TRA-s: Topological Residual Asymmetry with Smoothing

• The paper introduces TRA-s, a robust method for inferring causal direction in additive-noise models using geometric persistent homology of regressor-residual clouds.
• It employs a binning strategy on reverse residuals to induce a one-dimensional signal, restoring identifiability when noise does not vanish.
• Extensive experiments validate its theoretical consistency, computational efficiency, and superior performance over traditional causal inference methods.

Topological Residual Asymmetry with Smoothing (TRA-s) defines a robust criterion for inferring bivariate causal direction in additive-noise models (ANMs) under fixed noise, leveraging geometric signatures obtained from the persistent homology of regressor-residual clouds. It addresses limitations of the original TRA method, which loses discriminatory power when the noise level is fixed and does not vanish. TRA-s employs a binning strategy that induces a one-dimensional geometric signal in the reverse residual direction, restoring identifiability. It is theoretically consistent, computationally practical, and empirically validated across diverse synthetic and real-world benchmarks (Bouchattaoui, 31 Jan 2026).

1. Motivation and Formal Definition

In the classical Topological Residual Asymmetry (TRA) method, causal direction is inferred by comparing the geometric structure of two residual clouds after copula standardization: the forward direction $X \rightarrow Y$ produces residuals that are approximately independent, yielding a 2D bulk, while the reverse direction $Y \rightarrow X$ under small noise collapses to a one-dimensional tube. The separation is quantified by a 0D persistent-homology functional computed from the Euclidean minimum spanning tree (MST) on the residual cloud, with the raw TRA score $\Delta_n$ providing directionality.

However, under fixed additive noise, the reverse residuals do not collapse; their distribution remains thick, and the original TRA statistic loses its discrimination. TRA-s overcomes this by binning reverse residuals along the $Y$ copula axis and averaging, converting the "thick" cloud into a binned average that again collapses to a 1D signature at mesoscopic scale, while forward residuals retain a 2D structure.

Formally, let $\{(X_i, Y_i)\}_{i=1}^n$ be sampled from an additive-noise model $Y_i = f(X_i) + \varepsilon_i$ with $\varepsilon_i \indep X_i$, $\operatorname{Var}(\varepsilon) = \sigma^2 > 0$. TRA-s proceeds by cross-fitting regressors in both directions, computing residuals, copula-standardizing, binning the reverse direction, and calculating the MST-based persistence profiles over fixed mesoscopic windows. The TRA-s score $\widetilde{\Delta}_n$ is then the difference in windowed TP-profiles between the forward and binned-reverse clouds. If $\widetilde{\Delta}_n > 0$, the method infers $X \to Y$; if $\widetilde{\Delta}_n < 0$, $Y \to X$ (Bouchattaoui, 31 Jan 2026).

2. Binning and Copula Standardization Procedure

TRA-s modifies the reverse residual cloud via a binning strategy following rank-copula standardization:

• For the reverse direction ($X | Y$), only the $Y$ coordinate is rank-transformed: $U_i = \frac{\mathrm{rank}(Y_i)}{n+1}$.
• $[0,1]$ is partitioned into $B_n$ equal-width bins: $I_{n, b} = (\frac{b-1}{B_n}, \frac{b}{B_n}]$, $b = 1, \dotsc, B_n$.
• For each bin, empirical bin centers $\bar{u}_b$ and mean residuals $\bar{r}_b$ are computed over the points within the bin.
• The binned cloud is $$\widehat{\mathcal R}^{(n)}_{X\mid Y} = \{(\bar{u}_b, \bar{r}_b): b = 1, \dots, B_n\} \subset [0,1] \times \mathbb{R}$$.

The bin number $B_n$ is chosen such that $B_n \to \infty$, $B_n = o(n)$, and $B_n^{7/3} \log B_n / n \to 0$, ensuring asymptotic validity without over-smoothing or excessive bias (Bouchattaoui, 31 Jan 2026).

3. Persistent Homology Functional and TRA-s Statistic

The core geometric statistic is a normalized windowed profile of edge-lengths in the MST of the residual cloud, specialized to a "soft window" $[\alpha, \beta]$ that captures mesoscopic geometry:

$\overline{\TP_0^{[\alpha, \beta]}(\mathcal{R})} = \frac{1}{(M-1)(\beta-\alpha)} \sum_{e \in \operatorname{MST}(\mathcal{R})} \Psi_{\alpha, \beta}(\|e\|),$

with

$$\Psi_{\alpha, \beta}(t) = (\min\{t, \beta\} - \alpha)_+ = \begin{cases} 0, & t \le \alpha, \ t - \alpha, & \alpha < t < \beta, \ \beta - \alpha, & t \ge \beta. \end{cases}$$

• For the forward ($Y \mid X$) copula cloud, set $M = n$, $\alpha_n = \kappa n^{-2/3}$, and $\beta_n = c_\beta \alpha_n$.
• For the binned-reverse cloud ($X \mid Y$), $M = B_n$, $\widetilde\alpha_n = \kappa B_n^{-2/3}$, $\widetilde\beta_n = c_\beta \widetilde\alpha_n$.

The TRA-s test statistic is then: $\widetilde{\Delta}_n = \overline{\TP_0^{[\alpha_n, \beta_n]}(\widetilde{\mathcal R}^{(n)}_{Y\mid X})} - \overline{\TP_0^{[\widetilde\alpha_n, \widetilde\beta_n]}(\widehat{\mathcal R}^{(n)}_{X\mid Y})}.$ The direction is assigned according to the sign of $\widetilde{\Delta}_n$ (Bouchattaoui, 31 Jan 2026).

4. Theoretical Guarantees and Assumptions

Under the following conditions (cf. (Bouchattaoui, 31 Jan 2026), Assumptions 3.5–3.7):

• Forward model: $f \in C^1$, branchwise invertible, $|f'| \in [c_f, C_f]$, noise sub-Gaussian.
• Reverse conditional mean $m(y) = \mathbb{E}[X \mid Y = y]$ continuous.
• Regression errors: cross-fitted regression risk $o(1)$ and $\max_i |\widehat{g}(Y_i) - m(Y_i)| = o(\alpha_n)$.
• Binning: $B_n \to \infty$, $B_n = o(n)$, $B_n^{7/3} \log B_n / n \to 0$.
• Reverse fluctuations: $\|X - m(Y)\|_{\psi_2 | Y = y} \le K_0 \sigma$.

Theorem 3.2 (Bouchattaoui, 31 Jan 2026) establishes that, as $n \to \infty$,

$\overline{\TP_0^{[\alpha_n, \beta_n]}(\widetilde R_{Y|X})} \to 1, \qquad \overline{\TP_0^{[\widetilde\alpha_n, \widetilde\beta_n]}(\widehat R_{X|Y})} \to 0,\qquad \widetilde{\Delta}_n \to 1,$

in probability. Therefore, a rule that declares $X \to Y$ when $\widetilde\Delta_n > \tau_n \downarrow 0$ is consistent with vanishing abstention probability (Bouchattaoui, 31 Jan 2026).

5. Algorithmic Procedure and Computational Considerations

The TRA-s method is implemented as follows:

1. Cross-fit forward and reverse regressors using $K$-fold splits, producing residuals $r_i^{(Y|X)}$ and $r_i^{(X|Y)}$.
2. Copula-standardize the forward cloud using both $(X, r^{(Y|X)})$ coordinates.
3. Bin the reverse residuals along $Y$ copula with $B_n$ bins, compute bin mean coordinates $(\bar{u}_b, \bar{r}_b)$.
4. Compute MSTs and mesoscopic persistence profiles in both directions.
5. Form the TRA-s statistic and make a directional decision.

Computational complexity per $n$ samples is $O(n \log n)$ for ranking and MST computation (using Delaunay-based MST algorithms), $O(B_n \log B_n)$ for the binned reverse cloud, and regression complexity depends on the choice of estimator (e.g., smoothing splines).

Parameter choices for $B_n$ (recommended $n^{0.4}$) and mesoscopic scaling $(\kappa, c_\beta)$ are robust; $B_n$ must neither be too small (under-smoothing) nor too large (over-smoothing), ensuring resolution of the 1D collapse in the reverse cloud.

6. Empirical Validation and Performance

Extensive experiments illustrate the efficacy and stability of TRA-s:

• On synthetic ANMs with various forms (cubic, near-linear, heteroscedastic, non-monotone), TRA-s achieves low directed risk and matches theoretical predictions, while most established baselines (RESIT, IGCI, RECI, CDCI, COMIC, RCC, NCC) exhibit regular failure modes under stress (Bouchattaoui, 31 Jan 2026).
• Under confounding, TRA-s combined with the abstention/correction procedure (TRA-C) abstains appropriately, whereas alternatives tend to commit to a potentially incorrect direction.
• On the Tübingen real-world benchmark, TRA-s delivers optimal coverage and second-lowest risk, with the abstaining TRA-C attaining highest decided accuracy and minimal overall risk.
• Ablation studies confirm that smoothing of the reverse direction is necessary for robust separation under fixed noise, and performance is stable to tuning in the recommended $B_n$ and scaling parameter ranges.

7. Practical Usage and Limitations

TRA-s is applicable to bivariate causal inference under additive-noise settings, robust to moderate regression error, and computationally tractable for moderate to large $n$ given efficient MST computation. Binning is central: it restores the geometric signature required for correct causal inference when raw residual dispersion precludes a 1D collapse. The approach is grounded in persistent-homology theory, but performance may degrade for non-ANM or heavily confounded distributions (necessitating abstention via TRA-C).

A plausible implication is that TRA-s, by explicit geometric regularization of the residual cloud, defines a broadly applicable, nonparametric, and theoretically rigorous template for distribution-based causal inference that is robust to non-vanishing noise scales, provided the core bivariate ANM structure holds (Bouchattaoui, 31 Jan 2026).