Inferring the Invisible: Neuro-Symbolic Rule Discovery for Missing Value Imputation
Authors: Wendi Ren
Ke Wan
Junyu Leng
Shuang Li
Venue: ICLR 2026 Poster (2026)
Keywords: Neuro-symbolic Learning
Rule Discovery
Interpretable Reasoning
Primary area: interpretability and explainable AI

Abstract: One of the central challenges in artificial intelligence is reasoning under partial observability, where key values are missing but essential for understanding and modeling the system. This paper presents a neuro-symbolic framework for latent rule discovery and missing value imputation. In contrast to traditional latent variable models, our approach treats missing grounded values as latent predicates to be inferred through logical reasoning. By interleaving neural representation learning with symbolic rule induction, the model iteratively discovers—both conjunctive and disjunctive rules—that explain observed patterns and recover missing entries. Our framework seamlessly handles heterogeneous data, reasoning over both discrete and continuous features by learning soft predicates from continuous values. Crucially, the inferred values not only fill in gaps in the data but also serve as supporting evidence for further rule induction and inference—creating a feedback loop in which imputation and rule mining reinforce one another. Using a staged block-coordinate gradient descent, the system learns these rules end-to-end by iteratively optimizing over parameter blocks in an alternating fashion. Experiments on both synthetic and real-world datasets demonstrate that our method effectively imputes missing values while uncovering meaningful, human-interpretable rules that govern system dynamics.

Published as a conference paper at ICLR 2026
INFERRING THE INVISIBLE: NEURO-SYMBOLE RULE DISCOVERY FOR MISSING VALUE IMPUTATION
Wendi Ren ' Ke Wan Junyu Leng"; Shuang Li
School of Data Science. The Chinese University of Hong Kong (Shenzhen)
2 University of Virginia
Texas A&M University
wendiren& link. cuhk. edu. en, tbnSpjevirginn edu revilengetenn tamu. edu, lishuang@cuhk edu.cm
ABSTRACT
One of the central challenges in artificial intelligence is reasoning under partial observability, where key values are missing but essential for understanding and modeling the system. This paper presents neuro-symbolic framework for latent rule discovery and missing value imputation. In contrast to traditional latent vari- able models, our approach treats missing grounded values latent predicates to be inferred through logical reasoning. By interleaving neural representation learning with symbolic rule induction, the model iteratively discovers both conjunctive and disjunctive rules- explain observed patterns and recover missing entries. Our framework seamlessly handles heterogeneous data, reasoning over both dis- crete and continuous features by learning soft predicates from continuous values. Crucially, the inferred values not only fill in gaps the data but also serve as feedback
supporting evidence for further rule induction and inference-createnn loop in which imputation and rule mining reinforce one another. Using a staged block-coordinate gradient descent, the system learns these rules end-to-end by it- cratively optimizing over parameter blocks in an alternating fashion. Experiments both synthetic and real-world datasets demonstrate that our method effectively imputes missing values while uncovering meaningful, innerniiterppetable rules that govern system dynamics.
INTRODUCTION
Neural-symbolic reasoning integrates neural representation learning with symbolic structure, com- bining the noise tolerance of embeddings with the interpretability of symbolic reasoning (Hitzler Sarker 2022, Yang al. 2024). This hybrid paradigm enables Al systems to detect complex patterns in unstructured data while reasoning about them in structured and explainable manner.
Traditional rule induction methods extract explicit patterns from observed data but often fail when some observations are missing or incomplete (Campero et al., 2018 Claire Glanois 2022]. These approaches can effectively learn surface-level rules, yet their ability to fully explain the underlying system is limited when essential data points are absent. For example, in healthcare diagnostics, critical measurements may be missing, making accurate imputation necessary for reliable reasoning. Probabilistic models such as Markov Logic Networks (MLNs) (Richardson & Domingos, (006) handle missing data by treating unobserved facts as latent predicates. However, they typically rely on fixed rule base and expensive joint inference, limiting scalability and adaptability in large or heterogeneous datasets (Oltramari et al., 2020p. In contrast, we propose neuro-symbolic system that co-learns rules and imputations in single differentiable loop, enabling fast forward- inference and end-to-end learning.
We view each missing table entry an unobserved grounded fact. Starting from the observed facts, our differentiable forward-coe woward-chaining engine applies the current rule missing predicates. We then update rule embeddings, together with soft predicates for continuous attributes, by minimizing masked reconstruction objective on the observed entries, treating the partially observed table weak supervision. Better rules yield better imputations, and improved
infer soft truth values for
"Equal contribution "orresponniig author

---

Published as a conference paper at ICLR 2026
imputations provide stronger evidence for refining rules in subsequent passes, producing an iterative coupling between completion and induction.
Many targets require compasitional explanations, such as multi-hop chains and disjunctive rule sets, and the data are often heterogeneous with both discrete and continuous attributes. Our framework supports expressive rules over mixed data types by learning soft predicates for continuous features (sigmoid thresholds and slopes) and combining them with discrete predicates through smooth logical operators. In particular, we use soft-min as differentiable surrogate for AND and soft-max surrogate for OR, enabling uniform forward chaining without pre-discretization.
Learning in this expressive space must also remain tractable. To achieve this, we use a staged, block- coordinate optimization scheme, where we update one rule or clause at a time while holding others fixed. This approach reduces interference between candidate explanations and helps manage search complexity as the rule set grows. For disjunctive heads, a common challenge is that individual clauses may be underrepresented due to missing data. To address this, we first generate diverse set of clauses through sequential covering, then fine-tune them together using a soft-OR aggregator (LogSumExp), which captures interactions between clauses. This procedure scales efficiently to complex disjunctive rule sets, even under high missingness.
Contributions. (i) We introduce a differentiable neuro-symbolic framework that co-learns rules and imputations through an iterative feedhack loop, instead of treating imputation as separate pre- processing step. (ii) We develop staged training strategy based on asynchronous block-coordinate updates, sequential covering for discovering diverse clauses, and joint fine-tuning with a soft-OR (LogSumExp) aggregator for disjunctive heads. (iii) We design unified differentiable forward- chaining engine for heterogeneous tables, combining discrete predicates with learned soft predicates for continuous attributes (sigmoid thresholds and slopes) via smooth logical operators (soft-min for AND, soft-max for OR). (iv) We empirically validate the approach on synthetic chain and disjunction benchmarks and on real-world datasets (Birds, Heart, SPECT), demonstrating accurate imputation, strong downstream prediction, and recovery of human-interpretable rules [
2 RELATED WORK
Our work combines neuro-symbolic Inductive Logic Programming (II.P) with missing value impu- tation.
Neural Embedding--based ILP. Embedding-bbsed models such as TransE (Bordes et al. 2013). TransH (Wang et al.] 2014), and TransR (Lin et al., 2015) are widely used for Knowledge Base (KB) completion, learning vector representations for entities and relations. Complex (Trouiloon) ct al. 2016) extends these with complex-valued embeddings to handle asymmetric relations, and mopti-hop methods like Guu et al. (2015) use path-basce embeddings for reasoning over knowledge graphs While successful, these methods are limited in their ability to discover complex logical rules. Recent advances in ILP integrate symbolic logic with neural networks, including RRoktäschell & Riedel (2017)'s Neural Theorem Proving (NTP), differentiable backward-chaining method, and Campero et al. 2018)'s forward-chaining rule induction network. However, both rely on hand- designed templates, limiting flexibility. Claire Glanois (2022) improve flexibility with a hierarchical structure for more adaptive rule induction, but these models are still designed primarily for fully- observed data and struggle with missing values.
Interpretable Rule Learning. Learning interpretable logical rules has long been a goal in AL. Dash et al. (2018) propose BRCG, an integer programming approach that efficiently searches the space of candidate clauses, balancing accuracy with simplicity. |Wang et al. (202T) introduce RRL, which uses Gradient Grafting to learn non-fuzzy rule lists in deep learning framework, ensuring scalability. Qiao et al. (221)) present DR-NET, which learns decision rules in Disjunctive Normal Form (DNF) by optimizing both rule generation and weight learning. More recently, Barbiero et al. 2022 propose LEN, a differentiable neuro-symbolic method that extracts concise First-Order Logic (FOL) explanations from neural networks using an entropy-based criterion. In contrast to these methods, which focus primarily on classification tasks with complete data and binary features, our framework integrates rule learning directly with missing value imputation, enabling both tasks to reinforce each other in a feedback loop.
Rule-Based Missing Value Imputation. Traditional missing data imputation methods, ranging from statistical techniques such as MICE (Multivariate Imputation by Chained Equations) (van Bu- uren& & Groothuis- Oudshoorn 201 MissForest (Random barest- (Stekhoven & Bühlmann
Our code is available at httpp://// http://iii/ii////oo//ooooooo///a
www.a/oooommmeemeeoooommm.maaaae emeas /iieeessaiiineemmmree
2

---

Published as a conference paper at ICLR 2026
2012). and SOFT-IMPUTE
(Mazumder et al., 2010), to deep learning approaches like GAIN (GAN-
based) ( Yoon et al. 2018). diffusion-baite models (Ouyang et al. 2023). mDAE (DAE-basee based)
(upuy al. 2022). and VAE-based methods (Veldkamp et al. 2025), generally rely statisti- cal patterns to fill in missing values. However, these models do not explicitly use logical rules to govern inter-variable relationships (see Appendix fAror detailed overview).
In recent years, there has been increasing interest in combining rule-based reasoning with missing value imputation. Chen et al. (2023) apply interpretable machine learning to imputation, but their methods do not explicitly use cule-bbased reasoning. MINTY (Stempfle & Johansson, 2024) employs bile-based model, yet it lacks neuro-symbolie reasoning learn complex relationships between observed and missing values. contrast, our framework integrates neural representation learning with symbolic reasoning, enabling flexible discovery of interdependencies between observed and missing data. Non-neural approaches like (Wang et al. (2017) address missing values but lack the adaptability and representation learning of neural networks. Our method stands out by forming feedback loop where rule discovery and imputation mutually reinforee each other.
3 BACKGROUND
Predicates and Latent Predicates. A predicate a Boolean-valued relation that describes a prop- erty of an entity relationship between entities. Grounding predicate with concrete entities yields an atomic fact, which may be observed (true/false) or missing. For example, Wos_FFeve((2) indicates whether patient p has fever, and Use Drug(p) indicates whether drug is used for pa- tient p. We refer to missing grounding of predicate for particular entity (or entity tuple) latent predicate instance; importantly, the same predicate may be observed for other entities. In our differentiable setting, such instances can be assigned soft truth values to represent uncertainty. Rules and Forward Chaining. We model logical structure with rules in OR-of-ANDs form:
Qx + (P1 A P2) V (P3 A Pa) V
so that Qs is inferred when any clause is satisfied. Such rule sets support one-step inference when clause bodies involve only observed predicates, and multi-step inference when intermediate predi- cates are inferred first, e.g.. Q1 - PIA
P Qx + (Q V (
Given an evidence set of observed facts E, we perform forward chaining: whenever a clause (approximately) satisfied by facts in E, it produces its head predicate and adds the newly inferred fact back into Reusing inferred facts as evidence enables multi-hop reasoning through cascades of rule applications.
4 MODEL: BEURO-SYMBOLEC FORWARD CHAINING NETWORK (NS-FCN)
Consider problems where some in- formation or features are incom- plete. Given the observed predi- cate (denoted by X) and the miss- ing predicate (denoted by U), our goal is to learn a set of logical rules and use them to impute the missing ones (as illustrated in Figure
X5s
0
0
man
man
0
0
0
0
0
X,
X,
X
X
Xn
X₂
X;
man
BAn
man
Figure 1: Example of missing variables imputation with rule discovery. Feature with nan corresponds the predicates with missing information (belong to U), which can be inferred by the logic rules and other observed features.
We partition predicates as follows: X denotes predicates that are fully observed in the dataset, while U denotes predicates that have missing values for at least one entity (i.e., partially observed predi- cates). Thus a predicate belongs U as long as some of its groundings are missing, though other entities may have observed values. The predicate space considered by the model is the union X PUL U.
To summarize, our model learns logical rules to infer missing instances of latent predicates U by discovering hidden structures within data. This rule induction process identifies logical relation- ships among observed or partially observed predicates. By explicitly learning these structures, our approach enhances both inference capability and interpretability, offering clear insights into com- plex, otherwise hidden dependencies. The key method pipeline is summarized in Figure 2 with details presented in the following sections.
3

---

Published as a conference paper at ICLR 2026
Proparption
O,
Gel Via
Kx
(XAAX V
Prodicate Krsbeeddiigg
Simibartt Seare
Softmnia
Infore predicats valless
Laxp
Bd 12
Base lond
Softmax
Rule Embredding
Fine Tunn
0.00 0U01
Lear e)
Sequential Cavering
Got
Update
0.2 JU3
84
lafer Predicate Values
Caoedinate Gradicat Descant
OR Learning
Figure 2: Model framework. Rule embeddings are optimized using coordinate gradient descent. In each learning step, predicate values are inferred via the Softmin-Softmax operation (Eqs. 2 and 4). For disjunctive (OR) rule learning, sequential hard covering is applied, followed by fine-tuning of the learned rule embeddings (Section (..) Errors are back- propagato using MSE loss between inferred predicate values and the small portion of observed latent predicate samples, constituting a weak- weak-supervision setting.
4.1 MODEL PREPARATION: PRETRAINED PREDICATE EMBEDDINGS
We begin by defining two sets of predicates: X = (X1..... ,......,,,,, X,) represents the set of fully observed predicate variables, and U {UI..... Un} denotes the set predicate variables with missing information that the model aims to discover and define. Our framework is designed to handle both binary (categorical) and continuous features within a unified logical structure. Binary features are treated standard logical predicates. For continuous features, introduce mechanism to derive a "soft" truth value, effectively creating learnable predicates from them. This allows the model to reason over heterogeneous data types, detailed in Section 4.2
We initialize a fixed, unique embedding for each predicate, whether observable or missing. For example, these embeddings can be instantiated as one-hot vectors within an embedding space of di- mension d. We denote the collection of embeddings for observable predicates as Kx and incomplete predicates KU. These predicate embeddings remain frozen throughout the rule learning phase and serve as a foundational dictionary, enabling the interpretation the composition of learned rules by relating rule components back specific predicates.
With the predicate representations defined, we next describe the core of our model: the representa- tion of logical rules and the mechanism by which inferences are drawn.
4.2 MODEL BACKBONE: RULE REPRESENTATION AND INFERENCE
In our NS-FCN framework, logical rules are parameterized by learnable rule embeddings, which serve as the main trainable variables. We optimize these embeddings using an asynchronous block- coordinate descent scheme, updating one rule (or clause) at a time while holding the others fixed. This learning strategy well-suited for discovering expressive rule sets, including multi-hop com- positions (where one inferred predicate appears in the body of another rule) and disjunctive heads (where a predicate can be supported by multiple alternative clauses).
4.2.1 SPECIFICATION OF RULE EMBEDDINGS ④
Let F. be the of rules/clauses, and let θ = {Oy]seF be their embeddings. Each O, encodes the content of a rule whose head predicate is UR E U, and a single head predicate U2 E U can be supported by multiple rules (an OR-of-ANDs structure).
Conjunctive Rule Embedding. Suppose Up is supported by a single conjunctive clause (e.g.. Uk <- X. Xx), we associate the clause with a rule embedding matrix O [01..... .......,,, 0x] the number of body predicates in the conjunction (e.g.. dimension. Each column 0, Ied is trained match the embedding of one body predicate, that the learned O provides an interpretable representation of the clause
dd Here, for X, Axx)) and is the embedding
Disjunctive Rule Embeddings. Suppose Up is defined by a disjunction of R& distinct conjunctive clauses (e.g., UR RR, (clause,, it will be associated with set of Rg distinct rule embed-
4

---

Published as a conference paper at ICLR 2026
dings, denoted (().1..... RR J. Each individual rule embedding Ok,r is itself an d x h, matrix, representing the r-th conjunctive clause, where h, is the arity of that specific clause
All rule embeddings are initialized randomly prior to training and are subsequently optimized as described in Section []] Given these rule embeddings, the model infers the values (or continuous approximations thereof) of latent predicates through a carefully defined inference mechanism.
4.2.2 INFERRING PREDICATE VALUES
We infer latent predicate values from the current predicate state and the learned rule embeddings θ using differentiable forward-chainin procedure.
Predicate Matching, Each column 8 Rd of clause embedding Q = [ 0n] is intended to select one body predicate. Let K Kx U Ktr be the set of frozen predicate embeddings. We match by cosine similarity and recover the predicate index:
Kj arg max cos(K, 02), i, === ((K)) (tm)).
KEK
where I(-) maps an embedding back to its predicate index.
Barward-Chainiee Update. Let o' E [0. 1] 1|=+m denote the predicate state after & forward-chaining steps, containing observed values and current (soft) estimates for missing ones. We initialize od from the dataset (e.g., set unobserved entries 0.5). Let M {0, 10,1) indicate which entries
are observed. Denoting one- inference by g(v"; O), we update only unobserved entries:
w M v' + (1 M) g(v"; θ),
t= 0.1,...,T
(1)
where o is elementwise multiplication. In practice, we set 2 5 steps rather than iterating to fixed point to avoid gradient saturation. We next define g θ).
Clause Evaluation (Soft-AND). Given the matched indices (i, { the j-th body literal uses the current value of, and its matching score 8j cos( fj). For conjunctive clause f, we compute its value via differentiable AND implemented by soft-min over the 2h scalars (.1,..., Sh, -......::0 of
oy softmin ($1. ($1....,, Sh: 1
:T)
with
softmin(Z1. T) log
exp( (p:(-=//7)
2h 1=1
where - > 0 is temperature parameter and softmin - min as 7 0.
Disjunction over Clauses (Soft-OR), If latent head predicate Up is supported by Rd conjunctive clauses, Uk V flurr we aggregate their values (u/ "......., Je.,,
t+1 } using LogSumExp:
Ri
1
gx ())
log exp (xp() ttt))
(4)
where y controls the softness and Eq. approaches max, fur as ls + xc. After T forward- chaining steps, take v" as the final inferred predicate values, where observed entries remain fixed and unobserved entries are filled by the rule-based updates.
The model's 's ability to discover meaningful rules and infer latent predicate states accurately hinges on an effective learning procedure. We now outline the training methodology employed to optimize the rule embeddings O
MODEL LEARNING
Given nule embeddings the inference procedure in Section produces predicate state oT, where observed entries remain fixed, and missing entries are filled by forward chaining. Model learning updates so that the inferred values agree with the observed entries. Concretely, we minimize masked reconstruction loss (weak supervision): for each latent predicate, we compare its inferred values to the subset of instances that are observed, and ignore missing ones via a binary mask. To make optimization stable and scalable in expressive rule spaces (multi-hop and disjunctive rules), adopt a staged learning strategy: block-coordinate updates for rule embeddings, sequential covering to initialize disjunctive clauses, followed by joint fine-tuning.
5

---

Published as a conference paper at ICLR 2026
5.1 BLOCK- COOCD-CTRE GRADIENT DESCENT
We partition parameters by head predicate. For each latent predicate UR. let O₂ denote the set of embeddings associated with rules whose head is U3 (a single clause in the conjunctive case, or multiple clauses for disjunctive heads). We optimize @ by block-coordinate gradient descent: in each cycle, we iterate through latent predicates in a random order and update one block O while keeping all other blocks fixed.
During the update of block Os, we run the inference procedure to obtain the current prediction vector VU for all instances of Up- Let Uk,o be the observed values and let M2 E {0, 1) indicate which instances are observed. We minimize the masked MSE loss
Lr, (0x) mean ((VV, - obs @ Ma)²).
(5)
and update θ₂ using gradients from Lu. (Adam details are deferred to Appendix C.I This im- plements a Gauss-Seidel style block-coordinate method on the overall smooth objective [()) = E, Lüz (0). A brief convergence discussion is provided in Appendix B
For efficiency, a block reaches a near- sear--eefect fit on the observed entries (e.g.. imputation accuracy
0.99 and marginal loss improvement < 10-3 9), we freeze θ in subsequent cycles.
5.2 LEARNING DISJUNCTIVE HEADS VIA SEQUENTIAL. COVERING AND FINE-TUNING
When a head predicate Ug is supported by an OR-of-ANDs rule set with R₂ conjunctive clauses, UR= V fk.r, we learn the clause embeddings { Ox, } in two stages.
Stage Sequential covering (clause discovery). We learn clauses one at a time to encourage diversity. Starting with 1, we optimize Oe,r using the same masked loss in Eq. | but restricted to the currently uncovered observed-positee observed instance covered if the clause output exceeds confidence threshold (e.g., Vsr. > 0.99), and remove those covered instances from the active set. We repeat this process until Rg clauses are obtained.
instances UK. After training clause т, we mark an
Stage 2: Joint fine-tuning (interaction-aware refinement). After initialization, we jointly opti- mize all clause embeddings for Ug. by backpropagating through the soft-OR aggregation (Eq. (...44... Let Pt, denote the aggregated prediction and let M, indicate observed instances of Ug. We minimize
finetuue = mcan ((vu, - Uk,obs) O M²)²).
(6)
which refines clause embeddings while accounting for their interactions under the soft-OR operator. Optimization hyperparameters and normalization details are provided in Appendix C.I
DISCUSSION ON CONTINUOUS FEATURES
Our inference engine operates on predicate truth values in [0, 1]. Binary attributes are already pred- icates, but a continuous feature must be converted into predicate instances (soft bins/conditions) before it can participate in rule evaluation. For each continuous feature e and bin index E {I...., R we learn threshold co. and slope Berr and define
(v.)( (Be cc,)) E 10.
where v. is the raw feature value for given entity. During forward chaining, whenever a matched
body predicate corresponds to P..r. we use its value from Eq. 7 in clause evaluation. Let Up...
denote the inferred values for predicate Pe.r (the corresponding coordinates in uT), and let Pe.robs be the observed soft truth values computed from observed Ve via Eq. 7 With mask M, indicating entities where is observed, we use the masked reconstruction loss
LP..- = mean ((Up_. Pe.r. C )M)
which is optimized jointly with the losses for latent predicates Section 5 update both rule embeddings and the soft-discretization parameters {Ca,ry B.,r
Gradients therefore
7 EXPERIMENTS
We evaluate our framework on synthetic and real-world datasets to demonstrate its effectiveness in rule discovery and missing value imputation.
7.1 SYNTHETIC DATA EXPERIMENTS
We use synthetic datasets to evaluate our model's ability to learn chained and disjunctive rules under partial observability (Figure 3. Each dataset is built from observable Bernoulli variables, with miss-

---

Published as a conference paper at ICLR 2026
ing predicates defined by ground truth rules and made partially available (10%-30% observability) under an MCAR setting. The task is to learn rule embeddings that capture the ground truth logic, evaluated by Rule Discovery Accuracy (i.e. the proportion of runs which learn the truth rules) and Imputation Accuracy.
Our method also demonstrates robustness under MAR and MNAR missingness mechanisms (Ap- pendix B Detailed hyperparameter configurations are provided in Appendix P.4] For the two key temperature parameters of soft operators- in Softmin (Eq.. 3 and μ. in Softmax (Eqq. - - con- duct sensitivity analysis in Appendix H The results indicate that our framework does not depend
on careful hyperparameter tuning: moderately large for Soft-OR and small for Soft-AND
consistently achieve optimal performance. Accordingly, we fix t = 0.1 and μ = 10 across all experiments.
Xa Pa A
X3 Xo Xi)
: PL X
X3 Po xx)
X4 P2 Xs)
X, A
X; pp Xx)
Xa (XG X2)
X; Px X
X5 Pa Xt)
X, P (
X,s pe()))
Xs : X²)
X, Px, A XT) v ps px (x))
:polX A Xe) V (())) Vpp(X, Xx)
(a) Chain
Disjunctive (OR) Chain
(c) Lang Chain with Multiple Disjunetive (OR)
Figure 3: Example rule structures of synthetic experiments.
Training
Unnolsrrsed
Remaining Samples
Number of Sample
1200
Accuncy
X4
0000
Epucc
Epoch
Epuchs
Epoch
Figure 4: An example loss and imputation accuracy during coordinate optimization (Obs. Ratio 0.2, seed 42). We assume the training order is X3. X4, Xs. Epochs 0-19 correspond to rule learning for X3; epochs 20-39 for X4; and epochs 40-end for Xs. Remaining samples identified how many samples are "well-explained" during the hard covering phase.
Table Results for synthetic data example Figure b) with an observation ratio of 0.2. Metrics are averaged over 30 random seeds dataset of 20,000 samples. Ground truth rules are underlined. Full results of observation ratio at 0.1 and 0.3 are in Tables 21 22 of Appendix J.1
Imp. Acc. (Before FT)
Imp. Acc. (After FT)
Train Loss (Before FT)
Train Loss (After FT)
Learned Rules
Rule Acc.
Xs 1.000 + 0.000
/
/
0.001 + 0.000
X
1.00
X4 1.000 + 0.000
0.001 + 0.000
Xa
1.00
(Xo Xx) (Xx
Xs 0.893 + 0.072 0.954 + 0.053 0.087 + 0.045 0.060 + 0.039
(X3 A Xa)
0.53
(Xo A4)
representative example. The full results and corresponding analyses for settings (a) and (c) are provided in Appendix [.2] and Appendix J.3 Table I shows that our model achieves perfect recovery for simple conjunctive rules (X3, X,) and high imputation accuracy for the complex disjunctive rule (Xt). The top learned rule structures contain at least one correct clause, with the ground truth rule showing the highest accuracy.
Results and Analysis. We take setting (b) in Figure
Figure 4 ab))(b) illustrates stable training dynamics. For Xş. the model uses sequential covering (Figure 4 4e)) with "well-explained" examples reducing the remaining set. The fine-tuning (FT)
7

---

Published as a conference paper at ICLR 2026
Table Ablation Study: Effect of Fine-tuning on X5 (Disjunctive Rule) Learning
Metric for Xs
Before Fine-tuning
After Fine-tuning
Recovered Rule Structure
(Xo A X2) (
(Xs A Xo) V (Xo Xa)
Imputation Accuracy for Xs (Unobserved) 0.8729
1.0
phase is followed, which corrects the rule structure and boosts accuracy (Figure (c)) The corresponding ablation study (Table 2 confirms that fine-tuning critical for disjunctive rules, increasing unobserved imputation accuracy for X5 from 0.87 to 1.00.
Learning Efficiency. We further investigated the impact of dataset sample size, varying it from 500 to 20,000 samples. As shown in Figure of Appendix J.1] our method is data-efficient, recovering complex disjunctive rules for Xs with as few as 1,000 samples (Appendix Figure 99. Besides, while coordinate descent requires different cycle numbers, Table [ of Appendix F demonstrates minimal time and memory costs. Processing 50,000 samples in about minutes demonstrates strong efficiency for CPU-based execution.
Table 3:
Impact of rule optimization order on learning progress. Use the example (a) of Figure Note: / denotes successful learning for the respective predicate.
Cycle
Metric
Run
Run 2 X3. Xs, X4]
Run
Optimization Order
[Xs, X4,
X3, X4, Xs}
Cycle
Rule Accu.
X3 V, X4. Xs
X3 X4 X5
X3 V. X4 v Xs
Imputation Accu.,
X3 1.00, 0.005
X3: 1.00, 0.005
Xs 1.00, 0.005
Train Loss
X4 0.87, 0.074
X4 1.00, 0.004
X4 1.00, 0.004
0..44,
Xs 0.94, 0.035
Xs 1.00,0.000
Optimization Order
[X,,
Cycle
Rule Accu.
X4 V.
X3 X5
Imputation Accu.,
X3 I.00, 0.005
:,0, 0.005
1.00,
Train Loss
X4 1.00, 0.004
Xa 1.00, 0.004
Xs 0.94. 0.035
1 0.003
Optimization Order
[Xa, X4,
Cycle
Rule Accu.
X3 V. Vx v
Imputation Accu.,
X3 1.00, 0.005
Train Loss
Xa 1.00, 0.004
X5 : 1.00, 0.003
Convergence Analysis of Asynchronous Coordinate Descent. As discussed in Section 5.1 the learning order for sets of rule parameters, O₁. can be varied. This is particularly relevant in real- world scenarios where the true dependency structure among rules is unknown. To illustrate this "or- der effect", we present exemplary results (corresponding to example (a) in Figure in Table This table highlights that different learning trajectories affect training efficiency, while our method can learn the rule exactly. The complete results for example (a) and (b), including detailed learning curves with loss and gradient plots (Figures 8 are provided in Appendix J.2
We now analyze the convergence of our coordinate gradient descent design. Exact rulc-set induction reduces the minimum-set-covee problem (NP-hard), like any practical rule learner, do not claim global optimality. Instead, we frame search as asynchronous block-coordinate descent on smooth surrogate loss: each step, we update a single rule embedding in closed form, which guarantees the loss never increases yet keeps each move computationally cheap. To guard against poor local minima, we (i) freeze a rule only after this rule is perfectly learned, and (ii) launch diverse initializations. Across 30 runs on synthetic datasets (Appendix Tables 26). this strategy delivers tiny imputation performance variance, and the top-ranked learned rules consistently match ground truth rules. High Jaccard index scores (> 0.6) in Table of Appendix G. further support that the learned rules are structurally stable. More theoretical discussions are provided in Appendix B
7.2 REAL-WORLD DATA EXPERIMENTS
We validate approach three real-world datasets, comparing it with (i) statistical models (MICE(yan Buuren Ouddhhoo- 201T), MissForesst (Seeesttt & Bühlmann 2012 )
(ii) deep generative models (MLP, GAIN Yoon al. 2018). Oissain et al. 2023
8

---

Published as a conference paper at ICLR 2026
mDAE Dupuy et al. ) VAE Veldkamp et al. 2025)) )) and (iii) rule-based interpretable models (BRCG Dash et al. 2018). RRL Wang et al. 2021). DR-N et al.) 2021). LEN L Barbiero et al. 2.22)). For each dataset, we randomly miss some features. We then evaluated the models on their ability to impute these missing values, as well as their performance on a downstream target classification task. Preprocessing and baselines details are provided in Appendix and
For logical reasoning, we used the Birds dataset (Tafjord et al., 2021) with 90% missing ratio for two key predicates. As shown in Table 4 under some random seeds, NS-FCN achieves per- fect imputation accuracy (1.00) and, crucially, perfectly recovers the ground truth logical rules, highlighting its superior capability in deciphering underlying logical structures. Table 6 compares our approach with non-interpretable baselines. While an MLP achieves optimal performance given model remains highly competitive; more importantly, it
the simplicity of the Birds dataset, demonstrates robustness across diverse random initializations, successfully recovering the correct ground-truth rules in the majority of cases. Table 20 further shows that few hundred samples are sufficient for the model to converge to the correct logical truth.
Table 4: Comparison of imputation accuracy and learned rules on the Birds dataset.
Method
Imp Ace.
Learned Rules Example
0.57
abnormal
biird bird +- (ostrich A -wounded) V (bird A wounded)
LEN
0.55
can fly +- (hird A -ostrich) V (-ostrich A -wounded)
0.53
abnormal.bive + (bird -wounded) V (bird ostrich)
RRL
0.51
can_fly (-ostrich -wounded) V (bird A -ostrich)
0,50
abnormal + bird A ostrich
BRCG
0.47
can-fly + bird A aabnormal.bir
0.56
abnormal bird [bird -ostrich A wounded) V (bird A ostrich A -wounded)
DR-NET
0.53
can-fly t (bird -ostrich -abnormal V (bird -ostrich -wounded)
1.00
abnormal bird + ostrich V (bird h wounded)
NS-FCN
1.00
can fly + bird A aabnrrmal bird
Table 5: Comparison of imputation accuracy and learned rules on the Heart Disease dataset.
Method
Imp Ace.
Learned Rules Example
0.65
tresthps +- (-st cp etpaapppicallaaccmpp attyaicll_aagnngggg Langina) (chol A cp sp.ssymptomatic)
0.53
chol high +- (sex female A ca 2) (hp normal A cp asymptomatic)
LEN
0.62
hr high + (cp asymptomatic A target) V (chol low A ca 1)
0.70 st severe +- (cp non anginal A -sbs normal) V (age old A chol low)
0.28
tresthps high
(sex female A -cp typical angina) V. (exang yes -thal normal)
0.33
hr high + (age middle A sex male) V (-resteeg stt abnormality slope upsloping)
RRL
0.33
thalach <- (age < 60) A (restecg = 0)
0.32
stveee +- exang-yes A -lat V (chollow A up sp.asymptomatic)
0.53
trestbps high
age young -ca_4
0.35
chol high <- age young -restecg hesttegghyperrrpply
BRCG
0.33
high +- cp Jypiccical _ipiin A Acaa4
0.32
st.eeee 4- age-young A -slope.upsloping
0.53
trestbps high + (chol A. Jow A high_ V (slope.flat A ca_ that normal) chol high <- sex sex_male male slope upsloping A ca_3
0.33
DR-NET
0.33
hr high +- aage old A -cp-typical angina A fbs high
0.32
st.eee +- hr high A -sex male A -fbs normal
0.86
trestbps_hig
(age > 60) (chol 250)
0.85
chol high +- (sex = age > 55) V (trestbps > 150) hr high + (trestbps > 1 145) V (age > 37 A cp 3)
NS-FCN
0.90
0.76
st severe + (slope = 2) A (thalach < 150)
In medical diagnosis, we use Heart Disease (Detrano et al. 1989) and SPECT Heart (Kurgan al. 2001) datasets, introducing 30% missingness. We also vary the observation ratio from 0.3 to 0.9 and the results in Tables 18 and 19 show comparable performance with only 30% of the data observed. On the Heart Disease dataset, with its mix of continuous and categorical features, NS-FCN's direct handling of continuous values led to superior imputation (e.g., 90% accuracy for thal ach) and

---

Published as a conference paper at ICLR 2026
Imputation accuracy of missing feature value comparison across Heart Discase and Bird datasets on non-interpretable baselines. Results are over 10 random seeds.
Table 6:
Heart Disease
Birds
Method
frestbps
chol
thalach
oldpeak
abnormal
can fly
MICE
0.84100011
0.83.10.014
0.88110.011
0.87-10.015
0.0000
0.86.10.011
MissForest
0.8811.0015
0.8410.012
0.91-00.004
0.88110.016
0.38:10.123
0.68.10.086
MLP
0.88
0.8510.0116
0.88.10.014
0,80:10.025
0,96110.05
0.99110,003
GAIN
0.85=0.022
0.84=0.011
0.90+00.044
0.89:0.014
0,83=0,100 102
0,82=t0.083
MissDiff
0.82=#0.017
0.83=0.019
0.89=00018
0.84
0.83=0.020
0.86=000077
mDAE
0.88
0.84#0.012
0.90::0.155
0.87=00015
0.87=0.002
0.87
VAE-based
0.85++00015
0.84+000211
0.90+-0.015
0.86+:0.015
0.62+00.06
0.87+00.004
NS-FCN
0.87++00225
0.051
0.884
0.78+0
0.95++0.064
0.95+00064
Table 7: Medical diagnosis after missing value imputation. Results are over 10 random seeds.
Heart Disease
SPECT
Method
Accurucy
F1
Accuracy
F1
MICE (van Buuren & Ouushoomm
2011
0.831000100
0.811000112
0.78:10.019
0.871-00113
MissForest (Stekhoven & Bühlmann 2012)
0.83+000113
0.1110.014
0.79:00012
0.87110.008
MLP
0,8410.010
0.82:00012
0.92:00007
0.90+00000
GAIN Yoon al.
2018)
0.8410.004
0.82+00006
0.76:10.019
0.85+100113
MissDill (Ouyang al.
2023
0.84100010
0.8210.011
0.7710.023
0.861100116
mDAF (Dupuy ct al. 2024
0.84+00009
0.82+000110
0.80+000133
0.88+-0.009
VAE-hased Veldkamp ct al
2025)
0.83+0.009
0.81+0.009
0.75+0.016
0.85++0011
BRCG (Dash et al. 201
0.77+00.066
0.74+0.034
0.85+00.46
0.90+0.035
RRL (Wang al. 202T
0.7810.002
0.80100003
0.901100055
0,94.00.005
DR-NET (Qiao (RiNET(Qiao al. 2021
0.8510.005
0.8210.005
0.8910.025
0.9210.0017
LEN (Barbiero el al.
2022
0.691000077
0.801000000
0.76.10.035
0.85.10.017
NS-FCN
0.8910.018
0.89+0.018
0.92+00009 0.96+000009
the discovery of clinically relevant rules with numerical thresholds (e.g., age > 60, chol 250), as shown in Tables [5] and 30 NS-FCN attains imputation accuracy comparable to the advanced statistical and generative baselines, yet distinguishes itself by offering full interpretability. a criti- cal advantage over these black-box approaches. Compared with rule-based models, our evaluation highlights NS-FCN's unique ability to handle heterogeneous data types. A key distinction that NS-FCN directly models continuous features, whereas baseline methods are restricted to binary inputs, forcing discretization (e.g., for trestbps, binning values into < 120, 120 140, > 140 mmHg low, normal, and high).
On the binary SPECT dataset, we randomly miss all 22 features, thus we report the diagnosis ac- curacy after imputation. When the imputed features are used for diagnosis, NS-FCN outperforms all baselines on both Heart Disease and SPECT, as shown in Table 7 Unlike baseline models that train a classifier on previously imputed samples, where imputation errors inevitably propagate to the downstream task, our method jointly optimizes rule discovery and target inference. Furthermore, use of soft-logic relaxation prevents the model from overfitting to noise (such as incorrect fea- tures), enabling it to capture dominant logical structures. This robustness is further supported by the comprehensive noise sensitivity analysis in Appendix [.] (Tables T6 and 17p, which demonstrates that the model learns valid rule approximations (e.g., capturing one correct clause) and maintains strong predictive performance even as noise levels increase. Detailed rules and LLM assessments
are in Appendix Tables 28 29 and 30
CONCLUSION
Our NS-FCN framework effectively learns interpretable rules for missing value imputation, demon- strating strong performance across diverse range of synthetic and real-world datasets. A key strength is its ability to seamlessly reason over heterogencous data, handling both binary predicates (e.g., Birds) and continuous features in complex domains like medical diagnosis (e.g.. SPECT and Heart Disease). It successfully handles missing data and learns hierarchical rule structures, offering significant potential for trustworthy diagnostics and transparent Docision-making
10

---

Published as a conference paper at ICLR 2026
REPRODUCIBILITY STATEMENT
We have made extensive efforts to ensure the reproducibility our results. The complete descrip- tion of both the synthetic dataset generation and the real-world dataset preprocessing methods is illustrated in Appendix D.2 and J Details of the computational setup, including hardware con- figuration and software environment, well as the choice of hyperparameters, are documented in We also release our code.
Appendix and K.3
ACKNOWLEDGEMENTS
We thank the anonymous reviewers and the area chair for their constructive feedback, which helped improve the paper. This work was supported in part by the Key Program of the National Natural Science Foundation of China (NSFC) under Grant No. 72495131; the Shenzhen Stability Science Program 2023; the Shenzhen Science and Technology Program No. JCY12025000414103300 and the Longgang District Key Laboratory of Intelligent Digital Economy Security.
REFERENCES
Pietro Barbiero, Gabriele Ciravegna, Francesco Giannini, Pietro Lió, Marco Gori, and Stefano Melacci. Entropy- logic explanations of neural networks. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 36, pp. 6046-6054 2022.
Dimitri P Bertsekas. Nonlinear programming. Journal of the Operational Research Society, 48(3):
334-334, 1997.
Antoine Bordes, Nicolas Usunier, Alberto Garcia-Duran, Jason Weston, and Oksana Yakhnenko. Translating embeddings for modeling multi-relational data. In Advances in Neural Information Processing Systems, volume 26, 2013.
Andres Campero, Aldo Pareja, Tim Klinger, Josh Tenenbaum, and Sebastian Riedel. Logical rule induction and theory learning using neural theorem proving. arXiv preprint 1rXXvv1809.000 193, 2018.
Zhi Chen, Sarah Tan, Urszula Chajewska, Cynthia Rudin, and Rich Caruna. Missing values and imputation healthcare data: Can interpretable machine learning help? Conference on Health, Inference, and Learning, pp. 86-99, PMLR, 2023.
Xuening Feng Paul Weng Matthieu Zimmer Dong Li Wulong Liu Jianye Hao Claire Glanois, Zhao- hui Jiang. Neuro-symbolic hierarchical rule induction. In International Conference on Machine Learning (ICML), pp. 7583-7615. PMLR, 2022.
William W Cohen. Fast effective rule induction. In Machine learning proceedings 1995, pp. 115- 123. Elsevier, 1995.
Andrew Cropper and Rolf Morel. Learning programs by learning from failures. Machine Learning, 110(4):801-88) -856, 2021.
Sanjeeb Dash, Oktay Gunluk, and Dennis Wei. Boolean decision rules via column generation. Advances in neural information processing systems, 31, 2018
Robert Detrano, Andras Janosi, Walter Steinbrunn, Matthias Pfisterer, Johann-Jakob Schmid, Sar- bjit Sandhu, Kern H Guppy, Stella Lee, and Victor Froelicher. International application of new probability algorithm for the diagnosis of coronary artery disease. The American journal of cardiology. 64(5):304-310, 1989.
Honghua Dong, Jiayuan Mao, Tian Lin, Chong Wang, Lihong Li, and Denny Zhou. Neural logic machines. arXiv preprint arXiv:1904.1 1164,, 2019.
Mariette Dupuy, Marie Chavent, and Remi Dubois. mdae: modified denoising autoencoder for missing data imputation. arXiv preprint arXiv:2411. 2024.
Bradley Efron. Missing data, imputation, and the bootstrap. Journal of the American Statistical Association, 69(426):463-477 1994.
11

---

Published as a conference paper at ICLR 2026
Khaled M Fouad, Mahmoud M Ismail, Ahmad Taher Azar, and Mona M Arafa. Advanced methods for missing values imputation based on similarity learning. PeerJ Computer Science, 7:e619, 2021.
Kelvin Guu, John Miller, and Percy Liang. Traversing knowledge graphs in vector space. arXiv preprint arXiv:/506.01000 2015.
Pascal Hitzler and Md Kamruzzaman Sarker. Neuro-symbolic artificial intelligence: The state of the art. 2022.
Diederik P Kingma and Max Welling. arXiv:1312.6114, 2013.
Auto-encoding variational
bayes.
arXiv preprint
Lukasz A Kurgan, Krzysztof J Cios, Ryszard Tadeusiewicz, Marek Ogiela, and Lucy S Goodenday. Knowledge discovery approach to automated cardiac spect diagnosis. Artificial intelligence in medicine, 23(2):149-169 2001.
Linchao Li, Bowen Du Yonggang Wang, Lingqiao Qin, and Huachun Tan. Estimation of missing values in heterogeneous traffic data: Application of multimodal deep learning model. Knowledge- Based Systems, 1 194: 105592, 2020.
Yankai Lin, Zhiyuan Liu, Maosong Sun, Yang Liu, and Xuan Zhu. Learning entity and relation em- beddings for knowledge graph completion. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 29, 2015.
Robin Manhaeve, Sebastijan Dumancic, Angelika Kimmig, Thomas Demeester, and Luc De Raedt. Deepproblog: Neural probabilistic logic programming. Advances in neural information process- ing systems, 31, 2018.
Rahul Mazumder, Trevor Hastie, and Robert Tibshirani. Spectral regularization algorithms for learn- ing large incomplete matrices. The Journal of Machine Learning Research, 1 11:2287-2322, 2010.
Yurii Nesterov. Introductory lectures on convex optimization: A basic course, volume 87. Springer Science & Business Media, 2013.
Alessandro Oltramari, Jonathan Francis, Cory Henson, Kaixin Ma, and Ruwan Wickramarachchi. Neuro-symbolic architectures for context understanding. In Knowledge Graphs for Explainable Artificial Intelligence: Foundations, Applications and Challenges, pp. 143-160. IOS Press, 2020.
Yidong Ouyang, Liyan Xie, Chongxuan Li, and Guang Cheng. Missdiff: Training diffusion models on tabular data with missing values. arXiv preprint 2rXiv:2307.00467, 2023.
Leonardo Pellegrina and Fabio Vandin. Scalable rule lists learning with sampling. In Proceedings of the 30th ACM SIGKDD Conference Knowledge Discovery and Data Mining, pp. 2352-2363, 2024.
Litao Qiao, Weijia Wang, and Bill Lin. Learning accurate and interpretable decision rule sets from neural networks. In Proceedings of the AAAI Conference Artificial Intelligence, volume 35, pp. 4303-4311, 2021.
J. Ross Quinlan. Learning logical definitions from relations. Machine learning, 5:239-266, 1990.
Md Geaur Rahman and Md Zahidul Islam. A decision tree-based missing value imputation technique for data pre- vre-prceessing The 9th Australasian Data Mining Conference: AusDM 20/ pp. 41- 50. Australian Computer Society Inc, 2011.
Md Geaur Rahman and Md Zahidul Islam. Fimus: A framework for imputing missing values using co-appearance, correlation and similarity analysis. Knowledge-Basse Systems, 56:3 1-327, 2014.
Matthew Richardson and Pedro Domingos. Markov logic networks. Machine learning, 62: 12:107.136, 2006.
Peter Richtárik and Martin Takáč. Iteration complexity of randomized block-coordinate descent methods for minimizing composite function. Mathematical Programmingg 144(1):1-38, 2014.
12

---

Published as a conference paper at ICLR 2026
Tim Rocktäschel and Sebastian Riedel. End-to-end differentiable proving. Advances in neural information processing systems, 30, 2017.
Hikaru Shindo, Masaaki Nishino, and Akihiro Yamamoto. Differentiable inductive logic program- ming for structured examples. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, pp. 5034-5041, 2021.
Daniel Stekhoven and Peter Bühlmann. Missforest cocissttoooo--ppa con-paraettric missing value imputation for mixed-type data. Bioinformatics, 28(1):112-118, 2012.
Lena Stempfle and Fredrik Johansson. Minty: Rule-based models that minimize the need for im- puting features with missing values. In International Conference Artificial Intelligence and Statistics, pp. 962-972. PMLR, 2024.
Oyvind Tafjord, Bhavana Dalvi, and Peter Clark. Proofwriter: Generating implications, proof', and abductive statements over natural language. In Findings of the Association for Computational Lingnistics: ACL-IJCNLP 2021, pp. 3621-3634, 2021.
Théo Trouillon, Johannes Welbl, Sebastian Riedel, Éric Gaussier, and Guillaume Bouchard. Com- plex embeddings for simple link prediction. In International conference on machine learning, pD- 2071-2080. PMLR, 2016.
Paul Tseng. Convergence block coordinate descent method for nondiffcrentiable minimization. Journal of optimization theory and applications, 109(3):475-494 2001.
Stef van Buuren and Catharina Gerarda Maria Groothuis- mice: Multivariate imputation by chained equations in r. Journal of statistical software, 45(3), 2011.
Karel Veldkamp, Raoul Grasman, and Dylan Molenaar, Handling missing data in variational autoen- coder based item response theory. British Journal of Mathematical and Statistical Psychology, 78 (1):378-397, 2025.
Xinyu Wang, Isil Dillig, and Rishabh Singh. Synthesis of data completion scripts using finite tree automata. Proceedings of the ACM on Programming Languages, ((0OSSLA): 2017.
Zhen Wang, Jianwen Zhang, Jianlin Feng, and Zheng Chen. Knowledge graph embedding by trans- lating on hyperplanes. In Proceedings of the AAAI Conference Artificial Intelligence, vol- ume 28, 2014. doi: 10.1609/aaii wwt...8811..7777 URL httpp://aa....cg/111..600/ v28 8870
Zhuo Wang, Wei Zhang, Ning Liu, and Jianyong Wang. Scalable rule-based representation learning for interpretable classification. Advances in Neural Information Processing Systems, 34:30479- 30491, 2021.
Dennis Wei, Sanjeeb Dash, Tian Gao, and Oktay Gunluk. Generalized linear rule models. In International conference machine learning, pp. 6687-6696. PMLR, 2019.
Stephen J Wright. Coordinate descent algorithms. Mathematical programming. 151(1):3-34, 2015.
Yang Yang, Chao Yang, Boyang Li, Yinghao Fu, and Shuang Li. Neuro-symbolic temporal point processes. arXiv preprint 1xXi::2406003114 2024.
Jinsung Yoon, James Jordon, and Mihaela Schaar. Gain: Missing data imputation using generative adversarial nets. In International conference on machine learning, pp.
PMLR, 2018.
Chengqi Zhang, Yongsong Qin, Xiaofeng Zhu, Jilian Zhang, and Shichao Zhang. Clustering missing value imputation for data preprocessing. In 2006 4th IEEE International Conference on Industrial Informatics, pp. 1081-1086. IEEE, 2006.
13

---

Published as a conference paper at ICLR 2026
APPENDIX OVERVIEW
Section A
extends the related-work discussion and provides broader context on IIP and missing-dan imputation.
Section B gives a theoretical convergence view of our block coordinate optimization, in- cluding assumptions and stationarity statements.
Section C summarizes model-supplement details, including optimization implementation
notes (Section C.I
Section D describes datasets (Section D.I), preprocessing pipelines (Section D.2 and
baseline configurations used in experiments (Section D.3). (.3)).
Section reports performance under different missingness
mechanisms
(MCAR/MAR/NAR))
Section provides runtime and memory analyses synthetic and real-world settings.
Section
evaluates structural stability of learned rules (e.g., Jaccard-based analyses in
Section
Section
presents sensitivity analysis for soft-operator temperatures.
Section reports robustness under label noise (Section [.] and varying missing ratios
(Section 12
Section provides additional synthetic experiment results and extended case studies.
Section K
provides additional real-world experiment analyses and supplementary rule
sessments.
Section L
discusses limitations and broader impact considerations.
Section clarifies how I.LMs are used in manuscript polishing and rule-assessment nar- ration.
A RELATED WORK SUPPLEMENT
Traditional Inductive Logic Programming (ILP) Methods. Inductive Logic Programming learns logical rules from relational data. Cohen (1995) proposed RIPPER, a fast rule induction algorithm using ayparate-and-onngee
strategy. Quinlan (1900) developed FOIL, which generates clauses it- eratively. Dash et al.) (2018) introduced Boolean decision rules using column generation, Wei et al. (2019) proposed GLRM, integrating decision rules into linear models. Cropper & Morel (2021) presented LFF implemented in Popper. These approaches rely on heuristics but may not guarantee optimal solutions. Pellegrina & Vandin (2024) proposed SamRuLe for near-optimal rule lists via sampling.
Differentiable ILP Methods. Traditional ILP models struggle with noisy data and scalability. Dif- ferentiable approaches address these issues by integrating continuous relaxation, which allows gra- dient descent for optimization. (Shindo et al. (202T) proposed @IIP, which represents logic rules in a differentiable form and combines neural networks with symbolic logic. Manhaeve et al. (0188) in- troduced DeepProbLog, extending ProbLog with neural predicates. Neural Logic Machines (NLMs) Dong et al. 2019) combine MLPs with logic programming to improve computational efficiency but reduce interpretabilitty
Broader Missing Data Imputation Methods. Missing data imputation methods range from global model-based techniques to localized and hybrid strategies, extending to deep and ensemble frame- works.
At the global end, nonparametric bootstrap methods (Efron 1994) provide bias-corrected estimates repeated sampling, while spectral regularization approaches like SOFT-IPPUTE (Mazumder et al. 2010) solve a nuclear-norm minimization through iterative soft-thresholded SVD. Classi- call multivariate imputation schemes such as MICE (van Buuren & Groothuis-Qudshonk construct sequence of conditional models for each variable with missingness and iteratively sam- ple from these chained regressions until convergence, thereby approximating draws from the joint posterior and naturally propagating uncertainty across multiple imputations. Tree-based ensemble methods such as MissForest (Stekhoven Bühlmann, 2012) adopt an iterative refinement strategy in which random forests are trained per variable using the currently imputed data predictors, updating missing entries via out-of-buu bag predictions until changes stabilize, thus capturing complex nonlinearities and high-order interactions without requiring parametric distributional assumptions.
2011)
14

---

Published as a conference paper at ICLR 2026
Moving toward local adaptation, decision tree-based EM (DMI) (Rahman & Islam 2011 parti- tions complete cases via C4.5 and imputes within each leaf, and clustering based random imputation (CRI) (Zhang et al. (006) applies kemnel-weighted estimation in the nearest k- k-means cluster. Hybrid similarity learners, such as KI and its fuzzy extension FCKI (Fouad et al. 2021). refine this idea by dynamically selecting neighborhood sizes before multivariate imputation. For high-dimensional or heterogeneous data, deep architectures like GAIN (Yoon et al. 2018) cast imputation as generative adversarial game where generator proposes imputations conditioned on observed-mask vector and discriminator learns to distinguish observed from imputed components, while -AE-based im- puters (Veldkamp et al.) 2025) treat the complete feature matrix as generated from diw-dimnssional latent variables and learn to reconstruct missing entries via amortized variational inference under a probabilistic encoder-decoder architecture. Building on denoising autoencoders, mDAE (Dupuy) er al. 2024) modifies the reconstruction loss to ignore pre-imputed values at missing positions and couples this with an overcomplete hidden representation, which empirically improves RMSE over standard DAEs and several classical imputers across multiple UCI datasets (Dupuy al. 2024). In
the same spirit of generative modeling, Miss (Ouyang et al. 2023) trains a diftusion model on tabular data with missing values by injecting noise along forward stochastic process and learning a reverse denoising process that is explicitly conditioned on the observed-mask pattern, thereby pro- ducing imputations through iterative refinement from pure noise. Models such MMDL (Li al., 2020) align stacked autoencoder embeddings across modalities to exploit cross-view correlations. Ensemble schemes like FIMUS (Rahman & Islam 2014) combine no-appearance, correlation, and similarity in a weighted framework. Despite their varied ference to localized and multimodal learning these methods uniformly rely on statistical patterns and do not leverage explicit logical rules to govern inter-variable relationships.
from global in-
B CONVERGENCE ANALYSIS OF COORDINATE GRADIENT DESCENT
For clarity, we analyze simplified version of our learning algorithm in which each head predicate Uk is associated with a single parameter block θg. Let = (OI,..., ..)) collect all parameters. The global training objective is
((
Σ Cu (9), k=1
(e) = mean ( ((vv, (9)
(9)= @ M - Uk,abs C Mc))).
(8)
where vu. (0) is computed by forward chaining using the differentiable operators introduced in the main text (e.g.. Eqs. 2
B.1 ASSUMPTIONS
We make the following standard assumptions for smooth block coordinate descent (e.g., (Teeng 2001 Bertsekas
1997 Nesterov 2013))
Assumption The objective L Rd
Rd > is
1. bounded below: infe L(O) -x,
2. continuously differentiable θ and
3. has block-wiss wise Lipschitz-continuun gradients: for each there exists Typ < oo such that, for all θ and all his
Ve. L(O + eghi)) Ve, Ve. [()) S L& |lhall. where exhy denotes the vector obtained by changing only block k.
These conditions hold in our setting because L is built from smooth operations (e.g., linear maps, sigmoid, softmin, log- sum-exp) composed with a squared loss, and training is restricted bounded level sets.
B.2 IDEALIZED BILL-BATCH BLOCK COORDINATE GRADIENT DESCENT
Consider the following idealized algorithm. At iteration т we pick block index k, E {1..... m} (e.g.. by eycling through (1...., m)) and perform gradient step that block only:
θ r-1 0,,- 1) Ve., L(O").
(10)
O O for all { + kr.
(11)
where 17 > 0 is a step size. This matches the idealized version of the rule update in Section 5.1 when we update Uk, all other predicates Ug are kept fixed.
15

---

Published as a conference paper at ICLR 2026
Lemma (Monotone decrease for small steps) Suppose Assumption holds. If the step size sat- isfies < 17 S Lk, at iteration т, then
[( & [()7) " [V,,, [(er)]]
(12)
In particular, the sequence {f(0"))r20 is monotonically non- con-increasing and convergent.
Proof (Proof sketch) By block-wise Lipschitz continuity of VeL.C.
C( = =( + ck, (0 -)
(13)
Lke 10 -
- <((0) + Ve., ((07), 0r -
O 1,,
OR 112.
(14)
Substituting the update 0% 0 = 11 Ve,, [()7) and rearranging gives [( < ((07) - 7 nLu. Vo,,, [( If 1) S 1/,,,, then nL,, 2 1/2, yielding the claimed inequality.
(15)
Lemmal D Timplies that the loss decreases at every iteration and the gradients on updated blocks cannot stay large forever. Combined with mild assumption that each block is selected infinitely often, obtain convergence to a block-stationary point.
Proposition 1 (Convergence to block point) Assume Assumption 7 holds, the level ser (e : [()) < ]{ is bounded, each block lo is selected infinitely often, and the step sizes satisfy n ming L/L... Then any limit point Q* of the sequence (07) generated by the above block coordinate gradient method is block-stationary:
Ve,, = for all l....... Equivalently, no single block O₁ can be perturbed decrease C while all other blocks are fixed.
(16)
Proof 2 (Proof sketch) Summing the inequality from Lemmal 7 over T shows that
Elver,
<0,
so the block gradients must tend to zero along the subsequence where given block ls is updated. Since each block is selected infinitely often and the iterates remain in bounded level set, standard arguments for block coordinate descent (Tseng 2007) imply that any limit point has zero gradient in every block.
Thus, in the ideal full-batch setting with sufficiently small steps, our predicate-wis. wise coordinate up- dates produce non-increasing loss sequence (L(O*)} and converge to point where no single predicate block O₂ can further reduce the global objective.
B.3 STOCHASTIC MINI-BATCH VARIANT AND ADAM
In practice, implementation uses mini-batches and the Adam optimizer for each block update (as described in Section In this case, the gradient Ve, f. is replaced by a stochastic estimate com- puted on mini-batch, and the step block-coordinate gradient scheme: the loss is no longer guaranteed to decrease at every single up- date, but under standard assumptions, stochastic block-coordinate methods known approach neighborhood of stationary point in expectation (see, e.g., (Richtárik & Tak48 2014 2015
Adam's adaptive preconditioning. This yields stochastic
Wright
C C.1 OPTIMIZATION DETAILS
MODEL SUPPLEMENT DESCRIPTION
Throughout all training stages, each rule embedding (or set of embeddings during joint fine-tuning) is optimized using the Adam optimizer. A crucial step following each gradient update is the normal- ization of the rule embeddings. This involves applying Rectified Linear Unit (ReLU) activation to the embedding data (ensuring non non-negative values, which can aid interpretability for positive pred- icate contributions) followed by Lg normalization of each vector within the rule embedding matrix. This normalization helps stabilize the training process and maintains consistent magnitudes for the embedding components.
16

---

Published as a conference paper at ICLR 2026
D DATASETS AND BASELINES
D. DATASETS
Heart Disease. We use the widely-cited Cleveland Clinic dataset from the UCI Heart Disease database (Detrano et al. 1989). This dataset contains 303 patient records, each with 13 features- mix of continuous and categorical variables-sur such as age, cholesterol level, and resting blood pres- sure. The task is predict the presence of heart disease, which is indicated by the target variable on a scale from O (absence) to 4 (severe). Following standard practice, we simplify this into binary classification problem: predicting presence (values 1-4) versus absence (value 0).
SPECT. The SPECT (Single Proton Emission Computed Tomography) dataset presents a binary classification task to diagnose cardiac conditions (normal/abnormaal) based on 22 binary patient fea- tures. The dataset describes the diagnosis of cardiac SPECT images. Each of the patients is classified into two categories: normal and abnormal. The 267 SPECT image sets (patients) database was pro- cessed to extract features that summarize the original SPECT images. As a result, 44 continuous feature patterns were created for each patient. The pattern was further processed to obtain 22 binary feature patterns. The CLIP3 algorithm was used to generate classification rules from these patterns (Kurgan et al., 200T). The CLIP3 algorithm generated rules that were 84.0% accurate (as compared with cardiologists diagnoses). A key challenge in this domain is the prevalence of missing data, making it an ideal testhed for our model's imputation and rule-learning capabilities.
Birds. Bird's Rulebase is well-known logic problem designed to assess an Al's ability to learn and reason with hierarchical logical rules that mimic common- sense knowledge (Tafjord al. 202I). It has the ground truth single theory of six rules as follows.
can fly(X) <- bird(X), not abnormal (X)
bird(X) (X) <- XXt
abnormal_bin _iird X) <- ostrich(X)
not can_fly(X) <- oxtrich(X)
abnormal botormaabbbr((() bird(X) < wounded(XX)
not can_fly(X) < wX)
Figure 5 further illustrates the structure of these rules.
not abnornal bird(X)
bird(X)
Wounded(X)
ostrich(X)
OR
fly: p(bird( AND not abnormal birk(X))
abnormal bird: ((bird(X) AND wounded(X))
abnormal bird: P(astrich((2)
bird: plostrich()))
p(wounded(X) X) OR ostrich(X))
OR
abnormal bird: p(p(ostricb(X)) OR P[bird(X) AND wo)
Figure 5: Ground truth rules for Bird dataset.
D.2 PREPROCESSING OF DATASETS
Heart Disease. The UCI Heart Disease dataset contains a mix of 13 continuous and categorical fea- tures with 303 samples. To create a challenging imputation task, we introduced 30% missing ratio independently into four key continuous variables: resting blood pressure (trestbps), cholesterol (chol), maximum heart rate (thalach), and ST depression (oldpeak). Following the protocol in MissDift(Ouyang et al. 2023), we generate missing values under Missing Completely At Ran- dom (MCAR) mechanism. Let x E I denote the complete data vector. We generate a binary mask vector m E (0. 1)d, where m,- = 1 indicates that 24 is observed, and m, O indicates it is missing. The observed data is represented as x x ② m na O1 - m). where O denotes element-wise multiplication.
thtps:/wwww.doc.c.ac.uu hwwpa///weeeeeeeeoooooo////////e P. 1441-2x1.pdit p5
17

---

Published as a conference paper at ICLR 2026
For our NS-FCN framework, the task is to directly impute these missing continuous values. For deep learning baselines, continuous features are standardized using Z-score normalization, and categorical features are one-hot encoded. For tree-based and statistical baselines (MissForest, MICE), categorical variables are treated as factors. However, to accommodate the baseline mod- els which only support binary inputs, we first discretized these four variables into three cate- gorical bins based on clinical thresholds: blood pressure (< 120, 120 - 140, > 140), choles- terol (< 200, 200 240, 2 240), max heart rate (< 100, 100 160, 160), and ST depression (s 1.0, 1.0 - 2.0, 2.0). The baselines were then tasked with imputing the correct category. Con- sequently, we evaluate the imputation accuracy on the discretized bins.
SPECT Heart. The dataset's 22 binary features were randomly masked with 30% probability to simulate missing data. Our framework missing features, and second, to perform the final patient diagnosis based on the completed feature set. The diagnostic performance is compared against five baseline methods, including four rule- based approaches and an MLP.
then applied
two-stage task: first, to impute the
Birds. Following the ground truth logical rules, generated dataset of 1,500 samples. To create a difficult logical reasoning challenge, we introduced a 90% missing ratio for two crucial latent predicates: can_fly and abnormal matoormalbbirte _bird. The task for all models to impute these missing binary values based on the observed predicates. The imputation accuracy is compared against the same set of baselines.
D.3 BASELINE MODELS
To rigorously evaluate performance, we compare our method against 11 established baselines, rang- ing from classical statistical methods to advanced deep generative models and interpretable models.
Statistical Models.
MICE (van Buuren & Groothuis- 2011)
Multivariate Imputation by Chained Equations (MICE) is a widely used statistical method based on Fully Conditional Specification (FCS). It iteratively imputes missing values by modeling each feature with missing data as function of other features using linear re- gression (for continuous variables) or logistic regression (for categorical variables). We generate m imputed datasets and report results from the first completion.
MissForest (Stekhoven & Bühlmann] 2012)
MissForest is non- non--onnnneee parametric method that handles mixed-type data using an iterative Ran- dom Forest approach. It treats the missing data problem as a prediction task, training a ran- dom forest the observed parts of the data predict the missing values. It is particularly effective at capturing non-line non-linear interactions without explicit distributional assumptions.
Deep Generative Models.
MLP (Multilayer Perceptron)
We use a simple feed-forwart neural network with fully connected layers and ReLU acti- vations as deterministic imputation baseline. Given an input vector x E Rd and binary mask m E {0, 13d indicating observed entries (m, = if z is observed, 0 otherwise), we first obtain = x ④ m + na @(1 m), and use the observed mask for input gating:
he x O m.
The network fe takes he input and outputs reconstruction x = (bo). Training is performed under weak supervision by minimizing the Mean Squared Error (MSE) only on observed entries:
LMPP = ( - x) c m
so that gradients are propagated only through coordinates with ground-truth observations; at test time, the missing entries (m, = 0) are imputed using the corresponding components of &.
VAE (Variational Autoencoder)
Our VAE-based imputer follows the amortized inference framework of Kingma & Welling (2013), adapted to incomplete tabular data as in recent work on VAE with missingness (e.g. Veldkamp et al. (2025). Given (x, m), we construct gated and masked input
x ④
(1 m), hn x m,
18

---

Published as a conference paper at ICLR 2026
and feed the concatenated vector [hu, 1-m) mm] into the encoder to obtain a Gaussian posterior 4o (z x, m) ( diag(o)))
A latent sample is drawn via the reparameterization trick and passed through decoder Px z) to produce (z( The model is trained by maximizing the Evidence Lower Bound (ELBO), where the reconstruction term only involves observed entries:
LVAE (xxa(z) x) o ml +
(L x, m) ||p(z))
reconstraction on ohserved data At inference time, missing values are imputed by the decoder output {z(z) coordinates where m- = 0.
Kt. regularization
DAE/ / mDAE (modified Denoising Autoencoder)
For the autoencoder baseline, we adopt denoising autoencoder architecture with a modi- fication of the loss function proposed in the mDAE Dupuy et al. (2024). Given (x, m), we first perform simple pre- rreimputation obtain a complete input x, and then apply masking noise with rate only on originally observed entries:
X x m + na @(a@@11- - m).
c ~ Ber(p) x = (x c m) @ (1 (1-co m).
The corrupted input x (noisy) is fed into an encoder-decoder network gy that outputs a recon- struction & = Iw (m Crucially, following the modified-loss idea of mDAE (Dupuy et al. 2024 , the reconstruction loss is computed only on truly observed entries, and pre-
imputed missing values are ignored:
LIDDA = -x ④ ml|
This prevents the autoencoder from overfitting arbitrary pre-imputed values at missing po- sitions while still benefiting from denoising training; at test time, imputations for missing entries (m, = 0) are taken from the corresponding components of X.
GAIN (Generative Adversarial Imputation Nets) (Yoon et al. 2018
GAIN adapts the Generative Adversarial Network framework for imputation. The gen- erator G imputes missing components, while the discriminator /) attempts to distinguish between observed and imputed components. A hint mechanism is introduced to provide D with partial information about the mask distribution, forcing G to learn the true underlying data distribution. We utilize hybrid loss function combining adversarial loss with MSE for continuous features and cross-entropy for categorical features.
Miss (Diffusion Imputation Nets)(uuang et al., 2023)
We employ diffusion probabilistic model specifically adapted for tabular missing data. The model is trained to reverse noise-adding process. During inference (imputation). we utilize the guided sampling or conditioning strategy: at each denoising step t. the known observed values x oxs x @ m + na @ (1 - m)) are re-injected into the sample to ensure consistency with the ground truth. The model effectively samples iipp from the conditional distribution mixm (bb)).
Interpretable Models.
BRCG (Dash et al.
2018 an integer program designed trade classification accuracy for rule simplicity. It uses column generation to search over an exponential number of candidate clauses efficiently.
LEN (Barbiero et al.) 2022 is an end-to-end differentiable method for extracting logical explanations from neural networks using First-Order Logic.
DR-NET (Qiao et al.
2021 is method for learning independent logical rules in disjunc- tive standard form as an interpretable model for classification.
RRL (Wang et al. 2021) learns interpretable non-fuzzy rules for data representation and classification using a novel training method called Gradient Grafting.
PERFORMANCE UNDER DIFFERENT MISSINGNESS MECHANISMS
We compare three general missingness mechanisms for dataset generation:
MCR (Missing Completely at Random): The probability of being missing is the same for all cases, which is the missingness mechanism in other experiments in our paper.
19

---

Published as a conference paper at ICLR 2026
MAR (Missing at Random): Missingness depends observed variables. We can indicate which observed variable to use for missingness; the default Xo. Then, we set higher probability of missing when the dependency variable is 1.
MNAR (Missing Not at Random): Missingness depends on unobserved variables or the missing values themselves. Take X3 for example, we set it is more likely to be missing when X3 = 1 (positive values are harder observe).
We show an observation ratio 0.2 and sample size = 20,000 as a representative case in Table 8 We run 20 random seeds. Since the seeds are different from those used Tables 21 and 22 the results are slightly different.
Table 8: Comparison of inference accuracy and rule accuracy under different missing mechanisms.
MCAR
MAR
MNAR
Imputation Accu.
Rule Accu.
Imputation Accu.
Learned Rules
Imputation Accu.
Rule Accu.
X3
1.00 = 0.00
1.0
1.00 0.00
1.0000
1.00 + 0.00
1.0000
X1
1.00 = 0.00
1.0
1.00 0.00
1.0000
1.00 + 0.00
1.0000
Xs
0.95 0.07
0.6
0.95 0.07
0.6000
0.93 0.05
0.4000
The results show that MAR and MNAR show comparable results to MCAR, which demonstrates our method's effectiveness across the full spectrum of missing data scenarios.
F RUNNING TIME AND MEMORY COST ANALYSIS
F.1 SYNTHETIC DATASET
While coordinate descent requires different cycle numbers (Table method demonstrates effi- cient performance on standard CPU configurations. We conducted experiments using an Apple M4 chip with 10 cores and 16GB memory, taking observation ratio 0.2 as an example. Results over 20 runs on setting (b) of Figure 3
Table 9: Running time and memory cost of our model with varying sample sizes. Results over 20 seeds on the example (b) Figure 3
Sample size
2500
5000
10,000
25,000
50,000
100,000
Running time (s)
15. 15.66=33.88 30.17=2.12 54.49+15.98
130.36=45.80
194.59-98.55
493.99:1152.81
Memory cost (MB) 64.8 10.72 7..92:0.82 1.95 95.81=12.113 126.99+26.97 175.64133.93
Overall, we observe minimal time and memory costs. Time complexity scales near-linearly with increasing sample size, while memory requirements remain modest even for large datasets, Process- ing 100,000 samples in under 9 minutes demonstrates strong efficiency for CPU-based execution.
F.2 REAL-WORLD DATASET
Table 10: Comparison of running time and memory cost across different methods in SPECT dataset.
Method
Running time (s)
Memory cost (MB)
MILP
0.16 + 0.02
158.60 0.10
LEN
0.20 + 0.00
102.78 = 0.01
RRL
16.23 + 0.01
132.59 0.01
BRCG
2.65 0.27
135.11 = 0.08
DR-NET
89.01 = 0.06
45.93 0.30
NS-FCN (Ours)
10.34 1 0.30
61.42 1.02
We conducted a comparative analysis of our proposed NS-FCN model against baseline methods, focusing on computational efficiency. We take SPECT dataset example. The results in Table I0 demonstrate that US-FCN achieves a competitive balance between performance and resource consumption. While methods like MLP and LEEN offer the fastest execution times, they use higher
20

---

Published as a conference paper at ICLR 2026
memory costs. Our NS-FCN, though not the fastest, maintains considerably minimal memory cost and running time.
G ASSESSMENT OF RULE QUALITY
G.1 STRUCTURAL STABILITY.
To quantify the structural stability and reliability of the learned rules, we measure the consistency of rule predicates across different random seeds using the Jaccard index. For each rule, we treat the set of instances that satisfy its predicates in given run as a binary mask, and compute pairwise Jaccard indices between runs obtained under different random seeds and observation probabilities. The Jaccard index, defined as the intersection over union of two predicate sets, provides natural measure similarity between rule structures learned across independent runs. High mean Jaccard scores (close to 1.0) indicate that the learned rules are structurally stable and robust to stochasticity in training and sampling, whereas lower scores reveal predicates whose semantics are more sensitive noise initialization.
Synthetic Dataset (Figure (b)). As shown Table rules X3 and X1 achieve perfect Jac- card indices of 1.0 across all observation probabilities, demonstrating complete structural stability. In contrast, the aggregated X; rule exhibits more variability, reflecting the increased complexity of learning disjunctive rule structures. In this way, structural stability- measurrd via the Jaccard index of predicates across runs-prav providess a complementary notion of reliability that focuses on the consistency of the learned logical structure rather than solely on predictive performance.
Table 11: Jaccard index of learned rule predicates synthetic data under different observation probabilities. Example (b) of Figure 3 with 20,000 samples over 30 random seeds. These results correspond to the rule structures reported in Table 22
Obs. Ratio
X3
X4
X5
0.1
1.0000 0.0000
1.0000
0.0000
0.6707 = 0.2270
0.2
1.0000 + 0.0000
1.0000 + 0.0000
0.6568 + 0.2831
0.3
1.0000 0.0000
1.0000 0.0000
0.7124 + 0.2279
Table 12: Jaccard Index of learned predicates across different sample sizes on the Birds dataset. Results over 10 seeds.
Sample Size
abnormal clausel (ostrich)
abnormal.clan clause2 (bird A wounded)
can (bird A paahormal_bing
100
0.8000 +: 0.2449
0.5000 +: 0.3162
0.5000 +: 0.3162
500
0.8000 0.2449
0.8000 + 0.2449
0.7000 0.2449
1000
0,8000 = 0.2449
0.6000 = 0.3000
0.6000 = 0.3000
1500
0.8187 = 0.2404
0.6868 = 0.3024
0.7967 0.2670
2000
0.8000 = 0.2449
0.6000 = 0.3000
0.6000 + 0.3000
Birds Dataset. We analyze the consistency of learned rule structures in Birds Dataset (Figure 5 Table 12 presents the Jaccard indices across all pairwise comparisons between seeds for different sample sizes, where abnormal _clausel and abnormal_clause2 correspond to the two conjunctive clauses in the disjunctive rule for abnormal bird: abnormal bird + ostrich (bird wounded).
The results demonstrate that, with the exception model achieves good consistency (Jaccard index 0.60) across all rules and sample sizes. Overall, n = 1500 yields the best consistency, with abnormal clausel reaching 0.8187 and can flyfly reaching 0.7967, indicating that this sample size provides optimal balance between data availability and model stability.
100 where the sample size is insufficient, the
Heart Disease dataset. For this real-world dataset, where ground-trutt truth rules unknown, evaluate structural stability by computing the Jaccard index of selected features across all predic- tion rules learned under different random seeds. Table 13 shows that the model achieves moderate
21

---

Published as a conference paper at ICLR 2026
Table 13: Structural stability of learned prediction rules the Heart Disease dataset.
Metric
Value
Mean Pairwise Jaceard Index
0.4151 0.0994
Most Frequently Selected Features
restecg 1.0 (ST-T wave abnormality)
9/ 10 runs
thal_3.0 (normal thalassemia)
8/ 10 runs
ca_3.0 (3 major vessels colored)
8/10 runs
thalach (maximum heart rate achieved)
7/10 runs
consistency (Jaccard index 0.4151 + O. 09944, indicating that while different seeds may select vary- ing feature combinations, there is substantial overlap in the most important features. The most fre- quently selected features include restecg- (ST-T wave abnormality thal .3.0 (normal thalassemia, a blood disorder), ca 3.0 (three major vessels colored by fluoroscopy, indicating severe coronary artery disease), and thalach (maximum heart rate achieved during exer- cise). These features align with established clinical risk factors for heart disease, suggesting that the model successfully identifies medically relevant features despite the lack of explicit rule supervision.
resting clectrocardiogram),
G.2 RULE LENGTH ANALYSTS
To understand the sensitivity of our framework to the rule structure hyperparameters, we conduct ablation studies on the Heart Disease dataset, systematically varying the arity of conjunction (h) and the number of conjunctive clauses (R+).
We find that both h and Rg show optimal performance in wide range. For instance, [3, 9] and Rg E [5, 20), showing that except for very small h and Rg, our model is able to capture the logic structure within the dataset. Besides, the number of disjunctive clauses is more critical than the arity of individual conjunctions for this dataset. This aligns with the intuition that complex real-world decision boundaries often require multiple alternative rules rather than highly complex single rules.
Effect
Accuracy
Conjunctive
0.90
0.90
Accuracy
1est F1 Score
0.85
0.80
0.60
0.75
Test
0.75
Test
20
5
10
15
20
Number Conjunctive Clauses
Number Conjunctive Clauses
Figure 6: Classification accuracy for heart disease risk under the effect of the number of conjunction arity (h). Results are over 10 seeds.
Effect of Arity of Conjunction(h) Test Accuracy
Effect of Arity of Conjunction(h) on Test
0.92
0.92
stc
Accuracy
Score
0.88
0.88
0.86
0.86
0.84
Test Accurscy
0.82
13 std
0.80
0.80
10
Arity of Conjunction (h)
Arity of Conjunction (h)
Figure 7: Classification accuracy for heart disease risk under the effect of the number conjunctive Clauses (Rt), Results are over 10 seeds.
22

---

Published as a conference paper at ICLR 2026
H ANALYSIS OF TEMPERATURE IN SOFT OPERATORS
To validate robustness, we conducted a sensitivity analysis on Figure 3 with 20,000 samples and 0.2 observation ratio.
Table 14: Imputation accuracy for latent predicates X3, X1, X5 under different softmin temperatures (with fixed μ 10 over 20 random seeds).
in Eq.
Imputation Acc. X3
Imputation Acc. X4
Imputation Acc. X5
0.01
1.000 + 0.000
1.000 + 0.000
0.965 + 0.063
0.02
1.000 + 0.000
1.000 + 0.000
0.987 + 0.034
0.05
1.000 + 0.000
1.000 + 0.000
0.939 = 0.061
0.10
1.000 + 0.000
1.000 + 0.000
0.958 + 0.076
0.20
1.000 0.000
1.000 0.000
1.000 + 0.000
0.50
1.000 + 0.000
1.000 + 0.000
0.858 + 0.038
1.00
0.928 0.122
0.892 0.134
0.776 0.027
2.00
0.751 + 0.002
0.839 + 0.157
0.786=00.0 0.008
5.00
0.769 + 0.111
0.750 0.003
0.772 + 0.083
10.00
0.752 = 0.003
0.770 + 0.111
0.802 = 0.064
20.00
0.697 + 0.098
0.733 + 0.046
0.799 = 0.064
50.00
0.733 + 0.049
0.733 + 0.046
0.798 + 0.077
100.00
0.750 + 0.004
0.698 0.064
0.803 + 0.069
Table 15:
Imputation accuracy for latent predicates X3, X4, X5 under different constant temperature values (with fixed r = 0.1 over 20 random seeds).
f of Eq.
Imputation Acc. X3
Imputation Acc. X4
Imputation Acc. X5
0.1
1.000 + 0.000
1.000 = 0.000
0,218 = 0.002
0.2
1.000 + 0.000
1.000 0.000
0,218 + 0.002
0.5
1.000 + 0.000
1.000 = 0.000
0.218 + 0.002
1.000 = 0.000
1.000 = 0.000
0.217 = 0.002
1.000 = 0.000
1.000 0.000
0.870 0.117
1.000 +: 0.000
1.000 + 0.000
0.942 0.056
1.000 + 0.000
1.000 0.000
0.866 + 0.057
0.965 + 0.093
1,000 + 0.000
0.894 0.056
10
0.965 + 0.093
1.000 0.000
0.915
15
0.965 + 0.092
1.000 + 0.000
20
0.966 = 0.091
1.000 0.000
0.928 0.057
25
0.965 = 0.092
1.000 = 0.000
0.886 + 0.060
30
1.000 + 0.000
1.000 0.000
0.899 + 0.088
35
0.965 + 0.093
1.000 + 0.000
1.800
40
1.000 = 0.000
1.000 = 0.000
0.889 = 0.075
45
0.964 + 0.094
0.965 0.093
0.909 = 0.051
50
1.000 + 0.000
1.000 0.000
0.917 + 0.064
100
0.965 + 0.094
1.000 0.000
0.916 + 0.064
200
1.000 + 0.000
1.000 + 0.000
0.908 + 0.070
Soft-AND (r). Table 14 shows that the model maintains high accuracy when 7 is small (e.g., E (0.01, 0.20). This is expected because as - 0, Softmin approximates the hard min logic required for strict conjunctions. Performance degrades only when 7 becomes too large (r 2 1.0), where the operator becomes too "soft" to capture the decisive logical boundaries. Thus, small constant temperature (e.g.. = 0.1) a safe and effective default.
Soft-OR (1).. Table 15 reports the imputation accuracy under varying constant values. The sults demonstrate that our model is highly robust to μ: it achieves near-perfect accuracy for all latent predicates (X,, X4, Xs) across wide range, specifically for μ. 2 This aligns with the theoretical
23

---

Published as a conference paper at ICLR 2026
property that μ oc, the LogSumExp function approximates the hard max operator. In prac- tice, any sufficiently large provides a strong gradient signal for discrimination while maintaining differentiability.
Conclusion. Our framework does not rely careful hyperparameter tuning. A moderate to large μu. for Soft-OR and a small y for Soft-AND consistently yield optimal results. Thus, we use 0.1 and l 10 as temperature parameters for all experiments. Furthermore, complex scheduling strategies like cosine annealing can be employed if constant temperatures are not good enough.
I SENSITIVITY ANALYSIS WITH LABEL NOISE AND MISSING RATIO
111 ROBUSTNESS ANALYSIS WITH LABEL NOISE
To assess the robustness of our framework against data inconsistencies and imperfect logical depen- dencies, we conducted experiments by injecting label noise into the latent predicates.
Specifically, first generate the ground-truth latent predicates X3. X4,, Xs following the perfect logical rules (e.g., X3 = Xg A X1). Then, we introduce stochasticity by flipping the binary labels of these latent predicates with probability Proise E {0.0, 0.1, 0.2, 0.3, 0.4, 0.5). This setup simu- lates real-world scenarios where logical rules may have exceptions or where the observed data contains errors, directly challenging the model's ability to distill consistent symbolic rules from noisy supervision. Tables 16 and 17 present the learned rule structures and their corresponding im- putation accuracies under varying noise ratios. We use Figure (b) as the representative example with an observation ratio of 0.3 and sample sizes of 20,000.
In the noise-free setting (Pnoise 0.0), model perfectly recovers the ground ground-truth truth rules for the simpler conjunctive predicates X3 and X4 (with rule accuracy 1. 1.00), achieving perfect imputa- tion accuracy (1.000). For the more complex disjunctive rule Xg. the model achieves rule accu- racy of 0.50 and an imputation accuracy of 0.955 after fine-tuning, indicating that while the exact ground-truth structure harder to isolate, the learned approximations maintain strong predictive performance.
Remarkably, the model demonstrates strong robustness at low-to-moderate noise levels (Pnoise I 0.3). At Pnoise = 0.1 and 0.2, the ground-truth rules (underlined in the table) for X3 and Xa are perfectly recovered (rule accuracy 1.00) with near-perfect imputation accuracies; for the complex multi-hop rules of Xs. the ground-truth rules frequently emerge as the dominant learned structures (with rule accuracy above 0.5). Even at Proise 0.3, the model maintains high rule accuracy (0.85) for both X3 and X4, with imputation accuracies above 0.95; for Xs. the rule accuracy decreases to 0.2 Proise 0.3, but the imputation accuracy remains at 0.828, suggesting that the model learns valid approximations (e.g., capturing one correct disjunctive branch) that preserve predictive power.
As noise increases beyond 0.3, the performance degrades more significantly. At Pnoise = 0.4, rule accuracies drop to 0.85 and 0.6 for Xa and X4 respectively, while X5 fails recover the correct structure (rule accuracy 0.00). At Pnoise = 0.5, the model struggles to learn meaningful rules, with rule accuracies in [0.0, 0.1] for all predicates. However, the imputation accuracies remain above 0.70 even at these high noise levels, indicating that the learned approximations, while not perfectly matching the ground-truth rules, still provide useful predictive signals.
The imputation accuracy degrades gracefully as noise increases, rather than collapsing abruptly, indicating that the soft-logic relaxation effectively prevents the model from overfitting noise, allowing it to capture the dominant logical signals within the data. The fine-tuning step for X5 consistently improves imputation accuracy across all noise levels, demonstrating the effectiveness of the iterative refinement process.
1.2 MISSING RATIO
In three synthetic datasets, we have varied the missing ratio in {0.7, 0.8, 0.9] in the above results. In real-world datasets, to assess the model's robustness under different levels of data scarcity, we evaluated its performance on the SPECT and Heart Disease dataset while varying the observation ratio from 0.3 to 0.9 (i.e. missing ratio from 0.1 to 0.7).
As shown in Tables 18 and 19 the model's accuracy remains acceptable and improves consistently more data becomes available. Notably, in SPECT, even with only 30% of the data observed (a
24

---

Published as a conference paper at ICLR 2026
Table 16: Impact label noise on rule learning and missing value imputation performance. Results are over 20 random seeds.
Noise Ratio
Avg. Imputation Accu. (Before Fine-tune)
Avg. Imputation Accu.
Train Loss (Before Fine-tune)
Train Loss (After Fine-tune)
(After Fine-tune)
0.0
X3 1.000 0.000
X3 0.001 = 0.000
X4 1.000 0.000
X4 0.001 = 0.000
X5 0.907 + 0.050
X5 : 0.955 + 0.049
Xs : 0.089 = 0.035
X5 0.067 0.031
0.1
X3 1.000 = 0.000
/
X3 0.098 = 0.003
X4 1.000 0.000
0.099 = 0.004
0.948 0.042
X5 : 0.946 + 0.038
X; 0.168 0.026
X; 0.123 0.021
0.2
Xa : + 0.076
/ /
0.198 = 0,005
X4 : + 0.057
0.19. + 0.007
: 0.050
Xs 0.902 + 0.046
0.260 + 0.008
Xs 0.204 0.012
0.3
x3: 0.103
X + 0.006
0.987 0.056
X + 0.008
X5: :0822 + 0.046
Xs 0.824 + 0.066
: = 0.006
Xs. 0.266 0.006
0.4
0.863 0.127
0.360 0.009
0.862 0.128
X4 0.357 = 0.008
X5 : 0.792 1 0.078
Xs : 0.786 1 0.065
X5 : 0.370 - 0.006
Xs 0.310 0.006
0.5
X3: 0.745 1 0.085
X3 0.418 = 0.007
X4 0.725 1 0.077
X4 0.421 + 0.007
X5 0.761 0.057
Xs 0.767 + 0.072
Xs 0.421 + 0.007
Xs 0.349 0.007
Table 17: Learned rule structures under label noise. Ground truth rules are indicated with underlines. Results are over 20 random seeds.
Noise Ratio
Learned Rule Structure
Rule Accu.
0.0
Na No
Xa 1.00
Xe: X
Xs 1.00
V
Ns. X
X1 AT
X. 1.10
No (No. V Nol (Ng X,)
(X)
X, 0.55
0.2
Ns No Xi
Xa 1.00
X. X.
Xa 1.00
X.
A.
X
Xx
Xu A Xo
Xa 11.5
No (No Naj V (M (No
X Xol (N. v N.). (X).
X, : 0.20
0.4
Ys A Ni X, Xs
X
Xa 0.60
X X, Xy
X, X,
Xa 0.60
0.10 Xa
X1: A No. (NoA Naj (X, (a Xa) V XOL Xo) VIG Xe). A Xa) (X X... A (XG A.X))
X.
(X,
70% missing ratio), the model maintains a high FI score 0.751, demonstrating its capability to learn meaningful diagnostic rules from highly incomplete datasets.
For the Birds Dataset, we fix the observation ratio as 0.1 (i.e. 90% missingness) and show results over different number of training samples. Results in Table [0 show that a few hundred samples are sufficient for the model to converge to the correct logical truth.
Table 18: Performance on the SPECT dataset with varying observation ratios.
Observation Ratio
Imputation Acc.
Diagnosis Acc.
Diagnosis F1
0.3
0.501
0.679
0.751
0.5
0.630
0.765
0.808
0.7
0.763
0.920
0.958
0.9
0.791
0.929
0.960
25

---

Published as a conference paper at ICLR 2026
Table 19: Imputation accuracy for Heart Disease under different observation ratios.
Observation Ratio
Overall
trestbps
chol
thalach
oldpeak
0.3
0.6444
0.7129
0.6304
0.7393
0.4950
0.5
0.7434
0.8053
0.7558
0.7954
0.6172
0.7
0.8432
0.8647
0.8482
0,9043
0.7558
0.9
0.9439
0.9439
0.9307
0.9769
0.9241
Table 20: Impact of training sample size on the imputation accuracy of latent predicates (abnormal, fly) in the Birds domain. Results are reported mean = std over 10 random seeds, evaluated with 10% observation probability.
# Samples
Acc. Abnormal Bird
Acc. Can Fly
100
0.896 + 0.058
0.845 + 0.148
500
0.976 + 0.054
0.928 + 0.066
1000
0.951 + 0.067
0.928 + 0.066
1500
0.949 = 0.070
0.952 + 0.066
2000
0.951 = 0.067
0.928 + 0.066
J ADDITIONAL SYNTHETIC EXPERIMENTS RESULTS
J.1 MAIN RESULTS SUPPLEMENT OF EXAMPLE (B) OF FIGURE
Dataset Generation. The base variables (Xo. X1, X2, Xs. X7) are independently generated from Bernoulli distribution, each with p = 0.5. Subsequently, the values for (X3. X1. are determin- istically derived using the ground truth logical rules depicted in Figure Specifically, these rules are:
X3 + X
X4 X2
X V
Finally, to introduce missing information, portion of the values for X3. X4. and Xs randomly masked. These masked variables become the targets for imputation. In our experiments, we vary the level of missingness, applying masking probabilities of 70%, 80%, and 90% these target variables (corresponding to observation ratios of 30%, 20%, and 10%, respectively).
Main Results. As demonstrated in previous case study (Table which shows three runs using same seed but different internal rule optimization orders), variations the rule optimization sequence within single seed affect training efficiency. We thus show the coordinate descent training progress under a different random optimization order from Figure 4here here in Figure g this run, the optimization order is [Xs, X4, X3) for cycle and [X4, Xs, Xz) for eycle 2. Given such different learning trajectories, our model still discovers the correct rules successfully.
Furthermore, random initialization across different seeds can lead to the discovery of varied rule sets, and occasionally, model might converge to local optimum. However, as the analysis of convergence before. performing multiple runs with different initializations enhances the probability of identifying the global optimal solution. Our results show that the model identifies global optima in more than half of the 30 random-seed runs (Tables 2T and 22).
Learning Efficiency. As the observation ratio decreases, the guidance signal becomes less infor- mative, reducing both rule structure recovery and missing value imputation. We further investigated the impact of dataset sample size, varying from 500 to 20,000 samples. As shown in Figure 9 the most efficient setting we can recover all the AND/OR rules for X3, X4, andX, is to use an observation ratio of 0.1 and dataset of 1,000 samples.
J.2 RESULTS OF EXAMPLE (A) OF FIGURE
Dataset Generation. The base variables
Xa} independently generated from Bernoulli distribution, each with == 0.5. Subsequently, the values for X4, are determin- istically derived using the ground truth logical nules depicted in Figure Specifically, these rules
26

---

Published as a conference paper at ICLR 2026
[n] Predicate Rude mang Lnsses (Sed
Unooserued Predicate moutecion Arcuracies [Seed 12)
100
X4
0.15
X5 uss
0.9
1.22
5 0.10
2.07
6.2
Xs Unuloo Acc
Unstrarved
0.00
0.75
Predicate Triiiing Rpoch
Predicate Training Spoch
0/2 Predicate Rule Gradient (Geed
Overal Unobseeved Accurscy per Cycle (Seed 12)
0.10
x4 Grad Norn
0.68
35 drsd Narm
0.2
Graddent Marr
0.00
0.00
0.1
0.60
ExtiotCycteeee acc
Cydal
Predicate Training Rpoch
Cycle Number
Figure 8: Training dynamics for a representative run (Obs. Ratio 0.3) of Figure 3 (b). The optimization order: [x5, X4, for Cycle 1: 1X,,Xs, X3 ] for Cycle 2. Subplots display: (a) training losses, (b) unobserved imputation accuracies, and (c) gradient norms for rule embeddings; (d) overall imputation accuracies each cycle. Red dashed lines indicate the conclusion of training blocks for X3 or X4 (each allocated 30 epochs when active within cycle). Purple dashed lines delineate training phases for X5 (Rule 1, Rule 2, and Fine-tune); the epoch count for these X5 phases can vary per cycle due to the dynamic nature of the hard covering mechanism. Correct rule structures were learned for X3 by the end of Cycle I, and for X, and X5 by the end of Cycle 2.
Table 21: Summary of synthetic data experiment results for example (b) of the Figure Evaluated on 20,000 samples and results are averaged over 30 random seeds.
Obs. Ratio
Avg. Imputation Accu. (Before Fine-tune)
Avg. Imputation Accu. (After Fine-tune)
Train Loss (Before Fine-tune)
Train Loss
(After Fine-tunc)
0.3
X3 : 1.000 + 0.000
X3 0.001 0.000
/
X4 : 1.000 + 0.000
0.00. + 0.000
Xs : 0.889 + 0.071
Xs 0.956 + 0.052
Xs 0.090 + 0.046
Xs 0.060 + 0.037
0.2
X3 1.000 + 0.000
X₃ 0.001 + 0.000
X4 1.000 + 0.000
/
X4 0.001 0.000
X5 0.893 +: 0.072
Xs 0.954 1 0.053
Xs 0.087 0.045
Xs 0.060 + 0.039
0.1
Xa 1.000 + 0.000
X₃ 0.001 0.000
Xa 1.000 0.000
0.001 0.000
X5 : 0.879 + 0.064
Xs 0.950 0.050
Xs 0.090 0.052
Xs 0.066 + 0.039
are:
X3 + XOA
X4 X2
+ X
Finally, to introduce missing information, portion of the values for X3. X4, and X5 are randomly masked. These masked variables become the targets for imputation. In our experiments, we vary the level of missingness, applying masking probabilities of 70%, 80%, and 90% these target variables (corresponding to observation ratios of 30%, 20%, and 10%, respectively).
Main Results. We show the coordinate descent training progress under different random optimiza- tion order. Figure [0 demonstrates the convergence in two cycles, while Figure II requires three cycles complete training.
We summarize the results for example (a) of the Figure Tables and which demonstrate both the effectiveness of our rule discovery approach and the precision of missing variables imputa-
27

---

Published as a conference paper at ICLR 2026
Table 22: Summary of learned rule structures and accuracy for example (b) of Figure 3 Each
observation ratio was evaluated using 20,000 samples, with results averaged over 30 random seeds. We present the top 3 learned rule structures in order of discovery accuracy. Rule accuracy indicates the percentage of 30 runs in which rule was learned completely correctly.
Obs.Ratio
Learned Rule Structure
Rule Accu.
0.3
X3 Xo A X1
X3 1.00
X4
X2 A X
X4 1.00
Ns: (Xa V Xo),
[X3 X4) (X (6),
Xs 0.53
0.2
X3 Xo A
:
X2 X7
1.00
:s: (Xo A (Xa A X6), (Xx A (o).
Xs 0.53
0.1
X3 Xo A X1
Xg 1.00
X4: X2 X
X4 1.00
Xs A (X, (Xa Xo). (X, A X.) (X, Xo). (Xo. X.) (Xs X4)
Xs 0.43
X3 Imputation Accuracy
X4 Imputation Accuracy
X5 Imputation Accuracy
0.95
Accuracy
Accuracy
Accuracy
0.85
0.85
Scod
0.00
0.80
0.75
Seed
2003
5000 Jies 3160122-0011ee177700eeee
2030 5000
7000 200082220112e1ee00eeee
2030 seco 7000 10011200000001100000000000
I_samples
J.saamlles
s_sampless
Figure 9: Imputation accuracy versus dataset sample size for Figure 3 (b). For these experiments, 10% of the data was observed (i.e., a 90% missing ratio) for predicates in X3, X4, and Xs-
tion. Our analysis reveals that learning the multi-step chain structure presents significant challenges, primarily because the algorithm uses inferred predicate values of from previous steps to update the current values by Eq. 2 This creates dependency chain where suboptimal rule embeddings learned at earlier optimization steps can propagate errors to subsequent steps, potentially degrading overall performance. Despite these challenges, our model successfully identifies the correct rules the ma- jority of experimental runs. This robustness indicates that with multiple random initializations, the algorithm reliably converges to the optimal rule structures like the results from Figures [0] and [T] which effectively overcome the inherent difficulties of sequential dependency learning chain-like logical structures.
Table 23: Summary of synthetic data experiment results for example (a) of the Figure 3 Each observation ratio is evaluated using 20,000 samples and results are averaged over 30 random seeds. No fine-tune phase since we assume no disjunctive rules.
Obs. Ratio
Avg. Imputation Accu.
Train Loss
0.3
X; 0.88 + 0.13 X + 0.06
X₃ 0.06 + 0.07
X4 0.04 0.04
X590 00.95::00 0.03
Xs: 0.03 + 0.02
0.2
0.94
Xx 0.03 = 0.06
X.: +: 0.06
0.03 0.04
0.96
0.03 0.02
0.1
0.94.440..11
0.03 0.03
:494
X4 0.03 0.04
X, 0.96
X, 0.003 0.02
28

---

Published as a conference paper at ICLR 2026
(a] Predicate Training Losses (Seed 42)
(b) Unabserved Predicate Imputation Accuracies (Seed
1.0
0.05
0.04
Accuracy
0.6
0.03
Xb Loss
0.02
0.4
X3 Unobserved AcC. Unobserved Acc.
0.01
0.2
Unabserved Acc.
0.00
20
0.0
10
20
50
Predicate Training Epech (cccumulated)
Predicate Training Epech (Accumulated)
Prodicate Rule Gradient (Seed
Overall Unobserved Accuracy per Cycle (Seed 42)
Grad
1.0
MM
x4 Grad Norm
Grad Norm
Unobserved Accuracy
0.8
0.03
0.6
Gradient
0.4
X3 Ind-af-Cyclee Acc.
0.2
End Cycle
End-oo-Cycle
10
20
0.0
Predicate Training Epech (Accumulated)
Cycle Number
Figure 10: Training dynamics for a representative run (Observation Ratio = 0.2) of Figure 3 (a). Subplots display: (a) training losses. (b) unobserved imputation accuracies. and (c) gradient norms for rule embeddings; (d) overall imputation accuracies each cycle. Purple dashed lines indicate the conclusion of training blocks for one cycle (each allocated 30 epochs). The rule embedding optimization order: [ Xxs X3, X4 I for Cycle 1; [Xs, X4, ] for Cycle 2. Correct rule structures were learned for X3 and X4 by the end of Cycle for X5 by the end Cycle 2. The learned rules:
X3 + Xo A X1, X4 + X2 X3. X5 < X4 A X6.
Table 24: Summary of learned rule structures and accuracy for example (a) of Figure 3 Each observation ratio evaluated using 20,000 samples, with results averaged over 30 random seeds. We present the top 3 learned rule structures in order of discovery accuracy. The rules that are truth rules are indicated by underline. Rule accuracy indicates the percentage of 30 runs in which a rule was learned completely correctly.
Obs. Ratio
Learned Rule Structure
Rule Accuracy
0.3
X3 : Xo X1, X1 A X4, Xo Xa
X₃ 0.53 X4 0.40
X4 X2 A X3. Xo A X XI A X2
X₃ : X, X6. X3 X6. Xo X
X5 0.23
0.2
X3 Xo A X X1 A X5, X A
X3 0.77
Xa X2 A X3, X A Xs, X1 X
X4: 0.60
X5: X4 X,, X A X4, X
Xs 0.37
0.1
X3 Xo A X1, X, A X4 X X
X3 0.77
X4 : X A Xa. Xo A X1, Xo X2
X4 0.53
X5 X X6, X A X4, X3 XA
Xs 0.33
J.3
RESULTS OF EXAMPLE (C) OF FIGURE
Dataset Generation. The base variables {Xu, X1, X2, X7) are independently generated from a Bernoulli distribution, each with = 0.5. Subsequently, the values for X4, Xg. are deterministically derived using the ground truth logical rules depicted in Figure 3 Specifically,
29

---

Published as a conference paper at ICLR 2026
(a] Predicate Training Losses (Seed 42)
(b) Unabserved Predicate Imputation Accuracies (Soed
1.0
0.08
X4 Loes
Accuracy
0.6
Ban 4
0.4
X3 Unobserved AcC.
0.2
X4 Unobserved Acc.
Unabserved Acc.
20
80
0.0
20
40
Predicate Training Epech (tccumllated)
Predicate Training Epech (Accumulated)
Prodkcate Rule Gradient (Seed
Overall Unobserved Accuracy per Cycle (Seed 42)
1.0
Unobserved Accuracy
0.8
Mymming w
Gradient
0.00
N Mm wh
Grad Nom
Grad
0.6
x5 Grad Nom
0.4
0.02
X3 End-af--yyck Acc.
0.2
End Cycle
0.00
End-ot-Cyck
20
80
0.0
Predicate Training Epech (Accumalated)
Cycle Number
Figure 11: Training dynamics for a representative run (Observation Ratio = 0.2) of Figure 3 (a). Subplots display: (a) training losses. (b) unobserved imputation accuracies. and (c) gradient norms for rule embeddings; (d) overall imputation accuracies each cycle. Purple dashed lines indicate the conclusion of training blocks for one cycle (each allocated 30 epochs). The rule embedding optimization order: [ Xxs, X4, X3 for Cycle 1,2; [X3, Xs, X4 ] for Cycle 3. Correct rule structures were learned for X3 by the end of Cycle 1, for X4 by the end Cycle 2, and for X5, by the end of Cycle 3. The learned rules: X3 < Xo A X1, X4 +- X2 X3, X5 t X4 A X6.
these rules are:
X3 X
+ X X (X3 A A6) V (Xs) VXx
Finally, introduce missing information, portion of the values for X3, X,, Xs and X5 are ran- domly masked. These masked variables become the targets for imputation. In our experiments, we vary the level of missingness, applying masking probabilities of 70%, 80%, and 90% to these target variables (corresponding observation ratios of 30%, 20%, and 10%, respectively).
Main Results. We summarize the results for example (c) of the Figure 3 in the Tables 25 and 26 showcasing the effectiveness of rule discovery and the precision of missing variables imputation. We have random coordinate descent training order for rule optimization.
This task more challenging due to the chain-like structure of the disjunctive rules, particularly with three clauses for Xs, resulting in lower learning accuracy than in example (b). Nonetheless, our method achieves the highest rule discovery accuracy for the ground-truth rules while maintaining acceptable imputation accuracy. For the most difficult prediction task (Xş) and the chain-derived predicate Xs. we obtain about 90% accuracy across all three observation ratios. Other predicate predictions reach almost perfect accuracy. For the learned rules in Table 266 we can find most body predicates are correct even for the complex three-clause rules governing Xs. which include the chain-derived predicate Xg. We also show the loss plot for one run Figure
J.4 PARAMERS SETTING AND COMPUTING RESOURCE
Our model operates efficiently in CPU environment utilizing the PyTorch library. The hyperpa- rameters are configured as follows:
Rule Embedding and Fine-tuning Optimizer: Adam, learning rate: 0.01.
Temperature of softmin and softmax: 0.1 (for Eq. 3 and 10.0 for (Fq. 4
30

---

Published as a conference paper at ICLR 2026
Summary of synthetic data experiment results for example (c) of the Figure 3 Each observation ratio is evaluated using 20,000 samples and results are averaged over 30 random seeds.
Table 25:
Obs. Ratio
Avg. Imputation Accu. (Before Fine-tune)
Avg. Imputation Accu. (After Fine-tune)
Train Loss
Train Loss (After Fine-tune)
(Before Fine-tune)
0.3
X3 0.98 + 0.06
X3 0.013 + 0.045
: 0.00
X4 0.001 = 0.000
Xs 0.89 0.08
X5 0.91 + 0.08
Xs 0.117 = 0.077
Xs 0.077 + 0.055
Xg: 0 0.05
X3 0.053 = 0.037
0.2
: +.06
Xa 0.013 + 0.046
11: 0.00
/
X1 0.001 = 0.000
0.91
X3 0.93 0.08 /
X; 0.109 = 0.084
X5 : 0.061 = 0.052
: + 0.05
Xs : 0.068 + 0.035
0.1
X3 0.97
X3 0.020 = 0.057
: 0.00
/
X4 0.001 = 0.000
X5 0.90 0.08
X5 0.92 0.08
X5 0.103 + 0.081
X5 0.065 + 0.055
Xg 0.91 + 0.06
X 0.072 + 0.047
Table 26: Summary of learned rule structures and accuracy for example (c) of Figure 3 Each observation ratio is evaluated using 20,000 samples, with results averaged over 30 random seeds. We present the top learned rule structures in order of discovery accuracy. The rules that are truth rules indicated by underline. Rule accuracy indicates the percentage of 30 runs in which rule was learned completely correctly.
Obs. Ratio
Learned Rule Structure
Rule Accu.
0.3
Xs Xo A X1, X1 X2, Xo X
X3 0.93
X2 A
X4 1.00
Xs: (Xa A X6) Xs (Xa A X7), (Xg A Xo) V (XI A (XGAA)))
X; 0.27
(X3 X4) V (X3 X&) V (X, A X6)
X X3 A X4, X A XA, X. X
Xa 0.30
0.2
Xs Xo XI. Xo A Xo A X6
Xa 0.93
X,: XAXX
X4 1.00
X5: (Xa X6) V (N, A (X,, XX) V A VX
X3 0.37
(Xa X. V (XIA Xx) X 13: X,, X X7, XOA
Xs 0.17
0.1
X3 XOAX, Xo Xo. XOA X2 X X,A,; (N6 A X7), ( XAXXVV ( (X6A
X₃ 0.90
X4 1.00
Xs:
Xs 0.30
(Xa (X A Xo) A X+)
X Xo A X4, X A X X
Xs 0.23
"Well-explained" Threshold: 0.99 (for sequential hard covering disjunctive rule learn- ing).
Batch Size: 64.
K
ADDITIONAL REAL WORLD DATA EXPERIMENTS RESULTS
K.1
SPECT
We ask for expertise from cardiovascular surgery of a hospital to give domain knowledge, and then we try to explain the learned rules. We select several meaningful rules to demonstrate. The domain knowledge are as follows.
R1: The anterior wall and the septum of the left ventricle adjacent and often simulta- neously affected by the Left Anterior Descending artery (LAD). If both anterior wall and septum show infarction, it strongly suggests an issue with the LAD. If both apical anterior
31

---

Published as a conference paper at ICLR 2026
Training Loss
Unobserved Accuracy
Remaining Samples
50000
0.20
Number of Samples
45000
Accuracy
X3
0.15
XB
40000
0.4
37500
0.05
35000
32500
0.00
20
60
40 Epoch
Epoch
Epoch
Figure 12: An example of loss and imputation accuracy during coordinate optimization (Obs. Ratio
=0.1, seed = 88, from example (c) of Figure 3.. The training order is [X3, X4, Xs. Xs.. Epochs 0-19 correspond to rule learning for X3; epochs 20-39 for X4; epochs 40-59 for Xs. and epochs 60-end for X5. Remaining samples identified how many samples are "well-explained" during hard covering phase. As the imputation accuracy for missing X5 is 1.00, we do not go to the fine-tune phase. The learned rules: X3
Xo X,, X4 + X2AX7,Xx f [X3AXX V X A7), Xg < Xg X4.
Short Axis Apical
Short Axis Mid
Short Axis Basal
Horizontal Long Axis
Vertical Long Axis
ANT
APICAL
ANT
ANT LAT
ANT SEPTAL
SEPTAL
SEPTAL
INF
S SEPTAL
ANT
INF LAT
LAT
APICAL
BASAL
INF
BASAL
Figure 13: The five slices consists of 22 regions of interest (ROI) for SPECT Diagnosis. The slices are chosen according to the following: Three slices for short axis view-one slice near heart's 's apex, one in middle of the LV and one near the heart base; One slice corresponds to the center of the LV cavity for horizontal long axis view; One slice corresponds the center of the LV cavity for vertical long axis view (Kurgan al. 200T).
and mid-anterian show infarction, it indicates more extensive problem within the LAD territory, affecting both the apical and mid-portions of the anterior wall.
R2: The apical lateral wall (typically LCX territory) and the apical inferior wall (typically
RCA or LCX territory) are adjacent. Infarction both suggests problem this combined region.
R3: If both apical septal and apical septal show infarction, it indicates problem in the LAD territory, involving ischemia in multiple myocardial segments.
extensive
R4: If apical lateral and apical lateral show infarction, it indicates a more extensive is- chemic problem in the Left Cireumflex artery (LCX) territory.
R5: The apical anterior (ANT) and apical septal (SEPTAL) regions are primarily supplied by the Left Anterior Descending artery (LAD); the apical lateral (LAT) region is primarily supplied by the Left Circumflex artery (LCX); the apical inferior (INF) region primarily supplied by the Right Coronary Artery (RCA), but can sometimes be supplied by the LCX. depending on the coronary artery dominance pattern.
Refer to Figure [3] we can give some explanations of rules learned in Table 27 based on the domain knowledge RI to R5. For example,
Fs + F, A Fy: F, and F2 are features from the first slice near the heart's apex, while F5 is from the second slice at the middle of the left ventricle (LV). According clinical
32

---

Published as a conference paper at ICLR 2026
knowledge R1 and R5, the anterior and septal regions are primarily supplied by the Left Anterior Descending (LAD) artery. Therefore, this rule is clinically plausible: if partial diagnosis (labeled as 1) is present in both Fi and F2, it strongly suggests LAD artery problem. Since F5 is in the mid-anterior region, also supplied by the LAD, it has a high probability of being affected well.
Diagnosis +- F3 A F6: From RI, we know that the anterior wall and the septum the left ventricle are adjacent. Fs and Fe both from middle of the LV (left ventricular), and they are adjacent. Thus, if both these adjacent mid-ventricular regions (Fs, and F&) show signs of infarction, significantly increases the likelihood of an overall positive diagnosis.
Table 27: Example rules learned by NS-FCN for SPECT feature imputation and diagnosis.
Selected Feature Imputation Rules Learned by NS-FCN
F5 < F1 A F2: partial diagnosis of segment 1 and 2 causes the partial diagnosis of segment 5.
F6 - Fn A F19
F13
F22 F12
Learned Diagnosis Rule Structure
Diagnosis + (Fa F6) v (F2 A (F Fia)
As detailed in Table 28 the learned rules for diagnosing cardiac abnormalities correspond closely with established domain knowledge from cardiovaseular surgery experts. For instance, the model identified that infarcts in adjacent regions like F1 and F2 are indicative of issue in the Left Anterior Descending (LAD) artery territory. Furthermore, the model learned a composite rule for the final diagnosis, logically aggregating signals from multiple infarcted regions across different coronary artery territories (LAD, LCX, RCA). This ability to synthesize information from disparate features into a coherent diagnostic rule highlights the model's capacity for complex reasoning. The clinical relevance of these rules was further validated by Large Language Model (LLM), which confirmed their consistency with expert knowledge on ischemia propagation patterns.
Table 28: Analysis of learned rules for the SPECT dataset, evaluated by human experts and LLM.
Rules
Evaluation with Human Expert Knowledge
LLM Evaluation
Pe PIAP2
Matches RI & R5: Fi and F2 are in LAD territory. Infarction in both sug- gests LAD issue affecting apical and mid-anterior LV.
Plausible:
Both regions LAD- supplied and adjacent; mid-anterior (Fs) likely also affected if Fi & F2 show infarction. Clinically consistent.
Fo + Fi1 A
Related to R2 & R4: Fn and Fis are adjacent. Infarction implies LCX RCA/LXX combined territory issue.
Valid: Matches adjacency and vascu- lar territory logic (LCX-lateral, RCA- inferior). Supports ischemia propaga- tion in midventricular slices.
F19
F13 + F22
Partial link to R3 R5: Likely in- volves basal/apical septal (F22) and ad- jacent basal regions. Indicates LAD or multi-segment ischemia.
Reasonable: Suggests ischemia spread in basal-septal regions (LAD) adjacent to basal/anterior. Fits multi-segment LAD pathology.
F12
Diagnosis
<
Consistent with RI R4. Combines LAD (F)), LCX/RCA (F)), and adja- cent mixed regions. Multiple adjacent infarct pairs increase diagnosis likeli- hood.
Strong: Logical aggregation of ad- jacent infarcted regions across LAD, LCX, RCA territories. Matches expert ischemia propagation patterns.
(Fi A Fo) V
(22 A Fn)
(Fo A Fis)
Performance with varying missing ratios. To assess the model's robustness under different levels of data scarcity, we evaluated its performance on the SPECT dataset while varying the observation ratio from 0.3 to 0.9. As shown in Table 18 the model's accuracy remains strong and improves
33

---

Published as a conference paper at ICLR 2026
consistently as data becomes available. Notably, with only 30% of the data observed (a 70% missing ratio), the model maintains a high F1 score 0.751, demonstrating its capability to learn meaningful diagnostic rules from highly incomplete datasets.
K.2
HEART DISEASE
K.2.1
ASSESSMENT OF LEARNED RULES
For feature imputation, as shown in Table 29 our model discovers rules with clinically relevant numerical thresholds by directly modeling continuous data. For instance, learns to impute resting blood pressure (trestbps) based on conditions like age > 60 and chol > 250. Similarly, it links high cholesterol to factors like age 55 in males very high blood pressure (trestbps 150). The learned rule for ST depression (o (oldpeak) combines the slope of the ST segment with maximum heart rate threshold (thalach < 150), demonstrating the model's ability capture complex, non- linear relationships within the data.
Beyond imputation, NS-FCN learns interpretable rules for the final diagnosis, classifying patients into low-risk or high-risk categories.
Table 30 presents several of these diagnostic rules. For example, the model learns that com- bination of factors such as an upsloping ST segment sHope.upslopingg a fixed thallium defect laal_fixed_ddeee and exercise-induced angina (exang -ys) is strongly indicative of high risk. Con- versely, identifies that factors like the absence of exercise-induced angina (exang-no) and flat ST slope (slope. flat) in female patients suggest low risk of coronary artery disease. These diag- nostic rules were also evaluated by an LLM and deemed "Excellent" or "Strong," underscoring their consistency with clinical practice.
Table 29: Learned rules for feature imputation on the Heart dataset, with LLM assessments.
Feature
Imputation Acc.
Learned Example
Rule
LLM Assessment
trestbps
0.86
trestbps_hid _hihh +
Excellent: This rule captures the well- established link between age, high choles- terol, and hypertension. Both are primary risk factors for cardiovascular disease and often co-occur.
(age > 60) A
(chol > 250)
chol
0.85
chol high
The rule correctly identifies two key risk profiles for high cholesterol: middle-aged to elderly males, and individ- uals with significant hypertension. This aligns perfectly with clinical understand- ing of metabolic syndrome.
Excellent:
(sex A age 55) V (trestbps
150)
thalach
0.90
hr high (trestbps 145) V (age A cp =3)
Strong: This rule insightfully links fac- tors that limit exercise capacity the max- imum heart rate achieved. Both hyper- tension and severe asymptomatic coronary disease can prevent patient from reaching higher peak heart rate.
oldpeak
0.76
st_severe
This rule identifies a classic high-risk pattern. A downsloping ST seg- ment is a strong positive finding, and its oc- currence at uub-maximal heart rate indi- cates ischemia at a low workload, sign of severe coronary artery disease.
Excellent:
(slope
2) A
(thalach < 150)
K.3
HYPER-PARAMETER SETTING AND COMPUTING RESOURCE
For NS-FCN (Ours):
Rule embedding optimizer: Adam with learning rate of 0.01.
Fine-tune optimizer: Adam with learning rate of 0.01.
34

---

Published as a conference paper at ICLR 2026
Table 30: Learned rules examples for disease prediction on the Heart dataset, with LLM assess- ments.
Learned Rule Example
LLM Assessment
high_risk restecg_ste (tsssss___bbaa abnooommalityy A ca 3 A oldpeak 1.49
Excellent: This rule identifies high-risk profile by com- bining three critical indicators of severe coronary artery disease: significant ST depression, an abnormal resting ECG, and extensive vessel blockage.
high_risk & slope downsloping A / trestbps 145.68
Strong: A downsloping ST segment is a powerful predic- of ischemia. Combining this with hypertension identi- fies patients at high risk, even if their resting ECG appears normal, highlighting the importance of stress-test indica- tors.
high_risk + slope-fatt oldpeak 1.49 restecg hypertrophy
Excellent: This rule effectively combines signs of acute ischemia (a flat ST slope with significant depression) with evidence of chronic cardiac stress (left ventricular hyper- trophy). This profile is strongly indicative of advanced coronary artery disease.
Temperature softmin and softmax: 0.1 (for Eq. 3 and 10.0 for (Eq. 4
Our model can run efficiently CPU environment with the PyTorch package.
Baselines:
BRCG (Dash et al. 2018). LEN (Barbiero et al., 2022). , DR-NET (Qiao al. 2021
RRL (Wang et al.) 2021 are trained with the default hyperparameter settings specified in
the original paper.
MICE (van Buuren & Groothuis- -uthhuoorn 10:
We use 5 imputations and mazil = 5 iterations with the default imputation methods in the mice R package.
MissForest (Stekhoven & Bühlmann, 2012): We use the default hyperparameter settings in the mi missForest R package.
MLP: We train 3-layer fully connected network finput-128- 108-128-our outtutt with batch size 32, learning rate 0.001, and 100 epochs using Adam optimizer.
VAF "AE/Veldkamp et al. 2025): We use variational autoencoder with latent dimension 16, encoder architecture (input 2- batch size 32, learning rate 0.001, and 100 epochs.
1imput22-122 2-122-266666--- 1-8-64 (e86nn-l,tent,, decoder architecture (latent-64- 14-2-oo---1)
DAE (mDAE) (Dupuy et al., 2024): We use a denoising autoencoder with bottle- neck dimension 16. encoder architecture (input-128-64 1att1128-66-bbttleeekkk decoder architecture (bottleneck-64-122 14-0ooutt corruption rate p = 0.2, batch size 32, learning rate 0.001, and 100 epochs.
GAIN (Yoon et al. 2018): We use mini-batch size 128, hint rate Phint = 0.9, MSE loss weight 100.0, cross-entropy loss weight μ 100.0, learning rate 0.001, and 1000 epochs.
MissDiff Ouyang et al. 2023): We 1000 diffusion timesteps with Mlstrtt 10-4 and Hend 0.02, batch size 32, learning rate 0.001, and 100 epochs.
All baseline models run efficiently CPU environment with PyTorch package (for deep learning methods) or R packages (for statistical methods).
L LIMITATION
While model shows promising performance, the ethical implications, such potential over- reliance or misuse for inferring sensitive information, require careful consideration.
Despite its strengths, NS-FCN has limitations. While effective, the asynchronous coordinate gra- dient descent optimization can be computationally intensive. Besides, the negative predicates are not well explored (we consider negative predicates independent predicate from positive pred- icates). Furthermore, while our model can derive predicates from continuous features, the current implementation leams single threshold per feature, which may capture more complex relation-
35

---

Published as a conference paper at ICLR 2026
ships (e.g., intervals). Extending the framework to learn expressive predicates from continuous data a promising direction for future work.
M USE OF LLMS
In this paper, LLMs were used solely for writing polishing. The key idea, the model design, research study, and all substantive writing completed by human authors. In the assessment of discovered rules, we use LLM to write the evaluation of rule quality, which have mentioned in the paper.
36


Reviews:
[{"review_id": "sMpKzEGf0j", "reviewer": "Reviewer_KLE1", "summary": "This paper proposes a new Neuro-Symbolic framework called NS-FCN (Neuro-Symbolic Forward Chaining Network), designed to integrate missing value imputation with interpretable logical rule discovery. \nWhile existing approaches for missing data either rely purely on statistical imputation or require fully observed data for rule induction, \nNS-FCN treats missing values as latent predicates and learns to infer them within a differentiable NeSy framework. \nThe model establishes a closed-loop system where the imputed values feed back into rule discovery, promising to improve both imputation accuracy and interpretability over time. \nThe combination of symbolic logic is done via a differentiable forward-chaining mechanism that uses soft logical operators to handle both continuous and discrete features. \nEmpirical evaluations on synthetic and real-world datasets show that NS-FCN achieves high accuracy in imputing missing values while simultaneously finding human-interpretable rules that describe the data.", "questions": "I don't have further questions, I hope to see some discussion around the weaknesses I spotted.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "One strength of the paper, which is the main contribution by the authors, is proposing a closed feedback loop between imputation and rule discovery,  that allows the model to iteratively improve its reasoning and inference capabilities. \nAs a merit, the framework handles heterogeneous data types, both discrete and continuous, by transforming continuous features into soft predicates, which is valuable in cases where both real and categorical values are considered. \nThe model’s optimization method, based on coordinate gradient descent, seems to be a useful and scalable recipe for learning in this setting. \n\nThe experimental analysis also reveals that the framework generates logical rules that remain meaningful and consistent with domain knowledge. The empirical studies are good and cover a wide setting, showing that the model performs competitively against black-box neural architectures while maintaining explainability.", "weaknesses": "While the experiments involve also different black-box neural competitors, the range of baselines could be expanded to include more recent generative models, such as those based on diffusion processes. Due to my limited experience on this specific topic, I do not understand if the baselines are the effective SotA or one could have repurposed auto-encoders or diffusions.\n\n\nOne thing is not mentioned and discussed is the relation to reasoning shortcuts, where learning may result in incorrect predicates and rules, as in [1,2,3]. \nSince NS-FCN operates with learning latent discrete values, an analysis of whether it avoids such behavior would have strengthened its claims about interpretability and reliability. \nIn fact, the validation of interpretability remains largely qualitative; the paper reports interpretable rules, but lacks a systematic  assessment of their quality. \n\nAdditionally, the dependence of the model’s performance on temperature hyperparameters in the soft logical operators is not clear and not enough explored.\n\n------\n\n[1] Learning with Logical Constraints but without Shortcut Satisfaction, Li et al., 2023 \\\n[2] Not All Neuro-Symbolic Concepts Are Created Equal: Analysis and Mitigation of Reasoning Shortcuts, Marconato et al. 2023 \\\n[3] Shortcuts and identifiability in concept-based models from a neuro-symbolic lens, Bortolotti et al., 2025", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "6Vw1Ji473H", "reviewer": "Reviewer_GyS2", "summary": "Authors aim to combine neural embedding–based ILP with rule-based missing value imputation. However, the paper doesn’t clearly explain why their methodology should outperform the baselines they compare against. The only reason - apart from being the first to combine these three techniques - seems to be a more native and advanced way of handling continuous variables. But the baselines this method is compared against are not even introduced in the related works or other sections. It’s also not very well written and hard to follow.", "questions": "1. Please introduce and justify the baseline methods used in the comparison (in Table 4, 5 and 6).\n2. Are there other missing-value imputation methods that natively handle continuous features (rather than relying on hard thresholds)? This seems to be a key limitation of the baselines you compare against.", "limitations": "", "rating": 4, "confidence": 2, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "The paper presents a methodology that leverages neural embedding–based inductive logic programming to enhance rule-based methods for missing value imputation, which handles both binary and continuous variables. The method employs a coordinate gradient descent algorithm specifically tailored to this type of problem. Its novelty lies in being the first to combine ILP and rule-based approaches for improved missing value imputation. The integration of soft predicate learning enables native handling of continuous features, avoiding discretization and improving flexibility. The paper is well written and detailed. The empirical results on both synthetic and real datasets are\nstrong, demonstrating both interpretability and accuracy.", "weaknesses": "The paper does not clearly justify why the proposed method outperforms the baselines, particularly in real-world experiments. The other methods in Tables 4 and 5 are not clearly introduced or discussed in the main body of the paper. In particular, the baselines listed in Table 7 (with references) should be properly cited and briefly discussed in the Related Work section. Moreover, the primary reason for NS-FCN’s superior performance—aside from being the first approach to integrate ILP and rule-based imputation—appears to be its native support for continuous variables (lines 412–414). While the ablation studies are thorough, they could more clearly isolate the contribution of individual system components (e.g., the impact of the fine-tuning step versus differentiable forward chaining).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "Rth7tXi4Uj", "reviewer": "Reviewer_vtkH", "summary": "The paper introduces a novel neuro-symbolic framework (NS-FCN) for jointly performing missing-value imputation and interpretable rule discovery in heterogeneous datasets. The proposed model treats missing entries as latent predicates and combines neural embedding learning with differentiable logical reasoning in a closed-loop fashion—where imputation and rule induction reinforce one another. Using coordinate gradient descent, sequential covering, and soft logical operators, NS-FCN discovers both conjunctive and disjunctive rules and imputes missing data. Experiments on synthetic and real-world datasets show promising imputation accuracy and human-interpretable rules.", "questions": "1. Have you conducted a sensitivity analysis for the temperature parameters used in the soft-min and log-sum-exp approximations? How do these values impact the model's performance and stability across different datasets?\n\n\n2. The paper mentions that negative predicates are treated as independent predicates. ​ Have you explored alternative approaches to model negative predicates more effectively, such as incorporating negation directly into the rule learning process?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The framework’s closed-loop design elegantly unifies imputation and rule induction, a combination that is conceptually strong and rarely explored.\n\n2. The method produces explicit logical rules that are easy to interpret and verify. This property is valuable for domains where transparency is critical.\n\n3. The paper is clearly written, with intuitive figuresillustrating the reasoning and training process. The problem is generally well motivated and related-work coverage are comprehensive.", "weaknesses": "Despite its novelty, several issues limit the methodological soundness and clarity.\n\n* The paper claims that the coordinate gradient descent “never increases” the loss, but no proof, or convergence bound is presented. It is unclear whether this property holds under stochastic mini-batching or asynchronous updates when rules interact via shared predicates.\n* The freezing of “perfect rules” during training may lead to premature convergence or sub-optimal local minima, especially when data contain noise or imbalanced predicates. This could hinder exploration of alternative rule structures.\n* Equation (2) relies on a non-differentiable argmax to match rule components to predicate embeddings. The paper does not explain whether gradients are approximated or ignored. If predicate embeddings are frozen, this matching process may prevent end-to-end adaptation and yield inconsistent updates.\n* The soft-min and log-sum-exp approximations can distort logical semantics. Soft-min may amplify small numerical differences, resulting in unstable gradient propagation.\n* Although the experiments are diverse, comparisons are limited to interpretable baselines. It would be informative to benchmark against state-of-the-art deep imputation methods, e.g., MICE, GAIN, MissForest) to better contextualize performance gains. Sensitivity to missingness ratios and noise could also be analyzed more rigorously.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "OjmV90h94r", "reviewer": "Reviewer_PgSy", "summary": "The paper proposes a neuro-symbolic framework for missing value imputation that integrates neural feature learning with symbolic rule discovery. The key contributions are A hybrid imputation architecture that couples a neural network (for pattern recognition and latent representation) with a symbolic reasoning engine (for interpretable rule-based inference of missing values), and a a rule discovery mechanism that extracts symbolic if-then rules from latent representations to explain imputation decisions.", "questions": "1. How does the symbolic rule discovery scale with feature dimensionality and rule length? Could pruning or differentiable symbolic learning help?\n2. Have you conducted any human evaluation to assess whether the discovered rules are semantically meaningful and useful to domain experts?\n3. How does the framework explicitly handle MNAR data when missingness depends on the missing values themselves?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. The fusion of symbolic rule discovery with neural representation learning is both conceptually elegant and practically meaningful. It directly addresses a persistent problem in missing data research: neural models can impute accurately but are opaque, whereas symbolic models are interpretable but rigid.\n2. The ability to generate human-readable rules adds practical value for analysts, especially in regulated domains (e.g., healthcare, finance).\n3. The iterative mechanism that alternates between neural imputation and symbolic rule adjustment is well-motivated. It allows symbolic reasoning to guide neural learning and vice versa, enhancing both accuracy and transparency.", "weaknesses": "1. While the motivation is sound, the paper lacks theoretical guarantees for convergence or consistency of rule discovery. There’s no formal justification that the extracted rules remain faithful to the true data-generating process, especially under high missingness rates.\n2. The symbolic component introduces combinatorial complexity, especially when discovering multi-variable rules. The approach may not scale well to high-dimensional datasets with thousands of features.\n3. Missing data in real-world structured domains might behave differently, so generalizability remains unclear.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)