SingGraph: Hybrid GAN for Financial Time Series

• SingGraph is a generative adversarial network that unifies signature transforms, LSTM autoregression, and graph neural networks over visibility graphs to capture temporal order and complex dynamics in financial data.
• It transforms financial series into dynamic graphs to extract autoregressive, geometric, and higher-order path features, addressing challenges like non-stationarity, heavy tails, and volatility clustering.
• Empirical results on major indices such as the S&P 500 and Nikkei 225 demonstrate improved distributional fidelity, lower Earth Mover’s Distance, and superior replication of financial stylized facts.

SingGraph, introduced as Sig-Graph GAN, is a generative adversarial framework for synthetic financial time series that fuses three complementary representations and inductive biases: path signatures to summarize the temporal evolution hierarchically, recurrent neural networks (LSTM) to capture autoregressive structure, and graph neural networks (GNNs) operating on visibility graphs to exploit geometric and structural patterns in the time series. The model is designed for financial data exhibiting non-stationarity, heavy tails and skewness, time-varying and clustered volatility, short-memory in returns but long-memory in volatility, autocorrelation structures, and regime shifts, and is reported to outperform classical baselines while remaining competitive with deep time-series GAN baselines on distributional and stylized-fact criteria (Gregnanin et al., 21 May 2026).

1. Problem setting and modeling rationale

SingGraph is motivated by the observation that financial time series are difficult to model generatively because weak-stationarity assumptions often fail, empirical return distributions exhibit fat tails and jumps, conditional heteroskedasticity is prominent, returns are nearly uncorrelated at short lags while squared returns show long-range dependence, and market microstructure, macro events, or regulatory changes cause regime shifts (Gregnanin et al., 21 May 2026). In this setting, classical models such as ARIMA, GARCH, stochastic volatility, and Geometric Brownian Motion enforce parametric structure and stationarity assumptions that constrain expressiveness.

The framework addresses a specific gap in deep generative modeling. GANs can learn complex data distributions without explicit parametric forms, but they struggle with sequential dependence and geometry. GNNs, when fed a graph representation of a time series, encode multi-scale topological patterns and non-Euclidean structure, including fractal-like properties, while the signature transform offers an essentially universal feature set of a path capturing temporal order via iterated integrals. SingGraph therefore combines three components with distinct roles: LSTM for autoregressive dependence, visibility graph plus GNN for geometric and topological patterns, and signature-based losses to match distributions in expectation and higher-order temporal interactions (Gregnanin et al., 21 May 2026).

A plausible implication is that the method should be understood less as a conventional sequence-only GAN and more as a hybrid path–graph generator. Its central claim is not merely that adversarial training is applied to financial returns, but that temporal order, local autoregression, and graph-derived geometric structure are enforced simultaneously within one synthesis pipeline.

2. Time-series representation, visibility graphs, and signatures

SingGraph begins from a univariate time series $S = \{s_1, \ldots, s_T\}$ with discrete timestamps $\{t_1, \ldots, t_T\}$ and converts it to a graph using the standard Natural Visibility Graph. Two points $(t_a, y_a)$ and $(t_b, y_b)$ are connected if all intermediate points are visible along the straight line segment joining them, with criterion

$$y_c < y_b + \frac{y_a - y_b}{t_b - t_a}(t_b - t_c).$$

The implementation-equivalent form stated for the series is

$s_k < s_j + (s_j - s_i)(t_j - t_k)/(t_j - t_i).$

For completeness, the Horizontal Visibility Graph variant connects $(t_a, y_a)$ and $(t_b, y_b)$ if

$$y_c < \min(y_a, y_b), \quad \forall\, t_c \in (t_a, t_b).$$

SingGraph treats the choice of directed or undirected VG as a hyperparameter, and in the reported experiments the undirected NVG is adopted (Gregnanin et al., 21 May 2026).

Graph construction is performed per rolling window. A snapshot of $m+1$ consecutive observations $\{t_1, \ldots, t_T\}$0 forms a dynamic graph sequence $\{t_1, \ldots, t_T\}$1 with adjacency $\{t_1, \ldots, t_T\}$2. Nodes correspond to time indices in the window. For the discriminator, node features are the standardized log-returns $\{t_1, \ldots, t_T\}$3 with $\{t_1, \ldots, t_T\}$4 channel; for the generator, node features are Gaussian noise $\{t_1, \ldots, t_T\}$5 with $\{t_1, \ldots, t_T\}$6 channels sampled i.i.d. from $\{t_1, \ldots, t_T\}$7. Edges are binary under the VG criterion, and no explicit edge features are used. The return preprocessing pipeline is explicit:

$\{t_1, \ldots, t_T\}$8

followed by standardization to zero mean and unit variance, after which the inverse Lambert-W transform is applied to returns prior to modeling (Gregnanin et al., 21 May 2026).

The signature component supplies path-level statistics. For a $\{t_1, \ldots, t_T\}$9-dimensional path $(t_a, y_a)$0 with finite $(t_a, y_a)$1-variation, the $(t_a, y_a)$2-th order signature term is

$(t_a, y_a)$3

and the truncated signature up to order $(t_a, y_a)$4 is

$(t_a, y_a)$5

living in the tensor algebra $(t_a, y_a)$6 with retained dimension $(t_a, y_a)$7. In implementation, discrete series are converted to continuous paths using the lead-lag transform, which doubles dimensionality and encodes temporal ordering robustly. A truncated signature of degree $(t_a, y_a)$8 is computed for both the raw series and its cumulative sum using the iisignature or signatory libraries (Gregnanin et al., 21 May 2026).

The role of signatures is explicitly distributional rather than merely descriptive. They are described as invariant to time reparameterization, able to encode order via tensor products of increments, and capable of capturing higher-order interactions between path coordinates. Because expected signatures uniquely determine the law under mild conditions, matching signatures is intended to improve fidelity beyond first- and second-moment alignment (Gregnanin et al., 21 May 2026).

3. Architecture of Sig-Graph GAN

Sig-Graph GAN comprises identical generator and discriminator network topologies built from three blocks: Recurrent, Geometric, and Feedforward. Inputs are a length-$(t_a, y_a)$9 feature matrix $(t_b, y_b)$0 and the VG adjacency $(t_b, y_b)$1 for the window (Gregnanin et al., 21 May 2026).

The Recurrent Block is a multi-layer LSTM. Given $(t_b, y_b)$2, an $(t_b, y_b)$3-layer stack produces $(t_b, y_b)$4, followed by a linear head to output temporal features $(t_b, y_b)$5. The cell equations are

$(t_b, y_b)$6

$(t_b, y_b)$7

$(t_b, y_b)$8

with final temporal features

$(t_b, y_b)$9

The Geometric Block combines GCN message passing with an LSTM. Initial node states are $$y_c < y_b + \frac{y_a - y_b}{t_b - t_a}(t_b - t_c).$$0, and the generic message-passing form is

$$y_c < y_b + \frac{y_a - y_b}{t_b - t_a}(t_b - t_c).$$1

with permutation-invariant aggregation. The specific layer used is

$$y_c < y_b + \frac{y_a - y_b}{t_b - t_a}(t_b - t_c).$$2

where $$y_c < y_b + \frac{y_a - y_b}{t_b - t_a}(t_b - t_c).$$3, $$y_c < y_b + \frac{y_a - y_b}{t_b - t_a}(t_b - t_c).$$4 is the degree matrix of $$y_c < y_b + \frac{y_a - y_b}{t_b - t_a}(t_b - t_c).$$5, $$y_c < y_b + \frac{y_a - y_b}{t_b - t_a}(t_b - t_c).$$6 is $$y_c < y_b + \frac{y_a - y_b}{t_b - t_a}(t_b - t_c).$$7, and $$y_c < y_b + \frac{y_a - y_b}{t_b - t_a}(t_b - t_c).$$8 are learnable weights. After $$y_c < y_b + \frac{y_a - y_b}{t_b - t_a}(t_b - t_c).$$9 GCN layers, an LSTM ingests $s_k < s_j + (s_j - s_i)(t_j - t_k)/(t_j - t_i).$0 to extract spatio-temporal patterns, followed by a fully connected layer producing geometric features $s_k < s_j + (s_j - s_i)(t_j - t_k)/(t_j - t_i).$1 (Gregnanin et al., 21 May 2026).

Fusion is performed by simple summation:

$s_k < s_j + (s_j - s_i)(t_j - t_k)/(t_j - t_i).$2

An optional skip-layer concatenation with the original $s_k < s_j + (s_j - s_i)(t_j - t_k)/(t_j - t_i).$3 is controlled by a hyperparameter, after which a three-layer MLP with PReLU activations maps fused features to predictions $s_k < s_j + (s_j - s_i)(t_j - t_k)/(t_j - t_i).$4 with layer sizes $s_k < s_j + (s_j - s_i)(t_j - t_k)/(t_j - t_i).$5 and activation

$s_k < s_j + (s_j - s_i)(t_j - t_k)/(t_j - t_i).$6

The generator receives $s_k < s_j + (s_j - s_i)(t_j - t_k)/(t_j - t_i).$7, whereas the discriminator receives $s_k < s_j + (s_j - s_i)(t_j - t_k)/(t_j - t_i).$8, and both use the same adjacency $s_k < s_j + (s_j - s_i)(t_j - t_k)/(t_j - t_i).$9 built from the real series window, thereby enforcing a shared geometric context (Gregnanin et al., 21 May 2026).

4. Objective functions, optimization, and experimental protocol

The adversarial objective is presented conceptually as

$(t_a, y_a)$0

In the reported implementation, training is driven by signature-based losses that compare fake and real outputs after lead-lag and cumulative transforms. Two separate training runs are instantiated: one using a signature MSE loss and one using a signature KLD loss (Gregnanin et al., 21 May 2026).

The signature MSE loss is

$(t_a, y_a)$1

$(t_a, y_a)$2

while the signature KLD loss is

$(t_a, y_a)$3

$(t_a, y_a)$4

Here $(t_a, y_a)$5 is the number of signature components, $(t_a, y_a)$6, $(t_a, y_a)$7 is the lead-lag transform, and superscript $(t_a, y_a)$8 denotes cumulative-sum signature. The paper notes that one may combine adversarial and signature losses as an auxiliary generator loss, although the reported experiments use separate MSE or KLD runs. RMSProp is used for optimization; spectral normalization or gradient penalty can be added if instability arises, but were not required in the reported experiments (Gregnanin et al., 21 May 2026).

The empirical study uses daily closing prices for the S&P 500, Nasdaq Composite, and Nikkei 225 from 2010-01-04 to 2019-12-30, with approximately 2515 observations for each index. Rolling windows have length $(t_a, y_a)$9 with $(t_b, y_b)$0, so $(t_b, y_b)$1, and each window is converted to an undirected NVG using the ts2vg library, yielding $(t_b, y_b)$2. Baselines are QuantGAN, GARCH(1,1), and Monte Carlo Geometric Brownian Motion calibrated by maximizing log-likelihood for mean and variance. Training uses RMSProp, batch size 30, 100 epochs, and Optuna for hyperparameter search over learning rates, layer counts, hidden sizes, dropout, and VG directionality (Gregnanin et al., 21 May 2026).

Evaluation is organized around three criteria. Distributional fidelity is measured by Wasserstein-1, or Earth Mover’s Distance, between real and generated cumulative return distributions over horizons $(t_b, y_b)$3 days. Signature-RMSE measures the RMSE between truncated signatures of real and generated paths over the same horizons. Stylized-fact preservation is assessed through a leverage effect score, defined as in QuantGAN, quantifying the negative correlation between returns and subsequent volatility (Gregnanin et al., 21 May 2026).

5. Empirical behavior, ablations, and interpretation

The principal quantitative finding is that Sig-Graph GAN consistently achieves lower EMD than GARCH and GBM across all indices and horizons, and performs on par or better than QuantGAN in many settings, with notable wins on the S&P 500 across horizons and on the Nikkei 225 at short horizons. QuantGAN remains competitive for the Nasdaq Composite at the 1-day horizon. By contrast, Monte Carlo GBM sometimes attains smaller signature RMSE, which the paper interprets as reflecting its tendency to match first- and second-moment structure without capturing higher-order dependencies; Sig-Graph GAN remains competitive on this metric and generally superior to GARCH. On the leverage effect score, Sig-Graph GAN is reported to attain values close to real data and competitive with or better than QuantGAN, while outperforming GBM and GARCH (Gregnanin et al., 21 May 2026).

The ablation study is central to the model’s interpretation. Removing the Geometric Block degrades performance notably under both loss functions, particularly under KLD, which is presented as confirmation that the VG-derived geometric embedding captures non-linear dependencies and fractal-like structures. Skip-connections and dropout contribute consistent improvements, especially under MSE loss, and removing the Feedforward Block worsens EMD. These observations support the claim that performance is not attributable to any single component in isolation, but to the fusion of temporal and graph-derived representations (Gregnanin et al., 21 May 2026).

Qualitative analysis focuses on cumulative return distributions, especially for Nikkei 225 horizons of 5–10 days, where the model is reported to reproduce tail behavior well and track the body of the distribution closely. At the 1-day horizon, the tails remain well matched while the high-density region is described as challenging, which the paper identifies as common in financial GANs because of microstructure noise and near-normal central tails. This suggests that SingGraph is strongest in matching broader pathwise and tail-distribution structure rather than eliminating every short-horizon density mismatch (Gregnanin et al., 21 May 2026).

The paper’s explanation for these gains is threefold. Signatures enforce matching of higher-order path statistics and temporal-order invariants. Visibility graphs transform the series into a topology preserving trends, peaks, valleys, periodicity, randomness, and fractality, after which GNN message passing extracts multi-scale structural features that sequence-only models do not directly encode. LSTM then preserves autoregressive dynamics and short-range memory, while the shared MLP fuses temporal and geometric patterns. A plausible implication is that the framework’s improvements in EMD and leverage effect arise from this complementarity rather than from adversarial training alone.

6. Limitations, practical use, and terminological disambiguation

The reported limitations are explicit. Although visibility graphs and signatures reduce reliance on stationarity assumptions, abrupt regime shifts can still challenge training. GAN mode collapse remains possible; signature-based auxiliary losses mitigate but do not eliminate it. Results depend on window length $(t_b, y_b)$4, directed versus undirected VG choice, and sampling resolution. Higher signature truncation order $(t_b, y_b)$5 increases expressiveness but grows exponentially in cost, making $(t_b, y_b)$6 a practical compromise. The model was tested on three large indices, so micro-cap or intraday tick data may require different VG window sizes, sampling resolutions, and signature dimensionality. Computationally, dynamic graph construction per window and GNN–LSTM stacks are heavier than RNN-only GANs, and the reported experiments used A100 GPU resources (Gregnanin et al., 21 May 2026).

Practical guidance in the paper is correspondingly concrete. Exemplary Optuna-tuned settings include window length $(t_b, y_b)$7, LSTM depths between 2 and 7 layers, GCN depths between 3 and 7 layers, hidden sizes roughly 70–190 depending on block and loss, recurrent dropout around 0.31–0.35, feedforward layers $(t_b, y_b)$8, RMSProp learning rates around $(t_b, y_b)$9 for MSE and $$y_c < \min(y_a, y_b), \quad \forall\, t_c \in (t_a, t_b).$$0 for KLD, batch size 30, 100 epochs, undirected NVG with ts2vg default parameters, and lead-lag signatures with $$y_c < \min(y_a, y_b), \quad \forall\, t_c \in (t_a, t_b).$$1 computed on both raw and cumulative paths. Training advice includes starting with MSE signature loss, monitoring EMD across horizons and the leverage effect, and adding gradient clipping, spectral normalization, or WGAN-GP penalties if instability emerges (Gregnanin et al., 21 May 2026).

The term “SingGraph” also requires disambiguation. In the context of synthetic financial time series, it denotes Sig-Graph GAN (Gregnanin et al., 21 May 2026). Elsewhere, unrelated arXiv works use similar or confusingly adjacent names. “Signature-Graph Networks” uses the official abbreviation SGN and explicitly states that “SingGraph” does not appear in the paper (Hamdi et al., 2021). A different “SingGraph” designates a singing voice deepfake detector that combines MERT, wav2vec2.0, RawBoost, beat matching, and heterogeneous spectral–temporal graph attention for SingFake detection (Chen et al., 2024). In sign language research, “MixSignGraph” is the official method name, and “SingGraph” is described as a typo or informal misreference to SignGraph or MixSignGraph (Gan et al., 16 Apr 2025). In continuous-time signed-graph modeling, “SingGraph” is used to refer to SGODE, a signed graph neural ordinary differential equation framework (Chen et al., 2023). This naming overlap does not indicate methodological continuity; it reflects terminological collision across financial generation, audio forensics, sign language processing, and signed-graph dynamics.

Within its own domain, however, SingGraph has a precise meaning: a signature-graph-based GAN that unifies autoregression via LSTM, geometry via visibility graphs and GNNs, and distribution-level path features via signatures. Its reported contribution is not the introduction of any one of these components in isolation, but their integration into a single generative model for non-stationary financial returns that aims to preserve both stylized facts and higher-order path structure (Gregnanin et al., 21 May 2026).