Explicit RIP Matrices Beyond Square-Root Sparsity

Explicit RIP Matrices Beyond Square-Root Sparsity

Definitions and mathematical statement

Shared notation.

Standard number systems are denoted by $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$; index ranges, computational models and specialized notation follow the cited problem source and the definitions below.

A real $n\times p$ matrix $A$ has the restricted isometry property of order $s$ with distortion $\delta\in(0,1)$ if $(1-\delta)\|x\|_2^2\leq\|Ax\|_2^2\leq(1+\delta)\|x\|_2^2$ for every $x\in\mathbb R^p$ with at most $s$ nonzero coordinates. Here $n$ counts measurements and $p$ is the ambient dimension. The asymptotic question fixes a nontrivial distortion bound independently of $n,p$. An explicit construction produces the matrix deterministically with controlled construction time.

Statement recorded in the supplied source.

For fixed $\varepsilon>0$, construct explicit real-valued restricted-isometry matrices in the polynomial regime $n=p^{O(1)}$ with sparsity $s\geq n^{1/2+\varepsilon}$. [S2]

Scope qualification.

The source asks about the compressed regime $n<p$, with a fixed distortion below one. Sparsity must obey the feasibility restrictions of this regime; the supplied text does not state a range of exponents or a numerical distortion. Construction cost and representation precision must be stated in any answer.

Source status.

The supplied catalogue labels this record open. Its source-status metadata was checked against the supplied snapshot on 2026-10-11. This import does not independently establish current open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-22.

Short English statement

Construct real restricted-isometry matrices beating square-root sparsity when the number of measurements grows polynomially with the ambient dimension.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30004125 (OWR-16931-009). Catalogue curation and metadata: CC BY 4.0. The supplied source question is retained; definitions and the English summary are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Combinatorics, Probability and Computing $2020$, Oberwolfach Reports, DOI 10.4171/owr/2019/19, pp. 1113--1189 https://doi.org/10.4171/owr/2019/19.
• S3 Combinatorics, Probability and Computing, Oberwolfach Reports 19/2019, primary report PDF https://ems.press/content/serial-article-files/46798. Underlying Oberwolfach Reports material retains its source-specific licence, including CC BY-SA 4.0 where applicable; see THIRD_PARTY_NOTICES.md.

Preparation and review.

Original question attributed to the cited source and UnsolvedMath contributors. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, source inspection and formatting. No new human mathematical authorship or independent proof verification is asserted. Canonical ID 30004125 and source code OWR-16931-009 checked against the complete supplied registry and existing repository statements; known flagged repetitions and obvious identical conjectures excluded. Full mathematical-equivalence/literature review has not been performed. Statement preparation and final review followed contribution prompts C and D; automated checks verify structure and identity, not open status.

Research attempt

Literature, source context and scope

The full original catalogue body and source qualifications are preserved above. The relevant primary workshop passages were read in the cached publisher report. Historical claims and subsequent attributed results are distinguished from the mathematical work below. A restricted example, standard lemma or known theorem is not counted as new progress on the entire original target. Missing assumptions and repeated or rhetorical extracts are not silently turned into different solved problems.

Known realification and the exponent quantifier

The historical workshop discussion separates real and complex constructions. However, Bourgain, Dilworth, Ford, Konyagin and Kutzarova explicitly state realification in Section 1 of their primary paper. Their Theorem 1 supplies an effective $\varepsilon_0>0$ and explicit complex $m\times P$ matrices for sufficiently large $m$ and $m\le P\le m^{1+\varepsilon_0}$, with order $\lfloor m^{1/2+\varepsilon_0}\rfloor$ and distortion $m^{-\varepsilon_0}$. This is an attributed theorem, not reproved here. The transfer is elementary: if $\Phi=C+iD$ has complex RIP of order $s$, then \[ B=\begin{pmatrix}C&-D\\D&C\end{pmatrix} \] has real RIP of that order, because a real $s$-sparse vector $(u,v)$ gives a complex vector $u+iv$ with at most $s$ nonzero coordinates and the same input and output norms. Doubling dimensions preserves a beyond-square-root exponent after any fixed slack $0<\varepsilon<\varepsilon_0$, since $m^{\varepsilon_0-\varepsilon}\to\infty$. Consequently a blanket claim that no positive exponent is available over the reals is inappropriate. This known observation neither supplies every prescribed feasible $\varepsilon$ nor the separate polylogarithmic measurement regime. Realification adds constant-factor storage and arithmetic cost; we do not independently audit the full external construction's proof or precision complexity.

Testing an explicit two-basis route

Here is a complete obstruction to a simpler proposed construction. Let $m=2^r$ with integer $r\ge1$, set $n=m^2$, and let $H_m$ be the normalized Walsh matrix, with rows and columns indexed by $\mathbb F_2^r$ and entry $m^{-1/2}(-1)^{a\cdot b}$. Define \[ H=H_m\otimes H_m,\qquad A=[I_n\ H]. \] All entries of $H$ equal $\pm1/m$, so this is a rational, polynomial-time constructible real $n\times2n$ matrix. A direct parity computation writes its $O(n^2)$ entries in $O(n^2\log n)$ bit operations, with $O(\log n)$ bits per entry.

Verification and remaining gaps

Exact rational Walsh computations for $m=2,4,8$ check orthogonality, coherence and this sparse kernel. The route fails at $2\sqrt n$ for this family itself, not merely because its coherence proof is weak. It cannot achieve any fixed $n^{1/2+\varepsilon}$ eventually. The known external theorem above does cover some positive exponents; reproducing that fact is not new progress, and no construction covering the original exponent quantifier and all its unstated representation requirements is proved here.

References and source attribution

Primary workshop report: https://doi.org/10.4171/owr/2019/19. Relevant passages read: 2019/19 pp. 1132--1135, explicit RIP questions and polynomial versus polylogarithmic regimes.

https://arxiv.org/html/1008.4535. Read Section1 realification paragraph and Theorem1; known effective small-exponent RIP construction and its real transfer are attributed, not independently reproved.

All retrievals were checked on 11 October 2026. Original question attribution remains in the preserved Sources section. External theorems are attributed and not claimed as new discoveries or fully verified merely by citation.

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Completion Estimate

COMPLETION: 0% No substantiated new progress toward the entire original target is claimed. The worked results and failed route are delimited above; a known result, illustrative model, or auxiliary obstruction is not counted as original-target completion.

Final status and attribution

Attempt status: unresolved. This describes this research attempt, not a new ruling on the source catalogue's status or on external literature claims. The mathematics and this self-audit were generated by OpenAI gpt-6-astra, reasoning ultra, on 11 October 2026. Codex (GPT-6; exact serving revision not exposed) orchestrated the batch, reviewed the written mathematical arguments, and contributed formatting/repository review. This additional model review is not independent human verification. No human mathematical authorship, independent human proof checking, or formal verification is claimed. The historical preparation paragraphs in the reproduced catalogue body concern that body only.

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