Learning with Expected Signatures: Theory and Applications

We would like to thank the reviewer for the comments. We believe the reviewer well understood the main contributions of the paper, which they highlighted in "Claims and Evidence".

Methods And Evaluation Criteria

In practical applications the (expected) signature transform may face the curse of dimensionality. Nevertheless estimation of the expected signature is still a relevant topic as, at least for low dimensional time-series, it provides a theoretically motivated domain-agnostic baseline model. Moreover, ongoing work is being carried out to "a priori" select a transformation of the $\sum\_{k=0}^p d^k$ truncated (expected) signature feature vector retaining its essential characteristics. For example in [2] the authors introduce a linear state-space transformation such that redundant information can be easily removed from the signature vector by (a further) truncation.

Usually, a domain-specific architecture – for example, in the case of image/video data, a convolutional layer targeting the known topological structure of the input – will likely improve empirical model performance (cf. Remark E.1). In such cases, the (expected) signature can be used as a layer in a neural network, see [3]. When working with video streams, one might consider a sequential architecture with an initial embedding layer mapping the data to a lower dimensional space before computing the (expected) signature. The empirical applications discussed in this paper provide a set of simple benchmark tasks on which to evaluate the performance of the martingale correction (such simple models do not and should not be understood as trying to achieve SOTA results). The scope of this work was not to develop new expected signature-based architectures but rather, as well understood by the reviewer, to fill a gap in the theory and, under suitable conditions, provide an improved empirical estimator for the theoretical expected signature.

[2] Bayer and Redmann, Dimension reduction for path signatures, preprint, https://arxiv.org/pdf/2412.14723

[3] Bonnier et al., Deep Signature Transforms, NeurIPS 2019, https://arxiv.org/pdf/1905.08494

Other Strengths And Weaknesses

We shall add intuitive definitions of technical terms in the appendix, the following is an extract from said glossary.

Informal glossary

The $p$-variation of a path is a measure of its regularity. For the purpose of our discussion it suffices to note that paths that have finite $p$-variation for low $p$ are more regular. A bounded variation (BV) path is a path with finite 1-variation (also known as total variation), this regularity ensures there exists a well-defined notion of integral against this path (e.g. a piecewise linear paths or continuously differentiable path) and hence we can easily define its signature as in Equation (2). Many interesting stochastic processes (e.g. those driven by Brownian motion) have infinite 1-variation (i.e. are not BV) but have finite $p$-variation for all $p>2$ and hence defining their signature requires rough path theory.

The shuffle property of the signature is an algebraic property stating that the product of two signature terms is a linear combination of higher-order signature terms. More precisely, the product of the signature terms corresponding to words $I$ and $J$ is the sum of all signature terms indexed by words $K$ of length $|I|+|J|$ obtained by interleaving $I$ and $J$. In the context of the discussion on page 1 this means that all moments of the signature can be written as linear combinations of higher order expected signature terms.

A word $I=(i\_1, \ldots, i\_n)$ with $i\_1, \ldots, i\_n\in\\{1, \ldots, d\\}$ is a multi-index used to denote an entry of the signature, i.e. a real-valued number. The length of the word, i.e. $|I| = n$, denotes to which level ($n$-dimensional tensor) of the signature the entry belongs. For example $S^I(\mathbb{X})\_{[0,T]}$ where $I=(1,2)$ denotes the $(1, 2)$-entry of the second level of the signature (a matrix).

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Questions For Authors

Yes, we will contribute public code in the following two ways:

• An independent code repository containing methods to estimate expected signatures using both the simple empirical estimator and the martingale corrected estimator (compatible with numpy and torch arrays, using iisignature and signatory for signature computations). This directory will also contain the scripts needed to reproduce the option-pricing experiment of Section 3.2.2.
• Forks of the original repositories for the other two experiments adding the option to use the martingale corrected estimator, which can be used to reproduce the results of Sections 3.2.1 and 3.2.3.

We will refer to the public repositories in the final version of the paper. This information was redacted from the first submission in adherence with the conference's double blind review policy.

We would like to thank the reviewer for the comments. We believe the reviewer has well understood the main contributions of the paper, which they summarized in "Theoretical Claims".

Minor comments

• Line 37: typo, the informational content (i.e. the norm) of signature terms decays factorially.
• Line 39: we agree this was poorly worded, we shall update the references.
• Line 88: we will fix notation to be consistent.
• We will add references to the proof locations in the main text (e.g. Proof of Theorem 2.8. See Appendix A.1.)
• We will add short outlines to the proofs (e.g. the main idea in the proof of Theorem 2.8 is to show the sequence of signatures along $\pi\_n$ is a Cauchy sequence in $L^m$). We shall also add a better partitioning between sections of this proof that concern conditions $(i)$, $(ii)$, and $(iv)$.
• Line 296: we will spell out Multivariate Continuous-time Autoregressive Process (MCAR).
• Line 827: typo.

Questions For Authors

1. Yes, there is a theoretical barrier to progressing to $\alpha\leq 1/4$. This arises when applying Lemma A.1 to bound the sum of $Z^{\mathcal{I}}\_{[s,t]}$ with $i\_2=4$. More precisely, let $\alpha\leq 1/4$ and assume we follow the same proof strategy of Theorem 2.8 up to line 794 under assumption $(iv)$. Then, when $\alpha\leq 1/4$, we cannot apply (19) to bound the sum of the $Z^{\mathcal{I}}\_{[s,t]}$'s with $i\_2=4$ and so this term needs to be treated separately in the same way the $i\_2=2$ and the $i\_2=3$ had to be treated separately when $\alpha\leq 1/2$ and $\alpha\leq 1/3$ respectively. In a similar spirit to the $i\_2=2, 3$ cases we can thus attempt to apply Lemma A.1 to bound the sum of the $Z^{\mathcal{I}}\_{[s,t]}$'s. In this case though, even if we impose a condition similar to (A$\beta$) and (A$\gamma$) to bound the first term by $|\pi\_n|^{\zeta -1}$ for some $\zeta>1$, the second term, i.e. the sum of the $\\|Z^\mathcal{I}\_{s,t}\\|^2$, will be bounded by $|\pi\_n|^{8\alpha - 1/2}$ which does not necessarily vanish when $\alpha \leq 1/4$.
2. When the data generating process is not a martingale the corrected estimator introduces an additional bias which, depending on how "far" the process $\mathbb{X}$ is from being a martingale, might offset the benefits of the variance reduction. When $\mathbb{X}$ is "close" to being a martingale though, the corrected estimator is still theoretically better (in terms of MSE) than the classic estimator. Moreover, in practical applications, the corrected estimator might turn out to be a more informative quantity than the classic estimator. Intuitively, one might expect this to be the case when the drift of the underlying process is not relevant for the downstream task. In cases where the underlying process cannot be assumed to be a martingale we thus suggest to treat the martingale correction as a data transformation applicable in the learning pipeline (a model hyper-parameter in a similar spirit to the add-time or the lead-lag transform in the signature context) whose usefulness may be empirically ascertained via cross-validation. This is discussed in Appendix E.1 (paragraph starting on line 2436). If the reviewer feels this is an important point to include in the main text we would be happy to move the relevant discussion to Section 2.2.
3. Fixing the sampling rate leads to an irreducible lower bound on the precision with which we can learn the expected signature of the continuous-time process (in a similar spirit to the Nyquist–Shannon sampling theorem). If $\pi$ is fixed (e.g. because of constraints on the sampling rate) and we let $N\rightarrow \infty$ the empirical expected signature $\hat{\phi}\_{\mathbf{I}}^{\Pi(N)}(T)$ where $\Pi(N) = \pi \cup (T+ \pi) \cup \dots \cup((N-1)T+ \pi)$ is a consistent (and asymptotically normal) estimator for the expected signature $\phi^\pi\_{\mathbf{I}}(T) = \mathbb{E}[S^\mathbf{I}(\mathbb{X}^\pi)\_{[0,T]}]$ of the piecewise linear process $\mathbb{X}^\pi$ under the condition that $\\{\mathbb{X}^{n, \pi}, n\geq 1\\}$ – or sufficiently $\\{\mathbb{X}^{n}, n\geq 1\\}$ – is stationary and ergodic (and strongly mixing). This follows directly from the application of Birkhoff's ergodic theorem given in A.2.1 (and the dependent CLT in A.2.2) with $\mathbb{X}$ replaced by $\mathbb{X}^\pi$. The estimator $\hat{\phi}\_{\mathbf{I}}^{\Pi(N)}(T)$ is thus asymptotically (in the $N\rightarrow\infty$ limit) biased for the expected signature $\phi(T) = \mathbb{E}[S(\mathbb{X})\_{[0,T]}]$ of the underlying stochastic process $\mathbb{X}$, with bias $\phi^\pi(T) - \phi(T)$ (this is the "irreducible error" due to $\pi$-sampling). The in-fill asymptotic $|\pi|\downarrow 0$ is crucial to show such error vanishes, i.e. the empirical expected signature is a consistent estimator for the expected signature $\phi(T)$.

We would like to thank the reviewer for the comments. We believe the reviewer well understood the main contributions of the paper, which they highlighted in "Summary".

Other Strengths And Weaknesses

• Conclusions were not added due to space constraints, in case of acceptance we shall include the following short conclusion reiterating the main contributions of the paper. "In this paper we established new estimation results for the expected signature, a model-free embedding for collections of data streams. Our consistency and asymptotic normality results bridge the gap between the theoretically "optimal" continuous-time expected signature and the empirical discrete-time estimator that can be computed from data. Moreover, we introduced a simple modification of such estimator with significantly better finite sample properties under the assumption of martingale observations. Our empirical results suggest the modified estimator might improve the performance of models employing expected signature computations even when the underlying data generating process is not necessarily a martingale."
• Inspired by the provided reference [1], in case of acceptance, we shall include a simple diagram to visually illustrate the main contributions of the paper in the introduction (unfortunately we cannot upload the image in this rebuttal).
• We generally followed the convention that all the equations that are cross-referenced in the paper are numbered and the others are not.
• See answer to question 3. below.
• The results in the appendix provide the detailed proofs of the results discussed in the main paper. These details though are not strictly necessary to understand the main contributions of this paper but we still believe a good mathematical proof should leave as little details unexplained as possible. This is why we would like to keep the level of detail presented in the appendix as is.

Other Comments Or Suggestions

• We will move the discussion contained in this footnote to the main text as it provides the main motivation for using the martingale correction term, see also the answer to question 3. below.
• As discussed above, we will highlight key points and contributions further by adding an intuitive diagram and a short conclusion.
• This is addressed in the answer to question 3. below.

Questions For Authors

• In our notation $S^k(\mathbb{X})\_{[0,T]} \in (\mathbb{R}^d)^{\otimes k}$ is the $k$-th level of the signature, i.e. $S(\mathbb{X})\_{[0,T]} = (S^0(\mathbb{X})\_{[0,T]}, S^1(\mathbb{X})\_{[0,T]}, S^2(\mathbb{X})\_{[0,T]}, \ldots),$ where the signature $S(\mathbb{X})\_{[0,T]}$ lives in the tensor algebra $T((\mathbb{R}^d)) = \oplus\_{i=0}^\infty (\mathbb{R}^d)^{\otimes k}$. We believe the reviewer was referring to the notation used in [1] where $S^k(\mathbb{X})\_{[0,T]}$ denotes the $k$-th level truncated signature, i.e. the mathematical object living in $T^k((\mathbb{R}^d))= \oplus\_{i=0}^k (\mathbb{R}^d)^{\otimes i}$, and hence $S(\mathbb{X})\_{[0,T]}$ is indeed retrieved by taking the limit as $k\rightarrow\infty$.
• This equation was left unnumbered due to formatting reasons. We agree with the reviewer that this is quite an important equation and thus we will number it.
• This corrector (aka control) was chosen based on a control variate argument: by adding an appropriately signed (and scaled) mean-zero control to the estimator we preserve the original estimator's bias while reducing its variance (the stronger the correlation between the control and the original estimator, the larger the reduction in variance). The form of the signature of a martingale yields a natural candidate for the choice of such control variate: by substituting the outermost Stratonovich integral with an Ito integral we obtain a mean-zero term which, by construction, should be highly correlated with the original estimator, thus leading to a significant reduction in variance. This is the content of footnote 6 on page 6. As this provides the main motivation for choosing the form of the corrected estimator we shall move the discussion contained therein to the main text.