tensor-network-research

name

tensor-network-research

description

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 / PyTorch / JAX / TensorNetwork / ITensor / TeNPy / quimb / opt_einsum, or experiment design in the tensor-network area. Triggers on phrases like "tensor decomposition", "matrix product state", "tensor train", "tensor network", "PEPS", "MERA", "DMRG", "Born machine", "tensorized transformer", "Riemannian optimization on Grassmann", "TT rank", "CP rank", "HOSVD", "ALS for tensor", "tensor regression", "ground state search", "entanglement entropy bound", "tensor completion", "TT-SVD", "quantum-inspired machine learning".

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