Conditional Sig-Wasserstein GANs for Time Series Generation

• Off-the-shelf deep generative models struggle with temporal and spatial feature interactions, and long time series can make the joint distribution high-dimensional.High-fidelity synthetic time series can support product testing and validation and data sharing subject to privacy constraints.Source: §1 · p. 2
• The paper targets the conditional law of future time series given the past, a setting that includes classical autoregressive processes.The authors identify predictive modeling, causal modeling, and building a joint law from conditional laws as uses of this setting.Source: §1 · p. 2
• Learning the conditional distribution can be more desirable and more data-efficient than learning the joint law.The introduction motivates this comparison with prior work, Ng and Jordan (2002); Buehler et al. (2020).Source: §1 · p. 2
• A time-series learning problem over R^d×T can be reduced to learning conditional distributions over R^d.Under the time-invariant conditional-law setting, one trajectory provides T − p − 1 samples, versus one sample for direct learning of Law(X1:T).Source: §1 · p. 2

2. PROBLEM FORMULATION

• The formulation considers sliding-window samples from one time series as draws from an unknown path-space distribution for unconditional generation.Each segment has window size W, typically much smaller than the full series length T.Source: §2 · p. 3
• The target is the conditional law of a future path given a past path, with a generator mapping past observations and latent samples to future paths.The formulation assumes the joint distribution of past and future paths does not depend on time.Source: §2 · p. 3
• Conditional W1-GANs face difficult min-max optimization and costly conditional-expectation estimation, which can add bias and make training harder to tune.First-order methods may fail to converge even in convex-concave cases.Source: §2 · p. 4

2.1 Summary of the key results

• On regular paths, signatures provide a universal representation: continuous functions on signature space can be approximated by linear functionals of signature.The path signature determines the path on the specified regular path space.Source: §2.1 · p. 5
• The paper extends Sig-W1 from the l2 norm to general lp norms for p > 1.The earlier Sig-W1 formulation used the l2 norm of signature space.Source: §2.1 · p. 6, §2.2 · p. 9
• The conditional Sig-W1 discriminator has an analytic formula, and the framework reduces min-max training to supervised learning without loss of accuracy relative to path-space Wasserstein distance.The data-measure conditional expectation is computed only once; the resulting optimization can be estimated by linear regression.Source: §2.1 · p. 6
• On 2-dimensional VAR(1), SigCWGAN stabilizes training and accelerates convergence compared with CWGAN using the same conditional generator.Figure 1 compares the methods across three performance metrics.Source: §2.1 · p. 6, Figure 1 · p. 7

2.2 Related work

• SigCWGAN extends signature Wasserstein GANs from unconditional to conditional generation and can be viewed as combining unnormalized Sig-MMD with conditional MMD.It also addresses the conditional-WGAN discriminator bottleneck without training that discriminator.Source: §2.2 · p. 9

3.1 Tensor algebra space

• The tensor algebra represents tensor series, whose degree-n components are spanned by words of length n in the basis elements.The section also introduces truncated tensor algebra for working with finite-degree signatures.Source: §3.1 · pp. 9–11

3.2 Signature of time series

• Discrete time series are embedded as continuous, piecewise-linear paths, with an added time coordinate; the signature collects their iterated integrals.The signature uniquely determines the path, supporting its use as a time-series feature extractor.Source: §3.2 · pp. 11–12

Expected signature

• Under an infinite-radius-of-convergence condition, equal expected signatures imply equal path-space laws.The section notes that general conditions guaranteeing this convergence property are challenging to establish.Source: Expected signature · p. 13
• Expected signatures act like non-commutative moment-generating functions, but their law-characterizing role depends on regularity such as infinite radius of convergence.The section gives differing convergence examples: fractional Brownian motion for H ≥ 1/2 has infinite radius, whereas stopped Brownian motion up to a first exit has finite radius.Source: Expected signature · pp. 13–14

4. SIG-WASSERSTEIN METRIC

• Sig-W1 restricts the admissible discriminator to linear functionals on signatures and measures the supremum over those with Lipschitz norm at most one.The metric is presented for measures whose induced signature-space measures have compact support.Source: §4 · p. 14
• The authors conjecture that W Sig_1 and Sig-W1 generally differ, and say sufficient conditions for coincidence remain technically challenging to establish.Uniform approximation of a continuous function by a linear map does not guarantee closeness of their Lipschitz norms.Source: §4 · p. 14
• For an l_p distance with p > 1, the linear functional’s Lipschitz norm can be computed analytically, yielding an analytic Sig-W1 formula.The section states the simplification for q = 1 and D(x, y) = ||x − y||p.Source: §4 · pp. 14–15
• When p = 2, Sig-W1 equals the unnormalised Sig-MMD metric proposed by Chevyrev and Oberhauser (2022).The paper uses the l2 version by default and truncates the signature at degree M in practice.Source: §4 · p. 16

5. SIG-WASSERSTEIN GANS FOR CONDITIONAL LAW

• SigCWGAN learns the conditional future law from data using the Sig-W1 metric and a two-step framework.The framework first estimates the true conditional expected signature, then optimizes the conditional generator.Source: §5 · p. 16

5.3 The Conditional AR-FNN Generator

• The proposed AR-FNN generator extends classical autoregressive models with a feed-forward neural network and generates futures of arbitrary length by rolling its one-step generator forward.At each step, the generator uses the lagged generated values and a noise vector.Source: §5 · p. 16, §5.3 · pp. 18–19

5.1 Learning the conditional expected signature under the true measure

• The conditional expected signature is estimated by a one-off linear regression before generative learning, avoiding repeated discriminator expectation learning during conditional WGAN updates.The paper describes signature-space regression and notes that OLS is suitable under its stated noise assumptions.Source: §5.1 · pp. 16–17

5.2 Sig-Wasserstein GAN algorithm for conditional law

• The generator is optimized against a loss comparing the estimated conditional expected signature with the generator’s Monte Carlo estimate, using stochastic gradient descent.The empirical loss is formed from simulated future trajectories conditioned on past observations.Source: §5.2 · p. 17

6. NUMERICAL EXPERIMENTS

• The evaluation covers marginal distributions, temporal and feature dependence, and train-on-synthetic/test-on-real predictive usefulness across VAR, SPX/DJI, and Bitcoin-USD data.The experiments also compare SigCWGAN with CWGAN, TimeGAN, RCGAN, GMMN, and GARCH, using a shared generator architecture for the generative-model benchmarks.Source: §6 · p. 19

6.1 Synthetic data generated by Vector Autoregressive Model

• As VAR dimensionality increases, SigCWGAN retains very low relative TSTR R2 error while other models deteriorate, especially GMMN.Across dimensions, SigCWGAN is reported as comparable to or better than the baselines on the defined metrics.Source: §6.1 · p. 21, Figure 5 · p. 22
• With two minutes of training, SigCWGAN has the best performance on every reported metric except marginal distribution.The comparison uses a fixed training time across algorithms.Source: Table 2 · p. 22, §6.1 · p. 23
• The paper reports that, for a synthetic sampled path of 80,000 steps, SigCWGAN’s marginal density function is much closer to real data than the baselines.On the SPX/DJI task, its ACF loss and density test metrics evolve much more smoothly than those of other baselines. Its ACF plot also shows better autocorrelation fitting across various lag values.Source: §6.1 · p. 23, §6.2 · p. 24

6.2 SPX and DJI index dataset

• On SPX and SPX/DJI data, SigCWGAN is superior or comparable to baselines and outperforms GARCH on both datasets.For SPX alone, GMMN is slightly better on lag-1 autocorrelation and marginal distribution, with differences no greater than 0.0013, but has poorer predictive performance and feature correlation.Source: §6.2 · p. 24, Table 3 · p. 25

6.3 Bitcoin-USD dataset

• On Bitcoin-USD, SigCWGAN outperforms baselines on almost all test metrics; RCGAN’s R2 score is 0.3165, 0.0155 better, while SigCWGAN’s marginal-distribution metric is 2.0532 versus 2.803.The task predicts the next six hours of log returns from the previous 24 hours.Source: §6.3 · p. 26, Table 4 · p. 26

7. Conclusion

• Conditional Sig-Wasserstein GANs explicitly approximate W1 using signature features, avoiding a costly critic or discriminator and simplifying training.The paper reports state-of-the-art results on its synthetic and empirical datasets.Source: §7 · p. 26
• The method is effective for moderate-dimensional time series, but high path dimension may cause a curse of dimensionality.The authors also identify conditions under which path-space W1 coincides with Sig-W1 as a theoretical question for future work.Source: §7 · pp. 26–27

Appendix A. PRELIMINARY

• The preliminary section frames signature concepts as groundwork for analyzing signature approximations to Wasserstein-1 distance and proving Lemma 13..Source: Appendix A · p. 32

A.1 Signature of a path

• A path’s p-variation measures its roughness; continuous paths need not have finite 1-variation, as Brownian motion has infinite p-variation almost surely for p in [1, 2].For finite 1-variation paths, the signature has factorial decay.Source: §A.1 · p. 32

A.2 Expected Signature of stochastic processes

• The appendix defines an expected signature as having infinite radius of convergence when its expectation is well-defined and finite and satisfies the stated condition for every λ ≥ 0.The supplied passage does not include the condition’s displayed expression.Source: §A.2 · p. 33

A.3 The Signature Wasserstein-1 metric (Sig-W1)

• The Sig-W1 appendix proves the stated norm relations using constrained optimization and Lagrange multipliers, with Hölder’s inequality in the subsequent argument.The supplied proof passages refer to Equations (11) and (12).Source: §A.3 · pp. 33–34

B.1 Path transformations

• SigCWGAN lifts time series to signature features, which may be paired with time-jointed, cumulative-sum, lead-lag, or lag-added path transformations.The m-lag added transformation maps each time point to a tuple of the current value and the next m values.Source: §B.1 · pp. 34–35
• For Sig-W1 to distinguish measures on path space, the discrete-series embedding must preserve a bijection between time series and signatures.The paper gives lead-lag transformation as one way to ensure one-to-one correspondence.Source: §B.1 · p. 35

B.2 AR-FNN Architecture

• AR-FNN composes PReLUs, residual layers, and affine transformations, taking past p-lags and a noise vector as input.Its defined output is a d-dimensional vector.Source: §B.2 · p. 35
• The forecast procedure samples each Zi from an iid standard normal distribution and recursively applies G to generate the next values.It returns the generated sequence from the first through the q-th future step.Source: §B.2 · p. 36

Appendix C. NUMERICAL IMPLEMENTATIONS

• The VAR(1) CWGAN uses a 3-layer FNN discriminator with a gradient penalty to enforce 1-Lipschitzness.Source: Appendix C · p. 36
• SigCWGAN, TimeGAN, RCGAN, and GMMN share a 3-layer AR-FNN generator; the latter three use concatenated conditioning and synthetic paths as discriminator or MMD inputs.For GMMN, three kernels with bandwidths 0.1, 1, and 5 were used after a mixture of bandwidths performed best against the median heuristic.Source: Appendix C · p. 36
• Training uses 1,000 generator updates with Adam and a 0.001 generator learning rate; TimeGAN and RCGAN use TTUR and two discriminator updates per generator update.Source: Appendix C · p. 36
• Before signature computation, SigCWGAN combines the past path with its cumulative-sum path, then applies 1-lag-added and Lead-Lag transformations.The authors state that signatures of these transformed paths can capture marginal distributions, autocorrelations, and other temporal characteristics.Source: Appendix C · p. 37
• For evaluation, the generator produces forecasts for past-window inputs, and real and synthetic forecast samples are used to compute test metrics.Source: Appendix C · p. 37
• The appendix defines marginal-density, lag-1 autocorrelation, correlation-matrix, and TRTR/TSTR R2 evaluation procedures.Source: Appendix C · p. 37

D.1 VAR(1) dataset

• VAR(1) experiments vary d across 1, 2, and 3 and σ and ϕ across 0.2, 0.5, and 0.8, with test metrics reported in Tables D.1–D.3 and Figures D.1–D.4.Source: §D.1 · p. 38, Table D.1 · p. 39, Table D.2 · p. 40
• Figures D.5 and D.6 show SigCWGAN optimization stability relative to other baselines, including two using min-max game optimization.The figures show training distances and scores for 2- and 3-dimensional VAR(1) models with ϕ = 0.8 and σ = 0.8.Source: §D.1 · p. 38, Figure D.5 · p. 44, Figure D.6 · p. 45

D.2 ARCH(p)

• ARCH(p) experiments cover p = 2, 3, and 4, use signature degree 3, and summarize numerical results in Table D.4.Source: §D.2 · p. 46, Table D.4 · p. 46

D.3 SPX and DJI dataset

• Table D.5 summarizes test metrics for the stock datasets, while Figures D.7 and D.8 show metrics over training for SPX and SPX-plus-DJI data.Source: Table D.5 · p. 47, §D.3 · p. 47, Figure D.7 · p. 48
• Figure D.9 compares real and synthetic SPX cross-correlation matrices for log-returns and log-volatility; the historical negative correlation indicates leverage effects.The figure's axes index feature dimensions, and its color encodes pairwise correlations.Source: §D.3 · p. 47, Figure D.9 · p. 50
• Table D.6 reports stock-dataset R2 metrics and Table D.7 reports autocorrelation metrics across lag values, with separate SPX and SPX-plus-DJI results in each cell.Source: Table D.6 · p. 50, Table D.7 · p. 50

D.4 Bitcoin dataset

• Tables D.8 and D.9 report Bitcoin autocorrelation and R2 metrics, respectively, across lag values.Source: §D.4 · p. 50, Table D.8 · p. 51, Table D.9 · p. 51