Coding-Theoretic Methods for Sparse Recovery

We review connections between coding-theoretic objects and sparse learning problems. In particular, we show how seemingly different combinatorial objects such as error-correcting codes, combinatorial designs, spherical codes, compressed sensing matrices and group testing designs can be obtained from one another. The reductions enable one to translate upper and lower bounds on the parameters attainable by one object to another. We survey some of the well-known reductions in a unified presentation, and bring some existing gaps to attention. New reductions are also introduced; in particular, we bring up the notion of minimum “L-wise distance” of codes and show that this notion closely captures the combinatorial structure of RIP-2 matrices. Moreover, we show how this weaker variation of the minimum distance is related to combinatorial list-decoding properties of codes.

💡 Research Summary

The paper presents a unified framework that connects several combinatorial objects—error‑correcting codes, combinatorial designs, spherical codes, compressed‑sensing matrices, and group‑testing designs—showing how each can be derived from the others. It begins by formalizing the sparse recovery problems of compressed sensing (continuous domain) and non‑adaptive group testing (binary domain), emphasizing three practical requirements: explicit construction, efficient decoding, and robustness to noise.

The authors introduce the Restricted Isometry Property (RIP‑p, focusing on p = 2) and L‑disjunct matrices as the central combinatorial guarantees for these two settings. RIP‑2 ensures that every L‑column submatrix is nearly orthogonal, which yields stable recovery even with measurement noise, while L‑disjunctness guarantees that any set of L+1 columns contains a column whose support is not covered by the union of the other L, a property that exactly captures the distinguishability needed in group testing.

A substantial portion of the work revisits known constructions of RIP matrices from codes. Two embeddings are described: the spherical embedding Sph(c) that maps a q‑ary codeword c∈ℤ_q^n to a unit‑norm complex vector using a primitive q‑th root of unity, and the Boolean embedding Bool(c) that expands each symbol into a q‑dimensional standard basis vector. The paper shows that if a code is ε‑biased (i.e., the difference of any two codewords is statistically close to uniform), then its spherical embedding forms an ε‑coherent spherical code, which directly yields an RIP‑2 matrix with constant proportional to L·ε. Conversely, a code with relative minimum distance δ gives a Boolean embedding that is (1−δ)‑coherent, again leading to RIP‑2 matrices.

The novel contribution is the introduction of “minimum L‑wise distance,” a generalization of the classical minimum distance that measures the average correlation among any L distinct codewords rather than just pairs. The authors prove that small L‑wise distance implies a tighter RIP‑2 bound (α ≤ L·ε) than what can be obtained from pairwise distance alone, establishing a more precise link between coding parameters and compressed‑sensing performance.

Further, the paper connects L‑wise distance to list‑decoding. An ε‑biased code admits a list‑decoder with radius ε, meaning that matrices derived from such codes remain stable under noise and approximate sparsity. This bridges a gap between the combinatorial list‑decoding literature and the analytic RIP framework.

The authors also discuss how balanced codes (closed under addition of the all‑ones vector) can be quotient‑ed by the all‑ones word to produce sub‑codes whose columns directly yield L‑disjunct matrices, thus providing explicit constructions for non‑adaptive group testing.

Quantitative analysis uses the Gilbert‑Varshamov bound to show that, for a balanced q‑ary code with relative distance 1 − (1+ε)/q, the resulting RIP‑2 matrix has n = O(L²·log N) rows, while linear‑programming lower bounds (McEliece‑Rodemich‑Rumsey‑Welch) imply a necessary n = Ω(L²·log N / log L). Hence, code‑based constructions are within a factor of L of the optimal probabilistic constructions (which achieve n = O(L·log(N/L))).

In conclusion, the paper establishes a comprehensive set of reductions that translate parameters and guarantees across coding theory, design theory, and sparse recovery. By introducing the L‑wise distance notion and linking it to both RIP and list‑decoding, it opens new avenues for constructing deterministic measurement matrices with provable performance, and suggests future work on extending these ideas to higher‑order list decoding, non‑linear measurements, and practical algorithmic implementations.