Causal Sphere Hypergraph Transformer (CSHT)

• CSHT is a financial forecasting architecture that unifies Granger-causal hypergraph construction, Riemannian geometry, and causally masked Transformer attention.
• It constructs directional hyperedges from statistically validated Granger tests, ensuring temporal causality and reducing spurious dependencies.
• The model demonstrates superior performance and regime robustness with interpretable, traceable attention chains for asset return predictions.

Causal Sphere Hypergraph Transformer (CSHT) is a financial time-series forecasting architecture that unifies Granger-causal hypergraph structure, Riemannian geometry, and causally masked Transformer attention in order to model the directional influence of financial news and sentiment on asset returns. Introduced in “From News to Returns: A Granger-Causal Hypergraph Transformer on the Sphere” (Harit et al., 5 Oct 2025), CSHT extracts multivariate Granger-causal dependencies, encodes them as directional hyperedges on the surface of a hypersphere, and constrains attention by angular masks that preserve temporal directionality and geometric consistency. The model is presented as an interpretable framework for return prediction, market-regime classification, and top-asset ranking, with reported robustness across market regimes including the 2020 COVID-19 shock.

1. Formal definition and architectural scope

CSHT is defined by a six-stage pipeline. At each day $t$, the model collects three input families: news embeddings $x_t \in \mathbb R^{d_x}$, obtained with FinBERT on headlines; sentiment features $s_t \in \mathbb R^{d_s}$, given by daily average FinBERT polarity; and asset log-returns $r_t \in \mathbb R^{d_r}$. For each asset $i$, it then tests whether any subset of lagged inputs $$H_{(t-k)} = \{x_{t-k}, s_{t-k}, r_{t-k} : k=1\ldots K\}$$ Granger-causes $r_t^i$ via multivariate F-tests at FDR-adjusted $p<0.01$. The accepted lagged parents $C^i \subseteq H_{(t-k)}$ are those satisfying

$$P(r_t^i \mid C^i) \neq P(r_t^i \mid C^i \setminus \{j\}).$$

Each accepted parent set defines a directed hyperedge $x_t \in \mathbb R^{d_x}$0, yielding a directed hypergraph $x_t \in \mathbb R^{d_x}$1. The node set $x_t \in \mathbb R^{d_x}$2 consists of all lagged variables $x_t \in \mathbb R^{d_x}$3 that passed the Granger test, and the edge set $x_t \in \mathbb R^{d_x}$4 contains one hyperedge per asset target. Node embeddings are placed on the unit hypersphere $x_t \in \mathbb R^{d_x}$5, after which a Transformer computes causally masked geodesic attention and feeds a global root representation into prediction heads for next-day return regression, bull/bear regime classification, and top-asset ranking by predicted return score (Harit et al., 5 Oct 2025).

Within this formulation, the term “causal” is operationally tied to Granger-causal discovery and to a binary causal mask $x_t \in \mathbb R^{d_x}$6 defined by temporal and Granger-valid parentage. The architecture therefore narrows the admissible information flow before any attention weights are estimated. This design suggests that CSHT’s central claim is not merely that attention becomes more interpretable, but that interpretability is constrained at the level of the admissible computational graph itself.

2. Granger-causal hypergraph construction

The hypergraph layer is built from lagged predictive dependence. Given a maximum lag $x_t \in \mathbb R^{d_x}$7, the null hypothesis for variable $x_t \in \mathbb R^{d_x}$8 with respect to asset $x_t \in \mathbb R^{d_x}$9 is

$s_t \in \mathbb R^{d_s}$0

This null is rejected via an F-statistic when including $s_t \in \mathbb R^{d_s}$1 in the joint regression significantly reduces residual variance. Each accepted parent set $s_t \in \mathbb R^{d_s}$2 then becomes a directed hyperedge $s_t \in \mathbb R^{d_s}$3, which allows multiple lagged sources to jointly target a single asset return.

This construction is notable because the forecasting graph is not a simple pairwise adjacency structure. The use of directed hyperedges means that the input to a target node can be represented as a multivariate parent set rather than as a collection of independent dyadic links. In the paper’s formulation, $s_t \in \mathbb R^{d_s}$4 contains only lagged variables that passed the Granger test, and $s_t \in \mathbb R^{d_s}$5 contains one directed hyperedge per asset target. Causal sparsity therefore arises from the mask $s_t \in \mathbb R^{d_s}$6, and the paper explicitly states that no additional $s_t \in \mathbb R^{d_s}$7 or graph-Laplacian penalty is applied (Harit et al., 5 Oct 2025).

A plausible implication is that the hypergraph serves two simultaneous functions: it reduces the attention search space and encodes a multivariate view of temporal dependence that is closer to the underlying Granger-testing procedure than a conventional graph would be. The paper’s emphasis on “predictive causal structure” places this stage at the center of the model’s claim to trustworthy forecasting under uncertainty.

3. Hyperspherical geometry and masked attention

All node embeddings lie on the hypersphere

$s_t \in \mathbb R^{d_s}$8

For $s_t \in \mathbb R^{d_s}$9, the geodesic distance is

$r_t \in \mathbb R^{d_r}$0

and the angular similarity used by the model is

$r_t \in \mathbb R^{d_r}$1

Queries, keys, and values are formed by linear projections in $r_t \in \mathbb R^{d_r}$2: $r_t \in \mathbb R^{d_r}$3

The causal angular mask $r_t \in \mathbb R^{d_r}$4 is defined so that $r_t \in \mathbb R^{d_r}$5 if node $r_t \in \mathbb R^{d_r}$6 is in the parent set of node $r_t \in \mathbb R^{d_r}$7, and $r_t \in \mathbb R^{d_r}$8 otherwise. For query node $r_t \in \mathbb R^{d_r}$9 and key node $i$0, the masked attention score is

$i$1

with temperature $i$2. The normalized attention weight is

$i$3

The paper describes this as a “spherical softmax” over valid parents and states that, because all embeddings lie on $i$4 and $i$5 are linear projections in $i$6, the inner products $i$7 respect angular similarity (Harit et al., 5 Oct 2025).

The corresponding message-passing update for one CSHT layer is

$i$8

where $i$9, $$H_{(t-k)} = \{x_{t-k}, s_{t-k}, r_{t-k} : k=1\ldots K\}$$0 is learnable, and

$$H_{(t-k)} = \{x_{t-k}, s_{t-k}, r_{t-k} : k=1\ldots K\}$$1

projects the updated state back onto the sphere. Multiple heads operate in parallel, each with its own $$H_{(t-k)} = \{x_{t-k}, s_{t-k}, r_{t-k} : k=1\ldots K\}$$2, and the experiments use a stack of $$H_{(t-k)} = \{x_{t-k}, s_{t-k}, r_{t-k} : k=1\ldots K\}$$3 layers. Final node states $$H_{(t-k)} = \{x_{t-k}, s_{t-k}, r_{t-k} : k=1\ldots K\}$$4 are aggregated into a global “root” embedding for the asset, read out either by summation or by a special target node, and then passed to the downstream heads.

4. Objectives, optimization, and experimental protocol

CSHT is trained on two explicit prediction objectives and evaluated on a third task. For next-day return regression, the loss is mean squared error: $$H_{(t-k)} = \{x_{t-k}, s_{t-k}, r_{t-k} : k=1\ldots K\}$$5 For market-regime classification, the loss is cross-entropy on bull/bear labels $$H_{(t-k)} = \{x_{t-k}, s_{t-k}, r_{t-k} : k=1\ldots K\}$$6: $$H_{(t-k)} = \{x_{t-k}, s_{t-k}, r_{t-k} : k=1\ldots K\}$$7 Top-$$H_{(t-k)} = \{x_{t-k}, s_{t-k}, r_{t-k} : k=1\ldots K\}$$8 asset ranking is evaluated by NDCG@10, and the paper reports no direct surrogate ranking loss.

The optimization setup is specified as Adam with learning rate $$H_{(t-k)} = \{x_{t-k}, s_{t-k}, r_{t-k} : k=1\ldots K\}$$9, batch size 32 asset-day samples, early stopping on validation MAE/NDCG, and training for up to approximately 50 epochs with stopping by patience on validation loss. The Transformer configuration used in the experiments is 2 layers, 4 heads, hidden dimension 64, and angular scale $r_t^i$0. To maintain the manifold constraint, embeddings are re-projected onto $r_t^i$1 after every gradient step (Harit et al., 5 Oct 2025).

The evaluation universe comprises 450 actively traded S &P 500 stocks from 2018 to 2023, with 50 k ticker-matched tweets, daily FinBERT sentiment per stock, and price data derived from daily adjusted close transformed into log-returns. The temporal split is non-overlapping: training from 01 Jan 2018 to 31 Dec 2020 (756 trading days), validation from 01 Jan 2021 to 31 Dec 2021 (252 days), and test from 01 Jan 2022 to 30 Jun 2023 (376 days). Reported metrics are MAE for return regression, regime accuracy for bull/bear classification, NDCG@10 for top-10 return ranking, and causal alignment defined as the fraction of attention mass on true Granger edges.

5. Empirical performance and regime robustness

On the test set covering 2022 to Jun 2023, CSHT is reported to outperform the listed baselines across all three primary evaluation dimensions. The paper reports the following test metrics: FinBERT-RNN achieves MAE 0.0213, regime accuracy 65.4, and NDCG@10 0.612; CNN-LOB 0.0205, 67.2, and 0.625; Money 0.0198, 69.1, and 0.634; TEANet 0.0189, 70.6, and 0.639; FinGAT 0.0187, 71.3, and 0.641; HOT 0.0184, 69.0, and 0.636; and CSHT 0.0162, 74.6, and 0.683. For causal alignment, FinGAT is reported at 21.3, whereas CSHT reaches 66.4 (Harit et al., 5 Oct 2025).

The ablation study isolates the contributions of the causal mask and the spherical geometry. Validation MAE and accuracy are reported as follows: Full Mask + Spherical, 0.0171 / 72.3%; Causal Mask + Euclid. Attn, 0.0177 / 71.2%; CSHT without Mask, 0.0184 / 70.1%; CSHT w/o Sphere, 0.0175 / 71.4%; and Full CSHT, 0.0162 / 74.6%. In the paper’s own summary, this corresponds to outperformance over six strong baselines on return MAE $r_t^i$2 vs. FinGAT), regime accuracy $r_t^i$3 percentage points), and ranking quality $r_t^i$4 NDCG).

The regime-robustness claim is anchored in the March–June 2020 COVID crash. During that period, CSHT achieved 74.6% regime-classification accuracy, compared with 61–69% for baselines. The paper presents this as evidence of resilience to structural breaks and as support for the broader claim of robust generalisation across market regimes. Because the architecture combines causal sparsity, hyperspherical projection, and temporal masking, a plausible interpretation is that the reported gains are attributed not to a single design choice but to the interaction among structure discovery, geometry, and constrained attention.

6. Attribution pathways, case studies, and interpretability claims

The interpretability argument in CSHT is built directly into the attention mechanism. Because attention is strictly confined to Granger-validated parents, every nonzero $r_t^i$5 is said to have a known economic interpretation, exemplified by chains such as “news $r_t^i$6 sentiment $r_t^i$7 return $r_t^i$8.” For a multi-hop path $r_t^i$9, the influence score is defined as the product of edge-wise attention weights: $p<0.01$0 Summing or max-pooling over all paths from a macro node yields its total attribution to asset $p<0.01$1 (Harit et al., 5 Oct 2025).

Two case studies are used to illustrate this attribution mechanism. In the June 2022 Fed rate hike example, the highest-attention chain for JPMorgan is reported as: $p<0.01$2 The paper states that this mirrors the textbook transmission of monetary policy through sectoral sentiment to individual equity.

In the March 2020 COVID crash example, the selected path for ExxonMobil (XOM) is: $p<0.01$3 The predicted return is $p<0.01$4 versus realized $p<0.01$5, which the paper presents as evidence of both accuracy and semantic transparency.

These examples delimit the model’s notion of interpretability. The claim is not that the network is globally transparent in an unrestricted sense, but that its nonzero attention weights correspond to semantically valid edges in the hypergraph, enabling practitioners to trace which news or sentiment nodes drove a forecast. This suggests an interpretation regime in which attribution is structurally constrained rather than inferred solely after prediction.

7. Positioning and stated significance

The paper summarizes CSHT as principled and practical for five stated reasons: causal grounding, because hyperedges are derived from statistical Granger tests; geometric consistency, because spherical embeddings use angular distance to capture latent financial relations; structured attention, because causal masks prevent spurious dependencies; interpretability, because every nonzero attention weight corresponds to a semantically valid edge; and empirical superiority, because the model improves on the reported baselines across MAE, regime accuracy, and NDCG. It also states that causal sparsity and near-linear runtime scaling permit application to large portfolios in real time (Harit et al., 5 Oct 2025).

In conceptual terms, CSHT occupies the intersection of economic time-series testing, manifold-constrained representation learning, and Transformer-based sequence modeling. The model’s defining contribution lies in treating these not as modular additions but as a single constrained forecasting system: Granger tests determine the admissible hypergraph, hyperspherical geometry governs representation and projection, and masked attention restricts information flow to temporally and statistically validated paths. The result, as characterized by the paper, is a framework intended to deliver reliable, transparent, and high-performance financial forecasts under uncertainty.