Explicit constructions of RIP matrices and related problems (1008.4535v3)

Abstract: We give a new explicit construction of $n\times N$ matrices satisfying the Restricted Isometry Property (RIP). Namely, for some c>0, large N and any n satisfying N^{1-c} < n < N, we construct RIP matrices of order k^{1/2+c}. This overcomes the natural barrier k=O(n^{1/2}) for proofs based on small coherence, which are used in all previous explicit constructions of RIP matrices. Key ingredients in our proof are new estimates for sumsets in product sets and for exponential sums with the products of sets possessing special additive structure. We also give a construction of sets of n complex numbers whose k-th moments are uniformly small for 1\le k\le N (Turan's power sum problem), which improves upon known explicit constructions when (\log N)^{1+o(1)} \le n\le (\log N)^{4+o(1)}. This latter construction produces elementary explicit examples of n by N matrices that satisfy RIP and whose columns constitute a new spherical code; for those problems the parameters closely match those of existing constructions in the range (\log N)^{1+o(1)} \le n\le (\log N)^{5/2+o(1)}.

• The paper presents explicit construction methods for RIP matrices that support sparsity levels of k ≈ n^(1/2+ε), surpassing traditional √n bounds.
• It leverages advanced additive combinatorics and exponential sum estimates to overcome the limitations of coherence-based matrix designs.
• The constructions offer a robust framework for compressed sensing, with significant implications for signal processing and coding applications.

On Explicit Constructions of RIP Matrices and Related Problems

The paper presented by Bourgain, Dilworth, Kutzarova, and Konyagin explores explicit constructions of matrices that satisfy the Restricted Isometry Property (RIP) and addresses related mathematical challenges. This exploration provides significant advancements in understanding explicit RIP matrix constructions and fills gaps left by previous research methodologies, especially in overcoming barriers associated with small coherence in previous constructions.

Summary of Contributions

In this work, the authors propose new methods to construct $n \times N$ matrices that satisfy RIP, a fundamental property in compressed sensing, which states that for a sparse vector $x$, $$(1-\delta) \| x \|_2^2 \leq \| \Phi x \|_2^2 \leq (1+\delta) \| x \|_2^2$$ for a given $\delta$. The authors provide an explicit bound $\epsilon > 0$ which supports the construction of matrices with larger sparsity levels $k$ than those derived from matrices with small coherence parameters.

Key results demonstrate that their methods allow for the construction of matrices supporting the RIP for $k \approx n^{\frac{1}{2} + \epsilon}$, improving upon existing bounds that are limited to $k \leq \sqrt{n}$. Their approach utilizes advanced techniques from additive combinatorics, specifically estimates for sumsets and exponential sums over sets with special additive structures, to devise explicit constructions of such matrices.

Moreover, the discussion extends to the construction of complex number sets satisfying conditions relevant to Turán's power sum problem, which historically aimed for minimizing moments and has applications in RIP-related matrix construction. This particular development showcases that the columns of these matrices form a new spherical code with parameters comparable to existing constructions, thereby broadening potential applications in various coding and signal processing domains.

Numerical Results and Claims

The authors make prominent claims regarding the effectiveness and parameters of their constructions, notably achieving $n \times N$ matrices where $N$ scales as $n^{1+\epsilon}$ and recovering the RIP of order $k = \Theta(n^{1/2+\epsilon})$. Additionally, they argue that existing methods tied to coherence cannot achieve orders larger than $\sqrt{n}$ under classical constraints, thereby highlighting the potential and novelty of their approach.

Theoretical Implications

This paper's theoretical implications are manifold. Firstly, it lays a new framework for constructing RIP matrices beyond traditional coherence-based methodologies, which could have far-reaching implications in signal processing, randomized algorithms, and data compression fields. The use of additive combinatorics opens avenues for synergizing number theory and discrete mathematics with practical matrix construction challenges. The research also suggests further exploration into the relationship between Fourier coefficients and coherence, potentially unveiling new strategies to build more efficient, explicitly constructed matrices.

Practical Implications and Future Work

Practically, the explicit construction provided by this research offers a concrete methodology for developers of compressed sensing algorithms, where RIP matrices are critical. It potentially enhances robustness and recovery capabilities, especially in signal reconstruction tasks where deterministic guarantees are beneficial. The authors suggest that future work may refine and extend these frameworks to more general settings and explore additional combinatorial structures to further increase sparsity levels while maintaining the desired isometry properties.

In future works, it would be prudent to investigate the computational performance and implementation feasibility of these constructions in large-scale data scenarios and their adaptability to emerging computational models, such as quantum or parallel paradigms. Subsequently, further exploring combinatorial pathfinding and its impact on advanced linear algebra techniques remains an intellectually rich area for exploration.