L1 ⚛ L1-386 ⊙ Testnet

Compressed Sensing (random-projection linear inverse)

Signal Processing · Sparse linear inverse under RIP measurement

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⬇ L1-386.md

# ⚛ L1 Principle — Compressed Sensing (random-projection linear inverse) **ID:** `L1-386` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Signal Processing — *Sparse linear inverse under RIP measurement* > **🎯 Problem class:** sparse linear inverse · **🧮 Solution space:** sparse vector > **📡 Carrier:** electromagnetic · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555197 --- ## 🧠 1. Introduction **Compressed Sensing (random-projection linear inverse)** is a **sparse linear inverse problem** whose unknown lives in **sparse vector** space, within the **Sparse linear inverse under RIP measurement** sub-domain of **Signal Processing**. Measurements consist of electromagnetic field measurements via a **random projection** sensing mechanism. The forward operator applies, in order: L · project · random operator; pixel-level spatial averaging on the detector. Observations are corrupted by additive Gaussian noise. Well-posed under RIP (Candes-Tao 2006); kappa = (1+delta_2k)/(1-delta_2k). Failure when measurement_ratio < 2*k/N * log(N/k). Mismatch parameters: sparsity_overestimation, non-exact-sparsity (compressible). ## ⚙ 2. Forward Model Physical chain: **x** → L · project · random → Spatial integration → **y** (detector). ``` y = ∫_A dA `L.project.random` x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `L.project.random` | L · project · random operator | | `int.spatial` | Pixel-level spatial averaging on the detector | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Signal Processing | | Sub domain | Sparse linear inverse under RIP measurement | | Carrier | electromagnetic | | Problem class | sparse_linear_inverse | | Solution space | sparse_vector | | Noise model | gaussian | | Integration axis | measurement | | Difficulty delta | 3 | | L dag | 2.3 | ## 📡 4. Measurement Model Well-posed under RIP (Candes-Tao 2006); kappa = (1+delta_2k)/(1-delta_2k). Failure when measurement_ratio < 2*k/N * log(N/k). Mismatch parameters: sparsity_overestimation, non-exact-sparsity (compressible). | Metric | Value | |---|---| | Metric | NMSE | | Secondary | support_recovery_rate | ## 📏 5. Operating Range (Ω) **Center problem class:** `gaussian_cs` · **Forward operator:** `gaussian_random_matrix` **Center point:** | Parameter | Unit | Value | |---|---|---| | M | — | 256 | | N | — | 1024 | | K | — | 40 | | Basis | — | identity | | Sigma n | N | 0.01 | | Coherence | — | 0.05 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | M | — | 32 – 12288 | | N | — | 256 – 16384 | | K | — | 5 – 2000 | | Sigma n | N | 0.001 – 0.1 | | Coherence | — | 0.01 – 0.5 | | Basis mismatch | — | 0.0 – 0.3 | ## 🎯 6. Tolerance (ε) **Center tolerance:** NMSE 1e-3 | Metric | Range | |---|---| | Nmse | 1e-05 – 0.1 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **NMSE**, with κ = `2000` and δ = `3`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0x56ac025df0902b809009fb9e58314b49aea1607c76f5f7ff5cc16a2644fadce0` - **Chain tx hash:** `0x6e19ad14db087a1ecb927ec91321e75207ba3bb1d038eec43b5a0d1533424e21` - **Chain block:** `41555197` --- ## File Mapping This bundle consists of: `L1-386.md`, `L1-386.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-386.md` | Source of truth — edit this | Human or LLM | | `L1-386.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 `L1-386`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-386.md).
Regenerate L1-386.json so every field matches the Markdown.
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Output each file in its own fenced code block tagged with the filename.

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L2-386-001 Compressed Sensing Canonical Spec (Gaussian matrix + Lp recovery)

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📋 JSON metadata

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  "artifact_id": "L1-386",
  "chain_block": 41555197,
  "chain_hash": "0x56ac025df0902b809009fb9e58314b49aea1607c76f5f7ff5cc16a2644fadce0",
  "chain_tx_hash": "0x6e19ad14db087a1ecb927ec91321e75207ba3bb1d038eec43b5a0d1533424e21",
  "domain": "Signal Processing",
  "hardness_fn": {
    "delta": 3,
    "kappa": 2000,
    "metric": "NMSE",
    "type": "epsilon_fn"
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  "layer": "L1",
  "observable_profile": {
    "metric": "NMSE",
    "regime": "Well-posed under RIP (Candes-Tao 2006); kappa = (1+delta_2k)/(1-delta_2k). Failure when measurement_ratio \u003c 2*k/N * log(N/k). Mismatch parameters: sparsity_overestimation, non-exact-sparsity (compressible).",
    "secondary": "support_recovery_rate"
  },
  "physics_fingerprint": {
    "L_DAG": 2.3,
    "carrier": "electromagnetic",
    "difficulty_delta": 3,
    "domain": "Signal Processing",
    "integration_axis": "measurement",
    "noise_model": "gaussian",
    "primitives": [
      "L.project.random",
      "int.spatial"
    ],
    "problem_class": "sparse_linear_inverse",
    "sensing_mechanism": "random_projection",
    "solution_space": "sparse_vector",
    "sub_domain": "Sparse linear inverse under RIP measurement",
    "title": "Compressed Sensing (random-projection linear inverse)"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "gaussian_random_matrix",
      "partial_dft",
      "bernoulli",
      "sub_gaussian"
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    "allowed_omega_dimensions": [
      "N",
      "M",
      "k",
      "sigma_n",
      "basis",
      "coherence",
      "basis_mismatch"
    ],
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      "structured_cs",
      "block_sparse_cs"
    ],
    "center_spec": {
      "epsilon_fn_center": "NMSE 1e-3",
      "forward_operator": "gaussian_random_matrix",
      "input_format": "measurement_plus_matrix",
      "omega": {
        "M": 256,
        "N": 1024,
        "basis": "identity",
        "coherence": 0.05,
        "k": 40,
        "sigma_n": 0.01
      },
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    },
    "epsilon_bounds": {
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        1e-05,
        0.1
      ]
    },
    "omega_bounds": {
      "M": [
        32,
        12288
      ],
      "N": [
        256,
        16384
      ],
      "basis_mismatch": [
        0.0,
        0.3
      ],
      "coherence": [
        0.01,
        0.5
      ],
      "k": [
        5,
        2000
      ],
      "sigma_n": [
        0.001,
        0.1
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Sparse linear inverse under RIP measurement",
  "title": "Compressed Sensing (random-projection linear inverse)"
}

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