L1 ⚛ L1-028 ⊙ Testnet

Low-Rank Matrix Sensing (compressive matrix recovery)

Compressive Imaging · Low-rank linear inverse problems

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⬇ L1-028.md

# ⚛ L1 Principle — Low-Rank Matrix Sensing (compressive matrix recovery) **ID:** `L1-028` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Compressive Imaging — *Low-rank linear inverse problems* > **🎯 Problem class:** linear inverse · **🧮 Solution space:** low rank matrix > **📡 Carrier:** none · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41547811 --- ## 🧠 1. Introduction **Low-Rank Matrix Sensing (compressive matrix recovery)** is a **linear inverse problem** whose unknown lives in **low rank matrix** space, within the **Low-rank linear inverse problems** sub-domain of **Compressive Imaging**. Measurements consist of none via a **linear projection** sensing mechanism. The forward operator applies, in order: S · pattern · structured operator; L · trace inner product operator; detector accumulates flux over the exposure window; D · scalar operator. Observations are corrupted by additive Gaussian noise. Recovery guaranteed w.h.p. when m >= C * r * (n1 + n2) and A satisfies matrix-RIP; nuclear-norm minimization is exact for noiseless case; stability bound ||X_hat - X||_F <= C * sigma * sqrt(m) / smallest_singular_value. ## ⚙ 2. Forward Model Physical chain: **x** → S · pattern · structured → L · trace inner product → Temporal integration → D · scalar → **y** (detector). ``` y = `D.scalar` ∫_t dt `L.trace_inner_product` `S.pattern.structured` x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `S.pattern.structured` | S · pattern · structured operator | | `L.trace_inner_product` | L · trace inner product operator | | `int.temporal` | Detector accumulates flux over the exposure window | | `D.scalar` | D · scalar operator | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Compressive Imaging | | Sub domain | Low-rank linear inverse problems | | Carrier | none | | Problem class | linear_inverse | | Solution space | low_rank_matrix | | Noise model | gaussian | | Integration axis | basis | | Difficulty delta | 3 | | L dag | 3.2 | ## 📡 4. Measurement Model Recovery guaranteed w.h.p. when m >= C * r * (n1 + n2) and A satisfies matrix-RIP; nuclear-norm minimization is exact for noiseless case; stability bound ||X_hat - X||_F <= C * sigma * sqrt(m) / smallest_singular_value. | Metric | Value | |---|---| | Metric | relative_Frobenius_error | | Secondary | rank_recovery_accuracy | ## 📏 5. Operating Range (Ω) **Center problem class:** `low_rank_recovery` · **Forward operator:** `gaussian_matrix_sensing` **Center point:** | Parameter | Unit | Value | |---|---|---| | N1 | — | 256 | | N2 | — | 256 | | Rank r | — | 5 | | Noise level | — | 0.01 | | M over minmn | — | 6 | | Rank misspec | — | 0 | | Measurement noise tail | — | 0 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | N1 | — | 32 – 2048 | | N2 | — | 32 – 2048 | | Rank r | — | 1 – 50 | | Noise level | — | 0.001 – 0.1 | | M over minmn | — | 2 – 30 | | Rank misspec | — | 0 – 10 | | Measurement noise tail | — | 0.0 – 0.2 | ## 🎯 6. Tolerance (ε) **Center tolerance:** relative_error <= 0.05 | Metric | Range | |---|---| | Relative error | 0.01 – 0.5 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **relative_Frobenius_error**, with κ = `6000` and δ = `3`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0x7994df41ed4b3190fb25ac2b80c2be15623094842e8e7fcec64e622a3cd3d40f` - **Chain tx hash:** `0xec6ec05bec5dc423e2b09aa4c1f0b57ab8021e4bd727b30d6c7cace18f356493` - **Chain block:** `41547811` --- ## File Mapping This bundle consists of: `L1-028.md`, `L1-028.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-028.md` | Source of truth — edit this | Human or LLM | | `L1-028.json` | Structured metadata for the registry | LLM regenerates from the sections above | **Prompt for your LLM after editing this Markdown:** > Read the attached Markdown. Regenerate the sibling `.json` so every field matches. > Preserve the schema documented in the rows above. > Output each file in its own fenced code block tagged with the filename. > Output only the JSON object. _This Markdown was auto-synthesized from the catalog row for `L1-028`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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L2-028-001 Gaussian Matrix Sensing Mismatch-Only Spec

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📋 JSON metadata

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  "domain": "Compressive Imaging",
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    "title": "Low-Rank Matrix Sensing (compressive matrix recovery)"
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  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Low-rank linear inverse problems",
  "title": "Low-Rank Matrix Sensing (compressive matrix recovery)"
}

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