torch-structured 1.2.3

Butterfly, low-displacement-rank, and Monarch structured-matrix primitives in PyTorch

torch-structured

Consolidated PyTorch library of structured-matrix primitives:

• torch_structured (core) — butterfly matrices for exact fast linear transforms (FFT, iFFT, DCT, DST, Hadamard, circulant, Toeplitz) as learnable nn.Module drop-in replacements for nn.Linear.
• torch_structured.structured — low-displacement-rank layers ported from structured-nets: Toeplitz-like, Hankel, Vandermonde, Fastfood, Circulant, LDR subdiagonal / tridiagonal, Krylov utilities.
• torch_structured.monarch — Monarch / block-diagonal-butterfly primitives ported from m2: block-diagonal and block-diagonal-butterfly multiplies, structured linear layers, butterfly-factor helper, and Hyena implicit long filter.

See the NOTICE file for upstream attributions and citations.

Requirements

• Python >= 3.10
• PyTorch >= 2.0
• NumPy, SciPy, einops, opt_einsum
• A C++ compiler supporting C++14 (for building extensions)
• CUDA toolkit (optional, for GPU acceleration)

Installation

uv pip install .            # or: pip install .
uv pip install -e ".[dev]"  # development install

CUDA support

CUDA extensions are compiled automatically when a CUDA toolkit is detected. Override with env vars:

FORCE_CUDA=1 uv pip install .   # force CUDA compilation
FORCE_CPU=1 uv pip install .    # force CPU-only build

TORCH_CUDA_ARCH_LIST targets specific GPU architectures (default: "7.0 8.0 9.0+PTX").

Built extensions (CUDA builds):

• torch_structured._butterfly, torch_structured._version — core butterfly ops (torch.ops-style).
• torch_structured._hadamard_cuda — fast Walsh-Hadamard transform (pybind module).
• torch_structured._diag_mult_cuda — subdiagonal cycle-multiply helper (pybind module).

Quickstart

Core butterfly

import torch
from torch_structured import Butterfly
from torch_structured.special import fft, hadamard

layer = Butterfly(in_size=1024, out_size=1024)
fft_layer = fft(1024)
hadamard_layer = hadamard(1024)

Structured (LDR) layers

from torch_structured.structured.layers import ToeplitzLike, LDRSubdiagonal
from torch_structured.structured.hadamard import hadamard_transform_torch

toeplitz = ToeplitzLike(layer_size=256, r=2)
ldr_sd = LDRSubdiagonal(layer_size=256, r=2)
y = hadamard_transform_torch(torch.randn(4, 128))

Monarch primitives

import torch
from torch_structured.monarch.blockdiag_linear import BlockdiagLinear
from torch_structured.monarch.blockdiag_butterfly_multiply import (
    blockdiag_butterfly_multiply,
)

linear = BlockdiagLinear(in_features=512, out_features=512, nblocks=4)
# low-level multiply:
x = torch.randn(8, 64)
w1 = torch.randn(8, 8, 8)
w2 = torch.randn(8, 8, 8)
out = blockdiag_butterfly_multiply(x, w1, w2)

Triton backend (v1.2+)

Starting with v1.2, torch_structured ships a Triton-based GPU backend that replaces the CUDA C++ extensions for the main kernels (butterfly_multiply, diag_mult, hadamard_transform). The Triton path is the default when both a CUDA device and PyTorch >= 2.6 are available.

Hardware requirements

The Triton backend requires NVIDIA CUDA Compute Capability CC 8.0 or later (Ampere generation: RTX 30xx/40xx, A100, H100, etc.). Older GPUs are NOT supported on the Triton path:

• Volta (sm_70 — V100, Titan V): pin to v1.1 (pip install torch-structured==1.1.*) OR use the CUDA backend with a self-built .so.
• Turing (sm_75 — T4, RTX 20xx): same recommendation as Volta.

Switch to the CUDA backend (when the legacy .so is built) via:

export TORCH_STRUCTURED_BACKEND=cuda

Deterministic mode

By default, the Triton backward kernel uses atomic-add reductions for d_twiddle accumulation, which can produce slightly different results across runs (within documented tolerance, but not bit-identical).

For reproducible gradients, opt into deterministic mode:

import torch_structured

torch_structured.set_deterministic(True)
# ... training step ...
torch_structured.set_deterministic(False)

Under deterministic mode, the backward routes through the pure-PyTorch oracle (butterfly_multiply_torch) — slower, but deterministic by construction. Deterministic mode also activates automatically when torch.use_deterministic_algorithms(True) is set globally (additive OR composition with PyTorch's flag).

Switching backends

Use TORCH_STRUCTURED_BACKEND at import time OR torch_structured.set_backend() at runtime:

TORCH_STRUCTURED_BACKEND=triton  # default on Ampere+
TORCH_STRUCTURED_BACKEND=cuda    # legacy CUDA C++ path (requires built .so)
TORCH_STRUCTURED_BACKEND=torch   # pure-PyTorch fallback (CPU OK)
TORCH_STRUCTURED_BACKEND=auto    # try triton -> cuda -> torch

Runtime selector

On some shapes the Triton kernel may be slower than the legacy CUDA path. To avoid forcing users to choose between backends, the library ships a static routing table (torch_structured/_routing.json) baked from triton.testing.do_bench-style measurements at packaging time. When you call a routed shape with the Triton backend AND the legacy .so is available, the call transparently routes to CUDA. The selector is dormant when no cell is marked route_to_cuda — Triton handles every shape.

To regenerate the routing table on your hardware:

python tests/_baseline_butterfly.py            # regenerate forward perf grid
python tests/_baseline_butterfly_backward.py   # regenerate backward perf grid
python scripts/regenerate_routing_table.py     # rebake _routing.json

Measured performance

The numbers below were measured on an NVIDIA RTX 2000 Ada Generation Laptop GPU (sm_89) with PyTorch's CUDA 13.0 build, batch_size=64, nstacks=1, nblocks=1. Each cell is the p50 over a triton.testing.do_bench sweep (warmup=25ms, rep=100ms), in milliseconds — lower is better. "torch" is the pure-PyTorch oracle (butterfly_multiply_torch); "CUDA" is the legacy C++ backend; "Triton" is the v1.2 default.

Forward (butterfly_multiply):

size (n)	dtype	Triton (ms)	CUDA (ms)	torch (ms)	Triton vs torch	routed → CUDA
256	fp32	0.033	0.060	0.384	11.7×
256	complex64	0.043	0.054	0.380	8.8×
512	fp32	0.044	0.049	0.427	9.8×
512	complex64	0.074	0.087	0.571	7.8×
1024	fp32	0.072	0.076	0.464	6.5×
1024	complex64	0.125	0.071	0.473	3.8×	✓
2048	fp32	0.135	0.083	0.570	4.2×
2048	complex64	0.255	0.080	0.510	2.0×	✓

Backward (gradient, full autograd callback incl. trail recompute):

size (n)	dtype	Triton (ms)	CUDA (ms)	torch (ms)	Triton vs torch	routed → CUDA
256	fp32	0.303	0.483	2.421	8.0×
256	complex64	0.290	0.324	2.734	9.4×
512	fp32	0.265	0.330	2.601	9.8×
512	complex64	1.713	0.439	3.326	1.9×	✓
1024	fp32	2.723	0.829	5.842	2.1×	✓
1024	complex64	1.661	1.049	10.265	6.2×
2048	fp32	2.171	0.586	5.517	2.5×	✓
2048	complex64	2.132	0.462	5.684	2.7×	✓

Takeaways on this machine: Triton beats the pure-PyTorch oracle everywhere (~2–12×) and is competitive with — often faster than — the legacy CUDA kernel on forward and on the smaller backward shapes. On the larger/complex backward shapes the legacy CUDA kernel is still faster, so the shipped _routing.json transparently routes those cells to CUDA when the legacy .so is built (the ✓ rows above, matching the baked routing table).

Other GPUs will perform differently. These figures are specific to this laptop Ada GPU, this driver/toolkit, and these problem sizes — absolute times and the Triton-vs-CUDA crossover points shift with compute capability, memory bandwidth, PyTorch/Triton versions, and shape. The committed _routing.json reflects this dev host; regenerate it on your own hardware (see above) for routing decisions tuned to your GPU. Treat the table as illustrative, not as a performance guarantee.

Deprecation timeline

torch_structured ships Triton as the default backend in v1.2. The legacy CUDA C++ backend (csrc/) is being retired over a two-release deprecation cadence:

• v1.2 (current): Triton is the default. TORCH_STRUCTURED_BACKEND=cuda still works for users who built _butterfly.so / _diag_mult.so / _hadamard.so locally, but emits a one-time DeprecationWarning at import time pointing here. The Monarch Mixer MathDx kernel (previously vendored under csrc/) is removed entirely in v1.2; see the CHANGELOG for the full file list.
• v1.3 (next minor release, ~6 months out): CUDA build is default-disabled. csrc/ extensions stay in the source tree and can still be compiled via FORCE_CUDA=1, but the PyPI wheel does NOT include them. The DeprecationWarning still fires when a locally-built CUDA path is used.
• v1.4+ (post-milestone): csrc/ tree, setup.py CUDA extension code, and _cuda_legacy/ are deleted. The standard 2-release deprecation cadence gives users two minor releases to migrate.

Migration: most users should set nothing and let the Triton default take over. If you have a workload that needs the CUDA backend (e.g., Volta sm_70 / Turing sm_75 hardware that Triton doesn't fully support), see the "Triton backend" section above for hardware requirements; otherwise pin to v1.1.

Tests

pytest tests/

CUDA-only tests are automatically skipped when the corresponding extension is not built.

Citation

See NOTICE for full upstream attributions and BibTeX entries for:

• Dao, Gu, Eichhorn, Rudra, Ré, Learning Fast Algorithms for Linear Transforms Using Butterfly Factorizations, ICML 2019
• Dao et al., Kaleidoscope, ICLR 2020
• Thomas, Gu, Dao, Rudra, Ré, Learning Compressed Transforms with Low Displacement Rank, NeurIPS 2018
• Dao et al., Monarch: Expressive Structured Matrices for Efficient and Accurate Training, ICML 2022
• Fu, Arora, Grogan et al., Monarch Mixer: A Simple Sub-Quadratic GEMM-Based Architecture, NeurIPS 2023

License

Apache-2.0 (see LICENSE).