| # LDData: Low-Discrepancy Generating Vectors and Matrices |
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| A curated collection of **low-discrepancy point set parameters** including **lattice rules**, **digital nets**, **polynomial lattice rules**, **Sobol' nets**, and **RQMC randomizations**. This dataset enables reproducible research and high-performance Quasi–Monte Carlo (QMC) and Randomized QMC (RQMC) simulation. |
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| The [LDData repository](https://github.com/QMCSoftware/LDData) provides **standard text-based formats** for specifying structures used in QMC point generation across arbitrarily high dimensions. |
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| --- |
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| ## Dataset Summary |
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| LDData is a dataset of structured parameter files defining: |
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| - Rank-1 **lattice rules** |
| - Base-$b$ **digital nets** |
| - **Polynomial lattice rules** |
| - **Sobol' and Sobol–Joe-Kuo sequences** |
| - Various **randomizations** (shift modulo 1, digital shifts, nested uniform |
| scrambles, left matrix scrambles) |
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| Each file type follows a simple textual standard to ensure: |
| - Human readability |
| - Language-agnostic parsing |
| - Long-term reproducibility |
| - Extensibility to thousands of dimensions |
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| The dataset is motivated by the need for **standardized, compact, transparent formats** in [our QMC research and software](https://github.com/QMCSoftware). |
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| ## Motivation |
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| Many QMC constructions appear across scattered software packages, papers, or custom formats. LDData brings these formats together into a **consistent, unified, machine-readable** repository for: |
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| - Researchers developing new QMC methods |
| - Practitioners needing high-dimensional low-discrepancy point sets |
| - Developers of simulation libraries such as SSJ, QMCPy, and LatNet Builder |
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| This dataset is linked to the research works described in the Citation section below. **For detailed technical specifications and implementation details**, see |
| [LD_DATA.md](LD_DATA.md) |
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| --- |
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| ## Supported Tasks and Applications |
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| ### ✔️ Quasi-Monte Carlo (QMC) |
| Generate deterministic point sets with excellent equidistribution. |
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| ### ✔️ Randomized QMC (RQMC) |
| Use the included randomizations for variance estimation: |
| - Digital shifts |
| - Nested uniform scrambles |
| - Left-matrix scrambles |
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| ### ✔️ High-dimensional Integration and Simulation |
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| Used in: |
| - Bayesian computation |
| - Option pricing |
| - High-dimensional PDE solvers |
| - Uncertainty quantification |
| - Graphics and rendering research |
| - Machine learning sampling methods |
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| ### ✔️ Benchmarking |
| Standard formats help evaluate new constructions against established ones. |
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| --- |
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| ## Features |
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| - Simple `.txt` formats with **one line per dimension** |
| - Optional human-readable comments starting with `#` |
| - No binary encoding or word-size assumptions |
| - Supports extremely high dimensions (10,000+) |
| - Extensible constructions (e.g., Sobol or embedded nets) |
| - All formats interoperable with QMC software (SSJ, QMCPy, LatNet Builder) |
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| --- |
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| ## How to Use the Dataset |
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| ### Load files directly from Hugging Face |
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| ```python |
| from datasets import load_dataset |
| |
| ds = load_dataset("QMCSoftware/LDData") |
| ``` |
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| All data files are preserved in their directory structure and can be accessed |
| using: |
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| ```python |
| ds["train"] # or ds['default'] |
| ``` |
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| ### Typical workflow |
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| 1. Read a parameter file (e.g. `lattice_8d.txt`) |
| 2. Parse header (`# lattice`, dimensions, n, etc.) |
| 3. Parse one line per dimension for the generating vector or matrices |
| 4. Construct QMC point generator in your preferred library |
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| --- |
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| ## Dataset Structure |
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| The dataset includes multiple categories of files: |
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| ### 🔹 `lattice` |
| Rank-1 lattice generating vectors: |
| - Header: `# lattice` |
| - Parameters: |
| - Number of dimensions `s` |
| - Number of points `n` |
| - `s` lines of generating vector coefficients |
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| --- |
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| ### 🔹 `dnet` |
| General digital nets in base `b`: |
| - Header: `# dnet` |
| - Parameters: |
| - Base `b` |
| - Dimensions `s` |
| - Columns `k` |
| - Rows `r` |
| - Then `s` lines representing generating matrices |
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| Efficient for high-dimensional digital nets. |
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| --- |
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| ### 🔹 `plattice` |
| Polynomial lattice rules: |
| - Compact format using integer-encoded polynomials |
| - Base `b`, dimension `s`, polynomial degree `k`, and generating polynomials |
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| --- |
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| ### 🔹 `sobol` and `soboljk` |
| Parameters for Sobol' sequences: |
| - `soboljk`: Joe & Kuo format with primitive polynomials and direction numbers |
| - `sobol`: Simplified direction-number only format |
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| Used widely in QMC applications. |
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| --- |
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| ### 🔹 Randomization formats |
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| Includes: |
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| - `shiftmod1`: Shift modulo 1 |
| - `dshift`: Digital shift in base `b` |
| - `nuscramble`: Nested uniform scramble |
| - `lmscramble`: Left matrix scramble |
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| All formats are text-based and reproducible. |
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| --- |
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| ## Example: Parsing a Lattice Rule File |
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| Example file: |
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| ``` |
| # lattice |
| 8 |
| 65536 |
| 1 |
| 19463 |
| 17213 |
| 5895 |
| 14865 |
| 31925 |
| 30921 |
| 26671 |
| ``` |
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| Python pseudo-code: |
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| ```python |
| with open("lattice_8d.txt") as f: |
| lines = [l for l in f.readlines() if not l.startswith("#")] |
| |
| s = int(lines[0]) |
| n = int(lines[1]) |
| a = [int(x) for x in lines[2:2+s]] |
| ``` |
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| --- |
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| ## File Naming Recommendations |
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| To support discoverability and consistent tooling: |
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| - All files begin with their keyword (`lattice_`, `dnet_`, `sobol_`, etc.) |
| - Headers contain: |
| - Construction method |
| - Figure of merit (FOM) |
| - Weights |
| - Embedded range (if applicable) |
| - Comments allowed in headers only |
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| --- |
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| ## References |
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| This dataset incorporates formats and ideas from foundational work in QMC: |
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| - Bratley & Fox (1988) |
| - Joe & Kuo (2008) |
| - L’Ecuyer (2016) |
| - Goda & Dick (2015) |
| - Nuyens (2020) |
| - And others listed in the detailed specification [LD_DATA.md](LD_DATA.md). |
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| --- |
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| ## Citation |
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| If you use LDData in academic work, please cite: |
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| ``` |
| @article{sorokin.2025.ld_randomizations_ho_nets_fast_kernel_mats, |
| title = {{QMCPy}: a {P}ython software for randomized low-discrepancy sequences, quasi-{M}onte {C}arlo, and fast kernel methods}, |
| author = {Aleksei G. Sorokin}, |
| year = {2025}, |
| journal = {ArXiv preprint}, |
| volume = {abs/2502.14256}, |
| url = {https://arxiv.org/abs/2502.14256}, |
| } |
| |
| |
| @inproceedings{choi.QMC_software, |
| title = {Quasi-{M}onte {C}arlo software}, |
| author = {Choi, Sou-Cheng T. and Hickernell, Fred J. and Rathinavel, Jagadeeswaran and McCourt, Michael J. and Sorokin, Aleksei G.}, |
| year = {2022}, |
| booktitle = {{M}onte {C}arlo and Quasi-{M}onte {C}arlo Methods 2020}, |
| publisher = {Springer International Publishing}, |
| address = {Cham}, |
| pages = {23--47}, |
| isbn = {978-3-030-98319-2}, |
| editor = {Keller, Alexander}, |
| } |
| ``` |
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| --- |
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| ## License |
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| Apache 2 License. |
| See [`LICENSE`](LICENSE.txt) file for details. |
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| --- |
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| ## Acknowledgements |
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| This dataset is developed and maintained by: |
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| - **QMCSoftware team** |
| - Contributors to QMCPy, SSJ, and LatNet Builder |
| - Community contributions from QMC & RQMC researchers |
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| Special thanks to researchers providing widely used generating vectors and direction numbers used throughout the scientific computing community. |
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