No-Null-Space Leadership Condition

• No-Null-Space Leadership Condition is a criterion that defines when exact recovery of k-sparse signals is possible using l1-minimization in underdetermined linear systems.
• It involves computing the sparsity index αk, a combinatorial measure reflecting the geometric properties of the measurement matrix's null space and its extremal behavior.
• Advanced relaxations like the pick-l method and the sandwiching algorithm efficiently approximate αk, dramatically reducing computational complexity while ensuring recovery stability under noise.

The No-Null-Space Leadership Condition, more precisely known as the Null-Space Property (NSP) and its computational verifications, establishes necessary and sufficient criteria for the exact and stable recovery of sparse vectors from underdetermined linear systems in compressed sensing. Central to this analysis is the evaluation of the sparsity index $\alpha_k$, a combinatorial quantity encapsulating the geometry of a measurement matrix $A$ and the extremal behavior of its null space. Efficient and precise computation of this constant underlies the certification of recovery guarantees, motivating the development of relaxations and sandwiching algorithms to surmount the inherent complexity of combinatorial verification (Cho et al., 2013).

1. Definition and Significance of the Null-Space Property

Let $A \in \mathbb{R}^{(n-m)\times n}$ be a real, full row-rank measurement operator. The null-space is defined as $\mathcal{N}(A) = \{z \in \mathbb{R}^n : Az = 0\}$. The Null-Space Property of order $k$, $\text{NSP}(k)$, requires that for every nonzero $z \in \mathcal{N}(A)$ and every $K \subset \{1,\dots,n\}$ with $|K|\leq k$,

$\|z_K\|_1 < \|z_{K^c}\|_1.$

This is formalized via the null-space constant (sparsity index)

$$\alpha_k = \max_{\substack{z\neq 0,\,Az=0}}\, \max_{\substack{K\subset\{1,\ldots,n\}\|K|\leq k}}\; \frac{\|z_K\|_1}{\|z\|_1}.$$

$\text{NSP}(k)$ is equivalent to $\alpha_k < \frac{1}{2}$. If $\text{NSP}(k)$ holds, every $k$-sparse solution $x$ is uniquely recovered by $\ell_1$-minimization:

$\min \|u\|_1 \quad \text{subject to} \; Au = y.$

This condition further confers robustness: the $\frac{1}{2}$ gap is key in ensuring stable recovery under noise and nearly-sparse signals (Cho et al., 2013).

2. Combinatorial and Convex Formulations for $\alpha_k$

Exact computation of $\alpha_k$ is computationally prohibitive due to a double maximization over the (generally high-dimensional) null space and all $k$-subsets of indices. For a typical problem, exhaustive verification requires evaluating an exponential number of cases, rendering direct computation infeasible except for small $n$ and $k$.

To address this, the following relaxations are introduced:

• For $l \leq k$, consider all subsets $L$ of size $l$. Let $H \in \mathbb{R}^{n\times m}$ span $\mathcal{N}(A)$. Define

$$\beta_{l,L} = \max_{x\in\mathbb{R}^m} \| (Hx)_L \|_1 \quad \text{subject to} \; \| (Hx)_{L^c} \|_1 \leq 1,$$

$\alpha_{l,L} = \frac{\beta_{l,L}}{1+\beta_{l,L}}.$

Upper bounds on $\alpha_k$ are achieved by combining the largest $\alpha_{l,L}$ values over selected subsets $L$, yielding an efficiently-computable surrogate for $\alpha_k$ via a hierarchy of increasingly tight relaxations as $l$ increases (Cho et al., 2013).

3. Polynomial-Time “Pick-$l$” Relaxations: Algorithms and Improvements

The pick-$l$ method proceeds as follows:

• For each $L \subset \{1,\ldots,n\}$ of cardinality $l$, solve the convex program for $\beta_{l,L}$. This yields $\alpha_{l,L}$.
• Sort all $\alpha_{l,L}$ in descending order.
• Compute

$$\text{Pick-}l \text{ upper bound:} \quad \alpha_k \leq \frac{1}{C(k-1, l-1)} \sum_{j=1}^{C(k, l)} \alpha_{l, L_{(j)}}$$

where $C(a,b) = \binom{a}{b}$, and $L_{(j)}$ denotes the indices of the largest $\alpha_{l,L}$ (Cho et al., 2013).

An optimized version leverages non-uniform weights $\gamma_L$, optimizing over

$\max \sum_L \gamma_L \alpha_{l,L}$

subject to constraints on $\gamma_L$ determined by combinatorial coefficients, further tightening the upper bound. As $l$ grows, tightness improves at the expense of increased computational cost, since $\mathcal{O}(n^l)$ small convex programs must be solved (Cho et al., 2013).

4. The Sandwiching Algorithm for Exact $\alpha_k$ Computation

The sandwiching algorithm computes the exact $\alpha_k$ by integrating pick-$l$ bounds within a global bounding process. This method maintains a global upper bound (GUB) and lower bound (GLB) for $\alpha_k$ and proceeds as follows:

• Precompute for every $k$-subset $K$ a “cheap” upper bound

$$\mathrm{CUB}(K) = \frac{1}{C(k-1, l-1)} \sum_{L\subset K, |L|=l} \alpha_{l,L},$$

and sort all $K$ in descending order of $\mathrm{CUB}(K)$.

• For each $K$ in this order:
  • If $\mathrm{CUB}(K) \leq \mathrm{GLB}$, update $\mathrm{GUB} \leftarrow \mathrm{GLB}$ and terminate.
  • Else, compute a sharper local upper bound LPUB$(K)$ via a small linear program for $K$.
  • If LPUB$(K) > \mathrm{GLB}$, compute the exact $\alpha_{k,K}$ by enumerating $2^{k-1}$ sign patterns (requiring $k$ small LPs per $K$).
  • Update $$\mathrm{GLB} \leftarrow \max(\mathrm{GLB}, \alpha_{k,K})$$.
• Upon loop termination, $\mathrm{GUB} = \mathrm{GLB} = \alpha_k$.

This methodology offers a complexity-accuracy tradeoff parameterized by $l$: larger $l$ tightens bounds, reducing the number of exact solves at higher per-bound cost (Cho et al., 2013).

5. Complexity Analysis and Empirical Performance

The table below summarizes complexity characteristics:

Method	Complexity	Empirical (n=40, m=20, k=5)
Exhaustive Search	$O\big( \binom{n}{k} 2^k \, \mathrm{poly}(m) \big)$	$\approx16$ days
Pick-$l$ Upper B.	$O\big( \binom{n}{l} \mathrm{poly}(m)\big)$	---
Sandwiching	Precompute: $O\big(\binom{n}{k}\binom{k}{l}\big)$; Sort: $O\big(\binom{n}{k}\log \binom{n}{k})$	$\approx134$ min

Empirically, the sandwiching algorithm achieves dramatic runtime reduction: for $k=5$, exhaustive search requires approximately $23,\!000$ minutes, while the sandwiching algorithm (with $l=2$ or $3$) completes in $134$ minutes, and in practice visits only $3,900$ of $658,008$ $K$-subsets (a $170\times$ speed-up and $1/170$ subset reduction). Similar gains are reported for other $n$, $m$, $k$ combinations (Cho et al., 2013).

6. Implications for Sparse Recovery and Robustness

The NSP condition $\alpha_k < \frac{1}{2}$ is both necessary and sufficient for $\ell_1$-minimization to recover all $k$-sparse $x$ uniquely. Furthermore, this threshold extends to stability under noise and approximate sparsity: for measurements $y = Ax + w$, the $\ell_1$-minimizer $x^\star$ satisfies

$$\|x^\star - x\|_1 \leq C_1 \|x - x_k\|_1 + C_2\|w\|_2,$$

where $x_k$ is $x$'s best $k$-term approximation, and constants $C_1, C_2$ depend explicitly on $\alpha_k$. Thus, efficient computation or certification of $\alpha_k < \frac{1}{2}$ via the sandwiching algorithm enables practical and rigorous verification of recovery guarantees in compressed sensing (Cho et al., 2013).