High Rank Path Development: an approach to learning the filtration of stochastic processes
Authors: Jiajie Tao
Hao Ni
Chong Liu
Venue: NeurIPS 2024 poster (2024)
Keywords: adapted weak topology; stochastic process; synthetic time series generation; path development
Primary area: probabilistic_methods
TL;DR: This paper introduces the High Rank PCF Distance (HRPCFD) for metrizing extended weak convergence of stochastic processes, demonstrating its efficiency and effectiveness in numerical implementations such as conditional time series generation.

Abstract: Since the weak convergence for stochastic processes does not account for the growth of information over time which is represented by the underlying filtration, a slightly erroneous stochastic model in weak topology may cause huge loss in multi-periods decision making problems. To address such discontinuities, Aldous introduced the extended weak convergence, which can fully characterise all essential properties, including the filtration, of stochastic processes; however, it was considered to be hard to find efficient numerical implementations. In this paper, we introduce a novel metric called High Rank PCF Distance (HRPCFD) for extended weak convergence based on the high rank path development method from rough path theory, which also defines the characteristic function for measure-valued processes. We then show that such HRPCFD admits many favourable analytic properties which allows us to design an efficient algorithm for training HRPCFD from data and construct the HRPCF-GAN by using HRPCFD as the discriminator for conditional time series generation. Our numerical experiments on both hypothesis testing and generative modelling validate the out-performance of our approach compared with several state-of-the-art methods, highlighting its potential in broad applications of synthetic time series generation and in addressing classic financial and economic challenges, such as optimal stopping or utility maximisation problems. Code is available at https://github.com/DeepIntoStreams/High-Rank-PCF-GAN.git.

High Rank Path Development: an approach to learning the filtration of stochastic processes
Jiajie Tao Department of Mathematics University College London ucahjta@ucl.co ac. uk
Hao Ni Department of Mathematics University College London R.ni@ucl .cc. uk
Chong Liu" Institute of Mathematical Sciences ShanghaiTTech University 1 Liuchong@shonghaiite... edu. cn
Abstract
Since the weak convergence for stochastic processes does not account for the growth of information over time which is represented by the underlying filtration, a slightly erroneous stochastic model in weak topology may cause huge loss in multi-periodd decision making problems. To address such discontinuities Aldous in- troduced the extended weak convergence, which can fully characterise all essential properties, including the filtration, of stochastic processes; however was considered to be hard to find efficient numerical implementations. In this paper, we introduce novel metric called High Rank PCF Distance (HRPCFD) for extended weak con- vergence based on the high rank path development method from rough path theory, which also defines the characteristic function for measure-valued processes. We then show that such HRPCFD admits many favourable analytic properties which allows us to design an efficient algorithm for training HRPCFD from data and con- struct the HRPCF-GAN by using HRPCFD as the discriminator for conditional time series generation. Our numerical experiments on both hypothesis testing and genera- tive modelling validate the out- performance of our approach compared with several state-of- oftte-of-the-art methods, highlighting its potential in broad applications of synthetic time series generation and in addressing classic financial and economic challenges, such as optimal stopping or utility maximisation problems. Code is available at https: peetps://gite gightb....... //gpth/.///ttbb...... com commDeepIntoStreemee /Highh -iihh-aihg-h--h-ee-eea Rank- .git.
Introduction
popular criterion for measuring the differences between two stochastic processes is the weak convergence. In this framework, one views stochastic processes as path-valued random variables and then defines the convergence for their laws, which are distributions viewpoint ignores the filtration of stochastic processes, which models the evolution of information, and therefore such loss may have negative implications in multi-period optimisation problems. For example, for the American option pricing task, even if the two underlying processes are stochastic processes with very similar laws, the corresponding price of American options can completely different, see toy example A. in Appendix A. To address this shortcoming of weak conver- gence, D. Aldous I introduced the notion of extended weak convergence. The central object in
path space. However, this
"Corresponding author
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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this methodology is the so-called prediction process, which consists of conditional distributions of the underlying process based on available information at different time beings, and therefore reflects how the associated information flow (i.e., filtration) affects the prediction of the future evolution of the underlying process as time varies. Instead of considering the laws of processes (i.e., distributions on path space) in weak convergence, one compares the laws of prediction processes, which are distributions on the measure-valued path space, in extended weak convergence. Since the knowledge of filtration is captured through taking conditional distributions, it was shown in [3] the topology induced by extended weak convergence, which belongs to the so-called adapted weak topologies fully characterise essential properties of stochastic processes and endow multi- period optimisation problems with continuity, provided filtration is generated by the process itself. While the theoretical contributions to adapted weak topologies flourish in recent years (e.g., 31 (2], 4),, the related work on numerics is still very sparse because of the very complex nature of these topologies. In this pa- per, we propose novel metric called High Rank Puth Characteristic Func- tion (HRPCFD) which can metrise the extended weak convergence, and, more importantly, admits an efficient implementation algorithm. The core idea of this approach is built on top of the unitary feature of TR -alued paths (18, Zi, which exploits the non- commutativity and the group structure of the unitary developments to encode information on order of paths. Based on the same consideration, Lou et al. [19] introduced the Path Characteris- tic Function (PCF) for stochastic pro- cesses, which induces computable distance (namely, PCFD) to metrise the weak convergence. As extended weak convergence is defined in terms of laws of prediction processes which are measure- valued stochastic processes, the scheme of PCF remains valid in adapted weak topologies long as one can construct a PCF of measure-valued paths. One of the main contributions of the present work is give such suitable notion via the so-called high rank path development (see Figure [] for illustration); moreover, we can show that the induced distance (called HRPCFD) does not only characterise the more complicated extended weak convergence, but also inherits almost all favourable analytic properties of classical PCFD mentioned in [19.. Since the measure-value in an infinite dimensional nonlinear space, such a generalisation of the results in [19] from -valued paths to measure- paths is much technically involved and therefore significantly nontrivial.
00-2
Tims
PZ)FF)
Figure 1: The high-level illustration of the high rank path development. Here the prediction process X, = (F,) for all E [0, T). & 2x (M1) is the PCF of the prediction process and UM, MMg (X) is the high rank development of the path 1. ++ 4 x. (M1) under the linear map M2.
paths take values
On the numerical side, we design efficient algorithm to train HRPCFD from data and construct the HRPCF-GAN model with HRPCFD as the discriminator for time series generation. A key computational challenge in applying distances based on extended weak topology is the accurate and efficient estimation of the conditional probability measure. To address this issue, we have implemented sequence-to--equeece regression module that effectively resolves this bottleneck. Our work is the first of its kind to apply the adapted weak topology for generative models on time series generation. Moreover, to validate the effectiveness of our approach, we conduct experiments in (1) hypothesis testing to classify different stochastic processes, and (2) conditional time series generation to predict the future time series given the past time series. Our HIRPCF-GAN can be viewed as a natural generalisation of PCF-GAN [19 to the setting of extended weak convergence, so that the data generated by HRPCF-GAN possesses not only a similar law but also a similar filtration with the target model. The numerical experiments validate the out- nut-performance of this new approach based on HRPCFD compared with several state-of-the the-art GAN models for time series generation in terms of various test metrics.
In general, any topology on the space of stochastic processes which can reflect the differences of associated filtrations can be called adapted weak topology.
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Related work. So far most of existing statistical and numerical methods for handling stochastic pro-
cesses (e.g.. [ 16 19 are based on weak convergence, and the results on numerical implementation
of adapted weak topologies are rather limited. The most relevant work is [21]. whose theoretical foundation roots in [3] The present paper shares similar philosophy with 21 in the sense that both methods for defining metrics for extended weak convergence rely on the construction of feature of the measure-valued path by transforming it into a linear space-valued path. contrast to [21], where measure-valued path is lifted to an infinitediimensional Hilbert space, we reduce measure-valued paths into matrix-valued paths through unitary development which allows us to apply the techniques from [9] to design the algorithm. Another remarkable point is that in 221 one has to solve large family of PDEs to compute the distance, which can be avoided in the numerical estimation of the HRPCFD proposed here. On the other hand, as Wasserstein distances
metrise weak convergence, the so-called causal Wasserstein distances can be used to measure adapted weak topologies. One related work is [22] which can be seen
improved variant of the Sinkhorn divergence tailored to sequential data. Note that the discriminator (i.e., causal Wasserstein metric) used in [22] is slightly weaker than the HRPCFD, as the latter is actually equivalent the bi-causal Wasserstein distance.
2 Preliminaries
2.1 Prediction Processes and Extended Weak Convergence
Let = -Q...... T and X (Xt)ter be an vallue stochastic process defined on filtered stochastic basis (S** F, F = (Fe)eer,P) IP) such that X is adapted to the filtration F, i.c., X, is measurable with respect to Fr for all i E I. We call the five-tuple (Ωx F,, X, IP) a filtered process, and denote it by X. Throughout this paper, we will use FP denote the space of all (R--valued filtered processes on the discrete time interval J, and assume that F is the natural filtration in the Xt)-
sense that for every & F,
Since each discrete time path æ E (Rd)T+I can be uniquely extended to piecewise linear path on [0, 7) by linear interpolation, we will not distinguish the product space (R=))++ and the subspace X' := fæ [0, + Rd : x is piecewise linear) of ((,, IR (the space of all continuous functions in Rd with bounded variation Clearly each stochastic process X can be seen as X' v-llue random variable, and therefore the law of X, denoted by Px = IP o X-1, belongs to P(X'), the space of probability measures on the path space X. Recall that a sequence of filtered processes X" (S2", Fr F", X",,'") P") converges to limit X weakly or in the weak topology (in notation:
X W X ) if the laws Px== = P" p converges to Px in P(X)) weakly, i.e., for all continuous and bounded functions f E (x)) it holds that 1 Eg- [f(X))] = (x))
For each t E I, we denote X2 := P(X .(Fi) as the (regular) conditional distribution of X given Fr, which is a random measure taking values in P(2'). We call this measure-vall valuue process X = (Kt)ter the prediction process of the filtered process X. By definition it is clear that the state space of X is P(2)+++ and, again, by routine linear interpolation we can embed P(())++ into L {p: [0, T] + P(X): p piecewise linear). Thus the law of X. denoted by Px P X-1 belongs to P()) (the space of probability measures on the measure-valued path space X¹), where X' is endowed with the product topology and P()') is equipped with the corresponding weak topology.
Definition 2.1.
Two filtered processes X (ΩXx FF,R X, IP) and W (Ω 9, G, Y, Q) are called synonymous if their prediction processes X and Y have the same law in P(L), i.e. Px* = Py.
A sequence of filtered processes X (Ω² F" (Q",F",,",, "", IP"), N converges to another filtered process X = (Sxx 1 F,F,K,, P)) in the extended weak convergence if the law of their prediction processes Xn converges to the law of X in P(X) weakly, i.e., for all continuous and bounded functions f Co(d),, lim, Ep- If("m) Ex]((X). In
notation: X" EW X.
'This means that X is equipped with the topology induced by the total variation norm. 1-1
*For Po=to: Pey.... Pty=T E P(X) and t [0, T), define Pe 7, Pt.s1
4-1--
t411- Pt.-
3

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If F6 and Fo are the trivial o-algebra, then XØ Px" and Xo = Px are laws of X" and X respectively, so that x EW X certainly implies that X " X. This implies that extended weak convergence is stronger than weak convergence. Moreover, the extended weak convergence induces the correct topology in multi-period decision making problems, as the next theorem (see 1 shows.
Theorem 2.2. The extended weak convergence provides continuity for the value functions in multi- period optimisation problems (e.g.. optimal stopping problem, utility maximisation problem), as long as the reward function is continuous and bounded.
Admittedly, the above notions related to extended weak convergence (e.g., the spaces X and P(L). the weak convergence in ((X) etc.) are rather abstract. Therefore, we provide some simple examples in Appendix A. to explain these notions in more transparent way. We refer readers to [ and for more details on extended weak convergence.
2.2 Path Development and Path Characteristic Function (PCF)
In this subsection, we review some important notions and properties of -value path development and characteristic function (PCF) for Rd -aauuee stochastic processes, which will be used later to construct characteristic functions for measure-valued stochastic processes. More technical details
on PCF can be found in Appendix A.2 We also refer readers to [9 and [8] for more detailed discussion on this topic.
For m E N. let Cm be the space of m x m complex matrices, Im denote the identity matrix in Cmxm, and be conjugate transpose. Write (m) and u(m) for the Lie group of m m unitary matrices and its Lie algebra, resp.:
(mm) = {A E Cm
u(m) = (A E C : A + A = 0}.
Let C(R u(m)) denote the space of linear mappings from Rd to u(m).
Definition 2.3. Let æ E Cl ([0, TT, Rd)) be a continuous path with bounded variation and M
[(RA, u(m)) be linear map. The unitary feature of under M is the solution y [0, U (m) to the following equation:
dy: (dx.)), Yo0
Ims
(1)
where (dx:) denotes the usual matrix product. We write Unr := yr, i.e., the endpoint of the solution path, and by an abuse of notation, also call it the unitary feature of (under M).
The unitary feature is special case of the path development, for which one may consider paths taking values any Lie group G. It easy see that for piecewise linear path (æo. .... 2T) E X, holds Mas (x) = II (Axx)) for Az; = 2; x;-1 and exp denotes the matrix exponential. We now use the unitary feature to define the Path Characteristic Function (PCF) for valued stochastic processes:
Definition 2.4. Let X ",,,F,E,,,,,, be filtered process and Px be its law. The Path Characteristic Function (PCF) of X is the map Dx UMEN L(RR, u(m)) + UmEN Cmxm given by
T xx EX (x))
Remark 2.5. In the present work, only consider the discrete-time processes defined I O,..., ,T, and therefore the time index t appeared in the stochastic process X, and its filtration F, only takes values in O, O.... T. It is just convention in the rough path community that views discrete time path defined on I = O....,T T as piecewise linear path defined on the continuous time interval [0, T] by a routine linear interpolation, because such identification may make some formulations and computations easier (e.g., by doing so the unitary feature of a path can be formulated as the solution of an ODE [0, T)).
To distinguish Ox from the so-called high rank PCF which will be defined in the next subsection, we also call tx the rank 1 PCF The next theorem (see [19] Theorem 3.2]) justifies why ox defined in Definition 2.4 is called PCF for path-valued random variables.

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Theorem 2.6 (Characteristicity of laws). For X and Y two filtered processes, they have the same law (i.e., Px- Py) if and only 2x Øy.
The characteristicity of PCF allows to define novel distance FP which metrises the weak convergence (locally). This metric is called the PCF-based distance (PCFD), see [19 Definition 3.3]. Moreover, such PCFD possesses many nice analytic properties including boundedness (199 Lemma 3.5.), Maximum Mean Discrepancy (MMD, [9] Proposition B. B.10)) among others, see 19 Section 3.2], which ensures the feasibility of using PCFD in numerical aspect.
Remark 2.7. Rigorously speaking, need to add additional time component every -o-vel process X (i.e., consider X, = (t, X!....... x²)) always implicitly use such time-augmentation throughout the whole paper and still write X instead of X for simplicity of notations.
) to guarantee Theorem 2.6 holds true. We will
3 High Rank Path Development Embedding
We now want to construct a characteristic function for prediction processes and use it to metrise the ex- tended weak convergence just like PCFD metrises the weak convergence. Since prediction processes are X' valuee random variables, we first need to find a suitable notion of unitary feature/development for measure-valuet paths.
3.1 High Rank Development of Prediction Processes
Given a filtered process X (N* F,,, X),,, remember that its prediction process X satisfies X, = ( -(FF) for E Now, for linear operators M E ((RR,(() for E N, we take the conditional expectation of UM against X, = P(X E -(Fi) to obtain a C"x*_valued stochastic process & xx (M) Ep M (X) [FF] LE I. Then, for any M L(C"xn, u(m)) with some N, the unitary feature (t + + xx (M)) of
path (t $ xx (M)) well defined and takes values in the unitary group U (m). We call each pair (M, E L(R", u(n)) ) x L(C"x" .um)) for (n.m) m) N² an admissible pair of unitary representations, and the set of all admissible pairs of unitary representations is denoted by Aunitury.
Definition 3.1. For MM E Auniary with M e L(K", u(n)), M L(Cnxn u(m)) and (2** F, a filtered process with its prediction process X, we call
(M) Un(t + 1 xx (M))
the high rank development of the prediction process X under (M,M..
See Figure I for the schematic overview of the high rank development. From above we can see that the construction of (X) involves with taking finite dimensional path development in Section twice: first use the PCF under M E C(Rd, u(n)) to transform each conditional distribution P(X E ..F)) into a matrix $ x. (M), and then apply the unitary feature (- to the resulting matrix- valued path (t + 4 x, (M)) for M L(C"x", ,((m).
3.2 High Rank Path Characteristic Function
With the above notion of unitary feature of measure-valued paths, following Definition 2.4 we define the high rank Path Characteristic Function (HRPCF) for filtered processes.
Definition 3.2. For filtered process = (SX F,F X,
P. E FP, the function
2 Aunitary
U UCTXT X (M, + CM Ep(MM.(( = Ep (x)
(3)
m=1
called the High Rank Path Characteristic Function of (Abbreviation: HRPCF We use the superscript "2" in 2 emphasise that & is induced taking usual path development twice.
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6 is said to be HRPCF for X as it satisfies the following characteristicity of synonym for filtered processes (see Definition 2.T). For detailed proof please check the Appendix
Theorem 3.3 (Characteristicityy of synonym). Two filtered processes X and Y are synonymous if and only if they have the same high rank PCF, that is, 2 (M, M) (MM MM, E Aunitary
3.3 A New Distance induced by High Rank PCF
In this subsection, we will use the second rank PCF to define a distance on FP, which can (locally) characterize the extended weak convergence, as the classical PCFD introduced in subsection 2.2 can metrise the weak topology on FP.
Definition 3.4. For two filtered processes X and Y, let (M,M) be random admissible pair in Auniary with M E L(TR", u(n)) for some n, and M E L(Caxn, u(m)) for some m. The High Rank Path Characteristic Function-based distance, for short HRPCFD, between X and V with respect to PM and PM is defined by
& (MM) HRPCFD (X) / &
6 where d denotes the Hilbert-Schun Schmidt distance on Cmxm
As previously mentioned in the introduction, the so-defined HRPCED shares the same analytic properties the classical PCF, e.g., the separation of points, boundedness and the MMD property, whose proof can be found in Appendix [A] Moreover, it metrises a much stronger topology (the extended weak convergence). shown in the next theorem.
Theorem 3.5. Suppose (X')eN and % are filtered processes whose laws Px and Px are supported compact subset of X. Then X EW % iff HRPCFD(Z X) - O where HRM[1,,
(X', X)}
HRPCED(X X
where the sequence (M,, (Mj,M))nnn satisfies that for any (n, m) E N22 there is j E N such that Mj L(R4, u(n)) and M, E L(Cuxn, u(m)) and PMr,, PM2 have full supports for all j E N.
We provide concrete example in the last paragraph of Appendix A. verify the fact that HRPCFD really reflects the differences of filtrations via an explicit computation.
4
Methodology
In this section, let X and Y be two filtered processes with the law Px,Py E P(X), let X (#:) Px and (y:) ~ Py be sample paths.
4.1 Estimating conditional probability measure and HRPCF
A fundamental question is to estimate the conditional probability measure X, P(X E .FF from the finitely many data (#:) in particular the random variable & x, (M) = IE (M (X)](())F))) for any M L(Rd, u(n)). We solve this problem by conducting a regression. Fix M we learn sequence-to sequence-to-sequenec sequence model FØ (T+( (x*)x(T+1), where the input and output pairs (X N. Uar o)) More specifically, we optimize the model parameters of FX by minimizing the loss function:
RLoss (0; x, M)-
ΣΣ Eenn(Fp (ss.)) t=0 xEx
4
It is worth noting that the choice of F3 must be autoregressive models to prevent information leakage. A detailed pseudocode is shown in Algorithm [ Then, we approximate & using the trained regression model FXX following the Algorithm 2 We denote by d the estimation of Φ
For A, B E cm dhs (A, B) u((A (A B)

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4.2 Optimizing HRPCED
In most empirical applications as we will show in Section we employ HRPCED as a discriminator under the GAN setting. That is, we optimize the loss function supM. .M HRPCFD We would approximate the pair of random variables (M,M) by discrete random variables MR, = Ri ΣK M, and Mr, = R2 Eki Mi, parametrized by M, E ((B), and M; E L(C"X" u(()), K1, K2 E N and optimize so-called Empirical HRPCFD
V).
1
EHRPCED MAX Mx² (X, (X,Y=
( (MM, (2,M, K1K2 1=1==1
(5)
In practice, the joint training on both Mr, and Mr, is computationally expensive and prone to overfitting. We alleviate this problem by splitting the optimization procedure in the following three steps: 1) Optimize (M.)) K to maximize EPCFD (X,
, Di=1 dis (((xx((M)) (M;), Ty (M,)) K;
(xx (M) x E Mar(a;) 19 Section 3.3] denote by M (M
the optimized linear maps. 2) Train regression modules : Fg. for each MT using data sampled from Px and Py
Ki )=
respectively. 3) Optimize (M:) to maximize EHRPCED Mxg (X
The reason behind it natural: the optimal / captures the most relevant information that discriminates the distribution Px from Py. This difference is reflected in the design of higher rank expected path developments through regression models specifically trained for this purpose. Finally, Ki the HRPCFD based on [ tends be more significant among other choices of (M)) making it a stronger discriminator.
4.3 HRPCF-GAN for conditional time series generation
we consider the task of conditional time series generation to simulate the law of the future path Xuuure := X( given the past path Xpas == { from samples of X. To this end, propose the so-called HRPCF-GAN by leveraging the autoregressive gener- ator and the trainable HRPCED as the discriminator. See Figure 2 for the flowchart illustration.
Following [6
Conditional autoregressive generator To sim- ulate future time series of length T p. we construct generator Gg based on the step-1 conditional generator Gs following [6). This generator, go : Xpma x > RRd, aims to produce random variable upproximating IP(X+11|FF) By applying ge inductively, we can simulate fu- ture paths of arbitrary length. To address the limitation of AR-RNN generator proposed in [6). where P(X++1lF.) depends solely p- lagged values of X,, we incorporate an embed- ding module. This module efficiently extracts past path information into a low-dimensional latent space. The output of this embedding mod- ule, along with the noise vector, serves as the input for ge generate subsequent steps in the fake time series. Further details of our proposed generator are provided in Appendix B.2
Gg
(Rppstt
Pacurrl
X (xpaste
Xinture
PCFD (X.
asgmax
Reg. madel F
Mi.
Reg. model PA
E, (U. MC
HRPCFD Ni, Mx, (X.Y)
Figure 2: Flowchart of HRPCF-GAN for learning condition distribution P P(X (X ppsa)
High Rank development discriminator To capture the conditional law, we use the HRPCFD as the discriminator of joint law of (X X under true and fake measures. Here the empirical measures MR, and My, are model parameters the discriminator, which are optimized by the following maximization:
EHRPCED* M MMg
max
MX, MMx (x..7. (Xpo))) Go (X(Xx))) 2))).
In principle, one can generate the fake data by the generator via Monte Carlo and apply the training procedure outlined in Section 4.2 for training the generative model. However, it would be computa-
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tionally infeasible due to the need for recalibration of the regression module per generator update. To enhance the training efficiency for the regression module under the fake measure, we use the gradient descent method with efficient initialization obtained by the trained regression model under real data. For each generator; the corresponding regression model parameters are then updated to minimize the RLoss (Section 4.1 on batch of newly generated samples by Gg. The detailed algorithm is described in Algorithm
5 Numerical results
5.1 Hypothesis testing
To showcase the power of EHRPCFD in discriminating laws of stochastic processes, we use it the test statistic in the permutation test. Similar experiments have been done [2]] By regarding the permutation test decision rule, we assess its performance via computing its power (probability of correctly rejecting the null hypothesis) and rype-I error (probability of falsely rejecting the null hypothesis). Similar to [15], we compare the law of 3-dimensional Brownian motion B with the set of laws of 3-dimensional fractional Brownian motion = with Hurst parameter H ranging from [0.4, 0.6]. Details of the methodology and implementation can be found in Appendix C.1
Baselines We compare the performance HRPCFD with other test metries including 1) the linear and RBF signature MMDs [8 20) and its high-rank derivative, namely High Rank signature MMDs
[21] 2) Classical vector MMDs; 3) PCFD [5]
As shown in Table of the test power, HRPCFD consistently outperforms other models, especially when is close to 0.5. We do see an improvement from the vanilla PCFD by considering stronger topology. Furthermore, comparing HRPCFD and High Rank signature MMD, we observe distinct advantage for HRPCFD, This may be due to the challenge of capturing the conditional probability measure, as High Rank signature MMD) relies on linear regression for estimation, whereas we obtained better estimation using a non-linear approach. Additional test metries such type-I error and computational cost can be found in Appendix
Developments
Signature MMDs
Classical MMDs
H
High Rank PCFD
PCFD
Linear
RBF
High Rank
Linear
RBF
0.4
1
1
0.09 + 0.06
0.97 + 0.03
0.22 0.07
0.05 0.04
0.97 0.04
0.425
1 0
1+0
0.1 + 0.05
0.69
0.100
0.01 + 0.02
0.58 = 0.10
0.45
0.97 0.04
0.99 199:100 0.02
0.04 0.04
0.15
0.14
0.06 0.05
0.24 0.08
0.475
0.31 0.13
0.06 0.02
0.01 + 0.02
0.04
0.12 0.04
0.01 + 0.02
0.02 0.02
0,525
0.30 1 0.20
0.08 1: 0.02
0,05 0.02
0.07
0.19
0.08 0.04
0.09 = 0.04
0.55
0.99 0.02
0.95 0.03
0.13 0.05
0.18 0.17
0.00
0.19 0.11
0.575
1+0
1+0
0.07 + 0.02
0.1
0.14
0.10
0.48 = 0.15
0.6
1
1
0.05 1 0.03
0.75
0.22 + 0.05
0.06
0.67 = 0.14
Table 1: Test power of the distances when h A 0.5 in the form of mean std over 5 runs. After
careful grid search, set optimal v0.05 for the RBF signature MMD and classical RBF MMD,
whereas 01 = 52 = for High Rank signature MMD.
5.2 Generative modeling
To validate the effectiveness of our proposed HRPCF-GAN, we consider the task of learning the law of future time series conditional on its past time series.
Dataset We benchmark our model both synthetic and empirical datasets. 1) multivariate fractional Brownian Motion (fBM) with Hurst parameter H = 1/4: this dataset exhibits non-Markovian properties and high oscillation. 2) Stock dataset: We collected the daily log return of 5 representative stocks the U.S. market from 2010 to 2020, sourced from Yahoo Finance.
Baseline We compare the performance of HRPCF-GAN with well-known models for time-series generation such as RCGAN 100 and TimeGAN 23 Furthermore, we use PCFGAN [19] as benchmarking model to showcase the significant improvement by considering the higher rank
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Real
RCGANN
TimeGAN
ROrGAN
HAPCFGAN
t
t-6
t=7
t
t-9
Figure 3: Sample plots of the conditional distribution P(X[F;) on fBM conditioned on the same past path, using both true and GAN models (arranged from top bottom). Each column represents different /. The thick red /green line indicates the conditional mean of the future path estimated by model simulated samples/true models. The shaded area presents the region of std of model simulated samples, whereas the shaded area shown corresponds to the region of = theoretical std.
development as the discriminator. For fairness, we use the same generator structure (LSTM-based) for all these models.
Test metrics To assess the fidelity, usefulness, and diversity of synthetic time series, we consider test metrics, including Auto-Correlation, Tross-Correlation, Discriminative Score, Sig-W, Distance, and Conditional Expectation. For the stock dataset, we also consider test metric based on American option pricing. detailed definition of these test metrics be found in Appendix C.2
We summarize in Table the performance comparison between HRPCF-GAN and benchmarking models. For both datasels, HRPCF-GAN consistently outperforms the other models. Focusing the fBM dataset, HRPCF-GAN achieves the lowest Auto-Correlation (.082) and Cross-Correlatii (0.013), which is approximately 21.9 21.9%/72.3'? lower than the second-bess model. indicating better performance in fitting the dynamics the underlying process across time and feature dimensions. We also observe strong evidence in capturing the conditional probability measure HRPCFGAN achieves the lowest Conditional Expectation score (1-693 on fBM and 0.56 on Stock). Furthermore, observe average improvement of 34% 14% of HRPCF-GAN with respect to PCFGAN on /BM//tocc datasets respectively. The strong empirical results demonstrate the effectiveness of considering high rank path development to capture the filtration of stochastic processes. Finally, HRPCF-GAN attained the best estimation of an al-the-money American put option, which demon- strates its potential usage for optimal stopping problems in finance. Sample plots from all models conditioned on the same path are also shown in Figures 3 6 and 7 for qualitative analysis of generative quality. For additional test metries, we refer readers to Table
Dataset
Test Metrics
RCGAN
TimeGAN
PCEGAN
HRPCF-GAN
Auto-C.
105++0001
459+-0003
.125+-003
.082+-002
Cross-C.
.011+0001
.092+-001
.047+-001
.013++.001
fBM
Discriminative
.207+0008
480++.002
.65++006
.151+.006
Sig-W,
.512+-006
.341+-011
.199+.004
.169+-.009
Cond. Exp.
1.8224.003
2.265:1.029
2.278+.033
1.693:1.021
Auto-C. 22391.016
.228+.0100
.184..003
.1891.010
Cross-C.
.067+.011
.056+t.002
.0555.004
.053+1.05
Stock
Discriminative
.1341.058
.020+.021
.028+.017
.016
Sig-W1
.013+.002
.008:L.01
.005+-001
.0041.002
Cond. Exp.
.078+..033
.079+.001
.060+.001
.0561.002
Amer. Put
5564t.318
.21
202+.020
.1794.006
Table 2: Performance comparison HRPCF-GAN and baselines. The best for each task is shown in bold. Each test metric shown in the form of Lssd over runs.

---

Conclusion and Future work
Conclusion: In this paper, we apply the unitary feature from rough path theory define the CF for measure-valued paths, which further induces distance (HRPCFD) for metrising the extended weak convergence. Theoretically, we prove the key properties HRPCFD, such as characteristicity: uniform boundedness, etc. Additionally, the numerical experiments validate the out-performance of approach based on HRPCFD compared with several state-of-the-art GAN models for tasks such as hypothesis testing and synthetic time series generation.
Limitation and Future work:
The suitable choice of network architecture for generating data is crucial in the proposed HRPCF-GAN, which merits further investigation; in particular, it will be interesting to understand how the network architecture impacts the filtration structure the generated stochastic process. Furthermore, there is room for further improvement on the estimation method of conditional expectation in terms of accuracy and training stability. Possible routes include exploring the interplay between the regression module and the generator.
Broader impacts: Our approach based the extended weak convergence has the potential in many important financial and economic applications, such as optimal stopping, utility maximisa- tion and stochastic programming. Unlike classical methods built on top of parametric stochastic differential equations, our non-parametric and data-driven method alleviates the risk of the model mis-specification, providing better solution complex, real-world multi-period decision making problems. However, like other synthetic data generation models, it also poses risks of misuse, e.g.. misrepresenting the synthetic data real data.
Acknowledgments and Disclosure of Funding
HN is supported by the EPSRC under the program grant EP/S026347// and the Alan Turing Institute under the EPSRC grant EP/N510129/1. HN extends her gratitude to Terry Lyons and Hang Lou for insightful discussions. Moreover, HN is grateful to Jing Liu for her help with Figure 1. CL is supported by the National Key Research and Development Program China: Young Scientist Project 2023YFA1010901
10

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References
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[4] Daniel Bartl, Mathias Beiglboeck, and Gudmund Pammer. The Wasserstein spaces of stochastic processes, arXiv:2104. 14245, 2021.
[5] Patric Bonnier, Chong Liu, and Harald Oberhauser. Adapted topologies and higher rank signatures. The Annals of Applied Probability, 33(3), 2023.
[6] K. Chen. Integration of paths-a faithful representation of paths by noncommutative formal power series. Trans. Amer. Math. Soc., 89(2), 1958.
[7] Ilya Chevyrev and Terry Lyons. Characteristic functions of measures on geometric rough paths. Annals of Probability, 44(6), 2016.
[8] lya Chevyrev and Harald Oberhauser. Signature moments characterize laws of stochastic processes, Journal of Machine Learning Research, 2022.
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[10] Cristóbal Esteban, Stephanie L. Hyland, and Gunnar Rätsch. Real-valued (medical) time series generation with recurrent conditional GANs, 2017.
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future, learning evolving system. arXiv preprint
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[15] Siran Li, Zijiu Lyu, Hao Ni, and Jiajie Tao. On the determination of path signature from its unitary development, 2024.
[16] Shujian Liao, Hao Ni, Mare Sabate-Vidales, Lukasz Szpruch, Magnus Wiese, and Baoren Xiao. Sig-W Sig-Wasserstein GANs for conditional time series generation. Mathematical Finance, 34(2), 2024.
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[18] Hang Lou, Siran Li, and Hao Ni. Path development network with finite-diment dinte-dimesionall Lie group representation. arXiv:2204. 200.0222000000000, 2022.
[19] Hang Lou, Siran Li, and Hao Ni. PCF-GAN: generating sequential data via the characteristic function of measures on the path space. Advances in Neural Information Processing Systems, 36, 2023.
[20] Cristopher Salvi, Thomas Cass, James Foster, Terry Lyons, and Weixin Yang. The signature kernel is the solution of goursat pde. SIAM Journal on Mathematics of Data Science, 3(3):873-89, January 2021.

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[21] Cristopher Salvi, Maud Lemercier, Chong Liu. Blanka Hovarth, Theodoros Damoulas, and
Terry Lyons. Higher order kernel mean embeddings capture filtrations of stochastic processes. Advances in Neural Information Processing Systems, 34:1 1446561666667, 2021.
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12

---

A Examples and Proofs
A.1 Examples related to extended weak convergence
Prediction processes First let us give an explicit example for prediction processes of some simple filtered processes. Consider the two processes X (S2", F" F X ,p)) and X FX, IP). where
O" = { x2) oj = (z) = 1, (1) = 1 + n/ x2(2) = 2) and x = (2) =
1,)3(1) = 2. 23
= 0);
xp(())) ((] for 0, and = 1,2 2 is the coordinate process Ω (=()))= -x"
F" = F2) is the natural filtration generated by X": FD = = 32" and FP
F2 (XT', X3') is the power set of !?"
and
Ω (2), 2. (a)1 1, (1) 1, (2) 2) and æ2 (a:2(0) 1, (1)
12)) = 0);
Xx(x)) =1,(1) for = 0, and::nn: 1, is the coordinate process on ?2
P(x1) = F((22) 2/2
F = F1, is the natural filtration generated by X: Fo = Fi = [0,2) and
F² = oX1, X2) is the power set of f
We plot the sample paths of x and X in Fig.
p
p p= 0.5
p= 0.5
p p=1
p = 0.5
p
p 0.5
Figure 4: X" (left) converges to X (right) weakly, but the corresponding price of American options X cannot converge to counterpart on X, see Example A. below. Therefore the usage of slightly erroneous models in weak topology may cause significant loss in decision making problems. This example is taken from 3 and [5]
bello....
From the above, it is straightforward check that the prediction process X of X"
{x (>x X (a:)) PP" (F6) Pxt,
where Px" is the law of X under IP"; (x7())) Pn (X E (FF)) (=+?) ōwp; Xx Pm (X" (x)) 6ag
where 6Φp (1 = 1, denotes the Dirac measure at aF:; and
{x"²") = P"(X" E (FF))) day. x2(2a)) Pm (X" E (FF)) = 6ag.
Consequently, holds that the law of X satisfies
1
Px- pm (X8 (Px*, 6ay 0z;))
i 1,2.
2'
Similarly, the prediction process X of X is
8o(x1)= = X (I:2 = P(X E FF) = Px,
where Px is the law of X under P:
(x1x = P(X E (z))(x)) = Px, 81(22) = P(X E (F2) = Pxi
and
{x(+1) P(X E (Z)) = = 82(22) P(X E (Z2) 6pz.
so that the law of X reads
i= 1,2. Px P(X (Px. Px, 6w)) = 2'
13

---

Test functions for extended weak convergence For / {0, 1..... T) and filtered process X E FP on I, the typical test functions for defining the extended weak convergence have the following form:
f(8) =
where Sr E (x) are continuous bounded functions on the path space X and F E C% (, ). For instance, for the filtered processes X" and X in the above example, we have T = 2. and by choosing fo (o 21, x2) = 1, fi (Io, I1, #2) = 21 + 22 - 2. fa( 31, 222) = sin(12 - x1) and Fyy, M1, (22)22 exp(-Iæi
122. ) in view of the facts that JT = F2 are the power set of !?"
(see the last paragraph), we obtain that for each n,
f(x"(x)))) exp( - 1 [X] + X (x;])(;;)) (Ep.. (sin X - xT)
| (1) + (2) - 2| sin (x, x (1))) =xpp( exp(-la=f(1
= -cxp(- (11 + | - sin (1 )), i= 1,
and therefore Ep- = exp(- (1
)) - sin 2 (1 }).. On the other side, since Fo = Fi = {0, 2) are trivial o-algebra, for the prediction process X of X we have
[(x(=:)) = exp( |pp[ + X2 - 2(F))(())))))) - (Erlsinn (X2 - (1)1 -sin-sinn
= exp( - [x + X₂ - 2|| - sin (2) - x1)))
= sinp(=sin'pp (1)), 1,2, 1,2
as xp|X1 + X2- - 2] = 0; and therefore
Er[f(X)) exp(-sin?(1))
Clearly, as n + oc, we have Ep- [f(X)] exp( -in b) + exp(-1- *
sin(-sin"(1) = (x))( which shows that X" cannot converge to X in the extended weak convergence according to Definition 2.1 although it is easy to see that the laws of X" converges to the law of X in the weak topology.
In the example, while the unconditional law of the processes X converges to X, weak convergence fails to capture a key difference between the financial models X" and X. Specifically, if an agent believes the market dynamics as in x", he/she always knows the outcome of the last day in advance. granting a predictive advantage, whereas in the "fair" market X. the agent lacks this foresight. This crucial difference in the observed information flow- "No knowledge = Full knowledge + Full knowledge" for X" versus "No knowledge = No knowledge + Full knowledge" for X-is is not reflected in weak convergence alone.
Extended Weak Topology (EWT) is vital in this case, because it captures this difference through the conditional distributions. For markets X", where the agent has full information on day 1, the conditional distribution becomes a single Dirac measure, annihilating randomness. In contrast, X retains genuine randomness at day 1, as reflected by a linear combination of Dirac measures. Since EWT is based on conditional distributions, it effectively measures differences in information evolution styles, ensuring continuity in multi mulli-period decision-making making as agents update their actions based on continually evolving information.
Some important multi-periode optimisation problems The following multi-periods optimisation problems are very important in financial and economic applications, whose value functions are, in general, discontinuous with respect to the weak convergence, but continuous in the extended weak topology.
Example A.1 (Optimal Stopping Problem). Let g T - R be a continuous and bounded non-anticipative (i.e., for any
and x E X, the value of gtt, x)) only depends on To.-.. , X.a) function. For each filtered process X we set ST,, = {T: FF" -soppping time) be the collection of all stopping times with respect to the filtration Fn and similarly define for X. Then the value function
v)(e) in the Optimal Stopping Problem (OSP) with the reward g (in the context of mathematical finance, it is also called the price of American option) is defined by
0g (x(X")
sup Eg- (9(, TEST,
vg(X)
sup Ep]g(r,X)), TEST
14

---

x" , X, then (x(x + (X),,, whilst this continuity fails in the weak convergence: for the
processes X" and X considered as above (see Fig. 4 and the reward function g(t, x) == It, one has
X" X but
lim 1/g (Xx) (X)
n-000
Indeed, since X is martingale with initial value 1, it is obvious that for any stopping time E ST it always holds that Ep]g(T, X)] = [x[x,] = 1 which in turn implies that 1g (x) = 1; on the other hand, since the filtration of X satisfies that FP F2 (i.e.. the agent already knows everything at day 1). it is easy to check that T. 21 + lny is the optimal F" stoppiigg time for vgg (xx) and
consequently ug (X²) Ep- [X 1 x 2+ (1 :)) = 3
2 2n converges to 3
2 + 0 = V/9 (X).
Example A.2 (Utility Maximisation Problem). Let g R + R be continuous, bounded and concave utility function. For each filtered process x we set An := {pp = (P:t)tm...... is predictable w.r.t FT) be the collection of all predictable strategies (i.e., yr is Ft-1 ---emuuuurbblle for all t = I...., T) with respect to the filtration F" and similarly define A for K. Then the value function (-)-) in the utility maximisation Problem with the utility function g is defined by
Ug My(xx)) = sup Egn E [ol[[" p.ax3).. (x sup Ep (0 (/5% (dxx)] PEAr
where S`' pidX;=Eis - (X, - X(-1) is the stochastic integral. Xf EW X. then Ug (x()
-
(()), whilst this continuity fails in the weak convergence.
An example of HRPCF We still consider the example mentioned before (see the paragraph Prediction processes and Fig. 4 In the previous discussions we have known that X + X in the weak convergence (i.e., the laws Pxx converges to the law Px), but X" cannot converge to X in the extended weak convergence. Now we show that there exists an admissible pair (M,M) E Aunitary such that
lim dHs( (Mxx Sx (M, + 0, n-000
by an explicit calculation, which confirms that the HRPCFD does metrise the extended weak conver- gence and therefore reflect the differences of filtrations of stochastic processes.
Now pick a linear operator M C(R?, u(1) y which is given by M(t, y) := v( E u(() cC where denotes the imaginary unit in C. Since the prediction process X of X
Xo(x1) = Xo(x2) = PXX E (Fo) = Px,
where Px is the law of X under IP;
X = IP(X E (i)((())) = Px, 81(02) = P(X E E (222) = -Px
and
{=(+1)= = P(X E (Z1) = S0.1 (2(22) = P(X E (F2) = 601
we can check that
Ep (x) = (X) = 2/2 (c -e- 21) = 0,
Ep Ex Ex (x,
fe +c- F1 0,
and
= (2:) = 20 = i, Ee ( (20g) = (x0g) = gi i
which shows that the C-valued process (Eg [U, [Eg
(X)
satisfies that
(z) (1) 0,,
(6)
and
(Ee(Mss
-ii).
'Recall that in the unitary representation of a path x we actually always consider the time- augmented version (t, D.). here the domain of AM for real valued path is R
15

---

On the other hand, for every N, we have
P"IX" E = Px*,
P"[X" E = - =
=
and
P"[X" E - (.)) 6mz
which provides that
Ep LM (X
7 (e(1+ + (1+
Ep 15
= Ep [M
= (2r( = c (
and
Epn [xx" = Epn (o
= Unr(2,)) = -17tt))
Therefore, the C-valued process (Rpp (2 (X 100,1,2 satisfies that
(Epn
= 1 (1 (e
e ,
(8)
and
(Enn Mu (Xx"" ( + +e (1+
- (1+ , (1+
(9)
By viewing as IR² and only consider the imaginary part of the above two processes of generality assume that
in (6). 7 and (Bpn (Xx"))))( in 8) ( we may without loss
= (10)
= (0,0,-1)
and
1 क
1 ) ))
(Epa (X")))((((""))](((((()))))())) (i, +
sin((1
n 2
(Epa (Mm (m (0, - sin((1nn + )
1 π
) sin((1 })
Now, adding the additional time component to the above real valued paths, and choosing M E C(R?, u(2) via
M( !) = [ :]. mc()p [ 0
we can easily verify that
[1] ))
(XB(((()) = exp(MM
# exp(MM
= lim Um (p (aas x n
and
(XB(((())( = exp(MM
0 [xpp(M
+ exp(MM 1 expp((M
= lim (177))))))
n00
16

---

where exp denotes the matrix exponential on C2x2 From the above calculation, we can further derive that
lim Ep (( (L 2-000
exp(M
2
1 exp(M
+ 2
* exp(M
1
2
1
() 1) ))spp(M) [-1p)
exp(MM (M)
+ 2
Ex E (M (F (=0,1.))))))
because the matrix multiplication is non-commutative. Therefore,
lim dus( 4xx (M, 4x (M) + 0,
which coincides with our observation that x cannot converge to X for the extended weak conver- gence.
A.2 A Brief Introduction to Path Characteristic Functions
In this section we will summarise some crucial properties of the Path Characteristie Functions (PCF) of 1ddvalued stochastic processes which were obtained in [19] and [18. and briefly mention its connection with the signature theory.
Recall that for E C ((o, I, and M E ((B", u(m)), the unitary feature Us (x) (also called unitary path development) of under M is defined to be yr U(m) with y being the unique solution to the following linear ODE driven by M (dx;)):
dyt y:M (dx;), yo Lm-
If is an Rd-valued filtered process with sample paths in C2-var((O, [[, Rd), then its path characteristic function (PCF) is given by the expectation of the unitary feature of X:
+ xM = (x(()x(X))
where M E C(Rd, u(m)), m E N.
It is easy to see that the PCF of stochastic processes is natural generalisation of the classical characteristic functions for Rd valuue random variables. Indeed, for an Rdd valued random variable X, we may view it as linear path from 0 to 1, i... X, tX for t [o. 1]. Then, for 1, the 1-dimensional unitary Lie algebra u(1) is simple the real vector space spanned by the imaginary unit i, we know that every linear mapping M L(Rdd u(1) can be represented by
M(x) = (a, A)i
for some À Rd and (..)) denotes the Eulidean inner product. In this case it holds that the unique solution y(w) the ODE
dy:(w) = y. M (,) Ajidt, vo(ww) = 1
is simply y: (w) exp(t(X(w)) (w), AWi), which implies that UM = yi yi(w) exp((X(w)), AVi) and consequently
Φ x (M) (x) - = Ex(exp((X, X)T)].
which is exactly the classical characteristic function of X evaluated at À e Rdd
Connections of PCF with the Signature Theory Given continuous bounded variation path
E C12-aa (0, T7, Rd), its signature S(xx) (see e.g. (6) is given by formal series in the (dual of the)
tensor algebra (( = II=o(dd (gd)@n over dd:
S(x) (1, (z) (x)).....
17

---

ea &.., E (R") where e...... are the canonical basis of Rd and denotes the tensor product. Thanks the universal property of the tensor algebra )) every linear mapping M E L(R", u(m)) can be lifted to an algebra morphism M T(RR)) + Cmxm (where T T(IB)) is equipped with the tensor product, and Cm is endowed with the matrix multiplication).. can be shown that (see 18), (9) the unitary feature (x) is equal to the composition of M and the signature of æ. i.e. UM (x) ())))).. Moreover, the classical signature theory (see e.g. (7)) also tells that the signature S(x) belongs to the character group of ((R))) with respect to specified Hopf algebra structure. This algebraic property of signature together with the relation that Um (x) M (S(x)) does not only guarantees
where Sn(x) = E Ein..dh Joct <...
that MM (x) U ((m) takes values in the unitary group, but also the universality of unitary features of paths (see e.g. 19 Theorem A.8D:
Theorem A.3.
The linear functions unitary features are stable under multiplication and complex conjugation. More precisely, for any M1 E [(R", u(mi)) )). M2 E L(R", .u(m))), T, E L(C Xm,, (C) and 1.2 E L(C xma, C), there exist an M3 E L(R4, u(ma)), L3 L(Cmmamm,, C) an M4 E C(RR", .((ma) and Lg E =(0maxm C) such that
I (x:),,()))) (2(()M = (a(()Ms(x)));
and T, (((M),)())) T, (u (=))
Let KC C ((0 T), R2) be a compact subset. For any continuous and bounded functionn
f : K. + C and any > O, there is m. N and finitely many M1,-. MN C(R", u(m+)) as well as linear functionals Li....,Ly E C(U(m=), C) such that
N
[(x)
Σ II((Ms <E.
sup #EE
Clearly, the second statement of the above theorem follows immediately from the first statement together with the Stone Weierstarssss theorem for C-valued functions. Since the expectations of
continuous bounded functions determines the distributions on C1-var((O, T), Rd), as corollary of the universality theorem above, we obtain the following characteristicness of PCF as mentioned in Theorem 2.6. see also 19. Theorem B.1].
Other Properties of the unitary features and PCF Besides the universality and the characteristic- ness, in [19 and [15] one can find some other nice properties of the unitary featues and PCF, which we will list below without proof:
1. Since (a) takes values in the unitary group (mm) if M E L(Rd, u(m)), so the Hilbert- Schmidt norm of the PCF & M (X) of any stochastic process X (with continuous bounded variation sample paths) is always bounded by ym. In particular, P (X) can be defined for any stochastic process with no integrability requirement.
2. The unitary feature Um Cl-var (I0, T'I, Rd) - (m) is Lipschitz continuous with respect to the bounded variation norm, see 19 Proposition B.6].
3. If the laws of stochastic processes satisfies enough integrability condition (namely their expected signatures have infinite radius of convergence, see 7 for the definition), then one can use a special subclass of linear mappings M E L(Rdd u(m)) to determine the laws. More explicitly, for Px and Py two laws of stochastic processes with enough integrability, Px Py if and only if L Px (M) = Pp, (M) for all M e L(Rd, o(m) such that M only has possibly nonzero entries in M. with l 1-- jl
where My denotes the entry of M at the i-th row and j-th column, and (m)) is the orthogonal Lie algebra. Thanks to the significant sparsity, such M E L(Rd, ((m)) is easy to be implemented in the numerical application. See [15) for more details.
4.
The PCF induces metric which can (locally) characterise the weak convergence of the laws of stochastic processes, which is called the PCFD (see [9 Theorem 3.8)). In fact the HRPCFD defined the present paper can seen as counterpart of PCFD in the extended weak convergence.
18

---

A.3 Proof of Theorem 3.3
In this section we prove Theorem 3.3 a more general setting.
Definition A.4. For (M,M) E Aunitory with M E C(R"", ,u(n)), M E 2(C"x", u(m)) and p E & malsude-valued path, we call
MMsM(p) == =UM(t p(),, p, = (p) / Um(l))p.(z))
the high rank development of p under (M,M).
Definition A.5. For μ1 E (()) a probability measure on the measure-valued path space R, the function
$ Aunitary
C
[ pex UM (p)MMP))((p)) Sf Um - Pi
(M,
M pp(dp)
Spee
is called the high rank puth charucteristic fanction of (Abbreviation: HRPCF).
The next lemma straightforward, but will be helpful for us to construct the characteristicity for laws of measure-valued stochastic processes.
Lemma A.6. Let M (M; θi=1 L(TRA, u(j)) for some k N, Then, there exists an
M E C(R. u(n)) for (1 2 ..... k), such that for any measure-valued path p E and for
any E [0, TY, one has
/ (xr(
pdd
Sx Um (d)))
Sx (M
Sx (x)
E C
Proof. Given an M (M + L(RR) u(j)), we define M Rd - u(n) for n
1+2 ..... + & via
(1(x) Eu())
M2(x) E u2)
((x)
(10)
Mx(x) u(k)
which is obviously a linear mapping due the linearity of MI...., MM. For any Rd-valued path æ E X, we know that its unitary feature (x) is the unique solution y (evaluated at time 7) to the linear differential equation
dy: = (dxx,, yo = In.
On the other hand, let ze be curve in (nz defined by
(1) U1))
22(1) U(2)
2,
(t E (k)
where j = 1,.., & the unique solution to the linear differential equation
dy: (x yo 1,.
19

---

It is clear that x satisfies that
rdz (t)
dzz(t)
dz
(zx(t)
[2 (t)
M2
(t) (dxr))
(t)
M1 (dx,)
x2(1)
M2(dx;)
(1)
(x(d
M, (dx:).
Hence, by the uniqueness of the solution to the differential equation dy: = yc (dx:), and invoking that zj (T') = Uns, (x) for all j = 1,...,K ....,,, k, we must have
Uus, (x)
Uy, (z)
Uys = zT
Uare (x)
Now it follows immediately that
J (x.()))
Dt
[x Um, (dx))
Sx Ung (d))))
Sx UM, (d)))
Theorem 3.3 follows immediately from the next lemma by inserting μ = Px and v Py for prediction processes X and Y of filtered processes X and Y, respectively.
Lemma A.7. Let μ and v be two probability measures on measure-valued path space & (that is, ,, E P())). Then = v if and only if for every admissible pair of unitary representations (M, E Aunitary it holds that
& 4 (M,,) = (M,
Proof. Before we start rigorous proof, let first give an informal proof to provide some in- tuition: For each measure-valued path p (pt)te) L, first compute the PCF pMM (x) E ((m) for every / E 7, where M E L(Rd u(m)). By doing so, the measure- valued path is transformed to matrix-valued path MM in mm Thanks to the characteristic property of the PCF (see Theorem 2.6 each measure is represented by its PCF p,M therefore may study the matrix-valued path p M instead of the measure-valued path p. Under such iden- tification, the distributions, y and push-forward measure pj" and p M
the measure-valued path space & can be represented by the y respectively, which are distributions on the matrix-valued path is equivalent to showing that P μl P. v. But now using the
space. In other words, showing characteristic property of the PCF again, 2f = p holds and only their PCF under linear operator M E L(CmXm, (un))) coincide with each other, i.e., P (M) T pp (MM, and by
20

---

definition one has $ pp (M) = (MM, & (MM = (M,
Now we provide the rigorous proof of the theorem. Obviously we only need show the "if" part. Step 1: By hypothesis, for any admissible pair of unitary representations (M, Aunitory with M E C(R" u(n)) and M E L(C"x", (((")",u(m)) we have
(M, = S Un(t pt )lddp) - [ DEX
Un p()(p) (MM,M)
!PE
which means that & (pp-sM))(()) (M) - 1 (p-pp) (es) (M), where the push-forward measures (p + p²);; (()) and (p- + pd (a()) are probability measures on the -uxu-valued path space.
In fact, if we fix an arbitrary E N and an arbitrary M E C(R"" u(n)), and let M E [(C"X" u(m)
vary over all m N. we actually have the above equality & ((p-)p")x(00) (MM $ ((p-pp (M)
for all M E C(RR", u(m)), m N. Therefore, by applying the characteristicity of PCF of measures on finite dimensional vector space valued path spaces, see Theorem 2.6 we obtain that (p + p (()) = (p + pd1 (+()) for any n E N and any M E u(()).
Step 2: Fix an k E N and sequence of operators M = (M3) E D)= ((RR, u(])). Let n 1++2
k. By Lemma above, there exists an M E C(R u(n)) such that for any
p E X. one has
J x (x)())))(de))
Pi M
Sx Un, (dx))
Sx UMs (s))p((a))
Sx UM, (x) (d))
Now we take an arbitrary partition {t] < 12 <... ty} of the time interval [0, T]. Since we have shown in Step that (p p (+(()) = (p pd )=()), that is, the law of Cax*-valued stochastic process p,M = -x (ar E P())), we indeed have that the distributions of their marginals at fa....,tyy are same, that is,
( (dx). t E [0, T] under μ P(2) coincides with its law under
(p (p ..... Pix )au (p + (p.. M
Piy
M DEu P((CCXM)M)
Now, for each i 1, ,..., N and = l,..., k. pick arbitrary linear functions (, L(CIX) R) and continuous and bounded functions E (R). and use them to define function ğ: : Cuxn + R for L....,A N such that for any matrix A E Caxn (recall that n 1 + 2 form
2 ....... + k) written the
A C1xl
A E C2X2
*
A =
*
*
A E Ckxk
*
*
it holds that
(.(A) 9i o
Σ 1,
Obviously each function 9, is continuous and bounded.
Let g : (C=Xn)N + be the continuous and bounded function such that g(A), "..",...,44 AA = IN - (A') for every sequence A = (A'... (A'....AA) (C"XN)N, From the equality (p + PM. 1) P(( it follows that (p). M M 1a = (p- (p .p......,ppp PM
/ ((A)(p- + (p..... ...., P. )au(()) = 3(A)(p + (pM.... .pM.....PM)))) ((d(dd)
21

---

which can be reformulated as
J 1 gi Σ Ep. j=1
h
(L)(1) o uM, (ldd)= )
(11)
N
Σ Ep. [L, o u((d))
gi
where p. [L)() UMs,]
Smex L5(1) o ( (d)r).
Step 3: is a well known fact (see e.g. (7]) that the vector space generated by all real-valued linear
functionals of unitary representations on the path space X, namely
C = span o Mas : X - R: L(C))), R), M L(RR" u())), EN,
a sub-algebra in the space (a() of continuous and bounded (real-valued) functions X' which separates the points. Moreover, by picking Mo I u11) be the trivial representation (i.e., Mo(x) 0 C for all x IR we that for any path æ E X, MM(x)) 1 R. Therefore, by the Giles' Theorem (8 Theorem 9I) it follows that the set C is dense in (a())) related to the so called strict topology
Now, fix arbitrary continuous and bounded functions f. E C,(X), = 1,..., N. From the den- find sequence of unitary representations AT(()) (M )=
sity of C in Ca(2))
D k 1=1 C(R., u(j)), l E N together with sequence of linear operators (i))AEN, i I, with each (t) (L) (1))= E H5=1 [(()x), [R) such that for every i I,..., N it holds that
Σ 1 (1) o M
f: lim k-000
(12)
where the convergence happens in the strict topology. Furthermore, since every probability measure Pt; E ((X)) (i 1, ...... N) belongs the topological dual of (x())) equipped with the striet topology by the Giles" theorem, invoking the relation 12 we actually obtain that for every i = I.....N. N.
Ep, [fi] / f((x)p, lim k-000
Σ
E.,.
[1)h) (1) u
j=1
Then, as consequence of the result theorem to get that
(I) obtained in Step 2, we can apply the bounded convergence
N
E [fi))((d))
lim 100
II
Ep.
(1) Mygeol p(dp)
i=1
1=1
N
lim k
II 1=1
Σ Ep., [I, (8) (1l))ll.a oou uldp)
/ II g (T,, (dd))
(13)
On the other hand, by the Urysohn's lemma, for any i 1,... N1, any positive number R; > 0, the indicator function Rj. can be pointwise approximated by sequence of [0, 1]-valued continuous functions (gf kEN Hence, by replacing the functions 9: by of in (13) and then letting & - co, using the bounded convergence theorem we can derive that
f I (fl))(dpp) { (Ep, (dd) 1=1
For the definition of the strict topology, e.g. 8 Definition 8].
22

---

or, equivalently,
Je II 1o (- (Pt, (((p)) 1=
N
N
II 1 0% R.)) (Pr, (((p))
(14)
i=1
where 8 (Pt,) == Bp. [fi] denotes the evaluation map of f: E C2(2) against the measure Pt.
Step 4: By the very definition of weak topology on P(X). its Borel o-algebra is generated by the sets of the form that 07 ([- Rl) for f Cs(2') and It > 0. Consequently, the Borel o- algebra the product space P(x) X is generated by the measurable rectangles of the form that II 07 (1 Ri, Rel) for J: E (x()) and R. > O. From Eq. (4) we know that
p((P PIN) E II % 7 (.,,,, ((Pr............................ PeN) Of R:))) i=1
for all such measurable rectangles. Since the above equation holds for any partition {41 < ... IN of (0, 7) and the laws (continuous) stochastic processes are uniquely determined by their marginals on finitely many time points, we can conclude that f = y in (P)) by routine application of the monotone class theorem.
Now, for filtered processes X and Y, we note that the associated prediction processes X and y are stochastic processes taking values in (x2) which can be viewed as X'-valued random variable, which in turn implies that their laws Px and Py elements in P(2)). Hence, inserting μr Px and v = Py into the above Lemma A.7 we can easily deduce Theorem 3.3
A.4 Properties of HRPCFD
In this section we will mainly prove the properties recorded in section 3.3
First let us prove the property of HRPCFD on the separation of laws of prediction processes. To achieve this we need the following useful continuity lemma.
Lemma A.8. For any fixed μ P(()), any fixed n and m, the mapping
(M,M) E C(R", u(n)) x L(Caxe, u(m)) + (M,,) E mm
is continuous for the operator norm topology on C(R),u(n)) x L(C"x" (m(m) and the Hilbert- Schmidd norm topology on mm
Proof. For admissible pairs (M, M) and M'M from ((R4,u(n)) x L(Caxn ((C"x",u(m)), by the definition of HRPCF we have
/
| (M,,) " (M M =
Um <M(ttpp))pee p(((p) - Unr(t p. )u(pp) )
PEd
DEE
HS
Um(t P.N) - -u -.
lldpp)
pet
11S
um((+ PM -un(1 MM
u(dp)
HS
/ | |unlle + p. Un + M
u(dp).
(15)
HS
Let us first estimate the first integrand on the right hand side of (5) By 19 Proposition B.6] we know that for each measure-vas valued path p E &, one has
[[ +
(MM(t+ppM)) PM un + M 1|s < [|Mlllop1z M -D'' ||i-arr
where P pp, = Ux (dd) and p = p. (M') = S Unr (z)) (dx) are Cuxm_ valued paths. Since p E & is piecewise linear, these -axu-valued paths p + 4 (Md) and
23

---

pd = + D (M)) are also piecewise linear, say, they are linear on time subintervals [&,, 1441 Then we indeed have
for ..... 1.
N
1p pd
Mr
11(p
p M' ) ),, ls S
22>n i=0
P
M' Pt.
1|ss.
whence the estimates
N Σ
[um(t + ) UM P Mlss & MM
lpp - Pt,
Ilss.
(16)
1=0
Now we note that for each i = .,...,N,, we have
Uas (a) (dx)) [ JEE
Pi
Uus (2)) (dx)).
JaEX
Recalling that for each vd--ul path æ E X, one has Uas = yr where y M. and MM, M'." are the unique solutions to the linear ODEs
M. and Uxr VT
dy M Ma
Yi
M (dxx),
Yo
M.
In
and
dy: M' M' "M ('dx))
Vo M' In
Yt
respectively, by the continuity the flow of ODE (see e.g. III Theorem 3.15),, we obtain that for any x E EX
[[ - Un (x)) S C(n, var)| M'llop
In particular, if ||M' - M llop + O, then for all æ E X we have [| (x) - -urr( + 0. Then because Uus and Uys are unitary representations taking values the compact group Un)) by M'
the dominated convergence theorem we have llp Pt, |l - 0 as ||M Mllop - 0 for any
i N. As result, in view of (16) we obtain that
M Mlpp
[1 + p Um(L p Ilss
for p E X. Then, as UM is unitary representation taking values in the compact group U (m), by the dominated convergence theorem again we have
|M Mllop - 0 =
Um(t + p..)) Um(t + M'
(xdp)) - (17)
PEA
HS
Next we turn to bound the second integrand in 15 namely 11 + P.
Usr + p. Ills-
Again, invoking that Ut - p,M") y solutions (evaluated at 7) to the linear ODES
M. and uMr +n p.l vr
M' x the unique
dy M
M.p
M(dD."), yo M.p. Im
= y:
and
M' pp
M.M. M M'ddp.").
M'''" Im
dy. respectively, by the continuity of the flow of ODE, we obtain that for each p X,
[[M + p M' (t p. M' ll s C(m, Ip"" II MM'llop-
Since Uy takes values in the compact group U' (nn), it is easy to see that for piecewise linear path p.M =f (ahrr(w))) (x)p, (d(x)) holds that suparec(Rd..) u(()) E X and for any M' E L(Bdd u(n)),
II || 20, which implies that for any
IIM - Ml + + [M pM Urr(t p lnn + 0.
Again, since UM and UM are unitary features with values in compact group U(m), by the dominated convergence theorem we must have
S |un(e- pp') Uur (.
s(dp)) 0
(18)
HS
24

---

as long as M| - oM + O. Now, combining 18
and 13
we can conclude that
| M, (M MM' M''
as long as | - Mlop + 0, MI - -Mllop + O. which is the desired continuity claim.
Now we are able to prove the first property of HRPCFD.
Theorem A.9 (Separation of points), Let μ. V E P(x)) be two distributions on measurevalued path space such that Ps + v. Then there exists pair of integers (n,m) N such that for any PM E P(C(R", u(n))) with full support and any PM P(C(C"Xn, (P(C(C"X",,(m)) with full support, one has
HRPCFDM (H(,)) >
In particular, for filtered processes X. and V, if they are not synonymous, then with Jr Px and Py there exists a pair of integers (n, m) N2 such that for any PM P((((R u(n))) with full support and any PM E P(C(Cuua u(m))) with full support, one has
M2 Y)> 0.
Proof. Thanks to Lemma A.7, if u # then there must exist admissible pair of unitary represen- tations (Mo, Mo) E Aunitary with Mo E L(R u(n)) and Mo E L(C"X" ((m)) such that
(Mo, Mo) + " (M)
Then, by the continuity result proved in Lemma A.8 there exists 8 0 such that for all M L(R", u(n)) and all M E L(C"ka u(m)) with Mp Mllap I and [lMo Mllopp S 8, it holds that
dust (M, T (M,
Let BMM 8) c L(Rd, u(n)) and (MMo 8) c L(C"xM, u(m)) denote the ball centered Mo and Mo with radius 8 (with respect to the operator norms) respectively. Then if PM and PM have full supports, we have PM (B(BB.. d)) > 0 and PM 6)) > 0, which implies that
/
HRPCFD M.RCFDM.M M.M
M.M(M, (U,D)
(M,
( (M
B(Mow8) B(MM..))
claimed.
The boundedness HRPCED is easy to show by using the same arguments as in the proof [19 Lemma 3.5] for PCFD.
Lemma A.10. Let μ, E P(X) be two distributions measure-valued path space. Then for any given integers (n,m) m.) E N2, for any PM E P(C(RR u(n))) (P(((),,(())) and any PM E PP(C(Cnn, one has
HRPCHD S 2
Proof. By the triangle inequality, we have
HRPCFD M.M.M.MMMM m(,u)) = / d ( (M, PMM
/ /
+ MMM,M PPMM
Since (M,,)) = I Un(t + P.M (u(d)) and Um takes values in (m)) such that [ =
Vtr(Unll(H)= = (Im(In)) = vm, we indeed have
(M, & (F [um(t P.M (lssu(pp)))) S vm.
Similarly, it holds that [ S Vm. Combining all above together we can deduce that TRRCFD?MMM < 4m.
D
25

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Just like the classical PCFD (cf. [19 Proposition B.10J) we can also show that the HRPCFD is specific Maximum Mean Discrepancy (MMD). For the definition of MMD, we refer readers [9 Definition B.9].
Proposition A.11. For any m) E Nº. any PM E P(C(RA u(n))) and any PM E P(C(CuX* u(m))), the HRPCFD with respect to PM and PM is an MMD with the kernel function R: ; & given by
k(p P) = PP PM [MM(1 + pt") ),), + M Yrs)
& PM (r (p (f"))-1)
where: * denotes the concatenation operator paths and (p² denotes the reverse of the path t + Pt.
Proof. For U, E P(Z), easy to deduce that
HRPCFD M. (,,( /
T (M, (MM
EPMMPM $ (M, -EPM@PM (M,
22 C ( M), (M,,
Moreover, note that
PM (M, MM P, = / unloM) / (un ((d)) @ PM
/ | (ump)) (HSK((P)) & v(dp)d(PM
@ PM)
/ (fumlD).umtth (Hsd @ PM)) u(dp) @ v(d)),
where we used the Fubini's theorem in the last equality. Therefore actually obtain that for the kernel
k(p, p) SP (M > P ) Pi )us)
holds that
Ailppppppp.. [ /
HRPCFD
k(p, ()d
which implies that RRPCFDM.M is MMD with the kernel function A.
The last claim is obvious: for any p. P E &, one has, due to the fact that every A E (m) satisfies A
A*, that
<(p. p) = PPM@PM [UM(t P, ,, + P Hs
(PPreMM(t(lMvg (t + p. )M + pb )")
EPMSSP [t((M P. MM + p. )
E [trlUu(p" (
where we used the multiplicative property of the unitary features for C"x"_valued paths M and MMM see also [19 Lemma A.5].
Now we will construct metric from HRPCFD which can characterise the extended weak convergence on precompact subset FP.
Lemma A.12. Suppose that ((PM,, PM..) P(C(RR, u(n))) u(m))) n E m N3 is a double sequence of distributions with full supports After a re-numeration we label them as sequence ((PM, PM,)) seN such that each (M,,M,) a random admissibe pair in Agnitary Then the following defines a metric on P()):
min(1, Hww.HD M,, (1. v)]
HR
2
26

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Proof. The symmetry and the triangle inequality are easy to check. We only need to show that HRPCFD( v) - if and only μ = p. The "if" part is trivial. Now suppose that HRPCFD((, v) = holds but + v. Then by Theorem A.9 we know that there exists pair of integers (n, m) E N
such that for any PM E P(C(RR ( and any PM E P[[[C u(m))) with full supports, holds that HRPCHDM.M(11 ([C(RR) u(n))) P(C(C"Xm u(m))) we must have M, > O, which implies mln(L,HRPCFDDr,.M,, (k.))} >
> O. So, let us pick some E N such that (M,.M;) E
that HRPCFD((, v) 2
0, a contradiction. Hence we obtain that the
2¹
so-defined HRPCFD( v) is really a metric.
Theorem A.13. Fix a sequence of random admissible pairs (M,, Mj)JEN C Aunturr such that for every m) E N2 there exists j E N with M, E _(R u(n)) and M, E L(Cnxx u(m)) and their distributions PM2 E E(C((RR) u(n))) and PM; E E u(m))) are fully supported. Let HRPCFD be the metric defined as Lemma A.12 via this sequence (Mj;Mj)jes
1. Let K. C FP be compact subset in the space FP of filtered processes equipped with the topology induced by extended weak convergence. Then, for every sequence of filtered processes (x* (SF Fk, Fk, X ppt JeEN C. K. and X (,, FP, we have
X EW X + HRPCFD X) -00
as > 30.
2. Let K C X be compact subset. Let FP(K) be the space of all filtered pro- cesses taking values in K. Then, for every sequence of filtered processes (X (St Fkk ,,,,,, x* x*,IP*)))ee )REN C FP(K) and X (2, X, P)) E FP(KK, we have
Xh EW % > HRRCFF ( X) 0
as - 00.
Proof.
1. First suppose that x EW, X for sequence (X* REEN C. KC and X E K. Clearly, for any sequence of piecewise linear measure-valued paths ph k E N and (which are linear each subinterval [ē, í 1]. i = O,..., T 1) we have pd - p with respect to the product topology on & implies that for each fixed unitary representation M e C(R", u[n)), the C=xx-vvlued paths (p*)M = (SuM (x)))(( (dx)). converges to Dd = (f (2)r(()) (dæ)) with respect to the total variation norm as k: + co. Then, for every fixed unitary representation M L(C"xa, u(m)). by the continuity of unitary feature map Uy relative to the total variation norm (see e.g. [19 Proposition B.6]), we have (t + (p*) (p*) - Un + M. in C (relative to the Hilbert-Schmidt norm) + xc Hence, we actually have shown that the function p X + Um(t p.
mm is continuous and bounded for the product topology on d.. Now, X
EW X
means that xx* - Px weakly in (()) k + oo, we indeed have for all MM, Aunitary.
lim f [un(t P ) (dxp = / Unilt (M) PM)
M)
(+ -aa00t=uulttt
1=+00
that is, lim L (M, the boundedness of the HRPCFD (see Lemma A.10). allows us to apply the dominated convergence theorem derive that for every j E NNNN one has
2 (M, II HS = 0. This observation together with
HRPCED Xx
= - dis & MM, D (M) (dM) (dM) +
as + xo. Consequently, we can conclude that HRPCFD x x) - as h: - oo. Conversely, suppose that (xx keN a sequence of filtered processes in K. and X FP such that limg HRPCFD & X) 0. Since XC is compact, there is subsequence of (x*))EN keN
27

---

(without loss of generality, assume this subsequence is the sequence itself) converging to limit Y (2², .g. G, Q) E K in the extended weak topology. From the previous argument know that limg->oc HRPCED(X* Y) O. Therefore, actually obtain that HRPCFD(X, Y) = 0. In view of Theorem A.9 the equality HRPCFD(X, Y) 0 means that Px Py. i.e., X and are synonymous. The above reasoning reveals that any accumulation point Y of the sequence (x*))e in the extended weak convergence coincides with X. As consequence, we have X + in the extended weak topology as k co.
2. By 4 Theorem 1.7] the subspace FP(K) is precompact in FP for the extended weak topology, if K C X' is compact. Hence the claim follows immediately from the result contained in the statement with K. = FP(K) (the closure of (P( with respect to the extended weak convergence).
B Methodology and algorithm
B.1 Estimating the conditional probability measure
Algorithm Training algorithm seq-to-se regression model
Input: X real data; B batch size; ?r learning rate for the regression module; d path feature dimension: 7- lie degree; M E Ra w; T path length; 1), learning rate.
F iniitialize
2: for i .al....,iter,, do
3: Sample x: from X of size B
4: (t. &- Uar (T),)t)))) with E (0....,T].7 (1..... B)
5: RLoss(0; M) + BIT+IJ Et= E/= |1 (Z),),))) - - (t)ll
6:
0 +-0 --01-- Va (R(Looss(O;21,))) M))
7: end for
8: return FX
Return the optimal model
Algorithm 2 Sampling algorithm approximate Φ2
Input: F2 regression module; X (xx,) data sampled from distribution Px: fr learning
rate for the regression module; path feature dimension; n, m Lie degrees; M E RRdddlm u M IR U(n) Xdim ((m); I path length.
1: W.K.M
matrix of length N (T +1)
2: for t E (0, (O...., ... do
3:
for jrj (1, (L....... N) do MjsM. + Um (1,,10.4) Fx (2,,0,T))
4:
5:
Mg
8.N +- MM (1) MMJ.M.uummn (t)
6:
p j.t
7: end for
8:
end for
9: (M, MM - N MM
10: return $2(,M
B.2 HRPCF-GAN
In this section, we provide the mathematical formulation of HRPCF-GAN for conditional time series generation. Let X (X2) denote a va-uue time series of length with distribution Px Suppose that we have i..dd samples X (æx); from Px. We interested in generating synthetic
28

---

future paths to approximate the conditional distribution of the future path Xiutrre := (p.71 given the past path future == (o.))- For ease of notations, let Xpast Raxp and Xunure (T denote the space of the past path and future path, respectively.
Conditional generator One step generator Go Xmaa 2 + Rd, which maps (Apaat 2x) to samples of the next time step via the following formula:
Fe, (#past)pp
Fo, (h,
Fo. ..pust H. is the seubence-to-squennee embedding module to extract the key information of the path up to time and Fa, exhibits the autoregressive generator architecture. denote by where 8 (fe, 0a) the generator's parameter.
We then apply one step generator in a rolling window basis to generate future time series length
T p. More specifically, Gg (F(0:p) (2+)==+1)) + (oe) where we first set 00.p Jopp and for every 2 p. o++1 90(0t-p:
In the following, we summarise the training algorithm for HRPCF-GAN in Algorithm
Algorithm 3 Training algorithm for HRPCF-GAN
Input: p- past path length; T total path length; path feature dimension; X real data; lie degree for EPCFD; K1 number of linear maps; M E R xdx 4(x); m lie degree for EHRPCFD; K2 number of linear maps for EHRPCFD; M E IR Kg diin "(a) Xdim u(m); Gg generator; batch size; z noise dimension; iter, frequency of regression module fine-tuning; 7/. 1a generator and discriminator learning rates.
1: Vamilla PCFGAN training
2: while 0, M not converge do
3: Sample ~ A (0, of size B, sample from X of size B
4:
(p.T7] <- Go
5: Loss(0, M; 2) + EPCFD (*)0,)) ( *(nT)))
6: M M MM - Ma V
M;
Maximize the loss Minimize the loss
7:
0- " (o M; a.
8: end while
9: # Regression training for real measure
10: for i (1...., K1) do
Freal iinitialize
Train Freal using X described in Algorithm)
13: Ffake F
b Set as the initialization
14: end for
15: High-Rank PCF-GAN training
16: while M not converge do
Sample 2 ~ ~ N= (0, 1) of size B, sample from X of size B 18: (p.17 -- (G(
17:
for i (l....,K) do
20:
for L (0. T) do
21:
Ph. M. Usi (z) Freal (arpo.))
22:
P.t
M. +- UM, (310..)
F ((F #(,T))
23:
end for end for
24:
25:
Loss( M:x, 2, M) + EHRPCFD WARPCEDD... ((nr)))
b Use Algorithm 2 and
Equation 5 with p., Me and pi,t MM
26:
M +- A-M- na VM(-LLos(B, 2, M))
> Maximize the loss
27:
A с 0 7g (L(Los) M; M; 2, M))
28:
Do the following every iter, iterations:
29:
X + (a Go (X ((oop)..)) 2))
30:
Train Ffake using X as described in Algorithm for every i (L....,K)
31:
end while
29

---

B.3 Hypothesis testing for stochastic processes
We provide the following two algorithms for training ERHPCFD for the permutation test and computing the test power/Type error of the permutation test, respectively.
Algorithm 4 Training algorithm for the permutation test
Input: X samples from distribution μr: Y samples from distribution v/; m 0 sample size of X; n > 0 sample size of Y; lie degree for EPCFD; K₁ number of linear maps; M e R xdxdim ((n); m - lie degree for EHRPCFD; K₂ number of linear maps for EHRPCHD; M
R dim u(x) xdim ugm); B batch size; 1 learning rate; iter,, itery number of iterations.
1: # Vanilla PCFD optimization
2: for iter E (1,.... iter,) do
3: sample æ, y from X, Y of size B
4: Loss(M: xx y) t- EPCED (Mo.T)
5: M + M
V zo
Maximize the loss
6: end for
7: # Regression training for real measure
8: for i (1...., Ki do
9: FX b, inittillize
10: Train FX using X and M, described in Algorithm
11: Train F. using Y and M, described in Algorithm
12: end for
13: # High Rank PCFD optimization
14: for iter E (1,...,,tee) iterz) do
15: sample x. y from X, Y of size B
16: for E
K1) do
for & (0, (O....,,)) do
P.t .t
X.M. +- UM, (a))
FX (m)o.)
Y.M. UM, (H)))) (y)o.)
19:
P.t
20:
end for end for
21:
22:
Loss(M: xr y, M) EHRPCFD? M. (F(o,T) - Use Algorithm and Equation
with Pit X.M: and Pit
Y.M:
23:
M M Nd
>> Maximize the loss
end for
25:
return M.M
> Return learnt parameters
30

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Algorithm 5 Estimating the test power/Type-I error of the permutation test
# of experiments, M 0 of permutations: samples from distribution Y samples from distribution m > 0 sample size of X; n >> 0 sample size of Y: T test statistic function; Ho {1,0} whether the null hypothesis true false
Input: (0, 1) significance level;
(Ho = if μ otherwise Ho 0)
1: Z Concatenate Y)
2: num rem__reectioos- +
3: i<1
4: while - N do
5:
T < EmptyList
6:
j 5-1
7:
while M do
8:
~ Permutation( [1, ... m m²)
9:
T, --(((Zo()) (2)) Zo(m)) (zomm))):" Zo(m+n)))
10:
T.pppend(T.)
11:
jtj+1
12:
end while
13: if TXX, Y) > (1 - a quantile of T then
14:
num_rejections num_rejections
15:
end
i+i+1
17: end while
ratio num num_rejections)
19: if Hu then
Type _error + ratio
21: return I
else
23: test_power ratio
return test_power
25:
end if
C Numerical results
Code The code is written in Python 3.10.8 and Pytorch 1.11.0. The supplementary code is available at htttt::://// /tttss://iitubb
Soom/Deeplae Seeon/DeepIntSSiii www.Deeepptteeaaeeeeee IntoStreamss//.eooooooo.. harsstrrrmm/HHgh--eaa Saweaaams-shhhhaa---aSSaS- SACCFGGA..gg MANNgitti for ensuring full reproducibility. The experiments were performed on computational system running Ubuntu 22.04.2 I.TS, comprising five Quadro RTX 8000 GPUs with 48GB of memory each. The experiments are run on single GPU and the training time ranges from minutes to 4 hours.
C.1 Hypothesis testing
Description The permutation test is statistical method used to decide whether measures μr, are the same. The null hypothesis states Hg : whereas the alternative hypothesis H1 p F v. Given a metric T and sample data X {æ1,.... (n)), Y (, from μ and respectively, We construct the following distribution
T
T(Z T(Zo()))(m)) ():((()))mmm))) a Eatm
where Z (X, Y) and +m is the permutation group of n + elements. Given the significance level a, we reject the null hypothesis if ((X,Y) > (1 (1-aa) a)% quantile of
Methodology For each H. we sample the training dataset Dirain = (Buain BH and optimize the set [MK,,Mgz to maximize EHRPCED" between the pair of measures, a detailed procedure can be found Algorithm 4 Then, we sample two independent sets DMl = (B D D = (Brest B3 and calculate power and type-I error accordingly. We refer Algorithm 5 for the computation of test metrics.
31

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14
Ho H1
Ho
H1
Count
0.01
0.62
0.02
0.03
0.34
0.05
EPCTO"OX, Y1
EMARCPD"1X
Figure 5: Distributions of EPCFD (left) and EHRPCFD (right) under H, and H1 with Hurst parameter H = 0.475. The distribution consists of 100 runs under both hypotheses. For EPCFD, fix K₁ = and = 5. For High Rank PCFD fix K1 1, 3, K2 10, = 13.
Implementation details We provide the full details of the implementation the numerical exper- iment in Section [51] Adopting the notation in Algorithm 4 fix = m 13, K₁ 1 and K₂ = 10, these values are chosen via hyper-parameter fine-tuning. The regression model consists of 2-layer LSTM module. The model's 's parameter optimized using Adam optimizer with learning rates 0.001 (for regression) and 0.02 (for EHRPCFD).
Additional numerical results We provide comprehensive tables for summarising the Type-I error and the computational time involved Section 5.1
Developments
Signature MMDs
Classical MMDs
High Rank PCFD
PCFD
Lincar
RBF
High Rank
Lincar
RBF
0.4
0.04 0.04
0.04 1: 0.04
0,06 1: 0.05
0.04 1: 0.04
0.09 1 0.08
0.07 0.06
0.04 =: 0.04
0.425
0.07 0.04
0.08 0.07
0.03 0.03
0.04 + 0.04
0.14
0.01
0.03
0.45
0.06 0.05
0.08 0.02
0.05 1: 0.04
0.02 1 0.03
0.10 0.07
0.04 1 0.04
0.06 = 0.06
0.475
0.04 = 0.04
0.02 0.04
0.05 0.04
0.07 = 0.06
0.12
0.05 0.004
0.03 0.04
0.525
0.07 0.04
0.09 0.07
0.03 0.03
0.04 0.02
0.02
0.02
0.01
0.55
0.05 0.03
0.03 0.044
0.07 1: 0.06
0.05 0.05
0.17
1.004
0.02 = 0.02
0.575
0.06 = 0.04
0.04 0.04
0.02 0.03
0.06 0.07
0.07
0.06
0.06 0.05
0.10 0.07
0.06 0.04
0.05 1 0.04
1 0.04
0.06
0.06
Table 3: Type-I error of the distances when H * 0.5 in the form of mean = std over runs. For
PCFD, fix K₃ and 5. For High Rank PCFD fix K₁ RBF signature MMD and classical RBF MMD, fix 2n 0.1. For High Rank signature MMD, fix
3, K2 10, 13. For the
σ1 = =0
Developments
Signature MMDs
Classical MMDs
High Rank PCFD
PCFD
Linear
RBF
High Rank
Lincar
RBF
Interence (seconds)
10
3.17 0.02
2.38 2: 0.01
92.12 0.21
122.9 2 0.32
1211.6 12.41
0.13 1 0.01
0:28 2: 0.03
50
$2.32 4: 1.53
21.14 1.04
402 69 + 0.23
533.43 di 0.33
0.58 4 0.04
1.17 4: 0.16
100
111.25 2.92
89.13 2.19
1929.73 0.57
1780.18 0.29
1.16 0.04
2.64 1 0.11
Mini batch size
Training time (seconds over 500 iterations)
1024
695.18
8.57
73.51 6.21
Table 4: Inference time of the permutation test across different sample sizes (y = n)) and the training time of High Rank PCFD and PCFD before conducting the permutation test. The result is in the form of mean + std over runs. For PCFD, fix K₃ = 8 and n = 5. For High Rank PCFD fix K₁ = 1, 3. K₂ 10, m 13. For the RBF signature MMD and classical RBF MMD, fix 20 0.1.
Here fix h = 0.45.
32

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C.2
Generative modeling
Datasets construction (1) 3-dimensional fractional Brownian motion: we simulate samples using the publicly available Python package fbm. The total length of each sample is 11 (counting fixed initial point). The training and test data consists of two independent sampled sets of size 10000. (2) Stock: we select 5 representative stocks in the U.S. market, namely, Apple, Lockheed Martin, J.P. Morgan, Amazon, and P& G, and collect the daily return data from 2010 to 2020. The data collection is done using the Python package yfinance. We then construct the dataset using basis with length 10 (two weeks in real time) and stride 2. Finally, we split the dataset into training and test sets with a ratio of 0.8.
Baseline We compare the performance HRPCF-GAN with well-known models for time-series generation such as RCGAN [0) and TimeGAN (33 Furthermore, we use PCFGAN [19] as a benchmarking model to showcase the significant improvement by considering the higher rank development as the discriminator. For fairness, we use the same generator structure (LSTM-based) for all these models.
Conditional Generator The generator design is described in Appendix B.2 In particular, we choose Fe. and Fe.. to two independent 2-layer LSTM modules. The first module takes the past path and encodes the necessary information to the latent space. The final hidden and cell state will be used as the input for the second LSTM module and the latent noise vector to produce the output distribution of the next time step. Also, use the auto-regressive to simulate the future path recursively.
Implementation details The training procedure is described in Algorithm B] adopt the same notation in this section. For both datasets, we set T = 10 and = 5. We use the development layers on the unitary matrix [18] to calculate the PCFD distance, in particular, we fix K₁ 5. 5.
K₂ = 10, m = 13 for the discriminator design, these are obtained via hyper-parameter tuning. The regression model consists of 2-layer LSTM module. Finally, to train both the generator and discriminator with learning rates 0.0001 and 0.002 respectively. We fine-tune the regression every 500 generator optimization iterations. To improve the training stability of GAN, we employed three techniques. Firstly, we applied a constant exponential decay rate of 0.97 to the learning rate for every 500 generator training iterations. Secondly, clipped the norm of gradients in both generator and discriminator to 10.
the ADAM optimizer [12].
All benchmarking models are trained with 15000 training iterations. For HRPCF-GAN, we trained
the vanilla PCF-GAN with 10000 iterations then we switched to HRPCF discriminator and trained the model for further 5000 iterations.
Evaluation metrics
We list here the test metrics we used for generative model assessment.
Marginal score [6] the average of Wasserstein distance of the marginal distribution between real and fake data across each dimension.
Auto-correlation score 16 : the 1 norm of the difference in the ACF between real and fake
data
T d
ACF(X,Y) = ΣΣ
- Ö(: -
r=1
where Č(T; X) is the empirical cnto-coorreltion estimator of X, and X
Cross-correlation score [16]: the 4 norm of the difference in the correlation between real and fake data across each feature dimension.
T Σ 8,t=l 1
Corr(X,
X P(Y r.>
where is the empirical correlation estimator.
Discriminative score 23 we train post-hoc classifier to distinguish real data from fake data. Lower the score (absolute difference between classification accuracy and 0.5), meaning inability to classify, indicate better performance of the generative model.
33

---

Predictive score [23]: we train sequence- -equeencc model to predict the latter part of time series given the first part, using generated data and real data, resp. The trained models are then tested on the real data resp. The lower loss (ITSTR-TRTER means the better resemblance of synthetic data to real data for the predictive task.
S score [66] by embedding the time series the signature space, approximate the W1 distance by the l2 norm of the signature of the real and fake data.
Sigw; (X, Y) [[x (Sig( (X (o.r)) - By [Sig(X (x.(()u.
where Sig denotes the signature transform of path.
Conditional expectation score: estimate the conditional expectation of the future path the fake measure via Monte Carlo and compute the averaged pairwise l2 norm between real data.
Outgoing Nearest Neighbour Distance score 14 the ONND calculates for each example real data the distance between the nearest generated data. This score tests the model's capability to capture the diversity of the target distribution.
American put option score: we use Least-Square Monte Carlo method [77 to price an at- at-the-men thnym American put option using both real and generated data, We set the strike date T = 5 days and risk-free rate r 0.01. The score is computed the average of I differences of the estimated price across each stock.
For each test metric, lower value indicates better model performance. We provide the results on the additional metrics Table
Dataset
Test Metrics
RCGAN
TimeGAN
PCEGAN
HRPCE-GAN
Marginal
.0101.000
.041#.000
.0071.000
.0051.000
IBM
Predictive
.456:..004
.6862.013
.4741.003
.4462.002
ONND
.622+.002
.632..002
.654+.002
.622..002
Marginal (1+)
1.181=1.14
.6261.117
.4761.146
.2721.122
Stock
Predictive
.010#.000
.009=.000
.009:.000
.009
ONND
.017#.001
.017#.000
.016:.000
.016#.000
Table 5: Performance comparison of High Rank PCF-GAN and baselines. The best for each task is shown in bold. Each test metric is shown in the form mean over 5 runs.
In all the numerical experiments of GAN training, used moderate matrix order (1 S 30) to achieve satisfactory results. Specifically, our experiments were conducted on a single GPU, with the training time for HRPCF-GAN ranging from 30 minutes to 4 hours. Although HRPCF-GAN takes longer to train compared to other baselines, the total training time is kept at manageable level, while the HRPCF-GAN consistently delivers better performance, We summarize the computation time of each of the models over 100 training iterations in Table 6
Training Time (s)
TimeGAN
RCGAN
PCF-GAN
HRPCF-GAN
fBM
11.21 0.28
5.98 + 0.35
15.63 + 1.31
31.96 2.92
Stock
12.48 + 0.31
7.39 + 0.65
17.33 + 1.36
34.52 2.72
Table 6: Time measurement over 100 training iterations. The experiments are done using a single Quadro RTX 8000 GPU; each experiment is repeated times, with the mean and standard deviation recorded.
34

---

Hea
Figure 6: Sample plots from all models on fractional Brownian Motion conditioned on the same past path. The thick red line indicates the conditional mean of future estimated by fake samples, whereas the shaded red area presents the region of std. The thick green line corresponds to the theoretical value for the future expectation and the shaded area shown corresponds to the region of theoretical std.
Figure Sample plots from all models on Stock dataset conditioned on the same past path. The thick red line indicates the conditional mean of future estimated by fake samples, whereas the shaded red area presents the region of tstd.
35

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42


Reviews:
[{"review_id": "S54pQLBAQT", "reviewer": "Reviewer_zBhD", "summary": "This paper proposes High Rank Path method, motivated by the extended convergence notion and the rough path theory, to generate (conditioned) time-series data. A new metric HRPCFD is introduced, and experiments are conducted for Brownian motion, GANs with applications in finance.", "questions": "See weakness.", "limitations": "NA", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 1, "strengths": "The paper is rigorously written, which introduces a new metric HRPCFD on the path-valued processes based on various ideas from probability theory -- extended convergence, rough path, signature... I checked most proofs, and they are correct.", "weaknesses": "Weakness and comments:\n\n(1) The paper may be too heavy for the Neurips audience (though I enjoyed reading it). It seems to be more suitable for a rigorous mathematical or statistical journal (e.g., Annals of Statistics). \n\n(2) Many proofs of the results (e.g., Thm 3.3) are purely measure-theoretical, and I think the authors may shrink some proofs to keep the idea concise. \n\n(3) The authors may want to explain why the proposed HRPCFD outperforms others (e.g., signature...) Is there any possible theoretical guarantee?\n\n(4) The authors may have a discussion on the computational efforts of the proposed method (e.g., computational complexity and running time). The path-space optimization (or signature-type methods) often suffer from computational efficiency.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the helpful comments. We address each question in detail as follows. \n\n---\n\n**Answer to Q(1) / Why this work fits NeurIPS** \n\nOur paper addresses the crucial problem of defining the computationally feasible metric on the law of stochastic processes to capture the extended weak topology, which is ubiquitous for measuring the sensitivity w.r.t. models in decision making problems.  This has significant implications in machine learning, statistics and probability, particularly in hypothesis testing and generative models for time series data. As such, our work aligns well with the scope of NeurIPS.\n\nBesides, we would like to highlight substantial numerical contributions of our paper, which might be overlooked. In this paper, we design an efficient algorithm for training HRPCFD from data and construct the HRPCF-GAN by using HRPCFD as the discriminator for conditional time series generation. Numerical results show the promise of our proposed HRPCF-GAN for empirical time series data. We will make the code repository publicly available upon publication, which would be a useful tool for generative models for synthetic time series generation. The concrete algorithms and description of the corresponding methodology along with the open-source code repository, should have their own interests in the machine learning community. \n\nOur work balances the theoretical underpinnings of the HRPCFD metric with the practical implementation of the HRPCF-GAN algorithm for synthetic time series generation. Given the diversity and interdisciplinary nature of the NeurIPS audience, we believe that the NeurIPS conference is a suitable avenue for this work. \n  \n---\n\n**Answer to Q(2):** Now we adopt the reviewer's suggestion to add an ``informal proof'' for this theorem by emphasising the core idea and the main steps of the proof in an intuitive way so that the readers can easily understand the whole picture of the proof. \n\n---\n\n**Answer to Q(3):** \nBy the definition of the extended weak convergence (EWC) we can see that it induces a genuinely stronger topology than the weak convergence (i.e., **the EWC always implies weak convergence** while the converse fails in general, see Example A.1). Moreover, the EWC induces the coarsest topology for which the value function in any decision making problem is continuous.  Since the **HRPCFD metrises the EWC**,  it is natural to see an outperformance of HRPCFD in hypothesis testing and generative modeling than any other metrics based on the weak convergence, e.g., PCFD, signature MMD.\n\n---\n\n\n\n**Answer to Q(4):** \nWe summarize the computation complexity and running time of our proposed HRPCF method as follows. We will add Table 1 and a discussion on the computational complexity in the revised version.\n\n1. *Inference time*. As we use the same generator architecture across different GAN models for a fair comparison, the inference time of the PCF-GAN is the same as that of other baseline models. \n\n2. *Computational complexity \\& Training time*. Let $d$ and $T$ denote the feature dimension and time dimension of the target time series data. The training algorithm of HRPCF-GAN can be divided into three parts: 1) Vanilla PCF-GAN training, 2) Regression training, and 3) HRPCF-GAN training. For each part, the training time complexity per epoch is linear in both $d$ and $T$ when keeping the hyper-parameters the same. However, the evaluation of EPCFD/EHRPCFD might be costly when using a very large matrix order $l$ of the Lie algebra. It could alleviated by employing the scaling-and-squaring method for an efficient computation for matrix exponential. Detailed complexity of this operation is discussed in [1]. \n\nIn all the numerical experiments of GAN training, we used a moderate matrix order ($l \\leq 30$) to achieve satisfactory results. Specifically, our experiments were conducted on a single GPU, with the training time for HRPCF-GAN ranging from 30 minutes to 4 hours. Although HRPCF-GAN takes longer to train compared to other baselines, the total training time is kept at a manageable level, while the HRPCF-GAN consistently delivers better performance. We summarize the computation time of each of the models over 100 training iterations in Table 1. \n\n| Training Time (s) | TimeGAN           | RCGAN            | PCF-GAN           | HRPCF-GAN         |\n|-------------------|------------------|------------------|------------------|------------------|\n| fBM               | 11.21 ± 0.28      | 5.98 ± 0.35      | 15.63 ± 1.31      | 31.96 ± 2.92      |\n| Stock             | 12.48 ± 0.31      | 7.39 ± 0.65      | 17.33 ± 1.36      | 34.52 ± 2.72      |\n| RV                | 12.75 ± 0.41      | 6.37 ± 0.59      | 12.94 ± 1.21      | 32.77 ± 2.05      |\n\n*Table 1: Time measurement over 100 training iterations. The experiments are done using a single Quadro RTX 8000 GPU; each experiment is repeated 5 times, with the mean and standard deviation recorded.*\n\n3. *Significant computation reduction of HRPCFD over HR-SigMMD*. The dimension of PCF is independent with $d$, and the same applies to HRPCF. However, in contrast, SigMMD's dimension grows geometrically with $d$, leading to dimensionality issues. The kernel trick can be used to alleviate the curse of dimensionality issue for the signature-based method. However, it is worth noting that the training time of SigMMD computed via the signature kernel is quadratic in terms of the sample size $n$ and time dimension $T$. In comparison, the computation complexity of HRPCFD is linear w.r.t $n$ and $T$. \n\nThe significant computational advantage of PCF-based methods over signature-based methods is demonstrated in Table 4 of the appendix.\n\n[1] F. Longstaff and E. Schwartz. A new scaling and squaring algorithm for the matrix exponential. SIAM Journal on Matrix Analysis and Applications, 31(3):970–989, 2009.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "2p2hj9POQQ", "reviewer": "Reviewer_gzqb", "summary": "The paper constructs a computationally-implementable metric which metrizes an \"extended\" weak convergence for stochastic processes, which more plausibly accounts for the convergence of the process with respect to their filtrations.\nThe result can apparently more effectively account for similarities between controlled processes, at least in the class of linearly-interpolated stochastic paths considered here.", "questions": "l101: The prediction process seems to be introduced on a fixed finite set of times — $I = {0, \\dots, T }$ and $X = (X_t)_{t\\in I} $ — and yet we are concerned with continuously-indexed processes, so I would more naturally assume I is the interval $I=[0,T]$, and in fact we discuss linear interpolation in l109. Is this a notational confusion? What is the index $t$ of the filtration $\\mathcal{F}_t$?\n\nTitle: I'm not sure about the grammar of the paper's title. \"High Rank Path Development: an approach of learning the filtration of stochastic processes\" -> \"High Rank Path Development: an approach _to_ learning the filtration of stochastic processes\"?", "limitations": "No particular issues noted.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The method seems to construct high-rank analogues of classic tools, such as the characteristic function, such that the prediction processes arising from projection onto the filtration at each moment can be quantified and divergence between them metrized, without density evaluations, using empirical measures.\n\nThe classical SDE reasoning seems sound. I confess that I am less familiar with the rough path theory component, but there are no obvious red flags in the material.", "weaknesses": "The paper is long;\nThe results seem to be an improvement both theoretically and empirically over the main antecedents\n\n* [18] Hang Lou, Siran Li, and Hao Ni. PCF-GAN: generating sequential data via the characteristic  function of measures on the path space. Advances in Neural Information Processing Systems,  36, 2023. \n* [19] Cristopher Salvi, Thomas Cass, James Foster, Terry Lyons, and Weixin Yang. The signature 392 kernel is the solution of a goursat pde. SIAM Journal on Mathematics of Data Science, 3(3):873–899, January 2021.\n* [20] Cristopher Salvi, Maud Lemercier, Chong Liu, Blanka Hovarth, Theodoros Damoulas, and  Terry Lyons. Higher order kernel mean embeddings to capture filtrations of stochastic processes.  Advances in Neural Information Processing Systems, 34:16635–16647, 2021\n\nBut it is not clear whether the increment is \"important\" in practice; Is the increased performance \"worth\" the implementation effort and/or computational cost? The answer is probably problem-dependent.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We address the reviewer's comments and questions in detail as below.\n\n---\n\n**Comments:** *The results seem to be an improvement both theoretically and empirically over the main antecedents*\n\n*[18] Hang Lou, Siran Li, and Hao Ni. PCF-GAN: generating sequential data via the characteristic function of measures on the path space. Advances in Neural Information Processing Systems, 36, 2023.*\n\n*[19] Cristopher Salvi, Thomas Cass, James Foster, Terry Lyons, and Weixin Yang. The signature kernel is the solution of a Goursat PDE. SIAM Journal on Mathematics of Data Science, 3(3):873–899, January 2021.*\n\n*[20] Cristopher Salvi, Maud Lemercier, Chong Liu, Blanka Hovarth, Theodoros Damoulas, and Terry Lyons. Higher order kernel mean embeddings to capture filtrations of stochastic processes. Advances in Neural Information Processing Systems, 34:16635–16647, 2021.*\n\n*But it is not clear whether the increment is \"important\" in practice; Is the increased performance \"worth\" the implementation effort and/or computational cost? The answer is probably problem-dependent.*\n\n**Answer:**  \nWe agree with the referee that the answer is problem-dependent. We expect that our method would demonstrate more significant strength in solving multi-period optimization problems, such as optimal stopping and utility maximization, as well as in conditional time series tasks where the extended weak topology is crucial.\n\nHowever, the empirical performance is hard to estimate prior to experiments due to various factors regarding the dataset, models and training procedure (e.g., data size, model hyper-parameters, optimizer, etc). Therefore, it is very challenging to have such a cost-effective analysis before conducting the experiments. For practical applications, typically, given the computational budget, one would try a suite of machine learning methods and choose the best one among them. Given the moderate additional computational cost and robustness of our proposed HRPCF metric, we believe that the HRPCF metric and its corresponding GAN models are well-suited for practical applications.\n\n1. *Consistent performance improvement \\& moderate computational cost*. Our numerical experiments verified that our approach has a significantly better performance than several state-of-the-art methods (based on the weak convergence) in hypothesis testing and generative modelling. The computation complexity of HRPCFD is significantly reduced compared to the existing high rank methodologies such as HR-SigMMD, which is illustrated in Table 4 of the appendix. In comparison with popular GAN models (e.g., PCFGAN, TimeGAN and RCGAN), our HRPCF-GAN takes a longer time, but at the same magnitude. For example, in our numerical examples, the training time of HRPCF-GAN is kept at a manageable level across different datasets ranging from 30 minutes to 4 hours.\n\n2. *Open source codes*. The open-source codes for our proposed method will significantly reduce the implementation effort for its empirical applications. We will make the code repository publicly available upon publication. By doing so, it is readily available for reuse and adaptation by the research community.  In particular, our HRPCFGAN codes can be flexibly applied to general time series data without the need for re-implementation from scratch.\n\n3. *Robustness of HRPCFD metrics*. We would like to emphasize that our approach based on the extended weak convergence is **robust in any case**, as our Example A.1 shows: it can happen that sometimes weak topology provides a completely wrong criterion, whilst the extended weak convergence always works correctly, which was proved rigorously in many literature (see e.g., [1]). \n\nOn a related note, it is interesting to investigate the sufficient condition under which stochastic processes such that the induced extended weak topology is strictly stronger than weak topology. To our best knowledge, so far, there is no easy criterion to judge in which circumstance the weak convergence coincides with the extended weak convergence (and even when they are equivalent, it is very hard to get a quantitative comparison). This study will be instrumental for the theoretical underpinnings of the performance gain of the EWT-based distance and provide guidance on empirical applications. \n\n---\n\n**Question:** *The prediction process seems to be introduced on a fixed finite set of times--$I = 0,\\ldots,T$ and $X = (X_t)_{t \\in I}$--and* *yet we are concerned with continuously-indexed processes, so I would more naturally assume $I$ is the interval $I = [0,T]$*, *and in fact we discuss linear interpolation in l109. Is this a notational confusion? What is the index $t$ of the filtration $\\mathcal{F}_t$ ?*\n\n**Answer:** In the present work, we only consider the discrete-time processes defined on $I= 0, \\ldots, T$, and therefore the time index $t$ appeared in the filtration $\\mathcal{F}_t$ only takes values in $0,\\ldots,T$. It is just a convention in the rough path community that one views a discrete time path defined on $I= 0, \\ldots, T$ as a piecewise linear path defined on the continuous time interval $[0,T]$ by a routine linear interpolation, because such identification may make some formulations easier (e.g., by doing so the unitary feature of a path can be formulated as the solution of an ODE on $[0,T]$). We will leave a remark in the revised version to explain these notations explicitly to avoid confusion.\n\n---\n\n**Question:** *I'm not sure about the grammar of the paper's title. \"High Rank Path Development: an approach of learning the filtration* *of stochastic processes\" -> \"High Rank Path Development: an approach to learning the filtration of stochastic processes\"?*\n\n**Answer:** Thanks for this suggestion and we will change the title accordingly.\n\n---\n\n\n[1] Julio Backhoff-Veraguas, Daniel Bartl, Mathias Beiglboeck, and Manu Eder. All adapted topologies are equal. Probability Theory and Related Fields. 178(3), 2020.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "y6X3P20qa5", "reviewer": "Reviewer_9Q6u", "summary": "Time series is ubiquitous in machine learning. They are modeled as stochastic processes and therefore notions of distance between stochastic processes and more generally convergence of stochastic processes are fundamental ideas. Weak convergence of probability measures occupies a central position in this area, but for many settings, like optimal stopping, or the one studied in this paper, it is not sufficient. Extended weak convergence, defined via weak convergence of prediction processes, is the right notion. This paper introduces a metric on the space of filtered processes that metrizes the topology of extended weak convergence, proposes statistical procedures to compute it, and then tests these ideas on GANs for time series.", "questions": "None", "limitations": "N/A", "rating": 9, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "This is a paper on a very nice topic and I learnt a lot while reading it. Some of the ideas are very abstract and the paper does a good job of organizing the topics and defining everything precisely so that a meticulous reader is not left confused. It is welcoming to see a rigorous paper in ML conferences. The ideas introduced in the paper are novel and the proofs of all the claim are carefully done, although I can't claim to have read every section in appendix in detail.", "weaknesses": "Although as mentioned above the paper defines everything clearly, the exposition on PCF and HRPCF could be improved. It took me quite some time after re-reading the paper multiple times and some other referenced paper to develop an intuition for these concepts, even though I have some background on probabilistic notions like weak convergence and extended weak convergence. I understand this is difficult to do well in a conference paper with page limits, but I think having a more detailed appendix on PCF and HRPCF would help.\n\nI should mention here that I didn't find the experiments super convincing, but I am viewing this paper as a theoretical contribution, and thus any experiments it has as an added bonus and not a weakness.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We are pleased to know that the reviewer enjoyed reading our paper. We address the reviewer's comments as follows.\n\n**Comments:** *Although as mentioned above the paper defines everything clearly, the exposition on PCF and HRPCF could be improved.* *It took me quite some time after re-reading the paper multiple times and some other referenced paper to develop an intuition for these concepts*, *even though I have some background on probabilistic notions like weak convergence and extended weak convergence.* *I understand this is difficult to do well in a conference paper with page limits, but I think having a more detailed appendix on PCF and HRPCF would help.*\n\n**Answer:** We will add two additional sections in the appendix to provide a self-contained survey on the essential properties of extended weak convergence and PCF, and explain the construction of the HRPCF in a more transparent way.\n\n**Comments:** *I should mention here that I didn't find the experiments super convincing, but I am viewing this paper as a theoretical contribution, and thus any experiments it has as an added bonus and not a weakness.*\n\n**Answer:** We acknowledge that there is further room for improvement in numerical experiments. However, we would like to highlight that the proposed HRPCFD is significantly more computationally efficient than the existing high-rank signature MMD counterpart. This is empirically illustrated in Table 4 of Appendix C.1, which compares the inference times for the hypothesis testing example. Besides, the proposed HRPCFGAN overcomes the computational bottleneck of the HRSigMMD to enable its applications in generative modelling. We benchmark the proposed HRPCFGAN with strong GAN baselines specifically for time series generation, such as timeGAN and RCGAN, and show consistent performance improvement in 8 different test metrics to provide a comprehensive profile for the quality of synthetic time series data generation. The promising numerical results indicate the potential applications of HRPCFGAN on more challenging complex empirical datasets, which merits future research.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "UdDPBUIv3V", "reviewer": "Reviewer_Yt11", "summary": "The paper addresses the issue of weak convergence of stochastic processes, whereby evolving information is generally unaccounted for. This can lead to discontinuities when applying these processes to multi-period decision-making problems. Prior work has proposed the concept of extended weak convergence, as introduced by Aldous (1981), but practical numerical implementations have been challenging. To address this, the authors introduce a novel metric called High Rank PCF Distance (HRPCFD) which is shown to overcome computational issues encountered in previous attempts. The paper then demonstrates the utility of HRPCFD via experiments on hypothesis testing and generative modelling of time series data.", "questions": "See above.", "limitations": "N/A", "rating": 6, "confidence": 1, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Unfortunately this paper lies well outside my area of expertise and I am unable to review it effectively. The mathematical framework around extended weak convergence is not an area I’m familiar with, and I consequently found it challenging to grasp the nuances of the problem statement, the significance of the proposed HRPCFD metric, and the potential implications for applications in finance and economics.\n\nSo as not to negatively impact the paper’s chances of acceptance, I have defaulted to a mid-range score in my review, which reflects my assessment that the paper could nevertheless still be made more accessible for readers who are less familiar with the domain.", "weaknesses": "See above.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We address the reviewer's comments and questions in detail as below.\n\n---\n\n**Comments:** *Unfortunately this paper lies well outside my area of expertise and I am unable to review it effectively. The mathematical framework around extended weak convergence is not an area I’m familiar with, and I consequently found it challenging to grasp the nuances of the problem statement, the significance of the proposed HRPCFD metric, and the potential implications for applications in finance and economics.*\n\n**Answer:** \n\nWe appreciate the reviewer's kindness and positive score for our work. In the following, we would like to provide further explanation for our work, which hopefully would facilitate the reviewer's understanding.   \n\n1. *Importance and scope of our work*. Time series data, such as the stock price process, is ubiquitous and often exhibits randomness in finance and economics. Accurate modelling for the law of random time series (stochastic processes) plays an important role in decision marking. For example, training high-fidelity generative models to simulate synthetic time series that mimic future stock price movements can aid in developing more effective trading strategies and provide better estimators for assessing portfolio risk, with significant implications in finance and economics.\n\n    To tackle the challenge of time series modelling, a key question lies in defining a metric that quantifies the difference between two stochastic processes. We will explain the importance of the extended weak topology (EWT) in defining such a metric in the next bullet point. In this paper, we proposed the HRPCFD metric to capture EWT and design a computationally efficient algorithm to a computationally efficient algorithm that enables its direct application to enhance the discriminator of GANs to develop high-quality generative models for synthetic time series data.  \n\n2. *Why is EWT important?* We understand the difficulty for readers to grasp the meaning of EWT at first glance. To facilitate the understanding, we would like to use the toy example (Example A.1 in the paper) to illustrate the key concept of EWT and its usefulness. In the example, while the unconditional law of the processes $\\mathbb{X}^n$ converges to $\\mathbb{X}$, weak convergence fails to capture a key difference between the financial models $\\mathbb{X}^n$ and $\\mathbb{X}$. Specifically, if an agent believes the market dynamics as in $\\mathbb{X}^n$, he/she always knows the outcome of the last day in advance, granting a predictive advantage, whereas in the “fair” market $\\mathbb{X}$, the agent lacks this foresight. This crucial difference in the observed information flow-\"$\\text{No knowledge} \\Rightarrow \\text{Full knowledge} \\Rightarrow \\text{Full knowledge}$\" for $\\mathbb{X}^n$ versus \"$\\text{No knowledge} \\Rightarrow \\text{No knowledge} \\Rightarrow \\text{Full knowledge}$\" for $\\mathbb{X}$—is not reflected in weak convergence alone.\n\n    EWT is vital because it captures this difference through the conditional distributions. For markets $\\mathbb{X}^n$, where the agent has full information on day 1, the conditional distribution becomes a single Dirac measure, annihilating randomness. In contrast, $\\mathbb{X}$ retains genuine randomness at day 1, as reflected by a linear combination of Dirac measures. Since EWT is based on conditional distributions, it effectively measures differences in information evolution styles, ensuring continuity in multi-period decision-making as agents update their actions based on continually evolving information.\n\n3. *Significance of the HRPCFD metric.*  In machine learning, many popular metrics on stochastic processes, (e.g., the Wasserstein distance, the PCFD metric, the signature MMD) based on weak topology, fail to ensure that the closeness of the conditional law (filtration) of two processes. To this end, we propose a novel HRPCFD metric to tackle this issue and prove that it can characterise EWT. Therefore, by using *HRPCFD metric* as the discriminator,  we are able to detect those differences of filtrations between various stochastic processes which are hard to be measured by those classical tools based on the weak convergence, and this observation was verified by our Hypothesis testing in Section 5.1. \n\n    Besides, our proposed HRPCFD is much more computationally efficient than the existing HR-SigMMD, which can theoretically characterise EWT. This enables the feasibility of using the HRPCFD as the discriminator of GANs for synthetic time series in practice.   \n\n    The resulting *HRPCF-GAN* demonstrates the consistent outperformance of the strong and popular GAN model baselines specifically for time series generation, such as TimeGAN and RCGAN (see Section 5.2). The comprehensive evaluation of model performance using 8 different metrics indicates that the synthetic time series generated by our proposed HRPCF-GAN shares not only a similar law but also a similar filtration information with real-time series. These promising results highlight the potential of HRPCF-GAN for synthetic time series generation and its applications across various domains, including finance and economics. \n\n---\n\n**Comments:** *So as not to negatively impact the paper’s chances of acceptance, I have defaulted to a mid-range score in my review, which reflects my assessment that the paper could nevertheless still be made more accessible for readers who are less familiar with the domain.*\n\n**Answer:** We will try our best to explain the essential ideas behind these abstract terms  and simplified lots of presentations in the revised version to make sure that the mathematical content is accessible for readers without any prior specialised knowledge on this topic. For instance, we will add a very detailed explanation after Example A.1. (as above) to show why the EWT is important in the multi-periods optimisation problems.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We deeply appreciate all the reviewers for their helpful comments and constructive suggestions. We are pleased that all the reviewers find our work sound and well-presented. In the following, we provide detailed responses to the questions raised by each reviewer individually.

Meta-review: Most of this paper's reviews are positive. 

The main concern of the sole negative review - which comes from a reviewer who is a seasoned expert and said that they checked the paper's proofs - is that (a) the paper borders on too mathematical for NeurIPS, and (b) the emphasis is too much on technical aspects, many of which are already reasonably-well-understood to the experts, and too little on novel broad insights.

In discussion, however, that reviewer said they are not against the paper and are happy for it to be accepted if the other reviewers find it valuable and insightful. My understanding is that this is precisely what the other reviewers have said. I therefore recommend acceptance on basis of reviewer consensus.
Decision: Accept (poster)