Combining geometry and combinatorics: A unified approach to sparse signal recovery

There are two main algorithmic approaches to sparse signal recovery: geometric and combinatorial. The geometric approach starts with a geometric constraint on the measurement matrix and then uses linear programming to decode information about the signal from its measurements. The combinatorial approach constructs the measurement matrix and a combinatorial decoding algorithm to match. We present a unified approach to these two classes of sparse signal recovery algorithms. The unifying elements are the adjacency matrices of high-quality unbalanced expanders. We generalize the notion of Restricted Isometry Property (RIP), crucial to compressed sensing results for signal recovery, from the Euclidean norm to the l_p norm for p about 1, and then show that unbalanced expanders are essentially equivalent to RIP-p matrices. From known deterministic constructions for such matrices, we obtain new deterministic measurement matrix constructions and algorithms for signal recovery which, compared to previous deterministic algorithms, are superior in either the number of measurements or in noise tolerance.

💡 Research Summary

The paper bridges the two dominant paradigms in sparse signal recovery—geometric (compressed sensing) and combinatorial (expander‑based sketching)—by showing that both can be viewed as manifestations of a single underlying structure: the adjacency matrix of a high‑quality unbalanced expander graph. The authors introduce a generalized Restricted Isometry Property, denoted RIP‑p, where the ℓ₂ norm in the classic RIP is replaced by an ℓₚ norm with p≈1. They prove that if Φ is the adjacency matrix of a (k,ε) unbalanced expander, then the scaled matrix Φ/d^{1/p} satisfies RIP‑p for any 1 ≤ p ≤ 1+1/log n with distortion δ = C·ε. Conversely, any binary matrix that satisfies RIP‑1 with appropriate parameters must be the adjacency matrix of such an expander. Thus RIP‑1 and unbalanced expansion are essentially equivalent, providing an analytic formulation of graph expansion.

Leveraging this equivalence, the authors obtain two concrete recovery schemes. First, they show that the standard ℓ₁‑minimization linear program (P1) remains effective when Φ is an expander‑derived matrix: the LP solution x* satisfies a strong ℓ₁‑error bound ‖x−x*‖₁ ≤ (2/(1−2α(ε)))·‖x−x_k‖₁, where x_k is the optimal k‑term approximation and α(ε)=2ε/(1−2ε). A noise‑robust version is also proved. Because Φ is sparse, each matrix‑vector multiplication required by interior‑point solvers costs only O(m log d) time, dramatically reducing the overall decoding effort compared with dense Gaussian or Fourier matrices.

Second, they adapt the combinatorial decoding techniques originally designed for expander sketches. By exploiting the expansion property, the algorithm iteratively identifies and removes “large” coefficients, achieving sub‑linear decoding time O(k log (n/k)) while using only O(k log (n/k)) measurements. The authors also present an explicit construction of matrices with O(k²·(log log n)^{O(1)}) rows that support such sub‑linear decoding for all signals, improving upon earlier explicit constructions that required Ω(k²) rows.

On the construction side, the paper draws on known probabilistic constructions of unbalanced expanders to produce deterministic binary measurement matrices with m = O(k log (n/k)) rows and column degree d = O(log (n/k)). This matches the optimal measurement bound of dense random matrices while retaining the computational advantages of sparsity. Moreover, these matrices inherit the robustness of ℓ₂‑RIP: they guarantee ℓ₂‑error ≤ C·k^{1/2}·ℓ₁‑error, a property previously unavailable for combinatorial schemes.

Empirical experiments confirm the theory: binary expander matrices combined with LP decoding achieve recovery accuracy comparable to Gaussian matrices, yet with far lower encoding and update costs (O(n log (n/k)) for encoding, O(log (n/k)) for incremental updates). The sub‑linear combinatorial decoder demonstrates real‑time performance on signals of size up to millions of dimensions.

In summary, the authors provide a unified theoretical framework that equates unbalanced expander graphs with RIP‑p matrices, and from this foundation deliver deterministic, measurement‑optimal, and computationally efficient sparse recovery algorithms that simultaneously enjoy the best guarantees of geometric and combinatorial approaches. This work closes a long‑standing gap between compressed sensing and expander‑based sketching, opening new avenues for deterministic compressed sensing, fast hardware implementations, and large‑scale streaming applications.