# ⚛  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`

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## 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._