L1 ⚛ L1-391 ⊙ Testnet

Sparse Signal Recovery (analysis / synthesis sparsity)

Signal Processing · L1-minimization and greedy sparse approximation

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⬇ L1-391.md

# ⚛ L1 Principle — Sparse Signal Recovery (analysis / synthesis sparsity) **ID:** `L1-391` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Signal Processing — *L1-minimization and greedy sparse approximation* > **🎯 Problem class:** sparse linear inverse · **🧮 Solution space:** sparse vector > **📡 Carrier:** generic · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555198 --- ## 🧠 1. Introduction **Sparse Signal Recovery (analysis / synthesis sparsity)** is a **sparse linear inverse problem** whose unknown lives in **sparse vector** space, within the **L1-minimization and greedy sparse approximation** sub-domain of **Signal Processing**. Measurements consist of generic via a **sparse synthesis** sensing mechanism. The forward operator applies, in order: L · project · generic operator; L · synthesis · dictionary operator; pixel-level spatial averaging on the detector. Observations are corrupted by additive Gaussian noise. Unique sparsity under mutual-coherence or RIP bounds; relaxed L0->L1 equivalence under Donoho-Elad theorem. Mismatch: dictionary drift, non-exact sparsity, off-grid sparsity (basis mismatch). ## ⚙ 2. Forward Model Physical chain: **x** → L · project · generic → L · synthesis · dictionary → Spatial integration → **y** (detector). ``` y = ∫_A dA `L.synthesis.dictionary` `L.project.generic` x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `L.project.generic` | L · project · generic operator | | `L.synthesis.dictionary` | L · synthesis · dictionary operator | | `int.spatial` | Pixel-level spatial averaging on the detector | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Signal Processing | | Sub domain | L1-minimization and greedy sparse approximation | | Carrier | generic | | Problem class | sparse_linear_inverse | | Solution space | sparse_vector | | Noise model | gaussian | | Integration axis | none | | Difficulty delta | 3 | | L dag | 2.7 | ## 📡 4. Measurement Model Unique sparsity under mutual-coherence or RIP bounds; relaxed L0->L1 equivalence under Donoho-Elad theorem. Mismatch: dictionary drift, non-exact sparsity, off-grid sparsity (basis mismatch). | Metric | Value | |---|---| | Metric | NMSE | | Secondary | support_recovery_rate | ## 📏 5. Operating Range (Ω) **Center problem class:** `synthesis_sparse` · **Forward operator:** `dct_sparse` **Center point:** | Parameter | Unit | Value | |---|---|---| | M | — | 512 | | N | — | 1024 | | K | — | 50 | | Mu | — | 0.1 | | Sigma n | N | 0.01 | | Off grid | — | 0 | | Basis mismatch | — | 0 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | M | — | 32 – 16384 | | N | — | 128 – 16384 | | K | — | 1 – 2000 | | Mu | — | 0.01 – 0.8 | | Sigma n | N | 0.0 – 0.1 | | Off grid | — | 0.0 – 1.0 | | Basis mismatch | — | 0.0 – 0.5 | ## 🎯 6. Tolerance (ε) **Center tolerance:** NMSE 1e-3 | Metric | Range | |---|---| | Nmse | 1e-05 – 1.0 | ## ⚖ 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:** `0xc7a64eadaed8ed783367753f91dc8ae56439d5c257032eadab3bce406e4b0731` - **Chain tx hash:** `0xdd99c047577d0994992efa845befb13cf7e88cad33113121c7689ca4e184b237` - **Chain block:** `41555198` --- ## File Mapping This bundle consists of: `L1-391.md`, `L1-391.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-391.md` | Source of truth — edit this | Human or LLM | | `L1-391.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 `L1-391`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-391.md).
Regenerate L1-391.json so every field matches the Markdown.
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L2-391-001 Sparse Signal Recovery Synthesis-Model Spec (overcomplete dictionary)

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📋 JSON metadata

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  "chain_block": 41555198,
  "chain_hash": "0xc7a64eadaed8ed783367753f91dc8ae56439d5c257032eadab3bce406e4b0731",
  "chain_tx_hash": "0xdd99c047577d0994992efa845befb13cf7e88cad33113121c7689ca4e184b237",
  "domain": "Signal Processing",
  "hardness_fn": {
    "delta": 3,
    "kappa": 2000,
    "metric": "NMSE",
    "type": "epsilon_fn"
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      "name": "primary",
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  "layer": "L1",
  "observable_profile": {
    "metric": "NMSE",
    "regime": "Unique sparsity under mutual-coherence or RIP bounds; relaxed L0-\u003eL1 equivalence under Donoho-Elad theorem. Mismatch: dictionary drift, non-exact sparsity, off-grid sparsity (basis mismatch).",
    "secondary": "support_recovery_rate"
  },
  "physics_fingerprint": {
    "L_DAG": 2.7,
    "carrier": "generic",
    "difficulty_delta": 3,
    "domain": "Signal Processing",
    "integration_axis": "none",
    "noise_model": "gaussian",
    "primitives": [
      "L.project.generic",
      "L.synthesis.dictionary",
      "int.spatial"
    ],
    "problem_class": "sparse_linear_inverse",
    "sensing_mechanism": "sparse_synthesis",
    "solution_space": "sparse_vector",
    "sub_domain": "L1-minimization and greedy sparse approximation",
    "title": "Sparse Signal Recovery (analysis / synthesis sparsity)"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "dct_sparse",
      "wavelet_sparse",
      "learned_dictionary",
      "gabor_sparse"
    ],
    "allowed_omega_dimensions": [
      "N",
      "M",
      "k",
      "mu",
      "sigma_n",
      "basis_mismatch",
      "off_grid",
      "dictionary_type"
    ],
    "allowed_problem_classes": [
      "synthesis_sparse",
      "analysis_sparse",
      "cosparse_model"
    ],
    "center_spec": {
      "epsilon_fn_center": "NMSE 1e-3",
      "forward_operator": "dct_sparse",
      "input_format": "measurement_plus_dictionary",
      "omega": {
        "M": 512,
        "N": 1024,
        "basis_mismatch": 0.0,
        "k": 50,
        "mu": 0.1,
        "off_grid": 0.0,
        "sigma_n": 0.01
      },
      "problem_class": "synthesis_sparse"
    },
    "epsilon_bounds": {
      "nmse": [
        1e-05,
        1.0
      ]
    },
    "omega_bounds": {
      "M": [
        32,
        16384
      ],
      "N": [
        128,
        16384
      ],
      "basis_mismatch": [
        0.0,
        0.5
      ],
      "k": [
        1,
        2000
      ],
      "mu": [
        0.01,
        0.8
      ],
      "off_grid": [
        0.0,
        1.0
      ],
      "sigma_n": [
        0.0,
        0.1
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "L1-minimization and greedy sparse approximation",
  "title": "Sparse Signal Recovery (analysis / synthesis sparsity)"
}

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