tensordev

JAX-based tensor algebra library for signatures, free developments, Volterra signatures and inner product-kernels thereof.

Status

tensordev is under active development. The core tensor algebra, signature development, state-space kernels, Volterra signatures, and free/higher-order kernels are implemented and tested.

The current stable backend is JAX. Multi-backend support for PyTorch, TensorFlow, and Numba is planned, but not yet implemented.

The implemented JAX components are end-to-end differentiable — from elementary tensor operations and path signatures through to signature-kernel evaluations and Volterra kernel parameters.

Requirements

tensordev currently requires Python 3.10+ and JAX 0.10.0+.

The package is developed and tested primarily with the JAX backend.

Installation

pip install tensordev

For the latest development version:

pip install git+https://github.com/hagerpa/tensordev.git

License

tensordev is released under the Apache License 2.0. See LICENSE for details.

Quick start

import tensordev as td
from tensordev.util import random_trigonometric_polynomial_paths

X = random_trigonometric_polynomial_paths(batch=4, steps=32, dim=2, key=0)

sig = td.path_signature(X, trunc=4)
ip = td.tensor_inner_product(sig, sig)

print(td.tensor_to_flat(sig).shape)
print(ip.shape)

Package structure

tensordev/
├── core/           # Tensor algebra backends
├── development/    # Signature development, free and classical
├── sss/            # State-space signatures, aka Volterra signatures for finite state-space kernels
├── volterra/       # Volterra signatures: fractional, gamma, piecewise-constant kernels
├── kernel/         # Signature kernels: classical, free, FSSK, higher-order
└── util/           # Path generators and combinatorics

tensordev.core — tensor algebra operations

A core object exposes all tensor algebra operations over a chosen array backend. The default is Jax(). Operations are also available directly on the tensordev module:

# If desired, enable float64 — must be set before importing JAX or tensordev.
import jax
jax.config.update("jax_enable_x64", True)

import tensordev as td
import jax.numpy as jnp

# two elements of the truncated tensor algebra, levels 0..3
A = (jnp.ones((1,)), jnp.ones((2,)), jnp.ones((4,)), jnp.ones((8,)))
B = (jnp.ones((1,)), jnp.ones((2,)), jnp.ones((4,)), jnp.ones((8,)))

# Chen/tensor/concatenation product
C = td.tensor_product(A, B, trunc=3)

# level-wise sum
S = td.tensor_summation(A, B)

# truncated exponential and logarithm, inputs start at level 1
E = td.tensor_exponential(A[1:], trunc=3)
L = td.tensor_logarithm(A[1:], trunc=3)

# inner product
ip = td.tensor_inner_product(A, B)

For the shuffle, i.e. commutative, product, create a ShuffleCore for a fixed base dimension and truncation. All shuffle operators are precomputed at construction time as sparse arrays, so repeated calls are pure compute with no reallocation.

sc = td.shuffle_core(d=2, trunc=3)
C = sc.tensor_shuffle_product(A, B, trunc=3)

Use shuffle_core_expected_memory to check the memory budget before constructing at large d or trunc. It returns an upper bound; actual usage is typically 20–40% less.

td.shuffle_core_expected_memory(d=2, trunc=8)   # →   1.63 MB  (actual  0.52 MB)
td.shuffle_core_expected_memory(d=4, trunc=6)   # →   5.85 MB  (actual  4.07 MB)
td.shuffle_core_expected_memory(d=4, trunc=8)   # → 363.85 MB  (actual 218.93 MB)
td.shuffle_core_expected_memory(d=8, trunc=6)   # → 353.44 MB  (actual 297.57 MB)

tensordev.development — signature development

Compute truncated signatures with optional blocking. block_size splits the path into chunks and chains them via Chen's identity internally. This is useful for long sequences where the full path does not fit in memory at once.

import numpy as np
import jax.numpy as jnp
import tensordev as td
from tensordev.util import random_trigonometric_polynomial_paths

X = random_trigonometric_polynomial_paths(batch=4, steps=32, dim=2, key=0)

sig = td.path_signature(X, trunc=4)  # one shot

# signature on two consecutive intervals
sig_blocked = td.path_signature(X, trunc=4, block_size=16, accumulate=False)

# Chen's identity: sig = sig_a ⊗ sig_b
sig_a = td.tensor_slice(sig_blocked)[:, 0]
sig_b = td.tensor_slice(sig_blocked)[:, 1]
sig_c = td.tensor_product(sig_a, sig_b, trunc=4)

np.testing.assert_allclose(
    td.tensor_to_flat(sig),
    td.tensor_to_flat(sig_c),
    atol=1e-12,
)  # ✓

# shuffle identity: <sig, a ⊔ b> = <sig, a> · <sig, b>
# a, b are fixed basis vectors — broadcast over the batch dimension
e1 = td.tensor_densify((None, jnp.array([1., 0.])))
e2 = td.tensor_densify((None, jnp.array([0., 1.])))
sc = td.shuffle_core(d=2, trunc=4)

np.testing.assert_allclose(
    td.tensor_inner_product(sig, sc.tensor_shuffle_product(e1, e2, trunc=4)),
    td.tensor_inner_product(sig, e1) * td.tensor_inner_product(sig, e2),
    atol=1e-10,
)  # ✓

free_development generalises this to tensor-valued paths and adds block-level control. The example below computes per-block signatures and their tensor logarithms — the piecewise log-linear approximation that HigherOrderKernel uses internally:

from tensordev.development import free_development

# per-block signatures: (batch=4, n_blocks=4, dim^k) for block_size=8, steps=32
block_sigs = free_development((X,), trunc=3, block_size=8, accumulate=False)

# tensor log of each block signature → log-linear increments
log_sigs = td.tensor_logarithm(block_sigs[1:], trunc=3, output_zero_level=False)

# free development of the piecewise log-linear path
higher_order_sig = free_development(log_sigs, trunc=3, increment_input=True)

# piecewise log-linear sig recovers the original sig at the same truncation
sig = td.path_signature(X, trunc=3)

np.testing.assert_allclose(
    td.tensor_to_flat(sig),
    td.tensor_to_flat(higher_order_sig),
    atol=1e-12,
)  # ✓

tensordev.sss — state-space signatures

State-space signatures are Volterra signatures whose convolution kernel is a finite state-space kernel (FSSK), i.e. a matrix-exponential kernel of the form

$$K_{A,b}^\Lambda(t,s) = \sum_{r=1}^q \bigl(\mathbf{1}^\top e^{-\Lambda(t-s)} b_r\bigr) A_r .$$

with dense or Jordan state-space operators $\Lambda$. This package provides functionality for propagating and reading out the hidden state that evolves via an ODE, making online/streaming evaluation of such Volterra signatures exact and efficient.

import jax.numpy as jnp
from tensordev.sss import StateSpaceSignature

# Jordan kernel:
# one real exponential with rate 1.0
# plus one oscillatory pair with decay 0.5 and frequency 2π
# acting on R^2 paths via A = I_2 → state dim R = 1 + 2 = 3
sss = StateSpaceSignature.from_jordan(
    real_rates=jnp.array([1.0]),
    real_sizes=(1,),
    osc_decays=jnp.array([0.5]),
    osc_freqs=jnp.array([2 * jnp.pi]),
    osc_sizes=(1,),
    A=jnp.eye(2)[None],   # (n=1, m=2, d=2)
    b=jnp.ones((1, 3)),   # (n=1, R=3)
    trunc=3,
)

result = sss.vsig(X, dt=1.0 / 32)

StateSpaceSignature carries an optional persistent hidden state for streaming/online evaluation:

dt = 1.0 / 32

# consume the first half of the path — state is updated, not lost
sss_mid = sss.update_with_path(X[:, :17], dt=dt)
vsig_mid = sss_mid.readout()          # Volterra signature at t = 0.5

# continue with the second half
sss_end = sss_mid.update_with_path(X[:, 16:], dt=dt)
vsig_end = sss_end.readout()          # Volterra signature at t = 1.0

# equivalent to the one-shot call
np.testing.assert_allclose(
    td.tensor_to_flat(vsig_end),
    td.tensor_to_flat(sss.vsig(X, dt=dt)),
    atol=0,
)  # ✓

tensordev.volterra — Volterra signature

Volterra signatures for fractional, gamma, and piecewise-constant kernel families, computed via the quadratic triangular recursion with JAX-vectorized evaluation of the inner loop or, on uniform grids, by FFT acceleration.

import jax.numpy as jnp
from tensordev.volterra import ConvolutionKernel, VolterraSignature, vsig

A = jnp.eye(2)[None]  # (n=1, m=2, d=2)
dt = 1.0 / 32

# functional API — fractional kernel k(t,s) = (t-s)^{β-1} / Γ(β)
kernel = ConvolutionKernel.fractional(beta=jnp.array([0.8]), A=A)
result = vsig(X, kernel=kernel, dt=dt, trunc=3)

# class-based — Gamma kernel, adding exponential damping to the fractional kernel
kernel_g = ConvolutionKernel.gamma(
    beta=jnp.array([0.8]),
    rate=jnp.array([1.0]),
    scale=jnp.array([1.0]),
    A=A,
)
vsig_obj = VolterraSignature(kernel=kernel_g, trunc=3)
result = vsig_obj.vsig(X, dt=dt)

Available kernel constructors:

Constructor	Formula	Parameters
ConvolutionKernel.fractional	$k_p(t,s) = \Gamma(\beta_p)^{-1}(t-s)^{\beta_p-1}$	beta, A
ConvolutionKernel.gamma	$k(t,s) = \mathrm{scale}\cdot e^{-\mathrm{rate}(t-s)}\cdot\Gamma(\beta)^{-1}(t-s)^{\beta-1}$	beta, rate, scale, A
ConvolutionKernel.piecewise_constant	$k(i,j) = B_{p,i,j}$	B, A

Setting beta=1 with ConvolutionKernel.fractional recovers the classical iterated-integral signature.

tensordev.kernel — signature kernels

Kernel objects for empirical statistics: batchwise values, Gram matrices, MMD, and scoring rules. All inherit from BaseKernel.

Class	Description
SigKernel	Classical signature kernel for Euclidean paths
FreeKernel	Free signature kernel for tensor-valued paths
FSSKSigKernel	Kernel induced by the FSSK Volterra signature
HigherOrderKernel	Higher-order kernel via piecewise log-linear approximation
LinearKernel, RBFKernel, RBF_CEXP_Kernel, RBF_SQR_Kernel	Static pointwise kernels used as increment kernels

import numpy as np
from tensordev.util import random_trigonometric_polynomial_paths
from tensordev.kernel import SigKernel, RBFKernel

X = random_trigonometric_polynomial_paths(batch=4, steps=32, dim=2, key=0)
Y = random_trigonometric_polynomial_paths(batch=4, steps=32, dim=2, key=1)

k = SigKernel(dyadic_order=1)

vals = k.compute_kernel(X, Y)                  # (4,)   — batchwise k(X_i, Y_i)
gram = k.compute_Gram(X, Y)                    # (4, 4) — full cross Gram matrix
Kxx  = k.compute_Gram(X)                       # (4, 4) — symmetric Y=None shortcut
mmd  = k.compute_mmd(X, Y)                     # scalar — empirical MMD²
esr  = k.compute_expected_scoring_rule(X, Y)   # scalar — E_Y[S(X, y)]
sr   = k.compute_scoring_rule(X, Y[0])         # scalar — S(X, y) for a single y

# RBF increment kernel — replaces the default ⟨dx, dy⟩ inner product
k_rbf = SigKernel(dyadic_order=0, static_kernel=RBFKernel(sigma=1.0))
gram_rbf = k_rbf.compute_Gram(X, Y)            # (4, 4)

FreeKernel, HigherOrderKernel, and FSSKSigKernel share the same empirical API and are drop-in replacements for SigKernel:

from tensordev.kernel import FreeKernel, HigherOrderKernel, FSSKSigKernel

# free kernel — accepts tensor-valued paths; level-1 path reduces to SigKernel
k_free = FreeKernel(dyadic_order=1)
gram_free = k_free.compute_Gram(X, Y)          # (4, 4)

# higher-order kernel — log_steps must divide the number of intervals, 32 here
k_ho = HigherOrderKernel(log_steps=(2, 2), log_degree=(3, 3))
mmd_ho = k_ho.compute_mmd(X, Y)                # scalar

# FSSK kernel — wraps the sss.kernel of a StateSpaceSignature
k_fssk = FSSKSigKernel(kernel=sss.kernel, dt_x=1.0 / 32, dt_y=1.0 / 32)
mmd_fssk = k_fssk.compute_mmd(X, Y)            # scalar

tensordev.util — utilities

from tensordev.util import (
    path_to_increments,
    integrated_ou_first_on_path,
    random_trigonometric_polynomial_paths,
    unit_speed_paths,
    perturb_path_batch,
    deterministic_trigonometric_path_pair,
    bucket_pad_ragged_paths,
    velocity_to_increments,
)

Tests

pytest tests/

Tests are organized by subpackage:

tests/core/
tests/development/
tests/sss/
tests/volterra/
tests/kernel/

Backends

The core abstraction supports multiple array frameworks. Currently only the JAX backend is fully implemented.

Class	Backend	Status
Jax	JAX, JIT-compiled	stable
JaxSequentialCore	JAX scan / lax.associative_scan	stable
JaxShuffleCore	JAX sparse shuffle operators	stable
Universal	any array-API namespace	stable
Einsum	einsum-based base class	stable
Numba	Numba	stub
Torch / TensorFlow	PyTorch / TensorFlow	stub

The active backend is selected via the TENSORDEV_BACKEND environment variable. The default is "jax".

TENSORDEV_BACKEND=jax python my_script.py

from tensordev._backend import get_default_core, get_default_seq_core

core = get_default_core()
seq_core = get_default_seq_core()

Acknowledgements and theoretical background

tensordev is an independent implementation, but it was influenced by several excellent open-source projects in the signature-computation ecosystem:

• iisignature: a gold-standard reference for efficient signature and logsignature computation.
• signatory: inspired the fused Horner-style evaluation used for efficient tensor exponential / signature development routines.
• signax: provided the initial motivation for building a JAX-native tensor algebra and signature package.
• pySigLib: a high-performance CPU/GPU library for signatures and signature kernels, whose CUDA and JAX support provides an important contemporary reference point for accelerator-aware signature computation.
• sigkernel: inspired parts of the signature-kernel API and the second-order finite-difference stencil used for the standard signature kernel.
• high-order-sigkernel: inspired the predictor-corrector schemes for higher-order signature-kernel PDE systems, which are adapted and further developed in this package.

The main theoretical background for the algorithms implemented here is:

• J. Reizenstein and B. Graham, Algorithm 1004: The iisignature Library: Efficient Calculation of Iterated-Integral Signatures and Log Signatures, ACM Transactions on Mathematical Software, 2020.

• P. Kidger and T. Lyons, Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU, ICLR 2021.

• C. Salvi, T. Cass, J. Foster, T. Lyons and W. Yang, The Signature Kernel is the Solution of a Goursat PDE, SIAM Journal on Mathematics of Data Science, 2021.

• M. Lemercier, T. Lyons and C. Salvi, Log-PDE Methods for Rough Signature Kernels, arXiv preprint, 2024.

• P. K. Friz and P. P. Hager, Expected Signature Kernels for Lévy Rough Paths, arXiv preprint, 2025.

• P. P. Hager, F. N. Harang, L. Pelizzari and S. Tindel, The Volterra Signature, arXiv preprint, 2026.

• P. P. Hager, F. N. Harang, L. Pelizzari and S. Tindel, Computational Aspects of the Volterra Signature, manuscript.