Improved Sparsity Thresholds Through Dictionary Splitting

Known sparsity thresholds for basis pursuit to deliver the maximally sparse solution of the compressed sensing recovery problem typically depend on the dictionary’s coherence. While the coherence is easy to compute, it can lead to rather pessimistic thresholds as it captures only limited information about the dictionary. In this paper, we show that viewing the dictionary as the concatenation of two general sub-dictionaries leads to provably better sparsity thresholds–that are explicit in the coherence parameters of the dictionary and of the individual sub-dictionaries. Equivalently, our results can be interpreted as sparsity thresholds for dictionaries that are unions of two general (i.e., not necessarily orthonormal) sub-dictionaries.

💡 Research Summary

Compressed sensing (CS) seeks the sparsest representation of a measurement vector y in terms of a dictionary D. The exact sparsity problem (P0) is combinatorial, so practitioners replace it with the convex ℓ₁ relaxation (P1), known as basis pursuit. A central question is under which conditions the solutions of (P0) and (P1) coincide. Classical results express these conditions solely in terms of the dictionary’s coherence d (the maximum absolute inner product between distinct columns). From the coherence bound one obtains the well‑known spark lower bound spark(D) ≥ 1 + 1/d, which yields the sparsity threshold k₀ < (1 + 1/d)/2 for uniqueness of the ℓ₀ solution and, consequently, for equivalence of (P0) and (P1). While d is easy to compute, it captures only the worst‑case pairwise correlation and often leads to overly pessimistic thresholds.

The paper introduces a more refined analysis by viewing any dictionary D as the concatenation of two sub‑dictionaries A and B, i.e., D =