L1 ⚛ L1-392 ⊙ Testnet

Matrix Completion (low-rank recovery from partial entries)

Signal Processing · Low-rank matrix recovery via nuclear-norm minimization

⬇ Download bundle 🚀 Submit to mainnet — earn 100 PWM + 5% royalty

Markdown — human-readable source of truth

⚙ auto-generated

⬇ L1-392.md

# ⚛ L1 Principle — Matrix Completion (low-rank recovery from partial entries) **ID:** `L1-392` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Signal Processing — *Low-rank matrix recovery via nuclear-norm minimization* > **🎯 Problem class:** low rank inverse · **🧮 Solution space:** low rank matrix > **📡 Carrier:** abstract · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555198 --- ## 🧠 1. Introduction **Matrix Completion (low-rank recovery from partial entries)** is a **low rank inverse** whose unknown lives in **low rank matrix** space, within the **Low-rank matrix recovery via nuclear-norm minimization** sub-domain of **Signal Processing**. Measurements consist of abstract via a **entry masking** sensing mechanism. The forward operator applies, in order: D · mask · entry operator; pixel-level spatial averaging on the detector. Observations are corrupted by additive Gaussian noise. Unique recovery via nuclear-norm minimization under incoherence assumption. Fails when coherence is high (spiky matrices) or observation ratio is below sample-complexity threshold. ## ⚙ 2. Forward Model Physical chain: **x** → D · mask · entry → Spatial integration → **y** (detector). ``` y = ∫_A dA `D.mask.entry` x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `D.mask.entry` | D · mask · entry operator | | `int.spatial` | Pixel-level spatial averaging on the detector | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Signal Processing | | Sub domain | Low-rank matrix recovery via nuclear-norm minimization | | Carrier | abstract | | Problem class | low_rank_inverse | | Solution space | low_rank_matrix | | Noise model | gaussian | | Integration axis | none | | Difficulty delta | 3 | | L dag | 2 | ## 📡 4. Measurement Model Unique recovery via nuclear-norm minimization under incoherence assumption. Fails when coherence is high (spiky matrices) or observation ratio is below sample-complexity threshold. | Metric | Value | |---|---| | Metric | relative_frobenius_error | | Secondary | rank_recovery_accuracy | ## 📏 5. Operating Range (Ω) **Center problem class:** `low_rank_MC` · **Forward operator:** `uniform_random_sampling` **Center point:** | Parameter | Unit | Value | |---|---|---| | M | — | 500 | | N | — | 500 | | P | — | 0.3 | | R | — | 10 | | Mu 0 | — | 1 | | Sigma n | N | 0.001 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | M | — | 50 – 10000 | | N | — | 50 – 10000 | | P | — | 0.05 – 0.9 | | R | — | 1 – 500 | | Mu 0 | — | 1.0 – 10.0 | | Sigma n | N | 0.0 – 0.1 | | Sampling bias | — | 0.0 – 1.0 | ## 🎯 6. Tolerance (ε) **Center tolerance:** rel-Frob 1e-3 | Metric | Range | |---|---| | Rel frob | 1e-05 – 1.0 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **relative_frobenius_error**, with κ = `500` and δ = `3`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0x51bf33cda9829f321a2ed7a29fc60ce1bbaa070b892dab24ce53f272f5c09ce0` - **Chain tx hash:** `0x22e36a78932ea1433ae181070a4cd99433da1e54e0eee5d365fc2ce65ded1957` - **Chain block:** `41555198` --- ## File Mapping This bundle consists of: `L1-392.md`, `L1-392.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-392.md` | Source of truth — edit this | Human or LLM | | `L1-392.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 `L1-392`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

📝 Polish this artifact in your LLM

Download the files on the right.
Paste the Markdown into ChatGPT / Claude / Gemini and edit freely.
Send your edited Markdown back with the prompt below.
Save the regenerated JSON, then submit via /submit.

Prompt to give your LLM

Read the attached Markdown (L1-392.md).
Regenerate L1-392.json so every field matches the Markdown.
Preserve the schema documented in the File Mapping footer at the bottom of the MD.
Output each file in its own fenced code block tagged with the filename.

⌥ Hierarchy

Upstream lineage and downstream children

↓ L2 Spec 1 child ▾

L2-392-001 Matrix Completion Uniform-Sampling Spec (MovieLens family)

⬇ Download bundle

2 files · click to download

📋 JSON metadata

{
  "artifact_id": "L1-392",
  "chain_block": 41555198,
  "chain_hash": "0x51bf33cda9829f321a2ed7a29fc60ce1bbaa070b892dab24ce53f272f5c09ce0",
  "chain_tx_hash": "0x22e36a78932ea1433ae181070a4cd99433da1e54e0eee5d365fc2ce65ded1957",
  "domain": "Signal Processing",
  "hardness_fn": {
    "delta": 3,
    "kappa": 500,
    "metric": "relative_frobenius_error",
    "type": "epsilon_fn"
  },
  "initiator_dataset": [
    {
      "ipfs_cid": null,
      "license_hash": null,
      "name": "primary",
      "weight": 1.0
    }
  ],
  "layer": "L1",
  "observable_profile": {
    "metric": "relative_frobenius_error",
    "regime": "Unique recovery via nuclear-norm minimization under incoherence assumption. Fails when coherence is high (spiky matrices) or observation ratio is below sample-complexity threshold.",
    "secondary": "rank_recovery_accuracy"
  },
  "physics_fingerprint": {
    "L_DAG": 2.0,
    "carrier": "abstract",
    "difficulty_delta": 3,
    "domain": "Signal Processing",
    "integration_axis": "none",
    "noise_model": "gaussian",
    "primitives": [
      "D.mask.entry",
      "int.spatial"
    ],
    "problem_class": "low_rank_inverse",
    "sensing_mechanism": "entry_masking",
    "solution_space": "low_rank_matrix",
    "sub_domain": "Low-rank matrix recovery via nuclear-norm minimization",
    "title": "Matrix Completion (low-rank recovery from partial entries)"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "uniform_random_sampling",
      "power_law_sampling",
      "block_sampling"
    ],
    "allowed_omega_dimensions": [
      "m",
      "n",
      "r",
      "p",
      "mu_0",
      "sigma_n",
      "sampling_bias"
    ],
    "allowed_problem_classes": [
      "low_rank_MC",
      "robust_MC",
      "tensor_completion"
    ],
    "center_spec": {
      "epsilon_fn_center": "rel-Frob 1e-3",
      "forward_operator": "uniform_random_sampling",
      "input_format": "partial_observations",
      "omega": {
        "m": 500,
        "mu_0": 1.0,
        "n": 500,
        "p": 0.3,
        "r": 10,
        "sigma_n": 0.001
      },
      "problem_class": "low_rank_MC"
    },
    "epsilon_bounds": {
      "rel_frob": [
        1e-05,
        1.0
      ]
    },
    "omega_bounds": {
      "m": [
        50,
        10000
      ],
      "mu_0": [
        1.0,
        10.0
      ],
      "n": [
        50,
        10000
      ],
      "p": [
        0.05,
        0.9
      ],
      "r": [
        1,
        500
      ],
      "sampling_bias": [
        0.0,
        1.0
      ],
      "sigma_n": [
        0.0,
        0.1
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Low-rank matrix recovery via nuclear-norm minimization",
  "title": "Matrix Completion (low-rank recovery from partial entries)"
}

🎁

Earn PWM by promoting this to mainnet

• 100 PWM one-time grant on mainnet promotion
• 5% upstream royalty on every downstream L4 mining event, forever
• Your wallet is enrolled the moment the founder accepts

🚀 Submit to mainnet →

How: Download the bundle above → polish the Markdown in ChatGPT / Claude / Gemini → submit your version → founder reviews → mainnet.