# ⚛  L1 Principle — Robust Principal Component Analysis (low-rank plus sparse)

**ID:** `L1-393` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Signal Processing — *L+S decomposition / Principal Component Pursuit*
> **🎯 Problem class:** low rank plus sparse · **🧮 Solution space:** low rank plus sparse matrix
> **📡 Carrier:** abstract · **🌫 Noise:** gaussian plus outlier
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41555198

---

## 🧠 1. Introduction

**Robust Principal Component Analysis (low-rank plus sparse)** is a **low rank plus sparse** whose unknown lives in **low rank plus sparse matrix** space, within the **L+S decomposition / Principal Component Pursuit** sub-domain of **Signal Processing**.

Measurements consist of abstract via a **additive decomposition** sensing mechanism.

The forward operator applies, in order: L · synthesis · lowrank operator; L · synthesis · sparse operator; pixel-level spatial averaging on the detector.

Observations are corrupted by gaussian plus outlier. Identifiable under rank-sparsity incoherence: low-rank component must be incoherent w.r.t. standard basis; sparse component must be spread among entries. PCP (nuclear+L1) solves exactly under mild conditions.

## ⚙ 2. Forward Model

Physical chain: **x** → L · synthesis · lowrank → L · synthesis · sparse → Spatial integration → **y** (detector).

```
y = ∫_A dA `L.synthesis.sparse` `L.synthesis.lowrank` x
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.synthesis.lowrank` | L · synthesis · lowrank operator |
| `L.synthesis.sparse` | L · synthesis · sparse operator |
| `int.spatial` | Pixel-level spatial averaging on the detector |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Signal Processing |
| Sub domain | L+S decomposition / Principal Component Pursuit |
| Carrier | abstract |
| Problem class | low_rank_plus_sparse |
| Solution space | low_rank_plus_sparse_matrix |
| Noise model | gaussian_plus_outlier |
| Integration axis | none |
| Difficulty delta | 5 |
| L dag | 2.7 |

## 📡 4. Measurement Model

Identifiable under rank-sparsity incoherence: low-rank component must be incoherent w.r.t. standard basis; sparse component must be spread among entries. PCP (nuclear+L1) solves exactly under mild conditions.

| Metric | Value |
|---|---|
| Metric | relative_frobenius_error_L |
| Secondary | precision_recall_S |

## 📏 5. Operating Range (Ω)

**Center problem class:** `rpca_lowrank_sparse` · **Forward operator:** `additive_L_plus_S`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| M | — | 500 |
| N | — | 500 |
| R | — | 10 |
| Mu 0 | — | 1 |
| Lambda | — | 1 |
| Sigma n | N | 0.001 |
| S fraction | — | 0.1 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| M | — | 50 – 10000 |
| N | — | 50 – 10000 |
| R | — | 1 – 500 |
| Mu 0 | — | 1.0 – 10.0 |
| Lambda | — | 0.1 – 10 |
| Sigma n | N | 0.0 – 0.1 |
| S fraction | — | 0.0 – 0.5 |
| Column corruption | — | 0.0 – 0.3 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** rel-Frob 1e-4

| Metric | Range |
|---|---|
| Rel frob | 1e-06 – 1.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **relative_frobenius_error_L**, with κ = `1000` and δ = `5`.

## 💾 8. Reference Dataset

- **primary** · weight 1.0 · IPFS _(not pinned yet)_

## 9. On-chain Registration

- **Chain hash:** `0xa41b6f9388e8571d558943aae7f295ec69738963d691f8f13d97772977a18f6c`
- **Chain tx hash:** `0x10d318fe9c2c990b29e95eac8a40e9303d968a1bbe712451cada300aabca5e76`
- **Chain block:** `41555198`

---

## File Mapping

This bundle consists of: `L1-393.md`, `L1-393.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-393.md` | Source of truth — edit this | Human or LLM |
| `L1-393.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 `L1-393`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._