# Tensor Network Research > Use when the user asks about tensor networks, tensor decompositions (CP, Tucker, HOSVD, HOOI, Tensor Train / TT, Matrix Product State / MPS, Tensor Ring / TR, Hierarchical Tucker / HT, PEPS, MERA, TTN, LPS, PARAFAC, randomized / streaming / sparse / nonnegative / orthogonal decompositions), tensor optimization on manifolds (Riemannian, Grassmann, Stiefel, fixed-rank, quotient, ALS, block coordinate, proximal, trust-region, projected gradient, Łojasiewicz / KL convergence), numerical linear algebra for tensors (SVD, QR, eig, polar, randomized SVD, Lanczos, Arnoldi, preconditioning, matrix completion), tensor-network algorithms (DMRG, TEBD, TDVP, sweeps, canonical forms, orthogonality center, bond dimension, entanglement entropy), tensor-network machine learning (Born Machine, Neural Quantum States, tensorized neural networks, LoRA, tensorized transformers, LLM compression), proof writing (PRX / JMLR / NeurIPS / ICML / ICLR / Nature style), paper reading, paper writing, algorithm design, code in NumPy / SciPy / - Skill: `haotong-duan/tensor-network-research` (Agent Skill, multi-file: 4 files) - Install (CLI): `npx skillmds@latest add haotong-duan/tensor-network-research` - Raw SKILL.md: https://api.skillmd.com/api/skills/haotong-duan/tensor-network-research/raw - Safety review: pending - Works with: Claude Code, Claude.ai, OpenAI Codex - Category: AI & ML - Author: haotong-Duan (https://skillmd.com/u/haotong-duan) - Updated: 2026-09-22 - Page: https://skillmd.com/skills/haotong-duan/tensor-network-research --- # Tensor Network Research A research-grade assistant skill for tensor network science. Use this skill whenever a question, design task, paper draft, or implementation sits in the tensor-network / tensor-decomposition / tensor-manifold / tensor-network-ML neighborhood. ## What this skill provides 1. **Ten knowledge modules** under `references/` covering tensor algebra, decompositions, optimization, tensor networks, TN-ML, numerical linear algebra, scientific computing, proof assistance, paper reading, and paper writing. 2. **Eight workflows** under `workflows/` for reasoning, implementation, paper reading, paper writing, proof writing, algorithm design, coding, and experiment design. 3. **Reusable templates** under `templates/` for prompts, code, and reference patterns. 4. **Anti-patterns** under `mistakes/common-mistakes.md`. 5. **Resource index** covering notation, glossary, formula library, textbooks, seminal and recent papers, libraries, datasets, and repositories. ## When to load sub-files | User intent | Load first | | --------------------------------------------------- | ------------------------------------- | | Derive / verify a tensor identity | `references/01-tensor-algebra.md` | | Choose or critique a decomposition | `references/02-tensor-decompositions.md` | | Design an algorithm with convergence guarantees | `references/03-tensor-optimization.md` | | Reason about a quantum or TN-state algorithm | `references/04-tensor-networks.md` | | Apply TN to ML / LLMs | `references/05-tn-machine-learning.md` | | Discuss stability / complexity of linear algebra | `references/06-numerical-linear-algebra.md` | | Write / review / debug code | `references/07-scientific-computing.md` | | Write or check a mathematical proof | `references/08-proof-assistance.md` | | Read or summarize a paper | `references/09-paper-reading.md` + `workflows/paper-reading-workflow.md` | | Write a paper, rebuttal, or thesis chapter | `references/10-paper-writing.md` + `workflows/paper-writing-workflow.md` | | Design a new algorithm | `workflows/algorithm-design-workflow.md` | | Run numerical experiments | `workflows/experiment-workflow.md` | ## Core reasoning principles (apply to every response) - **Define symbols explicitly.** Every symbol that appears in a formula must be defined the first time it is used. Do not reuse `r`, `R`, `n`, `d` for different things in the same derivation. - **Verify dimensions / tensor shapes on both sides** of every equation. When a contraction is written, write the index bookkeeping explicitly. - **State assumptions.** Rank bounds, smoothness, regularity, genericity ("generic tensor"), separability, real vs. complex, and whether the objective is convex must be stated before they are used. - **Derive, do not hand-wave.** When a paper claims a step, derive it symbolically or with a small worked example. If a step cannot be derived, flag it as an unverified claim. - **Check complexity.** Every algorithm must come with a flops / memory estimate and, when meaningful, a per-iteration cost. - **Check numerical stability.** Comment on conditioning, orthogonality loss, catastrophic cancellation, and rank-deficiency. - **Be honest about limits.** If a question is open, say so. If a method only works under assumptions, list the assumptions. ## Output conventions - Use LaTeX math in `$...$` (inline) and `$$...$$` (display) in markdown. - Use code blocks with the right language tag (`python`, `latex`, `bash`). - Reference specific files with `path:line` or `path#section` so the user can navigate. - Prefer tables and bullets over prose when listing properties, options, or criteria. - When comparing two methods, always use a 2- or 3-column table covering the same axes (objective, complexity, convergence, requirements). ## Quick start 1. Identify the user's intent (see table above) and load the matching `references/` module first. 2. Cross-check against the relevant `workflows/` file. 3. Use templates from `templates/` when generating prompts or code. 4. Consult `mistakes/common-mistakes.md` before finalizing a recommendation — many "obvious" TN choices are subtly wrong. 5. Cite papers and books by `[Author Year]` style and look them up in `references/seminal-papers.md` / `references/recent-papers.md` / `references/textbooks.md`. ## File index ``` . ├── SKILL.md <- this file ├── README.md <- human-facing overview ├── capabilities.md <- one-page capability map ├── references/ <- deep knowledge modules │ ├── 01-tensor-algebra.md │ ├── 02-tensor-decompositions.md │ ├── 03-tensor-optimization.md │ ├── 04-tensor-networks.md │ ├── 05-tn-machine-learning.md │ ├── 06-numerical-linear-algebra.md │ ├── 07-scientific-computing.md │ ├── 08-proof-assistance.md │ ├── 09-paper-reading.md │ ├── 10-paper-writing.md │ ├── glossary.md │ ├── notation.md │ ├── formula-library.md │ ├── textbooks.md │ ├── seminal-papers.md │ ├── recent-papers.md │ ├── libraries.md │ ├── datasets.md │ └── repositories.md ├── workflows/ <- process definitions │ ├── reasoning-checklist.md │ ├── implementation-checklist.md │ ├── paper-reading-workflow.md │ ├── paper-writing-workflow.md │ ├── proof-workflow.md │ ├── algorithm-design-workflow.md │ ├── coding-workflow.md │ └── experiment-workflow.md ├── templates/ <- reusable patterns │ ├── prompt-templates.md │ ├── code-templates.md │ └── reference-templates.md └── mistakes/ └── common-mistakes.md ```