# ⚛  L1 Principle — Low-Rank Matrix Sensing (compressive matrix recovery)

**ID:** `L1-028` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Compressive Imaging — *Low-rank linear inverse problems*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** low rank matrix
> **📡 Carrier:** none · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41547811

---

## 🧠 1. Introduction

**Low-Rank Matrix Sensing (compressive matrix recovery)** is a **linear inverse problem** whose unknown lives in **low rank matrix** space, within the **Low-rank linear inverse problems** sub-domain of **Compressive Imaging**.

Measurements consist of none via a **linear projection** sensing mechanism.

The forward operator applies, in order: S · pattern · structured operator; L · trace inner product operator; detector accumulates flux over the exposure window; D · scalar operator.

Observations are corrupted by additive Gaussian noise. Recovery guaranteed w.h.p. when m >= C * r * (n1 + n2) and A satisfies matrix-RIP; nuclear-norm minimization is exact for noiseless case; stability bound ||X_hat - X||_F <= C * sigma * sqrt(m) / smallest_singular_value.

## ⚙ 2. Forward Model

Physical chain: **x** → S · pattern · structured → L · trace inner product → Temporal integration → D · scalar → **y** (detector).

```
y = `D.scalar` ∫_t dt `L.trace_inner_product` `S.pattern.structured` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `S.pattern.structured` | S · pattern · structured operator |
| `L.trace_inner_product` | L · trace inner product operator |
| `int.temporal` | Detector accumulates flux over the exposure window |
| `D.scalar` | D · scalar operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Compressive Imaging |
| Sub domain | Low-rank linear inverse problems |
| Carrier | none |
| Problem class | linear_inverse |
| Solution space | low_rank_matrix |
| Noise model | gaussian |
| Integration axis | basis |
| Difficulty delta | 3 |
| L dag | 3.2 |

## 📡 4. Measurement Model

Recovery guaranteed w.h.p. when m >= C * r * (n1 + n2) and A satisfies matrix-RIP; nuclear-norm minimization is exact for noiseless case; stability bound ||X_hat - X||_F <= C * sigma * sqrt(m) / smallest_singular_value.

| Metric | Value |
|---|---|
| Metric | relative_Frobenius_error |
| Secondary | rank_recovery_accuracy |

## 📏 5. Operating Range (Ω)

**Center problem class:** `low_rank_recovery` · **Forward operator:** `gaussian_matrix_sensing`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N1 | — | 256 |
| N2 | — | 256 |
| Rank r | — | 5 |
| Noise level | — | 0.01 |
| M over minmn | — | 6 |
| Rank misspec | — | 0 |
| Measurement noise tail | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N1 | — | 32 – 2048 |
| N2 | — | 32 – 2048 |
| Rank r | — | 1 – 50 |
| Noise level | — | 0.001 – 0.1 |
| M over minmn | — | 2 – 30 |
| Rank misspec | — | 0 – 10 |
| Measurement noise tail | — | 0.0 – 0.2 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** relative_error <= 0.05

| Metric | Range |
|---|---|
| Relative error | 0.01 – 0.5 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **relative_Frobenius_error**, with κ = `6000` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x7994df41ed4b3190fb25ac2b80c2be15623094842e8e7fcec64e622a3cd3d40f`
- **Chain tx hash:** `0xec6ec05bec5dc423e2b09aa4c1f0b57ab8021e4bd727b30d6c7cace18f356493`
- **Chain block:** `41547811`

---

## File Mapping

This bundle consists of: `L1-028.md`, `L1-028.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-028.md` | Source of truth — edit this | Human or LLM |
| `L1-028.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 `L1-028`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._