Venue: NeurIPS 2021 Conference

Reviews:

final_score: rating: 7: Good paper, accept
review_content: summary: This article study stochastic processes through the lens of kernel methods. The authors propose to use a generalization of kernel mean embeddings to define metrics between stochastic processes. In particular, they used these metrics to define a filtration-sensitive kernel two-sample test. Moreover, they showed how to use this new class of kernels in some learning tasks. 
main_review: This work is built upon existing theoretical tools (signature transform, signature kernel). Nevertheless, the use of this kernel seems to be novel in the ML community. From an empirical point of view, the use of this family of kernels seems to improve significantly upon existing methods in the field. However, the evaluation of the kernel is very consuming since it relies on the identity (5), which requires solving a PDE along with some heuristics such as identity (21). It would be interesting to comment in detail on the complexity of such an approximation scheme.  

I enjoyed very much reading the paper, and I have some questions: 

- In practice, can we go beyond 2nd order MMD with the approximation scheme based on the identity(5)?
-What is the intuition lying behind this transform? 
-Is there a toy example where the evaluation of the kernel has an analytical form?
limitations_and_societal_impact: No. The contribution is a general-purpose framework.
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: The paper proposes a generalized approach to kernel mean embeddings – higher-order mean embeddings --  and new two-sample test based on the newly proposed kernel which can be applied to stochastic processes and in particular is designed to capture filtration information in stochastic processes. Various theoretical results are derived, including consistency. 

main_review: The paper proposes an interesting new approach which can be used, in particular, in quantitative finance. The paper also presents an application to causal discovery. While I found the approach and motivation relevant, the applications described in the paper appeared to me rather limited. I believe more extensive experiments have to be performed to fully understand the advantages of the proposed method. The current experiments have to be explained in more detail and clearly motivated. 

Some detailed comments:
-	I found it a bit hard to follow at times, perhaps describing the outline of the paper in the end of the introduction with the main contributions could be helpful;
-	Finance application could use more description;  it is really not a classical finance model and since the finance audience is also not targeted audience of the paper it would be good to provide more extensive intuition about what it is used for, what are particular challenges etc. 
-	Related to the previous point: the authors do mention that the correlation parameter \rho is of particular interest; in the simulation studies the authors draw values for \rho from the interval [-0.4;0.4]. Is this reasonable? I would have expected a higher (positive and negative) correlation be possible as well. 
-	$h$ is not introduced (I suppose it is related to the frequency of sampling of the data)? And the choice of this parameter for the simulation study is not motivated. 
-	The choice of the kernels for the comparison seems to be not ideal; in particular I would indeed expect the RBF kernel to be not appropriate since it is high-frequency financial data (so it is unlikely to be smooth). 
-	Considering different numbers of rough volatility trajectories also could strengthen this application (see [6]).  In particular, if the advantage is the same over different numbers of trajectories would be of interest.
-	How is the choice of $h$ motivated in the simulation studies? 
-	In the causal discovery application I do not see a competitive method to which we could compare the proposed method. How does it perform in comparison to $K_{S}^{1}$, for example? Comparison to the state-of-the-art conditional independence test perhaps could highlight the strength of the approach.

UPD: Most of my original concerns were about empirical applications and the choices the authors made for the kernels/parameters range etc. During the rebuttal and discussion, the authors clarified the things I missed and provided additional results which are in line with the literature for the parameter choice of the financial model. These results confirmed earlier findings. Taking into account the addressed concerns and novelty of the proposed method I change my score to 7 and recommend the paper for acceptance. 
limitations_and_societal_impact: I do not see obvious negative societal impact.
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: The paper proposes (higher order) kernel mean embeddings (KMEs) for stochastic processes -- the novelty is in considering the filtration induced by the process rather than the sample path, which allows us to consider historical dependence.  

main_review: My overall impression of the paper is positive. 
The paper introduces an interesting way of summarising stochastic processes, reflecting their history information (given in the form of filtration). 
As demonstrated by the authors with examples, the maximum mean discrepancy (MMD) defined by the new embedding allows us to characterise differences between processes, which are not captured by the KME defined by the law of the path-valued random variable corresponding to the process. The latter approach is, to my knowledge, common in areas like functional data analysis, whereas we often need to exploit past information (dependence). 
The example applications make the KMEs look promising, although some of the applications are not fully established (e.g., a two-sample test is actually not proposed); the proposed approach would lead to new applications of the MMD (or the KMEs) for sequential data, as done for the original MMD. 

## Comments 
* Higher-order KMEs are defined in terms of conditional distributions. Is the space of the recursive RKHS regular enough so that regular conditional distributions are defined? It would be great this point (e.g., with kernel's continuity and separability of $V$ )  
* How does the process in (14) satisfy the assumption on $\mathcal{X}$ given in the first paragraph of Section 2? 
* The conditional independence testing with continuous conditioning variables is known as hard [Shah and Peters, 2020]  Although the experiment result in Section 4.3 looks reasonable, it might need to be replaced with some other results. 


## Minor comments: 
* L133: $\mathcal{H}(\mathbb{R}^d)$ -> $\mathcal{H}(V)$
* Some equations do not have punctuation marks, such as the period. 
* The Markov process figure (Figure 2) is missing the X-axis; adding time indices would help. 
* Numbering to equations would be better if it is limited to the ones that are actually referenced (I assume the authors did it for the reviewing purpose). 
* Is mentioning Eq. (12) necessary? On the RHS, the CME acts on the kernel; how is this defined? As done in Theorem 4 in [Song, Huang, Smola, and Fukumizu, 2009]? The operator definition is less general than the Bochner integral definition. 

References: 
* Rajen D. Shah, Jonas Peters "The hardness of conditional independence testing and the generalised covariance measure," The Annals of Statistics, Ann. Statist. 48(3), 1514-1538, (June 2020)
limitations_and_societal_impact: NA
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: This paper aims to construct a family of kernel mean embeddings that capture filtration information from stochastic processes. This is achieved through the use of the signature kernel, a characteristic kernel defined on pairs of paths that can be evaluated by solving a hyperbolic PDE, and by forming conditional kernel mean embeddings of the stochastic process conditioned on its own filtration up to some time in the past.

The resulting conditional kernel mean operator itself is a stochastic process, and thus this construction can be repeated to produce higher ordered conditional kernel mean embeddings of the original stochastic process. The corresponding MMD defined from the conditional kernel mean embeddings at each order can then be used to measure distances between two stochastic processes at each order.

The first major contribution shows that higher ordered MMDs are stronger discrepancy measures than lower ordered MMDs (Theorem 2 and 4). They can then be used to perform a kernel two-sample test between paths from two stochastic processes. In particular, the paper focuses on using the second order MMD and derive consistent estimators for it (Theorem 3).

The second major contribution applies these constructions for distribution regression on stochastic processes, where the input is a collection of sample paths and the output is a vector of scalars. This is done by embedding the sampled paths from the stochastic process into a kernel mean embedding through the signature kernel, then learning the mapping between this kernel mean embedding to the vector of scalar outputs. The authors pay strong attention to the application in financial markets where the functions of interest are often discontinuous with respect to the first order mean embedding, and show they can still be learned by proving that the RKHS induced in higher orders is dense in the space of functions of interest (Theorem 5).
main_review: Overall, the contributions of the paper is well motivated, well articulated, and technically interesting. The developments presented in the paper is clear and well ordered. It is clear why the signature kernel was chosen and the motivation behind constructing conditional kernel mean embeddings of stochastic processes conditioned on their filtrations. The subsequent developments in constructing higher ordered kernel mean embeddings and their applications to kernel two-sample testing and distribution regression is also easily to follow.

To me, the major novelty presented in this paper is the notion of recursively constructing the conditional kernel mean embedding of stochastic processes conditioned on its own filtration into high ordered versions of the original, by making use of the fact that the resulting conditional kernel mean embedding (or more exactly the conditional kernel mean operator, with the conditioned variable left as a stochastic process not as a realised path) is itself a stochastic processes. Combined with the fact that discrepancies become stronger at higher orders, as measured by the MMDs of the respective order, this is a very general notion for learning tasks and relationships involving stochastic processes with an ordered nature to its entirety, without losing information from filtration.

The two example applications presented in this paper, kernel two-sample testing and distribution regression, are rather straight forward in consequence by applying what we know about these applications of KMEs in general to this specific setting where KMEs are constructed as previously described. However, they are well developed nonetheless.

This paper can be further improved by some discussion on how the kernel hyperparameters can be learned or selected. Especially at higher orders, hyperparameter learning does not seem trivial and presumably performance can be quite sensitive to the kernel hyperparameters unless we can argue otherwise.

Overall, the notions introduced in this paper can be quite valuable to the wider community in extending adoption of KME based methods to also tasks involving stochastic processes where filtration information is important to capture. I would tend towards acceptance.
limitations_and_societal_impact: The work is motivated by applications in financial markets. However, its contributions are general in nature in terms of the technique and not limited to financial markets.

The paper can be improved by highlighting potential wider impacts in being able to capture filtration information through kernel mean embeddings. I believe these concepts are more general and can be applied to settings other than kernel two-sample tests and distribution regression.
review_title: null

Meta-review: Our opinion is that this paper should be accepted though there are some weaknesses: it would have been easy in the experiment section to use real-world data from quantitative finance instead of draws from a fractional BM; comparisons to CI tests from the causal inference area; and readability.  On a more detailed level it would be good if the authors would cite a disintegration theorem for their family of measure P_{X|F_{X_t}}.
Decision: Accept (Poster)