# ⚛  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._