Path Signatures

What Are Path Signatures?

Path signatures are a mathematical tool from rough path theory that provide a universal descriptor of the shape of a trajectory. Given a path through space, the signature captures its geometry through a sequence of iterated integrals — encoding displacement, curvature, volatility, and arbitrarily complex patterns into a single, fixed-dimensional vector.

For ChronosVector, path signatures unlock a fundamentally new capability: trajectory similarity search. Traditional vector kNN finds entities with similar positions at a given time. Signature-based kNN finds entities with similar evolution patterns — a query type no existing vector database supports.

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Mathematical Definition

Formally, the path signature of a continuous path $X: [0,T] \to \mathbb{R}^d$ is the collection of iterated integrals:

$$S(X)^{i_1, \ldots, i_k} = \int_{0 < u_1 < \cdots < u_k < T} dX^{i_1}(u_1) \otimes \cdots \otimes dX^{i_k}(u_k)$$

The full signature is an infinite series organized by depth (the number of iterated integrals):

$$S(X) = \left(1, \; S^{(1)}(X), \; S^{(2)}(X), \; S^{(3)}(X), \; \ldots \right)$$

where $S^{(k)}(X)$ is a tensor of rank $k$ containing all $d^k$ iterated integrals at that depth. In practice, we truncate at a finite depth, which is sufficient for most applications.

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Key Properties

Path signatures have remarkable mathematical properties that make them ideal for trajectory analysis:

Property	Description	Why it matters for CVX
Universality	Any continuous function on paths can be approximated as a linear function of the signature	Signatures are sufficient statistics for trajectory classification
Reparametrization invariance	The signature depends only on the shape of the path, not the speed of traversal	Robust to different sampling rates across entities
Hierarchical structure	Each depth level captures increasingly complex geometric features	Can truncate to desired level of detail
Uniqueness	Up to tree-like equivalences, the signature uniquely determines the path	No information loss at infinite depth
Multiplicativity	$S(X\vert_{[0,T]}) = S(X\vert_{[0,s]}) \otimes S(X\vert_{[s,T]})$ (Chen’s identity)	Signatures can be updated incrementally as new data arrives

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Intuition by Level

The hierarchical structure of signatures provides a natural interpretation at each depth:

Level 1: Displacement

$$S^i(X) = X^i(T) - X^i(0)$$

The net displacement in each dimension. This answers: “Where did the entity end up relative to where it started?” For a 768-dimensional embedding, level 1 is a 768-dimensional vector — the same information as subtracting the first vector from the last.

Level 2: Area and Volatility

$$S^{i,j}(X) = \int_0^T \bigl(X^i(t) - X^i(0)\bigr) \, dX^j(t)$$

The signed area between pairs of dimensions. This encodes rotation, correlation structure, and quadratic variation (volatility). Level 2 captures information that displacement alone misses: two trajectories can have identical start and end points but very different level-2 signatures if one followed a straight line while the other oscillated wildly.

Level 3+: Complex Patterns

Higher-order interactions capturing complex path geometry — analogous to higher moments of a probability distribution. Level 3 terms capture how the signed areas themselves evolve, encoding asymmetries, skewness of the path, and interaction effects between three or more dimensions.

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Practical Computation for CVX

The Dimensionality Challenge

For $D = 768$ dimensional embeddings, computing the full signature quickly becomes intractable:

Depth $k$	Number of terms $d^k$ (with $d = 768$)	Feasible?
1	768	Yes
2	589,824	Marginal
3	452,984,832	No

Solution: PCA Reduction + Truncated Signature

CVX uses a two-stage approach:

1. Project the trajectory onto its principal components, reducing to $d_{\text{reduced}} = 5\text{-}10$ dimensions. This retains the dominant modes of variation while making signature computation tractable.
2. Compute the truncated signature at depth 2-3 on the reduced trajectory.

The resulting signature is a fixed-dimensional vector of manageable size:

$d_{\text{reduced}}$	Depth	Signature dimensions	Log-signature dimensions
5	2	30	15
5	3	155	35
10	2	110	55
10	3	1,110	175

The PCA step is computed per-entity on its trajectory matrix $X \in \mathbb{R}^{T \times D}$, and CVX reports the variance explained so users can assess the quality of the projection.

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Log-Signature: The Compact Alternative

The log-signature is a more compact representation that removes redundant terms via the Baker-Campbell-Hausdorff (BCH) formula. It lives in the free Lie algebra rather than the tensor algebra, and contains the same information in fewer dimensions.

For practical purposes: where a depth-3 signature of a 5-dimensional path has 155 components, the corresponding log-signature has only 35. This makes the log-signature the preferred representation for storage and similarity search, with the full signature available when needed for specific analyses.

CVX defaults to computing the log-signature (use_log_signature: true in the configuration).

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Time Augmentation

By default, signatures are reparametrization-invariant: a trajectory traversed quickly produces the same signature as one traversed slowly. This is often desirable (robustness to sampling rate), but sometimes the timing of movement matters.

Adding time as an extra dimension — $\tilde{X}(t) = (t, X(t))$ — breaks reparametrization invariance but captures speed information. CVX supports this via the time_augmentation configuration option. Use it when:

• You care about when changes happened, not just what changed
• Comparing trajectories with very different temporal extents
• The velocity profile (fast vs. slow change) is part of the pattern you want to match

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Trajectory Similarity Search

This is the most transformative capability that path signatures bring to CVX.

Traditional vs. Signature-Based kNN

Query type	What it finds	Example
Position kNN	Entities near entity X at time $t$	“What concepts are similar to ‘AI’ right now?”
Trajectory kNN	Entities that followed a similar path	”What concepts evolved like ‘AI’ did over the last 5 years?”

Two entities can be far apart in embedding space but have identical signatures — they underwent the same type of evolution (same direction, speed pattern, volatility structure) in different regions of the space. Conversely, two nearby entities might have very different signatures.

How It Works

1. Compute signatures for all entities (or a filtered subset) over a specified time window
2. Store signatures as vectors in a dedicated signature index
3. Query with a reference signature to find entities with similar trajectories
4. Return results ranked by signature distance (L2 or cosine on the signature vectors)

This enables queries that are impossible with position-based search: finding historical precedents, detecting recurring patterns, and classifying trajectory types across a corpus.

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References

• Lyons, T.J. (1998). Differential Equations Driven by Rough Signals. Revista Matematica Iberoamericana. The foundational paper on rough path theory and path signatures.
• Chevyrev, I. & Kormilitzin, A. (2016). A Primer on the Signature Method in Machine Learning. arXiv:1603.03788. An accessible introduction to signatures for ML practitioners.
• Kidger, P. & Lyons, T.J. (2021). Signatory: Differentiable Computations of the Signature and Logsignature Transforms, on Both CPU and GPU. ICLR 2021. The reference implementation for efficient signature computation.
• Arribas, I.P., Goodwin, G.M., Geddes, J.R., Lyons, T.J., & Saunders, K.E.A. (2020). A Signature-Based Machine Learning Model for Distinguishing Bipolar Disorder and Borderline Personality Disorder. Translational Psychiatry. Demonstrates signatures for clinical trajectory classification.
• Chen, K.T. (1958). Integration of Paths, Geometric Invariants and a Generalized Baker-Hausdorff Formula. Annals of Mathematics. The original work on iterated integrals and Chen’s identity.