Causal Ordering for Structure Learning from Time Series
Authors: Pedro Sanchez
Damian Machlanski
Steven McDonagh
Sotirios A. Tsaftaris
Venue: Accepted by TMLR ()

Abstract: Predicting causal structure from time series data is crucial for understanding complex phenomena in physiology, brain connectivity, climate dynamics, and socio-economic behaviour. Causal discovery in time series is hindered by the combinatorial complexity of identifying true causal relationships, especially as the number of variables and time points grows. A common approach to simplify the task is the so-called ordering-based methods. Traditional ordering methods inherently limit the representational capacity of the resulting model. In this work, we fix this issue by leveraging multiple valid causal orderings, instead of a single one as standard practice. We propose DOTS (Diffusion Ordered Temporal Structure), using diffusion-based causal discovery for temporal data. By integrating multiple orderings, DOTS effectively recovers the transitive closure of the underlying directed acyclic graph (DAG), mitigating spurious artifacts inherent in single-ordering approaches. We formalise the problem under standard assumptions such as stationarity and the additive noise model, and leverage score matching with diffusion processes to enable efficient Hessian estimation. Extensive experiments validate the approach. Empirical evaluations on synthetic and real-world datasets demonstrate that DOTS outperforms state-of-the-art baselines, offering a scalable and robust approach to temporal causal discovery. On synthetic benchmarks spanning $d{=}3{-}6$ variables, $T{=}200{-}5{,}000$ samples and up to three lags, DOTS improves mean window‑graph $F1$ from $0.63$ (best baseline) to $0.81$. On the CausalTime real‑world benchmark (Medical, AQI, Traffic; $d{=}20{-}36$), while baselines remain the best on individual datasets, DOTS attains the highest average summary‑graph $F1$ while halving runtime relative to graph‑optimisation methods. These results establish DOTS as a scalable and accurate solution for temporal causal discovery. Code is available at \url{https://github.com/CHAI-UK/DOTS}.

Published in Transactions on Machine Learning Research (11/2025)
Causal Ordering for Structure Learning from Time Series
Pedro P. Sanchez'
pedro.sanchez4edd.coo
School of Engineering, University of Edinburgh, UK
Damian Machlanski"
d.machlansk@@ed..cm wk
School of Engineering, University of Edinburgh, UK
Causality Healthoare AT Hub (CHAT), UK
Steven McDonagh
mmoonnggh@dd.cco ackuk
School of Engineering, University of Edinburgh, UK Causality in Healthoare AT Hub (CHAT), UK
Sotirios A. Tsaftaris
s. s.tsattaris@ed
School of Engineering, University of Edinburgh,
Causality in Healtheare AI Hub (CHAT), UK
Reviewed on OpenReview: https: openreview. net/ forum? id- hWuTagggS
Abstract
Predicting causal structure from time series data is crucial for understanding complex phc- nomena in physiology, brain connectivity, climate dynamics, and socio-economie behaviour. Causal discovery in time series is hindered by the combinatorial complexity of identify- ing true causal relationships, especially as the number of variables and time points grows. common approach to simplify the task is the so-called ordering-based methods. Tra- ditional ordering methods inherently limit the representational capacity of the resulting model. In this work, we fix this issue by leveraging multiple valid causal orderings, instead of single one as standard practice. We propose DOTS (Diffusion Ordered Temporal Structure), using diffusion-based causal discovery for temporal data. By integrating mul- tiple orderings, DOTS effectively recovers the transitive closure of the underlying directed acyclic graph (DAG), mitigating spurious artifacts inherent in single-ordering approaches. We formalise the problem under standard assumptions such as stationarity and the addit- ive noise model, and leverage score matching with diffusion processes to enable efficient Hessian estimation. Extensive experiments validate the approach. Empirical evaluations on synthetic and real-world datasets demonstrate that DOTS outperforms state-of-the-art baselines, offering scalable and robust approach to temporal causal discovery. On syn- thetic benchmarks spanning d=3 - variables, T'=200 5,000 samples and up to three lags, DOTS improves mean window-graph F1 from 0.63 (best bascline) to 0.81. On the CausalTime real-world benchmark (Medicall, AQI, Traffic; 2-20-366 while baselines re- main the best on individual datasets, DOTS attains the highest average summary-graph F1 while halving runtime relative to graph-optimisation methods. These results establish DOTS as scalable and accurate solution for temporal causal discovery. Code is available htt.thb..cm//HHATU///DTTSS
"Equal contribution.

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Published in Transactions on Machine Learning Research (11/2025)
1
Introduction
Understanding cause-effect relationships from time series data essential in fields like biology (Marbach
al.
2009
neuroscience (Friston al. 2003), climate science (Runge et al. 2019), and economics (Pamfi)
et al.
2020),
where uncovering how event influences another can lead to valuable insights and better
prodictions. A key challenge in temporal causal discovery [Granger] 1969. Peters et al.) 2013 Nauta et al.
2019 Runge 2020) is the combinatorial complexity- there are many possible ways in which variables can inftuence each other, making it difficult to identify the true causal structure. The goal is to discover a
temporal Directed Acyclic Graph (DAG), 9, representing these relationships. The core problem is illustrated in Figure
Causal ordering approaches (Verma & Pearl 1990 Friedman & Koller 2003 Bühlmann et al. 2014 Rolland
et al. 2022 Sanchez et al. 2023) offer scalable alternative to direct graph estimation by constraining the search space, reducing it from a full adjacency matrix to a set of ordered permutations. While this reduction scales efficiently with respect to the number of variables and samples, it compromises representational power: a single causal ordering can imply extra edges, spurious artifacts, that are absent in the original DAG, other words, commitling to a single arbitrary ordering has an inherent downside: every node deemed a potential ancestor of all subsequent nodes, creating spurious edges that must be pruned heuristically. To mitigate this, existing methods employ two-step process, wherein feature selection post-processing step prunes spurious artifacts introduced by the ordering.
Crucially, a DAG generally admits multiple valid orderings consistent with edge directions. Each ordering contributes complementary information about the underlying causal structure. By systcmatically generating and aggregating information from diverse orderings, we can filter out the spurious artifacts specific to any single, arbitrary ordering choice. As we aggregate more orderings, we converge towards the transitive closure (G+) of the underlying temporal DAG. G contains all true direct edges present in G. While G also includes edges representing indirect causal pathways, it represents the stable, necessary ancestral relationships to recover the true graph. Therefore, harnessing several orderings increases both precision and recall and does so without incurring the combinatorial cost of full graph scarch. Our central insight is to embrace this multiplicity rather than fight it.
The benefit of multiple orderings hinges on access to a diverse collection of valid orderings. Naäve resampling or greody houristies tend to revisit ncar-duplicate permutations. We address this limitation by drawing on denoising diffusion models (Song & Ermon 2019 Ho al. 2020 Sanchez et al., 2023). Once trained to approximate the data score, single diffusion network delivers Hessian estimates at many noise scales; applying leaf-detection rule at cach scale yields fresh ordering. Because different scales emphasisc different frequency bands of the data, the resulting orderings cover the search space more uniformly. Moreover, diffusion training amortises computation after one network fit we can sample hundreds of orderings without
Input x Txd
Target G
aj
2
L+2
Learn structure
4+1
442
a
2
I+2
Time
Figure 1: Temporal causal discovery estimates, from raw time series data (left), the underlying temporal causal DAG G (right).
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retraining- serrining-mmaking the approach viable for datasets that are large w.r.t. variables and samples. Finally, causal ordering approaches have not been extensively explored for temporal causal discovery, but their principles naturally extend to time-dependent data. Our focus on the temporal setting is motivated by two koy advantages: (i) temporal constraints provide additional structural information that enhances ordering reliability through the temporal priority principle, and (ii) the temporal priority principle offers a natural filtering mechanism for spurious edges. The temporal dimension provides an inherent set of causal orderings, each contributing incremental insights.
Building on these ideas, we introduce DOIS (Diffusion-Ordeee Temporal Structure), a theoretically motivated ensemble-based approach for temporal causal discovery that systematically aggregates multiple diffusion-generaled orderings. Our contributions are thrre-fold:
We establish theoretically and empirically that aggregating multiple causal orderings to estimate the transitive closure (g+) provides more robust foundation for temporal cansal discovery than tradi- lional single-ordering approaches, effectively filtering spurious edges arising from arbitrary ordering choices.
(ii) We introduce DOTS, a novel algorithm leveraging multi-scale diffusion models to efficiently generate the diverse set of causal orderings required for robust aggregation in the temporal domain.
(iii We provide extensive benchmarks, adapting several static ordering methods for temporal data and demonstrating the superior performance of the multi-ordering strategy implemented by DOTS against both these adapted methods and state-of-the-arl. lemporal causal discovery baselines on synthetic and real-world datasets.
The rest of the paper is organised as follows. Section reviews notation and identifiability assumptions. Section 3 estaabiihes the theoretical foundations of multi-rrdering aggregation, and Section 4 presents the DOTS algorithm in detail. Section shows experimental results. Relevant literature is discussed in Section Scction 7 concludes the paper.
2
Preliminaries
2.1
Notation
We present in Table D description of all the symbols in the manuscript.
2.2
Problem Definition
We aim to discover the causal structure among d variables arranged in temporal setting. Let x' Rd denote veclor-valued random variable at time I, with components (xi, ...,.... x.). A temporal directed acyclic groph (temporal DAG) g then describes the causal relationships among these variables, where each node corresponds to a component xf, and the directed edges represent causal links between variables at the same time or across time steps (Eichler 2012). We now introduce standard assumptions for causal discovery from time series:
Assumption 1 (Temporal DAG). The true causal structure of the data can be represented by a temporal DAG g.. This graph includes lagged edges of the form x + x5 (for F > 0) and contemporaneous edges x - xf. This assumption effectively captures the temporal priority principle (Hume) 1904 Rankin & McCormack
2013 Assaad al. 2022) where causes precede effects in time. For any pair (x] x)), an edge x; x
indiaates that the value of x; at time (f-r) influences X, at time t. This forbids backward-in-timm causation and helps simplify structure learning since the search space is smaller.
Assumption (Stationarity). The causal relationships in G remain invariant over time; that is, both the causal links and their strengths remain unchanged for all values of t.
Assumption 3 (Time Scries Models with Independent Noisc (TiMINo)). The structural causal model follows temporal additive noise formulation:
x5 = =s5( (Pa (x1)) + c/p
(1)
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Table Summary of notation used throughout the paper.
Symbol
Mcaning / role in the paper
x E I2d
Random vector of d variables; component i is x,.
rls x!
Value / vector at time index t.
d
Number of variables (dimensionality).
Number of observed time steps.
Time lag; x +xj.
Tinax
Maximum lag included in A.
k E {0, kmmx)
Diffussio (noise-scale) timestep.
kmax
Final diffusion step (fully noised).
Causal ordering (topological permutation). Variable at position in ordering π. (Temporal) causal DAG.
G+
Transitive closure of G.
E {0, 1} d(max xd +1)
Temporal adjacency matrix to be learned. Data distribution (density).
p(x)
(x,x, k))
Neural network estimating Vx log p(x) at scale k.
x
Noisy version of x at diffusion step
g(*") (x,
Forward noising distribution.
Structural function generating x, from its parents.
Independent noise term for x².
Page (x))
Parent set x in the temporal DAG.
Ch((s)
Children of node Xy.
H (log (logp(x))
(inj)-entry of Hessian of logp(x) (score Jacobian). Edge set implied by ordering w.
(())
Elm)
Intersection edge sets from m sampled orderings. Set of temporally valid edges (causes precede effects).
T
(m,.))
Aggregated edge set restricted to T.
Wij
Fraction of sampled orderings where edge i-t appears (vote matrix).
Aij
Soft transitive-closure entry after thresholding W.
0
Threshold for vole-matrix aggregation (0 0 < 1).
on
Variance- perfance-preserving coefficient in the forward diffusion process.
where Page (xf) denotes the set of parent. variables of x in ge, ʃj is nonlinear function, and e is an independent noise term. TiMINo (Ptters al. 2013) essentially extends the additive noise model (ANM) framework to time scries.
Assumption (Causal Sufficiency). All common causes of observed variables are measured; that is, there are no unobserved confounders that influence multiple components simultaneously.
Assumption 5 (Noise Distribution Regularity). The exogenous noise terms e/ in Equation have densities p" such that 8² loge" is constant. This includes Gaussian noise special case but also encompasses other distributions with quadratic log-densities. This regularity condition is required for the score-matching approach to satisfy the leaf-detection criterion in Equation 3 and extends the assumptions of Sanchez et al. (2023) to the temporal domain using the TiMINo identifiability framework (Peters et. al. 2013).
2.3 Objective
Given an observational multivariate time series dataset E Rxx containing T timesteps of d variables, our goal to learn the temporal adjacency structure A E {0, 1] 1) maxinum lag that is being captured in the adjacency matrix. Each entry of A encodes whether there is a directed edge from x!-r to x for 0 S 7 Tmax- Note that indicates contemporaneous link x xf.
where Tmax denotes the
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2.4 Key Identifiability Results
Under stationarily, the additive noise assumplion, and causal sufficiency, Peters el. al. (2013) show that temporal causal relationships become identifiable if the data follow restricted structural equation model in which cach noise term is statistically independent and no directed cycles cxist within a single time slice. Specifically, their Time Series Models with Independent Noise (TiMINo) framework demonstrates that both lagged and instantaneous effects can be recovered uniquely, provided the functional form and noise distribu- tions meet certain identifiability criteria (e.g. linear Con-Gausiaan or nonlinear Gaussian).
2.5 Causal ordering
Causal search over the space of DAGs is an NP-hard problem (Chickering 1996). Traditional approaches leverage heuristic search strategies to navigate the combinatorial space of potential DAG structures. Order- based search offers simpler and more effective alternative. By shifting the search from graph structures to node orderings, the strategy exploits the fact that, for given ordering, identifying the highest-scoring network is not NP-hard. Such causal ordering approaches reduce the search space and inherently satisfy the acyclicity constraints. This bypasses the need for explicit, acyclicity checks during the search. These methods find a particular causal ordering of the nodes, i.c. a list of nodes such that a node in the ordering can be a parent only of the nodes appearing after in the exact ordering. Causal ordering is also known as topological ordering or a causal list in the causal discovery literature (Peters et al., (017). Formally, causal ordering of DAG G is defined as non-unique permutation of d nodes. Hence, a given node in always appears before its descendants in the list. Or more formally, n; T; < j E De(x;) where De(x;) are the descendants of the ilh node in G (Appendix B in Peters et al. 2017
2.6 Causal ordering via score matching
Rolland et al. (022) propose that the score of an ANM with distribution p(x) can be used to estimate the causal ordering by finding leaves. Leaves are nodes of DAG g that do not have children. RRoland et al. (2022) propose a method to find leaves based on the derivative of the ANM log density (also called score). An analytical expression for the score of an ANM from Equation is
Xx, logp(x) ① loggp" (x, (5) Ox,
05. 2 Ologggg (x; Si)
0x,
dr
(2)
(ECM())
where Ch(xx) denotes the children of Xy. Using this analytical equation of Vx, log p(x), Rolland et al. (2022) derive the following condition used to find leaf nodes. Given a nonlinear ANM with a noise distribution pd and a leaf node j, assuming that 8 logp" a, where is constant, then
D="
Vard [ j (log p(x)) 0.
(3)
This rule is based on the score's Jacobian (or Hessian of the log distribution). Hjg (log is used in Rolland et al. (2022) to propose a causal ordering algorithm that iteratively finds and removes leaf nodes from the dataset. Rolland et al. (2022) re-compute the score's Jacobian with a kernel-based estimation method at each iteration.
2.7 Approximating the score's Jacobian via diffusion training
Instead of computing l(x) via a kernel-based estimation (Li & Turner 2018 Rolland et al.) 2022). wo
follow Sanchez et al. (2023) and estimate the score's Jacobian with diffusion models (Song & Ermon 2019 Ho et al. 2020). This estimation is based on a diffusion process that progressively corrupts x with Gaussian k.max.
noise over timesteps E (0....,kmex). J. Let & be the noisy version of x at diffusion step k. A neural network co(**, k) is trained to denoise x back to x, thereby approximating the true score Vx p(x). Formally, this can be written as:
Expp((x)) stng(R" xx.)) (6 (x* - Vga log [R)]"²
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where q(** x, k) defines the forward noising process. Once trained, co effectively yields Vx logog(()) various noise scales k; which can be used to estimate the Hessian for causal discovery. Noise at multiple scales explores regions of low data density (Song & Ermon 2019
The score's Jacobian can approximated by learning the score Eo with denoising diffusion training of neural networks and back-propagating (Rumelhart al. 1986 from the output to the input variables. The quantity can be written, for an input data point x E R as
Hi ((x) (o,,o
where Ea, k)) means the ith output of Eg is backpropagated to the jth input. The diagonal of the Hessian in Equation can be used find leaf nodes as in Equation We use masking (Sanchez et al. 2023 to iteratively find and remove leaf nodes, without retraining the score.
3 Theoretical motivation: temporal structure from multiple causal orderings
Causal ordering methods, illustrated in Figure traditionally rely on single ordering to infer the full DAG in causal discovery. A single causal graph does not contain sufficient information to reliably infer the DAG, since each node is considered a cause for all subsequent nodes. Therefore, pruning methods are used to remove spurious edges (Bühlmann et al., 2014). With infinite data perfect oracle could delete the spurious edges and retain the true ones. In practice we face finite samples, noisy tests and high dimensionality. Starting from an over-dense candidate set inflates both kinds of statistical error: (i) false posilives remain whenever a test. fails reject a truly absent. edge (Lype-IT error), and (ii) false negatives appear when a test mistakenly deletes true edge (type-1 error) because that edge co-varies with many irrelevant ancestors in the initial ordering. Hence single ordering often yiclds fragile estimate whose quality varies wildly with sample size and noise level. In contrast, leveraging multiple causal orderings provides a richer representation of the underlying structure. Exploiting multiple valid causal orderings from data naturally follows from the fact that a given DAG typically admits more than one linear ordering, consistent with its structure.
X
DAG ordered list
X2
X
X2
X4
X
X1
X4
Figure 2: A DAG can be represented as an ordered list following the causal direction. A node in the ordering can cause any subsequent node. Scarching over the space of permutations is more efficient than scarching over the 2D space of matrices. However, topologically sorting nodes reduces the amount of information in the representation of causal relationships.
Rather than committing to single topological sort, estimating multiple valid orderings can offer more complete representation of ancsstor-descendant relationships. In doing so, we can aggregate local adjacency constraints from each ordering and thereby recover, asymplotiaally, the trunsitive closure of the DAG the minimal set of edges that preserves causal reachability. This perspective avoids over-specifying the order of variables that are not causally linked and reduces the risk of introducing extra edges that do not exist in the underlying temporal DAG.
The Jacobian of neural network can efficiently computed with auto-differentiatiio libraries such as functorch Hor-
ace He 2021

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3.1 Recovering DAG structure from complete set of causal orderings
We now investigate how the topological orderings of DAG relate to its underlying structure. The central takeaway is that the collection of all topological orderings of a DAG uniquely determines the transitive closure g+, which is a robust representation of the graph's reachability structure. Topological orderings capture the reachability relation of the graph-a partial order- but multiple DAGs can share the same reachability relation and, consequently, the same set of topological orderings.
In DAG G, every topological ordering is a linear extension of the reachability relation R. A key insight from order theory is formalised in Hiraguchi's theorem (Hiraguchi 1955), which establishes that any finite partial order can be exactly recovered as the intersection of all its linear extensions (i.e. total orders consistent with it). In our setting, this implies that aggregating multiple topological orderings of a DAG asymptotically recovers its transitive closure. This idea is formalized in the following proposition:
Proposition 1 (Reconstruction of the Transitive Closure). Let G = (V, be DAG, and let & be the set of all its topological orderings. Define a binary relation < on V by:
x x
=
x. appears before y every a L.
Then:
(1) < is strict partial order on V (trreftexive and transitive).
(i) For all n. y E V,
x
there erists divected path from x to y in G with x # y.
Thus, < matches the edges of the transitive closure g+.
(iii) Consequently, aggregating all topological orderings in L recovers G+, but not necessarily the original DAG G.
Justification. We show part (2) by establishing the equivalence:
(=)) If there is a directed path from & to y in G with x # y, then every topological ordering π E C must place æ before y to respect the direction of edges. Hence, x - y.
(=): Suppose < y but no directed path from x to exists in G with * y. Since G is DAG and no path exists from x to y, adding the edge (y, to G does not introduce a cycle (otherwise, a path from x to would exist, contradicting the assumption). In this modified DAG, there exists a topological ordering with y before æ, contradicting x - y. Thus, a directed path from x to y must. exist in G.
This shows that < corresponds to the strict reachability relation, i.e. the edges of g+. Since distinct DAGs can share the same g+, the original G cannot be uniquely recovered from L.
Example 1. Let G1 = (V, with V (a, c} and Fn ((a,b), (b,c)), and let G2 = (V, with Es (),, (a, Both DAGs share the same set of topological orderings: ba, In G1, the ordering respects a + b and b + c; in G2, the additional edge (a, c) is consistent with the order. The transitive closure for both is G G = (V, [a, (), (b, c), (a, c))). Thus, from the common ordering alone, we recover G (or G+ but cannot distinguish between G1 and G2.
Aggregating all topological orderings of DAG G yiclds its transitive closure g+ because the relation x captures all pairs of nodes connected by a directed path. However, since distinct DAGs can share the same transitive closure, the original edge set E remains ambiguous. This highlights a fundamental limit: while topological orderings reveal the reachability structure of the graph, they do not specify the precise topology of the DAG. It is crucial understand that the transitive closure G+ includes both direct edges from the original DAG and indirect relationships (edges representing multi-step causal pathways). While this

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means that G is not identical to the true causal DAG G, it serves as robust intermediate representation that captures all ancestral relationships. The subsequent pruning step (Section 4.4) is specifically designed to distinguish between direct and indirect relationships, refining G to recover the sparse structure of G. Nevertheless, recovering the transitive closure G+ a stronger claim than what has been previously achieved by single ordering methods (CAM, SCORE, DAS, NoGAM, DisTAN). A single ordering recovers only a weak approximation of the causal structure because, for any given position in the ordering, all subsequent positions are considered as potential descendants, introducing spurious edges in addition to possible indirect edges.
3.2 Ordering aggregation and recovery of temporal structure
In causal discovery, we do not know priori the total number of valid causal orderings that a resulting DAG will admit. The number of causal orderings is not only related to the number of variables but also to the edge topology and density. Therefore, we next explore how aggregating subset of the total orderings improves structure estimation. Empirical evidence, shown in Figure 3 suggests that enumerating all topological sorts is unnecessary to achieve strong performance, as a relatively small subset of randomly sampled orderings can suffice for accurate structure recovery.
To formalize the benefit of multiple orderings, assume that for cach valid causal ordering of DAG g, we obtain an edge set E(x) that corresponds to the directed edges implied by that ordering. Define the aggregated edge set over the intersection of orderings
((m)
n E(ms).
Then, under the assumption that each valid ordering provides complementary information about the true ancestral relations, following Proposition 1
we have
lim TR
((m) M8 ugg
g+
where G+ denotes the transitive closure of the true edge set E of and where M is the number of possible orderings for G.
To validate this idea, we generate random temporal DAGs and use Kahn's algorithm Kahn 1962 to list all possible orderings with lopological sorting. Then, we estimate the transitive closure for each DAG and
compare it with the estimated (m)) as increase m for each DAG. We compare the estimated and true transitive closure via Fl score. In Figure 3 we show the percentage of orderings M * 100 used with respect to the F1 score. For our data, we observe that 400% of all orderings is generally enough to recover the transitive closure with high Fl score.
3.3 Incorporating Temporal Constraints
Temporal constraints are critical when recovering causal structure from time series data. Following the temporal priorily principle; causes precede effects, in time. This principle forbids backward-in-tine causation and simplifies structure learning.
To incorporate this constraint into edge aggregation, restrict the aggregated edge set to include only temporal edges satislying ( 1- T < L. Formally, let.
T= (xt 12 (x) t t
denote the set of all temporally valid edges. Then, define the temporally constrained aggregated edge set as
F((.T) F((m) NT.
This filtering ensures that only edges adhering to the temporal priority principle are retained, thereby excluding any spurious backward-in-time connections and further enhancing the reliability of the recovered DAG. Empirically, as illustrated in Figure B adding the temporal constraint decreases the number of causal orderings required to estimate the causal structure for a given F1 score.
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F1 of multiple orderings with and without temporal constraint
1.00
0.95
0.90
Score
0.85
0.80
0.75
Temporal
Unconstrained
0.70
Constrained
10%
40%
70%
100%
Percentage of orderings
Figure 3: Impact of multiple causal orderings on DAG recovery. A single (or few) ordering (left) may include extra or spurions edges, whereas aggregating multiple orderings (right) more accurately recovers the full transitive closure of the underlying DAG.
In summary, integrating temporal constraints into the aggregation of multiple causal orderings is both a natural extension of, and an efficient strategy for, structure learning in time series data. By leveraging the inherent temporal priority principle-wlest where causes always precede effects we effectively filter and refine the aggregated edge set, ensuring that only temporally valid connections are retained. This dual approach not only enhances the robustness of the recovered causal structure by capturing complementary ancestral information across orderings but also reamlinee the learning process, as il reduces the effective search space and mitigates spurious dependencies.
While our empirical validation demonstrates the theoretical potential aggregating multiple causal orderings for recovering the transitive closure of DAG, is important to note that this evaluation was performed known graphs where all valid causal orderings were available and the total number of orderings predetermined. Orderings were selected uniformly at random from this complcte set. We further note that for practical applications, an algorithm may generate very similar orderings, thereby limiting the diversity necessary for robust structure recovery. In the following section, we explore methods to induce sufficient variability in the generated orderings, enabling the aggregation process to remain both effective and efficient in recovering the true causal structure.
4 Dased Approach for Temporal Discovery
We introduce DOTS, a method that utilises diffusion processes recover multiple valid causal orderings in temporal data. Our approach, illustrated in Figure 4 integrates both frequency domain perspective and multi-scale causal ordering to capture the complex structure of temporal relationships. We refer to our notation (Section 2.1 distinguishing between time lags (T), diffusion timesteps (k), and the indices for causal orderings (T))
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Lag-Embedded Time-Series Data
x2
Diffusion training + causal ordering
Denoising training
Hesslan Computation
Causal ordering per timestep (k)
Multiple Orderings
Single Ordering per Noise Scales de
x_
X3
x/
48
X2
X3
x1
X2
X3
X]
x1
Aggregation
Combine Orderings
CAM Pruning
Single Adjacency
Final Temporal DAG
Figure 4: DOTS pipeline for temporal causal discovery. We start with lag-embeddded lime-series data, apply diffusion-basse single-order discovery, then extend to multiple orderings and aggregate them. The final temporal DAG below shows an example with three variables over three timesteps.
4.1 Why do diffusion steps capture different frequency components?
Different diffusion steps capture distinct aspects of the data, which can be understood from frequency perspective. Consider forward diffusion process (e.g. DDPM (Ho al. 2020)), which decomposes each observation step as
Tk
Van To -ie a e~ ((,
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k + 0
1: - kmaz
1o (w))
10 (w0)
Pzo (v (vrda))
exp( ae W (?)
2en (v (rrw))
exp( L- 2-211/1 (w)P)
Figure 5: Frequency emphasis of diffusion steps. A forward diffusion step decomposes X VagXo + V1 - O&k e. As can be seen in the Fourier domain, high values emphasize learning of low-frequency while low values of k force the network to focus on high-frequency components.
Since zo and independent, the characteristic function of zµ becomes:
2z (w) = 2b Taran (w) @ (w) 2zo (Vorw) dc(V1 -ww).
For Gaussian 6, we have (w) exp(- exp(-$/w1²), the noise component contributes exp(-1=a4 (w|²), which acts low-pass filter with bandwidth inversely related to vI- ag- As k increases, aę decreases, making the filter narrower and emphasizing lower frequencies. This creates natural multi-scale decomposition where large & values capture coarse, low-frequency structure while small & values preserve fine, high-frequency details.
Figure 5 schematically illustrates how each i emphasizes different frequency components; large reveals coarse causal links and small highlights finer edges. This multi-scale view enables the network to focus spectral components most impacted by noise at each scale. Similar conclusions were drawn based observations in the imaging domain [ascenas al. 2023
4.2 Lag-Embedded Representation of Time Series
The diffusion model is trained on the lag-embedded representation of the time series. Lag-embedding is common technique in dynamical systems theory to capture temporal dependencies (Takens 1981). Let -xx'...,x7]] E Txx denote the raw sequence with variables and T time steps. We construct a lag- embedded matrix
D = [x', x x-1 x² Tmax T-1
t E RR('- (I' Tiaae x
Each column of now refers a specific variable lag pair x; -!-T. This representation feeds the diffusion network; temporal precedence is enforced later when we discard any edge that points backwards in time.
Incorporating temporal information For temporal data with lags T, cach variable is indexed of. and palential edges include both lagged (x:-T + x²) and contemporaneous (xf + x²)) relationships. A single diffusion model is trained on the lag-enbedded dataset DD xx and the leaf-finding process is applied in the same manner, ensuring that each node is treated a time-indexed variable. This strategy enforces temporal DAG that captures both lagged and instantaneous dependencies.
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4.3 Multi-Scaal Causal Orderings
Each diffusion step & corresponds to a distinct noise regime, and consequently, the Hessian Hx log p(x) com- puted at each k reveals different adjacency constraints. Large & values tend to highlight broad, low-frequency cause-effect relationships, while small & values accentuate fine-grained, high-frequency interactions.
Multiple orderings at different k. Instead of relying on a single noise scale, we execute a causal ordering Each execution yields a causal ordering, denoted by и, which reflects the partial order implied by that particular This process generates multiple orderings W1...., ws thereby capturing the multi-scale structure inherent in the data. We then identify leaf nodes via the diagonal of the Hessian, following the approach of Rolland et al. (0022) and (Sanchez et. al. 2023
algorithm at several discrete diffiusion steps (Ai...., ks}.
Causal ordering with Hessian from diffusion training. we approximate the partial derivatives as
After training diffusion model E ex(x, k),
á dxy [x)
Hi, log
for each k. The diagonal entries H., exhibit lower variance for leaf nodes than for non-leaf nodes (Rolland
eet al.] 2022 Sanchez et al. 2023). To identify leaf node, we: (i) Estimate Hx. k) on mini-batch of data D. Identify the variable with the lowest diagonal variance Var Hr,e). (iii) Remove 25 from the
distribution by masking out the variables in the input done with DiffAN masking (Sanchez et al. 2023
This procedure is repeated until all variables are assigned an order, yielding a complete causal ordering Repeating this process for each chosen & produces the set {TI,...,,s}.
After this procedure, the set mT} can be aggregated as described Section 4.4 In essence, each Ts represents a valid causal ordering that reflects the partial order constraints emphasized at its respective diffusion timestep kg By uniting these multi-scale perspectives, the DOTS algorithm produces final temporal DAG that captures both coarse (low-frequenc)) and fine (bigh-frequency) causal interactions.
Algorithm Estimating Multi-Scale Causal Orderings.
Require: D DERd
observational time-series > largest lag to consider
Tmax
k)
trained diffusion model approximating
V log p(x) at k [0, kmax] > selected noise scales List of valid causal orderings
K (ki,...,ks) ks)
Ensure: orders
1: function DOTS(D, K, Tmax)
2: orders <- Ø
3: for all E K do
V+ {x]
1...d. d, r 0... T
lag-embedded nodes ordering for this scale
π I
while V # 2 do
H diag f HESSIANDIAGVAN(E, D(: V).
8:
I+ arg min Haing [v]
leaf(ss)
9:
- L π
10:
V V+VV V
end while 12: orders + orders U {m}
11:
13:
end for
14: return orders
15:
end function
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4.4 Aggregating Multiple Orderings
Section 3 showed that taking the intersection of all topological sorts of DAG & yields its transitive closure G+, which is in general superset of g. Because cnumerating every ordering is infeasible, combine finite sample of orderings in two simple steps.
Soft voting. From S orderings "i,....,TT"" obtained at diffusion steps kiSs we form a vote matrix Wi S. [((2))) E n. Throsholding at 0 E (0, 1] produces the soft transitive closure A. = ([W, 2 0}. The extremes and 0 = 1 reduce to the plain union and the hard intersection, respectively; intermediate values let us balance recall against precision.
CAM pruning. The matrix À can still contain indirect or spurious edges. We refine it with the likeli- pruning routine of Bühhmann et al. (2014), removing edges that do not improve the predictive loss of the child variable given its other parents. The result is our final estimate A. This procedure is commonly used across most causal ordering approaches from Section and constitutes an integral part of our method as it allows us to recover DAG g from G by eliminating indirect links.
Practical reliability. Soft voting has the appealing property that any edge appearing in every sampled ordering is retained, whereas edges that never appear are discarded automatically. When (i) the sampler generales a diverse set of valid orderings and (ii) the sample size is large enough for CAM's tests to be informative, Á approximates G+ increasingly well and the pruning step tends to eliminate the remaining indirect links, often recovering G exactly in practice.
5
Experiments
5.1 Setup
Our experimental framework prioritizes replication of results, modularity, and ease of extension. These are features found in the Snakemake (Mölder et al.) 2021) workflow management system, that forms a base for our experimental setup. Snakemake has previously been used for benchmarking purposes in (non-temporal) causal discovery (Rios et al. 2025), from which we draw inspiration. Our codebase is accessible online
5.2 Data
Simulations. Our synthetic Data Generating Process (DGP) is based on the work of Beaunont et al.
(2021) and Lawrence et al.] 2020). Our experimental setup involves DGPs with the following properties: sample size (observed time steps) T E (200, 1000, 2000, 5000]. number of graph nodes E (3, 4, 5, 6), lag size I E {1, 33.. In addition, all setups use non-linear causal mechanisms (piecewise linear and trigonometric) and incorporate the same noise distribution
(O, [0.01, 0.05)). Each setting has been repeated 10 times to obtain robust results. Causal mechanisms and relationships are invariant across time (i.e. they are stationary).
Borrowing the notation from Lawrence et al. 2020 a set of temporal causal links T generated in the DGP is defined as follows:
Te. = (x,(t -) 7) X,(0),, E (1,...,dd).
(5)
with 1 denoting time index. Note that we do not consider instantaneous links in our experiments (r > 0), but do allow for autoregressive relationships (i = j). We also fix the lag size T across all relationships within any single dataset to isolate and study the influence of r on algorithmic performance.
Real datasets. We also perform experiments on datasets closer to real-life complexities. To achieve this, we incorporate CausalTime (Cheng et al. 2024), a realistic benchmark for time series causal discovery. CausalTime provides three datasets: Air Quality Index (AQI), Traffic, and Medical. The AQI data consist of 36 variables, whereas Traffic and Medical have 20. In terms of sample size, all datascts consist of 480 samples,
hhttps//giithbb www./CAAT/U///TAAA
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with time length of T=40. Since DOTS requires a large sample size to perform well, combine all samples (all 480 samples of length T=40 stacked vertically), plus one row of zero to avoid cross-ontaamnationn into single dataset of length T 480 x (40 + 1) - 1=19 679. We apply the samn pre-processing procedure to all three datasets.
AQI: hourly PM2.5 readings from -=36 monitoring stations across China (T=8760). A geographical distance kernel supplies sparse prior graph.
Traffic: average speed measured every min N=20 loop detectors in the San- -rancicnc Bay Area (T=52 116); the prior graph again follows pairwise distance.
Medical: N=20 vital-sign and chart-event channels extracted from 1000 MIMIC-IV ICU slays, re- sampled to 2-h resolution (T-=600 average).
5.3 Algorithms
We compare our proposed method to the following baselines:
Temporal:
Dummy: Returns a fully-connected temporal DAG as naive estimation.
PCMCI (Runge 2020 Employs lagged conditional-indepeneene tests for constraint-basel discov-
ery in autocorrelated time series.
PCMCI+ |Runge 2020): Extends PCMCI with additional conditioning to control false positives.
VARLINGAR (Hyvärinen et al. 2010 Combines linear VAR modeling with non-Ganssian 1CA to recover a unique causal order.
DYNOTEARS (Panfil et al. 2020 Casts temporal DAG learning single continuous optim- ization with an acyclicity constraint.
TCDF (Nauta et al.
2019:: Trains temporal convolutional networks and validates edges via in-silico
interventions.
-TMMNNo (Peters et al.
2013 Applics additivo-noise regressions with independence tests residuals
for both lagged and instant elfects.
Non-temporal:
CAM Bühhmann al., 2014h: Fits additive noise models with restricted maximum likelihood and
basss pruning.
SCORE [Rolland et al.
2022 Uses score-matching to estimate the Hessian variance for iterative
leaf removal.
DAS (Montagna al. 2023c
Scalable ANM ordering via efficient Hessian diagonal estimation.
NoGAM (Montagna et al. 2023b): Generalizes ANM ordering without Gaussian noise assumptions,
leveraging kernelized score estimates.
DiffAN (Sanchez et al. 2023 : Uses denoising diffusion models to approximate the score Jacobian
for fast, retraining-free free causal ordering.
While the ordering- basedd algorithms (CAM, SCORE, DAS, NoGAM, DiffAN were not developed for tem- poral lasks, we still include them our experiments due to our proposed method's 's st roots topological ordering. To make the use of the ordering- methods more appropriate in this temporal setting, we post- process their prodicted graphs by removing the edges that defy the arrow of time. This mild addition is indicated by the 'CC' suffix added to the name of the methods in question (e.g. CAM-C) when reporting the results in Section 5.6 Comparing to temporal methods, however, remains the main validation target for us.
"Separating
needed the end of the data, hence minus the formula.
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5.4 Graph Representation
Temporal causal graphs can be represented al dillerent granularities (Assaad et al. 2022
Window Causal Graph: Restricts the full time causal graph to a finite lag window Tmax: rep- resenting only edges from x--" xi for r Tmax- This trades off completeness for computational leasibility.
Summary Causal Graph: Aggregates causal relationships across time without specifying exact lag indices, creating more compact but less detailed representation as lagged and contemporaneous edges are represented the same way. Autocorrelated variables are represented with self-loops.
Window graphs naturally more suitable to method's case. However, also include summary graphs in study build upon TiMINo that outputs only this type of graphs. Including summary graphs allows us to directly compare to TiMINo.
5.5 Evaluation
The assumption that SCMs are invariant across time (stationarity) results in repeated causal links. Therefore, focus the correctness of the predicted edges that terminate at (non-lagged) time t, that is Tr We then compare predicted edges to the ground truth and calculate True Positives (TP), False Positives (FP)
and False Negatives (FN), from which we obtain Recall, Precision and Fl metrics as follows:
TP
Precision = TP
F1 x Recall x Precision
Recall TP FN
(6)
Recall + Precision
as per Assaad al. 2022). We report F1 on window and summary graphs (Flw and Fis, respoctively) the main metric of interest, but also supplement our results with Recall and Precision.
5.6
Results
5.6.1 Simulations
Figures and 7 show the main results (averages and 95% confidence intervals). In both cases (window and summary graphs), DOTS shows very strong and robust performance across different sample sizes, numbers of features and lag sizes. Apart from clear scparation from the competition, DOTS is also one of the few methods that keeps improving in larger sample sizes (T = 5000). Ordering-basse based mot and PCMCT family perform comparatively, with another diffusion based method (DiffAN-C) coming out on top among these, showing the advantage of diffusion models strongly nonlinear tasks. VARLINGAM, DYNOTEARS and TCDF struggle to outperform the naive baseline that predicts folly-comnected DAGs, suggesting their leniency towards linear settings.
0.8
dummy
CAM-C
SCORE-C
DAS-C
NoGAM-C
POWO
0.3
VARLINGAM
DYNOTEARS
0.2
TCDF
0.1
DOTS (ouns)
200
1000
2000
5000
sample size
number features
number of lags
Figure 6: Fl scores simulated window graphs (Flw). TiMINo does not provide Flw.
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Gummy
CAM-C
0.5
DAS-C
NoGAM-C
PCMCI
PCMCI+
VARL IGAM
0.3
DYNOTEARS
TIMING
DOTS (ouns)
230
sample size
number of features
number of lags
Figure 7: F1 scores on simulated summary graphs (Fls).
Figure 8 shows average running times of all methods across the simulations. DOTS places in the middle among the competitors, providing strong prediction performance at extra computational costs compared to baselines. The runtime of DOTS also scales well that shows its promisc in high-dimensional tasks.
10¹
10
10"
10
SCORE C
runtmmo [8]
10"
10
10"
NoGAM-C
10'
10
VARI INGAM
PCMC
10
10
10
TCDF
10
10"
-0
DIIAN-C
DOTS (ours)
2000
sample
number
lags
Figure 8: Average runtime of each algorithm in seconds obtained on synthetic data. Note the logarithmic scale on the y axis.
5.6.2 Real Datasets
Table
summarises the main findings. Two broad trends emerge.
1. Overall difficulty, Average Fls rarely exceeds 0.45, far below synthetic baselines, underscoring the gap between toy DGPs and real dynamics. Constraint-based PCMCI loses precision the noisier Medical subset, while score-based DYNOTEAM RSS collapses AQI potentially due to its linearity assumplion. On average, however, many temporal methods are competitive except for TCDF and DYNOTEARS.
2. Best-in-class methods. Among approaches with functional assumptions, TEMINo, which leverages the same function assumptions as DOTS, excels on the Medical data. VARLINGAM dominates the spatial datasets, hinting that weak non-Gaussianities suffice for identifiability when the linear VAR fit is adequate, though both TiMINo and PCMCI are not far behind. Our method (DOTS), despite not achieving the top rank on individual datasets, within one standard error of the leader on every subset and achieves the highest aggregate mean, validating the multi-ordering strategy.
Qualitatively, failures tend cluster around long-range edges (large spatial distance in AQI/Traffic or cross-system interactions in Medical). Future work should explore explicit distance-aware regularisation and non-stationary mechanisms to close this performance gap.
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Table 2: Fls (higher is better) on CausalTime. Best and second best per column are highlighted. SCORE,
DAS and NoGAM (non-temporal) have been excluded due to exceeded memory allocation limits.
Type
Method
Medical
AQI
Traffic
Avg. t
non-temporal
CAM-C
0.413
0.351
0.304
0.356
DifIAN-C
0.313
0.356
0.282
0.317
temporal
PCMCI/PCMER
0.427
0.382
0.372
0.393
VARLINGAM
0,457
0.464
0.391
0.437
DYNOTEARS
0.107
0.010
0.315
0.144
TCDF
0.286
0.218
0.000
0.168
TiMINo
0.553
0.429
0.340
0.411
DOTS (ours)
0.548
0.457
0.349
0.451
5.6.3 Ablation on the number of orderings
We now study how the number of causal orderings influences the predictive performance of DOTS. Our theoretical results in Section consider the case where all valid causal orderings are present. In Figure 3 study the impact of the nunber of orderings in ideal scenario where all causal orderings are known. DOTS generates a small, finite set of orderings based on a heuristic: using different diffusion timesteps (k). Therefore, we run this experiment to validate that, indeed, orderings derived from different k are effective in improving performance.
To investigate this, we run DOTS on synthetic data (T 5000) generated from a 3-node graph with 7 1. We repeat the experiment 500 times. The results shown in Figure D Sampling single causal ordering is underperforming
more instances as four orderings show substantial performance improvement, which is on the same performance level ten orderings. Moving further to 15 instances shows a mild improvement, beyond which the performance platcaus. Overall, selecting the number of sampled causal orderings clcarly has a large effect on mothod predictive performance, and while exploring more orderings may guarantee beiter performance, slaying in the range of 10 may often strike the right balance between computational cost and delivered performance. The asymptotic gains are in agreement with the theoretical results presented in Figure 3
needed to arrive at the right graph. On the other hand, few
False posilives (spurious edges) are pruned more effectively because conflicting orderings rarely vole for the same incorrect link, while false negatiues (missed true edges) are rescued when at least one ordering captures the correct ancestor descendant relation. These findings empirically substantiate our theoretical claim from Section 3 aggregating multiple orderings provides a robust consensus estimate of the transitive closure, correcting errors that any single ordering may introduce.
We also study the influence of the max lag Tmax and soft-voting threshold 0 hyperparameters on performance of DOTS, as well the diversity of orderings obtained at different noise scales k. The results for these be found in Appendix C
5.7 Limitations
We would like to highlight that the robustness of assumption violations tested Montagna et al. (2023a such as faithfulness, sufficiency, functional, and others, are not covered in our empirical evaluation. Instead, assuming stationarity and causal sufficiency, our robustness checks investigate variations in sample size, number of variables and size of lagged relationships. As result, any robustness claims we make throughout this work pertain specifically to those data characteristies we tested, and not the ones investigated by Montagna et al. 2023a). Future work should investigate our method's robustness to non-stationary and confounded settings.
Another important consideration is that DOTS relies on CAM pruning and hence inherits its limitations. However, by first recovering the theoretically grounded transitive closure, we provide stronger foundation for subsequent pruning than single-ordering methods can achieve.
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0.925
0.900
0.875
0.825
0.800
0.775
0.750
2
4
5
6
910
15
20
25
30
Number of orderings
Figure 9: The relationship between prediction performance (Flw) and the number of explored causal orderings in DOTS.
6
Related work
6.1 Ordering-Band Based Causal Discovery
Representing a DAG by one of its valid topological orderings (Verma & Pearl 1990 reduces structure learn- ing permutation search followed by edge selection. Early work framed this idea as discrete optimisation problem: greedy MCMC over permutations (Friedman & Koller 2003), hill-climbing with dynamic program-
ming [Teyssier & Koller 2005 arc-reversal scarches (Park & Klabjan 2017), and restrictad-MLE procedures such CAM (Bühlmann et al. 2014). More recent combinatorial schemes enforce sparsity via fo-penalised
likelihoods, yielding the "sparsest permutation" estimators with provable consistency guarantees (Raskutti
& Uhler 2018 Solus et al. 2021 Lam et al.) 2022). Reinforcement-leaminn formulations further cast the
permutation search as a sequential decision process, amortising exploration across datasets (Wang el. al. 2021
Beyond score optimisation, identifiability be strengthened by exploiting distributional or algebraic asym- metries. In linear additive models, sequentially peeling off leaf nodes from the precision matrix recovers the causal order under heteroscedastic noise assumptions (Ghoshal & Honorio 2018 Chen et al. 2019). De- terministic functional constraints are handled by Determinism-awware GES (DGES), which first clusters exact relations and then performs exact search within each cluster, using determinism itself as an unambiguous ordering cue [Li al. 2024). Con-Gaussianity offers an alternative route: L.iGAAM identifies unique ordering via independent-componee analysis, a principle extended functional data in Func-LiNGAM Shimizu et al. 2006).
Scalability to dign-diiennionnll, nonlinear settings has recently been advanced through continuous relaxations and deep generative models. CaPS estimates Hessian diagonals of the log-likelihood to iteratively detect leaves, unifying linear and nonlinear mechanisms and accelerating pruning with a "parent score" metric (u et al. 2024). Recent advances in ordering- balering-based causal discovery further refine and generalise score- basod estimation stratogics. SCORE (Rolland et al. (2022) leverages score matching techniques iteratively identify and remove leaf nodes, specifically utilizing variance estimates of the Hessian diagonal. Building upon similar principles, DAS (Montagna et al. 2023c) enhances scalability by efficiently estimating Hessian
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diagonals, significantly reducing computational overhead. DiffAN trains denoising diffusion model to approximate the score-function Jacobian, introducing a deciduous update rule that circumvents network retraining during iterative leaf removal, thus scaling ordering discovery to hundreds of variables (Sanchez ct al. 2023
More recently, there have been notable efforts to generalise the score-matching framework. For instance, NoGAM (Montagna et al. 2023b) generalises ordering- based approaches beyond Gaussian noise assumptions, employing kernelised score estimates to accommodate a wider range of data distributions. Furthermore, [Liu] et al. 2024 relaxes the assumption that all models must be either linear nonlinear and considers problems with mixed models, notably also leveraging parallel processing to improve scalability.
Together, these developments illustrate a shift from discrete combinatorics to differentiable optimisation, while preserving the core insight that a well-chosen causal ordering sharply narrows the search for a faithful DAG. None of these works explore combining multiple valid causal orderings to recover the full adjacency matrix.
6.2 Hessian of the log-likelihood
Estimating (log (l)) is the most expensive task of the ordering algorithm. Our bascline [Rolland et al., 2022) proposes an extension of Li & Turner (2018) which utilises Stein's identity over an RBF kernel (Schölkopf & Smola 2002). Rolland al.Js method cannot obtain gradient estimates positions out of the training samples. Therefore, evaluating the Hessian over subsample of the training dataset is impossible. Other promising kernel-based approaches rely on spectral decomposition to solve this (Shi et al. 2018 issue and constitute promising future directions. Most importantly, computing the kernel matrix is expensive for
memory and computation on n. There are, however, methods (Achlioptas et al.) 2001 Halko et al. 2011 Si
iet al. (017)) that help to scale kernel techniques not considered in the present work. Other approaches are
also possible with deep likelihood methods such as normalising flows (Durkan al., 2019 Dinh et al 2016)
and further computing the Hessian via backpropagation. This would require two backpropagation passes giving Odd²) complexity and be less scalable than denoising diffusion. Indeed, preliminary experiments proved impractical in our high-dimensional settings.
We use DPMs because they can efficiently approximate the Hessian with a single backpropagation pass while allowing Hessian evaluation on a subsample of the training dataset. It has been shown (Song & Ermon 2019)
that denoising diffusion can better capture the score than simple denoising (Vincent 2011) because noise at multiple scales explores regions of low data density.
6.3 Causal Discovery for Time Series
Granger's seminal definition of causality for time series past X improves the prediction of future Y-stiii underpins most modern approaches (Granger 1969). Structural causal models (SCMs) extend this idea to permit intervention semantics, latent variables, and cycles (Bongers et al. 2021 Constraint- algorithms such as PCMCI and its refinement PCMCI+ adapt the PC procedure to lagged conditional- independence testing, enabling scalable false-discovery control in autocorrelated, high-dimensional settings [Runge 2020). Functional-form assumptions provide stronger identifiability: TiMINo employs additive-noise regressions with independence tests on residuals (Peters et al. 2013 while VARLINGAM couples a linear VAR with non-Gaussian errors and independent-componeet analysis to recover causal ordering (Hyvärinen) ct al. 2010
Score-based and deep-learning methods further relax linearity and stationarity. DYNOTEARS casts struc- ture learning as single differentiable optimisation problem over lagged and contemporaneous edges with an acyclicity constraint (Pamfil et al., 2020). TCDF trains attention-based CNNs and validates candidates through in-silico interventions (Nauta et al. 2019), whereas CausalFormer augments Transformers with causality-awarr attention to handle long sequences (Kogg & Lu 2024). Contimuous dynamics can now be unveiled with sparse Neural ODEs that yield interpretable differential systems from irregular samples (Alice al. 2023). Information-theorret criteria such as transfer entropy generalise Granger tests to non- linear interactions, though density estimation remains costly in high dimensions (Schreiber) 2000 Most
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recently, PICK introduces score-matching algorithms to temporal causal discovery while notably reducing pruning time by exploiting the variance of the scores, offering an alternative path to efficiency (Chen et al. 2024).. Recent surveys also emphasise persistent challenges hidden confounders, non-stationarity, and com- putation at scale (Gong et al. 2023; Assaad ct al. 2022 Moraffah et al. 2021). Our method advances the field by combining the stalistical rigour of functional approaches with the scalability of continuous optimisation while accommodating nonlincarities typical of real-world data.
7 Conclusion
In this work, we introduced DOTS, diffusion-based approach leveraging multiple causal orderings to address the challenge of temporal causal discovery. While previous single-ordering methods were primarily developed in the context of static causal discovery, our work extends the causal ordering framework explicitly to the temporal setting. This temporal context inherently incorporates the causal temporality principle, where variables can only causally influence future variables, not past ones. Unlike traditional single-ordering meth- ods, DOIS effectively captures complementary information by aggregating multiple valid causal orderings, thereby roconstructing the transitive closure of the underlying temporal DAG. We formalized the theoretical benefits of this multi-ordering stralegy, demonstrating its capacity to miligate spurious dependencies and enhance robustness in causal inference. Our empirical results on synthetic datasots clearly illustrate the su- periority of DOTS compared to existing state-of-the-ar baselines in terms of accuracy and scalability, while staying competitive and the best on average on real data. By exploiting the inherent frequency domain characteristics of diffusson steps, our method provides nuanced insights into both coarse and fine-grained temporal causal interactions. Future research directions include exploring additional aggregation strategies for causal orderings, exlending our method to non-slationary enviromments, and further optimizing compu- tational efficiency for large-scale applications.
Acknowledgments
We would like to thank Ricardo Silva and anonymous reviewers for valuable comments and suggestions. We acknowledge the support of the UKRI AT programme, and the Engineering and Physical Sciences Research Council (EPSRC), for the Causality in Healthcare AI Hub igrant number EP 2P/Y028856/11
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A Order theory definitions
Here, we establish the foundational definitions used in order theory.
Definition (Partial Order). Let V be set. A binary relation on V is partial order if, for all H, U,, E V, it salisfies:
(i) Reflexivity: x xSS.
(ii) Antisymmetry: If x S and S x, then x y,
(iii Transitivity: If & y and y S 2, then S 2.
The pair (V,5) is called a partially ordered set (or poset).
Definition 2 (Strict Partial Order). A binary relation X on V is a striet partial order it is:
(i) Irreflexive: For all V, R x.
(ii) Transitive: If y and y - 2, then x - z.
Definition 3 (Linear Extension). For poset (V, linear extension is total order 3 on V such that if x S y, then x 3 y for all æ, E V. That is, 3 extends S into a total ordering consistent with the partial order.
Definition 4 (Reachability Relation). For directed graph G = (V, E), the reachability relation R VVV is defined as:
(r,8)= E R
there exists a directed path from æ to y G.
If is a DAG, R partial order on V (reflexive, antisymmetric, and transitive), as every vertex is reachable from itself via trivial path.
Definition (Transitive Closure). For a directed graph G (V, E) with reachability relation R, the
tronsilive closure of G is the graph:
=+ -V,
where:
E+ = ((x,)) E V x V (x.y) E R and x + y)-
Thus, g+ contains an edge (x, 3) if and only if there non-trivial directed path from x to y in G, excluding self-loops.
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B Supplementary Results
B.1 Alternative metrics
Figures 10 and supplement our main simulation results. In these, we report precision and recall on both window and summary graphs. The most important observation here is that the lop-performing mel hods almost never provide non-existent edges (i.e. high precision), but some undetected edges still remain (i.e. rocall lower than 1). Future work could focus on improving edge detection while maintaining high prccision levels.
1.0
0.8
SCORE-C
DAS-C
0.6
NoGAM-C
PGMCI
0.4
VARL
TC2F
0.2
DINAN
DOTS (ouns)
200
1000
2000
5000
sample size
number features
number of
(a) Precision (precision,))
0.7
SCORE-C
DAS-C
NoGAM-C
0.4
PCMQ
VARLINGAM
0.2
TCDF
DOTS (ours)
200
1000
2000
5000
number features
numbar of lags
(b) Recall (recallw)
Figure 10: Precision and recall (higher is better) on simulated window graphs. TiMINo does not provide window graphs.
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1.0
CAM-C
SCORE-C
DASIC
PCMCI
POMCI+
VARLINGAM
DYNOTEARS
TIMINo
DOTS (ours)
200
1000
2000
5000
sample size
number features
number of lags
(a) Precision (precisiong)
CAM-C
0.8
DAS-C
POMCI
IGRLINAAM
DYNOTEARS
0.2
TIMING
DOTS (ouns
200
1000
2000
5000
sample size
number of features
number lags
(b) Recall (recalls)
Figure 11: Precision and recall (higher better) on simulated summary graphs.
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B.2 Effect of the temporal constraint
Figure 12 shows the influence of the temporal constraint applied to ordering-based methods to ensure they return graph that respects the arrow of time. Most methods show some mild performance improvement and mostly in higher precision. Intcrestingly, diffusion-based methods (DiffAN and DOTS) do not exhíbit such a benefit, which could be partly due to already high precision achieved even without the constraint (i.e. not much room for further improvement).
con strainn
CAM
no
yes
DAS
NoGAM
DOTS
0.0
0.2
0.4
0.6
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.0
0.2
0.4
0.6
0.8
1.0
F1w
recally
precisiony
Window graphs
constraint
CAM
yes
SCORE
DAS
DiRAN
DOTS (ours)
F1s
recalls
precisions
(b) Summary graphs
Figure 12: The effect of the temporal constraint the performance of score-matching algorithms.
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C
Ablations
C.1 Maximum lag
DOTS has a hyperparameter that sets the maximum lag (Tmax) to be considered in the predicted graph. Here we study its influence on predictive performance. To this end, we run sinulations with 3-node graphs T=5000 and 7 E (1,2,3) 3), all repeated 100 times. Figure 13 presents the results.
Two main observations emerge. First, selecting Tmax shorter than ground truth can have an obvious significant negative impact performance it prevents DOTS from identifying the necessary longer relationships. However, setting Tmax to values larger than the truth maintains good performance the hyperparameter matches the lag size in the data. This has important practical implications, suggesting that setting Tmax to larger values may provide more robust results since the right lag size is unknown when processing real data.
compared to when
data lag
data lag
data lag 3
1.0
0.9
0.7
0.5
0.3
lag
max la
lag
Figure 13: The relationship between prediction performance (Flw) and the maximum lag (mar lag) hyper- parameter in DOTS. Each setting (subfigure) includes one type of lag in the data (data lag), which also highlighted by the vertical ground truth line. Error bars denote 95% confidence intervals.
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C.2 Soft-voting threshold
Here investigate the sensitivity of DOTS to the soft-voling hyperparameter 0. We run simulations with 3-node graphs, T=5000, = L, all repeated 100 times. Figure 14 presents the results.
The main observation is a clear trend, in which performance consistently degrades as we increase the threshold. This is consistent. with the behaviour we expect as increasing the threshold requires more voles from each ordering for graph edge to be present, which inevitably leads increased false negatives and decreased recall (middle subfigure).
Interestingly, union of all orderings (0 = 0) provides the best results, and which is notably free from false positives (i.c. perfect precision as in the right-hand-side subfigure). This result shows importance and effectiveness of CAM pruning since without it the number of false positives would likely be very high. This observation also makes f = the recommended default value.
10
1.-
1.0
0.9
0.9
0.9
orecistonw
0.7
0.6
0.6
0.6
0.5
0.5
0.5
0.3
8.
0.6
6.
0.
0.9
Figure 14: The relationship between prediction performance and the soft-voting threshold 0 hyperparamcter in DOTS. Error bars denote 95% confidence intervals.
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C.3 Diversity of orderings
To quantify the diversity of orderings produced during training run, computed pairwise distances between all permutations using the Kendall tau metric. This distance reflects the proportion of pairwise disggreements in item ordering between two permutations and is widcly used for comparing ranked lists. By converting each permutation into its corresponding rank vector and computing normalized Kendall distances, we obtain a symmetric distance matrix that captures how similar or dissimilar each ordering is to every other within the same run.
Figure 15 visualises this matrix heatmap. Blocks of darker regions indicate sets of orderings that remain relatively consistent, while lighter bands correspond to greater variability across orderings. The structured patterns suggest that the model explores multiple, partially overlapping ranking modes rather than collapsing to a single consensus. In particular, similar k values yield similar orderings as shown by the block patterns on the diagonals.
In Figure 16 we further investigate predictive performance of individual orderings different noise scale values k. Each ordering was passed through CAM pruning lo get the final performance measure against the true DAG. The figure implies that different noise scales provide different edge-detection capabilities, especially in the [0, 30) range. Importantly, however, none of the orderings achieves individually as high of a performance as when all orderings at different noise scales are combined into single robust prediction.
Kendall distance heatmap
10
20
30
0.5
values
40
50
0.3
60
70
80
0.1
90
10
80
k values
Figure 15: Kendal distance between different orderings obtained different noise scale values k.
combined orderings
1.04
0.8
0.7
1.00
0.65
0.60
0.58
0.96
10
70
19
8
30
90
90
Figure 16: Performance of individual orderings obtained different noise scales compared to the perform- ance achieved with combined orderings. Results averages over 100 repetitions. Error bars and shaded grey areas denote 95% confidence intervals.
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D Experimental Details
D.1 Hyperparameters
We summarisc the most important hyperparameters used in conjunction with the methods in Table 3 For the rest of the hyperparameters not mentioned here, we deferred to their default values, which can be found in their respective implementations as per Table
In addition, most methods have a hyperparaaeete that corresponds to the maximum lag size, though can be named differently in their implementations (Tmaz in PCMCI/PCMCI+, lags in VARLINGAM, p in DYNOTEARS, mar_aag lag in TiMINo). This hyperparameter formance. In order to decrease our experiments' dependence on hyperparameter tuning, we set this hy- perparameter to the true lag value in the simulations since we have access to the ground truth. In real data settings (CausalTime), we set the value to 1. Note that some methods (CAM, SCORE, DAS, No- GAM, DifEAN, DOTS) implement this hyperparameter implicitly by creating T lagged variables based on provided data.
have non-trivial effect method's per-
While sampling 10 causal orderings in DOTS is reasonable default (n_ord), increase to 20 when processing CausalTime to better showcase method's potential.
Table 3: Summary of hyperparameters of all methods used in the experiments.
Method
Hyperparameters
CAM
alpha 0.05
SCORE
0.05, ra = 0.001, πH = 0.001
DAS
0.05, fı = 0.001, TM = 0.001
NoGAM
a = 0.05, n = 0.001, 7и = 0.001
ridge, 0.01, ridge, 0.1
PCMCI/PCMCI-
a 0.05, test (par _corr, cmi kmn] Tinn 0.05, criterion bic, prune true Aw 0.05, A₂ 0.05, twweshooooodoo 0.01
VARLIGAM
DYNOTEARS
TCDF
epochs 5000, layers 2, = 0.01
kernel size = 4, dilation = 4, significance = 0.8
DiffAN
steps = 100, nn pasnn_neepp _en_hhehhhhheee = 3, batch size = 1024
early stop = 300,
0.001
TiMINo DOTS
0.05
steps 100, nn depth 3, batch size 1024
carly_ 300, Ir 0.001, nord 10
D.2 Implementations of algorithms
Table 4: Summary of source code used to run the methods in the experiments.
Method
Source
CAM/SCORE/DAS/NGGAM
dodiscover: httppe//sttabb.cccooorryyyloolll
PCMCI/PCMCI+
tigramite: htwrammitttttteee/ttttbbbb.oo http::///chab..ccc/////bbornggge
VARLINGAM
1 lingam: httpss//ggittub.cc/////t////////
DYNOTEARS
causalnex: http:///w.//p/seo.aooo/oo//////a
TCDF
http://stthuubccoo/ttNNttataaaa
DifTAN
httep:://github.coo//iit sDD:fFA
TiMINo
stggt://gtthbb. awm///aannad/nnni.ccccooo htg.://ww.........e.eeeeeeeeeeee
32


Reviews:
[{"review_id": "6SH1cPG248", "reviewer": "Reviewer_eFk5", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**\n\n- The paper's core idea to aggregate multiple causal orderings, rather than relying on a single one, is a significant and innovative contribution. This approach directly addresses the fragility of traditional single-ordering methods, which can introduce spurious artifacts.\n- The method cleverly employs a multi-scale diffusion model to efficiently generate a diverse set of causal orderings from a single trained network. The justification that different noise scales emphasize different frequency components of the data provides a compelling rationale for the diversity of the generated orderings.\n- The paper is well-written, clearly structured, and easy to follow. The methodology is explained logically, and figures such as the pipeline diagram (Figure 4) effectively illustrate the algorithm's workflow.\n\n**Weaknesses**\n- The theoretical framework aims to recover the transitive closure ($G^+$) of the DAG, which by definition cannot distinguish between direct and indirect causal relationships. This makes the method heavily dependent on a final, heuristic pruning step (CAM pruning) to obtain the final sparse graph, and the overall performance is thus a hybrid of the novel framework and this external module.\n- Training diffusion models, especially on the high-dimensional lag-embedded data required by DOTS, can be computationally very expensive. Furthermore, the method's performance is sensitive to key hyperparameters, such as the voting threshold $\\theta$ and the maximum lag $\\tau_{max}$, for which the paper provides limited practical guidance on selection.\n- The method relies on strong assumptions such as causal sufficiency and stationarity. While this is standard for the field, the paper does not empirically evaluate the method's performance in scenarios where these critical assumptions are violated (e.g., in the presence of hidden confounders or non-stationary data), making its real-world robustness an open question.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper introduces DOTS, a novel method for temporal causal discovery. The core contribution is a paradigm shift from traditional methods that rely on a single, often fragile, causal ordering. Instead, DOTS proposes to generate and aggregate *multiple* valid causal orderings to enhance the robustness and accuracy of structure learning. Theoretically, the authors argue that by integrating information from multiple orderings, their method effectively recovers the **transitive closure** ($G^+$) of the underlying Directed Acyclic Graph (DAG), which helps mitigate spurious artifacts inherent in single-ordering approaches. To achieve this, the paper presents a key technical innovation: the use of a **multi-scale denoising diffusion model**. This allows for the efficient sampling of a diverse set of causal orderings from a single trained model by leveraging different noise scales, which are argued to correspond to different frequency components of the data. Extensive experiments on different benchmark datasets demonstrate that DOTS outperforms state-of-the-art baselines in terms of causal structure accuracy and offers a scalable solution to the problem.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. The paper makes strong claims about the robustness of DOTS. Recent large-scale benchmark studies (Montagna et al., 2023), have demonstrated that violations of these assumptions are critical failure points for many causal discovery algorithms, often leading to performance comparable to a random baseline. However, the experiments do not test the method's performance against critical and common violations of its underlying assumptions, such as non-stationary processes. It is critical that the authors add a dedicated discussion in the limitations or experiments section to:\n    * Acknowledge that the current empirical evaluation does not cover scenarios where these assumptions are violated.\n    * Contextualize their robustness claims accordingly, clarifying that they pertain to variations in sample size and dimensionality rather than fundamental assumption violations.\n\n2. The core innovation (diffusion-based multi-ordering aggregation) recovers the transitive closure ($G^+$), which is fundamentally unable to distinguish direct from indirect causes. The final, crucial step of producing a sparse DAG is entirely delegated to an external, heuristic method (CAM pruning). The authors should clarify this in the methodology and discussion sections, explicitly stating that the final performance is a hybrid of their novel framework and the chosen pruning module, and that DOTS inherits the limitations of that module.\n\n3. The soft-voting threshold, $\\theta$, is a critical hyperparameter that directly controls the precision-recall trade-off of the resulting graph. The paper would be significantly strengthened by an ablation study that shows how the F1 score, precision, and recall metrics vary with different values of $\\theta \\in (0, 1]$. This would provide invaluable practical guidance for users and a deeper understanding of the method's behavior.\n\n4. The related work section provides a good overview, but it would be strengthened by positioning the paper's contributions in the context of several highly relevant, recent advancements in score-matching/diffusion-based causal discovery. We suggest the authors add and discuss the following works:\n- The work of Chen et al. (2024) is directly and critically relevant, as it also proposes a scalable, score-matching-based method specifically designed for temporal data. This paper presents an alternative strategy for improving efficiency. \n- The paper should discuss recent efforts to generalize the score-matching framework. For instance, the work of Montagna et al. (2023a) introduces an algorithm for models with arbitrary noise distributions, while Liu et al. (2024) addresses scalable discovery in mixed linear and nonlinear models. \n- Any claims of robustness should be contextualized by the findings in Montagna et al. (2023b). This paper provides a large-scale benchmark on how score-matching methods perform when core assumptions (like causal sufficiency) are violated. \n\n\n**References**\n\nChen, H., Yi, K., Liu, L., & Wang, Y. G. (2024). Score-matching-based Structure Learning for Temporal Data on Networks. arXiv preprint arXiv:2412.07469.\n\nLiu, W., Huang, B., Gao, E., Ke, Q., Bondell, H., & Gong, M. (2024). Causal Discovery with Mixed Linear and Nonlinear Additive Noise Models: A Scalable Approach. Proceedings of the Third Conference on Causal Learning and Reasoning (CLeaR), PMLR 236:1237-1263.\n\nMontagna, F., Noceti, N., Rosasco, L., Zhang, K., & Locatello, F. (2023a). Causal Discovery with Score Matching on Additive Models with Arbitrary Noise. Proceedings of the 2nd Conference on Causal Learning and Reasoning (CLeaR), PMLR 213:726-751.\n\nMontagna, F., Mastakouri, A. A., Eulig, E., Noceti, N., Rosasco,L., Janzing, D., Aragam, B., & Locatello, F. (2023b). Assumption violations in causal discovery and the robustness of score matching. Advances in Neural Information Processing Systems (NeurIPS), 36.", "broader_impact_concerns": "The research presented in this paper raises no ethical concerns or broader impact issues.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "uWScErhLPH", "reviewer": "Reviewer_Tqog", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths.** The main strength of the paper is its main idea: Multiple total orders are more useful than a single one, while also being cheaper than obtaining the partial order. Very neat observation that could be useful across many causal discovery tasks -- or any DAG learning task in general too. The approach is well-motivated, the way of sampling multiple total orders is very intriguing, and algorithm is overall well-presented and clean.\n\n**Weaknesses.**\nThere are multiple _classes_ of weaknesses. First, **theory problems**:\n- The masking-based causal discovery method (Sanchez et al., 2023) operates on the true data distribution. Therefore, in the noisy data model the authors use, the trick used to derive the score of the marginal distributions using Hessians **is not** (necessarily) **valid**. I acknowledge that the paper is mostly an algorithmic contribution, regardless, this constitutes a big gap in the \"correctness\" of the proposed algorithm.\n- Frequency domain explanation (Section 4.1): Fourier transformation is taken incorrectly. Density of sum of independent variables is the convolution of individual densities, Fourier transform of which is the product. This tracks, since adding independent Gaussian noise is equivalent to applying a Gaussian filter to the densities. While this is still a variable-width filter applied on the \"frequency\" data and therefore the high level description of the procedure still is well-founded, the technical details are simply **incorrect**. Similarly for Fig. 5.\n- Overall, the proposed framework is \"motivated\" and not really \"proved\" with any rigor. Given how related literature generally provides correctness statements together with the algorithms, it would help the paper to have them as well to situate it more strongly in the literature.\n- The paper, as it stands, does not make it clear whether said proofs exist or not. For example, having formal assumption statements like in Section 2.2 usually suggests -- but does not *directly* imply -- there is a formal theorem using them somewhere. Having a \"theory\" section implies similarly. These are not wrong per se, but the language needs to be adjusted so that readers don't get the wrong idea.\n\nSecond class is **presentation/framing** issues:\n- While the idea of using multiple total orders at the same time is a very neat concept, I do not see the reason why the authors made the jump of going directly to time series modeling rather than plain causal discovery task. Yes, the idea of using the temporal dependence structure to further refine the edge set is a neat trick, but it is literally the only additional step beyond the causal discovery algorithms. More constructively, why not present the causal discovery variant as well?\n- The paper belabors graph theory/terminology for a long time. These are well established concepts, simply use them. You can add references to relevant _textbooks_ if need be, but please don't overexplain.\n- The literature on score estimation could be consolidated. I liked how the authors tailor each nuanced discussion to the context, but this also causes some repetition and back-and-forth when trying to see what is available at all. You could try lightening in-text descriptions in favor of extending the relevant related work section.\n\nOn experiments.\n- The summary graph formulation doesn't fit well with the proposed algorithm's use cases. Please justify how they fit or remove/move to appendix otherwise.\n- In page 14, CAM is not a score-matching method; it predates (Rolland 2022).\n- You can add additional plain causal discovery algorithms for comparison. Adding at least 1 or 2 more is important for completeness, adding more state-of-the-art models would help your case.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes a diffusion model-based multi-level linear extension based method to estimate the topological closure of the temporal DAG of a multi-time lag dynamic Bayesian network with contemporaneous relations. It provides empirical evaluations demonstrating improved performance above baselines in both synthetic and real world data sets.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- **Critical:** The theoretical exposition needs to be **corrected**.\n- **Critical:** Please explain, in your response, why the authors chose to jump directly to time-series causal discovery and not also claim that their approach is useful for plain causal discovery as well. Depending on their own framing, I expect the relevant claims and/or clarifications to be incorporated in the revision. A logical leap to this extent weakens my trust in the correctness of other parts of this paper.\n- **Critical:** Adding on top of this, _if_ the method is applicable to causal discovery as well, then comparisons on this setting as well. Specifically, add synthetic and real-data experiments comparing against non-time-series methods.\n- I don't expect a full-scale theoretical analysis of the proposed approach to be provided in the next revision, but I would like it if the authors manage to do so.\n- Restructure the assumptions. See weaknesses for details.\n- Add the Gaussianity assumption on exogenous noises to the set of assumptions. You mostly use results/methods of (Sanchez et al., 2023) which requires this AFAIK.\n- Other two formatting issues are mostly suggestions. I expect the authors to accommodate them as much as possible, but they are not critical for my acceptance by themselves.\n- Add a reference to Hiraguchi's theorem: For all partial orders, there exists $\\lceil n/2 \\rceil$ total orders that fully specify it, i.e., the \"valid edge set\" equals the transitive closure using graph terminology. This observation -- while not directly relevant when sampling the total orders randomly -- justifies the multi-total order framework very strongly.\n- Verify that recovering the topological order too is NP-hard. Chickering (1994) probably already implies this, but add another citation that explicitly states it.\n- Section 3.2, using limit in that equation is slightly overcomplicating it. It looks fine, but there is probably a more professional and cleaner way to describe it.\n- Figure 3 lacks the experiment setting description, and its explanation is spread across two subsections. Tighten this and add missing context.\n- Minor typo on pg. 11: it should be $\\mathbb{D} \\in \\mathbb{R}^{(T - \\tau\\_{max}) \\times d \\times (\\tau\\_{max} + 1)}$ ($\\tau$'s missing the $+1$ currently)\n- In many places, $k$ and $\\alpha\\_k$ are conceptually confused with each other. Proofread and make them consistent.\n- Rolland (2022) reference is duplicated\n- Pg. 13: Description of how you reach $T =19 200$ samples overall is slightly vague. Can you clarify?\n- Pg. 14: CAM is not a score-matching method; it predates (Rolland 2022).\n- You have included CAM already, you can add additional plain causal discovery algorithms here as well for comparison. Adding at least 1 or 2 more is important for completeness, adding more state-of-the-art models would help your case.\n- The summary graph formulation doesn't fit well with the proposed algorithm's use cases. Please justify how they fit or remove/move to appendix otherwise.", "broader_impact_concerns": "No ethical/impact concerns", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "XFy91RAIyW", "reviewer": "Reviewer_t6jd", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths\n\n1. The proposed method, DOTS, demonstrates good performance on synthetic benchmarks, consistently outperforming the baselines. It also achieves competitive, state-of-the-art results on the real-world CausalTime benchmark. \n\n2. The central idea of aggregating multiple causal orderings to improve robustness is intuitive and well-motivated. This ensemble-like approach proves to be highly effective in practice, as demonstrated by the strong results and the ablation study (Fig. 9).\n\n3. The paper presents a creative application of diffusion models for a causal discovery task. Using different noise scales as a heuristic to generate a diverse set of causal orderings is a novel mechanism that connects generative modeling with structural learning.\n\n4. The experimental setup is comprehensive. The authors validate their method across synthetic datasets with varying sample sizes, feature dimensions, and lag structures, as well as on multiple real-world datasets. \n\n\n\nMajor Weaknesses\n\n1. The central claim is that leveraging multiple orderings is a novel contribution. However, aggregating the output of a model run with different initializations or hyperparameters is a standard ensemble technique. The paper frames this as a fundamental insight into causal discovery, but it can also be seen as an application of ensembling to an existing ordering-based discovery algorithm (DiffAN). \n\n2. The paper argues that recovering the transitive closure $\\mathcal{G}^{+}$ is a desirable outcome. The goal of causal discovery is to find the true causal graph $\\mathcal{G}$, which represents direct causal relationships. The transitive closure $\\mathcal{G}^{+}$  includes all indirect causal pathways, which can be misleading. For example, if A→B→C, the transitive closure includes the edge A→C, which is an indirect effect. In many scientific applications, distinguishing direct from indirect causes is important. While the authors use a final pruning step (CAM pruning) to remove indirect edges, the core theoretical argument and the aggregation method are aimed at recovering $\\mathcal{G}^{+}$. \n\n3. The method for generating different orderings relies on running the leaf-finding algorithm at different noise scales k of the diffusion model. The paper provides a high-level intuition that different scales capture different frequency components of the data. However, this link can be more rigorously established. Are the orderings generated from different k values truly diverse and complementary? Or are they minor variations of each other? The ablation study in Figure 9 shows that performance increases with the number of orderings, but this could be due to getting more \"shots\" at a good ordering. \n\n4. Several of the baselines (CAM, SCORE, DAS, NOGAM, DiffAN) are methods for static data, which the authors adapt to the temporal setting by applying a post-hoc filter to remove edges that violate temporal precedence (\"-C\" suffix). These methods were not designed for time series and may perform poorly due to their inability to properly handle autocorrelation and other temporal dependencies. The strong performance of DOTS relative to these \"constrained\" baselines is therefore not surprising. \n\n5. While the paper includes strong time-series baselines like PCMCI+ and TiMINo, the overall narrative is built around outperforming adapted static methods. On the real-world CausalTime benchmark, the results are much closer. DOTS achieves the highest average F1, but TiMINo is better on the Medical dataset and VARLINGAM is better on AQI and Traffic. \n\n\nMinor Weaknesses\n\n- The paper could better motivate why a diffusion model is necessary here. The core machinery for finding leaves is the Hessian of the log-likelihood. The diffusion model is used as one way to estimate this Hessian. The authors should clarify if other Hessian estimation methods could be used and what the specific advantages of the diffusion-based approach are.\n\n- The ablation in Figure 9 is useful, but it would be strengthened by also showing the performance of individual orderings sampled from different k.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper introduces DOTS, a method for causal discovery in time series data. The core idea is to move beyond ordering-based methods that rely on a single causal ordering. Instead, DOTS generates multiple causal orderings and aggregates them to improve robustness of the discovered graph. The authors claim that this aggregation process effectively recovers the transitive closure of the true underlying DAG, which helps filter out spurious edges that often arise from single-ordering approaches. \n\nTo generate a diverse set of orderings, the method uses a denoising diffusion model, where different noise scales are used to produce distinct orderings. The authors provide a theoretical argument that aggregating all valid topological orderings of a DAG recovers its transitive closure. Empirically, the paper shows that DOTS outperforms several methods on both synthetic and real-world time series datasets.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1.  The claims regarding the novelty of using multiple orderings might need to be toned down. The work can be framed as an effective ensemble-based method for an existing ordering-based algorithm (DiffAN).\n\n2.  The paper can include a discussion on the goal of recovering the transitive closure ($\\mathcal{G}^{+}$). The authors might want to explicitly state that the ultimate goal is the true DAG ($\\mathcal{G}$) and that $\\mathcal{G}^{+}$ contains indirect relationships, which can be a limitation. The role of the final pruning step in moving from $\\mathcal{G}^{+}$ to an estimate of $\\mathcal{G}$ should be made clearer and more central to the method's description.\n\n3.  The link between different diffusion noise scales (k) and the generation of diverse, complementary causal orderings must be substantiated. This requires more than the current high-level intuition. An analysis measuring the diversity of the permutations generated at different scales and linking this diversity to performance would be useful.\n\n4.  The experimental section can be revised to focus on fairer comparisons. The primary comparison should be against methods explicitly designed for time series. The results from adapted static methods should be presented with the clear caveat that they are not designed for this setting. Furthermore, the discussion of real-world results needs to acknowledge datasets where baselines like TiMINo and VARLINGAM outperform DOTS.\n\n\nMinor changes\n\n- The paper would be improved by a clearer justification for using a diffusion model to estimate the score's Hessian over other potential methods (e.g., normalizing flows). A brief discussion of the specific advantages (scalability, stability, the multi-scale property) would strengthen the motivation.\n\n- The ablation study in Figure 9 is good, but it could be improved. I recommend adding a comparison between the aggregated result and the performance of the best single ordering chosen from the generated set.", "broader_impact_concerns": "I don't have any concerns on the ethical implications of the work that would require adding a Broader Impact Statement.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)