuracy. For instance, with weight $w$ assigned on presumed support $\widetilde T$ of accuracy $\alpha$, the optimal recovery is achieved with $w^* = 1-\alpha$, and the corresponding NSP constant relaxes with fewer support errors (1412.1565).
• In the design theory and coding context, NSP constants manifest as the minimal Hamming weight of nontrivial null space vectors. For example, for Wilson-type subspace incidence matrices, the minimum weight of a $k$-uniform $\mathbb{Z}_2$-null design of strength $t$ is exactly $2^{t+1}$, and for $q>2$ bounded by $1 + \frac{q^{t+1} - 1}{q-1}$ below and $q^{t+1}$ above (2012.00037).

3. NSP in Structured and Weighted Settings

Weighted, block, or dictionary/informed recovery introduces distinct NSP constants which can be weaker or strictly less restrictive than those for unweighted, basis-based models.

• Weighted NSP guarantees enable uniform recovery for all $k$-sparse signals with an accurate support estimate using weighted $\ell_1$, requiring as few as $m \gtrsim k + s\log(N/s)$ measurements—fewer than plain $\ell_1$ minimization ($m \gtrsim k\log(N/k)$) (1412.1565, 2410.06794).
• Dictionary-based NSP constants certify recovery even when the signal model is a highly redundant/coherent frame. The D-NSP constant governs whether universality is possible: in full-spark, highly coherent dictionaries, D-NSP collapses to NSP for $AD$, making compression impossible unless $AD$ itself is NSP (1302.7074).
• Null space constants with nonconvex, nonseparable penalties (e.g., sorted $\ell_1$, $\ell_1-\ell_2$) are sometimes even less stringent than those for $\ell_1$: often, replacing strict inequality with non-strict suffices for uniform recovery, excluding certain "equal-height" vectors (1710.07348).
• Weak NSP constants are obtainable via properties such as column coherence; small mutual coherence ensures weak-NSP for most supports, and quantitative constants depend polynomially on the coherence parameter (1606.09193).

4. NSP, Restricted Isometry, and Robust Recovery

NSP is deeply connected to other geometric properties such as the restricted isometry property (RIP), but is strictly weaker. RIP constants are easier to interpret in terms of norm preservation but far more restrictive to satisfy.

• RIP $\implies$ NSP: Any matrix with suitable $\delta_{2s}<1/\sqrt{2}$ necessarily has NSP with explicit constant $$\gamma = 1 - \frac{\delta_{2s}}{\sqrt{1-\delta_{2s}^2}}$$ (1506.03040, 2410.06794).
• RIP-NSP (or $\omega$-RIP-NSP): A matrix has RIP-NSP if its null space matches some matrix with RIP; such matrices inherit robust, stable error bounds—mirroring those for strictly RIP systems—while depending only on the null space (1506.03040, 2410.06794).
• Hierarchy of Properties:

| Property | Stronger Than | Invariant Under | |------------------|-------------------|----------------------| | (Weighted) RIP | RIP-NSP, NSP | Scaling, col. action | | (Weighted) RIP-NSP | NSP | Row ops, invertible | | NSP | --- | Row ops/invertible |

• Explicit constructions show the inclusions above are strict: there exist matrices with NSP, but no scaling yields RIP (1506.03040).

5. Applications: Signal Processing, Coding, System ID, and Design Theory

NSP constants underpin rigorous guarantees in a broad range of applications.

• Compressed Sensing: NSP-based criteria dictate whether basis pursuit or its weighted/structured/composite variants recover all sparse signals, compressed even with arbitrary dictionaries or redundant frames (1302.7074).
• Coding Theory and Design: Minimal null space weights for subspace incidence matrices determine minimal codeword distance, error-correcting performance, and combinatorial limits in secret sharing and affine/projective design (2012.00037).
• Dynamical Systems: System identification under adversarial corruptions hinges on specialized group or weighted NSP constants, governing conditions for exact and robust model recovery (2210.01421).
• Sparse Regression and Variable Selection: Sequential and robust restricted NSP constants validate modern variable selection schemes (such as iSCRA-TL1), enabling consistent recovery under weaker conditions than classical Lasso REC/NSP (2411.01237).

6. Advances and Outlook on Null Space Property Constants

Recent research emphasizes the development, generalization, and quantitative refinement of NSP constants for modern problems:

• Phase transitions for NSP are now characterized with explicit, non-asymptotic probabilistic bounds in random models, coinciding with those for geometry of high-dimensional polytopes and sparse coding (1405.6417).
• The introduction of $\omega$-RIP-NSP bridges the gap between robust, stable guarantees and invariance under invertible transformations, enabling flexible and reliable recovery in weighted frameworks (2410.06794).
• For nonconvex and nonseparable models, NSP constants have been unified and sometimes relaxed, resulting in improved or matching recovery guarantees compared to classic $\ell_1$ theory (1710.07348).
• New domains, such as adversarially robust identification or high-dimensional adaptive variable selection, leverage modified NSP constants (e.g., robust sequential or restricted null space properties) to obtain support recovery and oracle estimation under conditions unattainable by traditional methods (2210.01421, 2411.01237).

7. Summary Table of Key NSP Variants

NSP Variant	Context / Application	Constant Characterizes
Classical NSP	$\ell_1$ sparse recovery (basis)	$\gamma<1$, smallest "leakage" from sparse support
Weighted NSP	Weighted $\ell_1$ with prior/support est.	$C<1$, incorporates weights and accuracy
Dictionary-based D-NSP	Overcomplete/redundant dictionaries/frames	Recovery thresholds in frame-sparse models
$\omega$-RIP-NSP	Robust/stable recovery w/ null space focus	Inherits robust error bounds from $\omega$-RIP
Strong/robust NSP	Recovery with noise/compressible vectors	$c>0$, controls $l_2$ error bound
Weak NSP	High-dimensional random or low-coherence settings	Typical NSP constants for "most" supports
Restricted/Sequential RNSP	Adaptive selection, sequential estimation	Min/max error bounds at successive recovery stages

The paper of null space property constants reveals the critical geometric bottlenecks for sparse and structured recovery across discrete mathematics, signal processing, system theory, and statistical learning. Current research continues to refine these constants, expose their limits, and leverage them for new algorithmic developments.