L1 ⚛ L1-393 ⊙ Testnet

Robust Principal Component Analysis (low-rank plus sparse)

Signal Processing · L+S decomposition / Principal Component Pursuit

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⬇ L1-393.md

# ⚛ 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._

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📋 JSON metadata

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  "status": "testnet",
  "sub_domain": "L+S decomposition / Principal Component Pursuit",
  "title": "Robust Principal Component Analysis (low-rank plus sparse)"
}

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