Third-Order Signature Tensors

• Third-order signature tensors are three-way arrays derived from iterated integrals of bounded variation paths that capture noncommutative geometric invariants.
• They decompose into irreducible GL(d)-modules and obey shuffle relations, providing a structured framework for algebraic and computational analysis.
• These tensors have practical applications in inverse problems, stochastic analysis, and high-energy physics through efficient tensor learning and recovery algorithms.

A third-order signature tensor is a three-way array arising from the iterated integrals of paths in $\mathbb{R}^d$. These tensors encode the noncommutative geometric content of a path and serve as algebraic invariants for path reconstruction, stochastic analysis, and event shape identification. In various domains, third-order signature tensors play a central role in inverse problems, algebraic geometry, representation theory, and high-energy physics.

1. Definition and Fundamental Properties

Let $X: [0,1] \to \mathbb{R}^d$ be a path of bounded variation. The third-order signature tensor (denoted $S^{(3)}(X)$ or $\sigma^{(3)}(X)$) is defined by

$$S_{ijk}^{(3)}(X) = \int_{0 \le t_1 \le t_2 \le t_3 \le 1} dX_i(t_1)\,dX_j(t_2)\,dX_k(t_3)$$

for indices $1\le i,j,k\le d$. This order-3 tensor is multilinear in the increments of $X$ and summarizes all third-degree iterated integrals of the path. For $C^1$ paths, it can alternatively be written via nested integrals of the velocity: $$S_{ijk}^{(3)}(X) = \int_0^1\int_0^{t_3}\int_0^{t_2} \dot X_i(t_1)\,\dot X_j(t_2)\,\dot X_k(t_3)\,dt_1\,dt_2\,dt_3$$ These tensors obey shuffle relations (quadratic identities) and possess a congruence equivariance under linear transformations: if $A\in GL_d$, then the action $X: [0,1] \to \mathbb{R}^d$0 induces

$X: [0,1] \to \mathbb{R}^d$1

where the tensor product is mapped by the modewise action of $X: [0,1] \to \mathbb{R}^d$2 on each slot (Schmitz, 16 Dec 2025, Pfeffer et al., 2018).

2. Algebraic and Representation-Theoretic Structure

The space $X: [0,1] \to \mathbb{R}^d$3 admits a natural decomposition into irreducible $X: [0,1] \to \mathbb{R}^d$4-modules, often termed "Thrall modules": $X: [0,1] \to \mathbb{R}^d$5 with the correspondences:

• $X: [0,1] \to \mathbb{R}^d$6: the fully symmetric part ($X: [0,1] \to \mathbb{R}^d$7),
• $X: [0,1] \to \mathbb{R}^d$8: mixed symmetry (Lie commutators and symmetrized products),
• $X: [0,1] \to \mathbb{R}^d$9: the free-Lie (alternating) component (Améndola et al., 2023).

The explicit decomposition of the third signature tensor for a general $S^{(3)}(X)$0 in terms of the final increment $S^{(3)}(X)$1, a skew-symmetric matrix $S^{(3)}(X)$2, and the free-Lie component $S^{(3)}(X)$3, is given by

$S^{(3)}(X)$4

This decomposition governs the algebraic appearance of symmetry and the possibility for certain invariants (such as the alternating volume form for $S^{(3)}(X)$5) (Améndola et al., 2023, Galuppi et al., 2024).

Symmetry restrictions on $S^{(3)}(X)$6 are strict: there are no nonzero third-order signature tensors with full skew-symmetry, and the only nontrivial partial symmetry involves either swapping the first two or last two indices. Full symmetry characterizes signatures of straight-line paths (Galuppi et al., 2024).

3. Rank, Conciseness, and Algebraic Varieties

For paths made up of $S^{(3)}(X)$7 straight segments, the sharp upper bound on the rank is

$S^{(3)}(X)$8

and this is attained for generic choices of $S^{(3)}(X)$9 increments in $\sigma^{(3)}(X)$0 (Galuppi et al., 2024).

Signature tensors are non-concise (i.e., contained in $\sigma^{(3)}(X)$1 for a proper subspace $\sigma^{(3)}(X)$2) if and only if the underlying path is confined to a hyperplane parallel to $\sigma^{(3)}(X)$3. Thus, the non-conciseness is a direct geometric witness of path degeneracy (Galuppi et al., 2024).

The set of all third-order signature tensors of $\sigma^{(3)}(X)$4-segment piecewise-linear paths in $\sigma^{(3)}(X)$5 forms an algebraic variety, denoted $\sigma^{(3)}(X)$6, whose dimension, degree, and defining equations (quadratics arising from shuffle identities and minors) are explicit for small $\sigma^{(3)}(X)$7. The universal variety $\sigma^{(3)}(X)$8 comprises all third-order signatures for generic paths (Améndola et al., 2018).

4. Inverse Problems and Identifiability

Given a third-order signature tensor $\sigma^{(3)}(X)$9 (typically observed), the inverse problem is to recover the underlying path, up to tree-like equivalence or congruence class. For piecewise-linear paths with $$S_{ijk}^{(3)}(X) = \int_{0 \le t_1 \le t_2 \le t_3 \le 1} dX_i(t_1)\,dX_j(t_2)\,dX_k(t_3)$$0 segments, this reduces to the problem: given $$S_{ijk}^{(3)}(X) = \int_{0 \le t_1 \le t_2 \le t_3 \le 1} dX_i(t_1)\,dX_j(t_2)\,dX_k(t_3)$$1 for a known "core" tensor $$S_{ijk}^{(3)}(X) = \int_{0 \le t_1 \le t_2 \le t_3 \le 1} dX_i(t_1)\,dX_j(t_2)\,dX_k(t_3)$$2 (e.g., axis path), find $$S_{ijk}^{(3)}(X) = \int_{0 \le t_1 \le t_2 \le t_3 \le 1} dX_i(t_1)\,dX_j(t_2)\,dX_k(t_3)$$3. The system $$S_{ijk}^{(3)}(X) = \int_{0 \le t_1 \le t_2 \le t_3 \le 1} dX_i(t_1)\,dX_j(t_2)\,dX_k(t_3)$$4 is cubic in $$S_{ijk}^{(3)}(X) = \int_{0 \le t_1 \le t_2 \le t_3 \le 1} dX_i(t_1)\,dX_j(t_2)\,dX_k(t_3)$$5 with $$S_{ijk}^{(3)}(X) = \int_{0 \le t_1 \le t_2 \le t_3 \le 1} dX_i(t_1)\,dX_j(t_2)\,dX_k(t_3)$$6 equations in $$S_{ijk}^{(3)}(X) = \int_{0 \le t_1 \le t_2 \le t_3 \le 1} dX_i(t_1)\,dX_j(t_2)\,dX_k(t_3)$$7 unknowns. Identifiability is generically guaranteed: for axis-path cores, the stabilizer is trivial and $$S_{ijk}^{(3)}(X) = \int_{0 \le t_1 \le t_2 \le t_3 \le 1} dX_i(t_1)\,dX_j(t_2)\,dX_k(t_3)$$8 is uniquely recoverable (Pfeffer et al., 2018, Schmitz, 16 Dec 2025).

Polynomial-path cores retain finite stabilizers up to moderate $$S_{ijk}^{(3)}(X) = \int_{0 \le t_1 \le t_2 \le t_3 \le 1} dX_i(t_1)\,dX_j(t_2)\,dX_k(t_3)$$9; highly generic cores are also uniquely identifiable (Pfeffer et al., 2018).

5. Computational Algorithms for Tensor Learning

Prior to 2025, recovery used polynomial system solving (often Gröbner-basis methods), which scale doubly-exponentially in $1\le i,j,k\le d$0 and are infeasible for $1\le i,j,k\le d$1 (Schmitz, 16 Dec 2025). Recent advances provide symbolic, exact, non-iterative $1\le i,j,k\le d$2 algorithms based on multilinear algebra and congruence orbits. The main steps are:

• Modewise Gaussian-type updates (upper, lower, diagonal) to achieve canonical forms slice-by-slice,
• Exploiting stabilizer and orbit characterization to reduce dimensionality at each step,
• Randomized coordinate changes to sidestep degeneracy.

Empirical data shows that this approach solves generic instances up to $1\le i,j,k\le d$3 in under a minute, while Gröbner-basis solvers fail at $1\le i,j,k\le d$4. Implementations in OSCAR (Julia-based CAS) leverage efficient mode-multiplication and sparse linear algebra (Schmitz, 16 Dec 2025).

Optimization-based frameworks are also available: minimizing $1\le i,j,k\le d$5 over $1\le i,j,k\le d$6 via BFGS and trust-region Newton methods allows for robust handling of noise and the shortest-path constraints in overparameterized settings (Pfeffer et al., 2018).

6. Applications and Physical Signatures

In particle physics, an analogous third-order normalized momentum tensor, constructed from event distributions in jet analyses, enables efficient signatures for three-jet topologies. Here, quadratic contractions of the rank-3 momentum tensor produce invariants whose eigenvalues, together with those from the rank-2 tensor, yield discriminants for event shape and jet counting. These methods are robust at high energy and do not require explicit jet clustering assignments (Dagan et al., 2011).

In stochastic analysis, expected third-order signature tensors for Brownian motion encode high-order noncommutative moments, refining classical moment varieties. For mixtures of paths, these invariants populate secant varieties of the signature variety and are accessible through shuffle algebra (Améndola et al., 2018).

7. Open Directions and Further Developments

Open questions remain for higher-level signatures (order $1\le i,j,k\le d$7), paths with more segments than ambient dimension, and efficient inversion algorithms for streaming or approximate (noisy) data. Representation-theoretic understanding continues to reveal structural constraints on symmetries and rank, with direct computational implications for identifiability and complexity. Another promising direction is the extension of these algebraic geometric recipes to statistical learning, optimal transport, and noncommutative mixture models (Schmitz, 16 Dec 2025, Améndola et al., 2018).