arXiv subjects

Sagar Malhotra

Publications and source records attributed to Sagar Malhotra.

At least 19 recordsLinked to original sources

arXiv2026

GNN Explanations that do not Explain and How to find Them

Explanations provided by Self-explainable Graph Neural Networks (SE-GNNs) are fundamental for understanding the model's inner workings and for identifying potential misuse of sensitive attributes. Although recent works have highlighted that these explanations can be suboptimal and potentially misleading, a characterization of their failure cases is unavailable. In this work, we identify a critical failure of SE-GNN explanations: explanations can be unambiguously unrelated to how the SE-GNNs infer labels. We show that, on the one hand, many SE-GNNs can achieve optimal true risk while producing these degenerate explanations, and on the other, most faithfulness metrics can fail to identify these failure modes. Our empirical analysis reveals that degenerate explanations can be maliciously planted (allowing an attacker to hide the use of sensitive attributes) and can also emerge naturally, highlighting the need for reliable auditing. To address this, we introduce a novel faithfulness metric that reliably marks degenerate explanations as unfaithful, in both malicious and natural settings. Our code is available at https://github.com/steveazzolin/gnn_deg_expl.

cs.LG↗

arXiv2026

Is BatchEnsemble a Single Model? On Calibration and Diversity of Efficient Ensembles

In resource-constrained and low-latency settings, uncertainty estimates must be efficiently obtained. Deep Ensembles provide robust epistemic uncertainty (EU) but require training multiple full-size models. BatchEnsemble aims to deliver ensemble-like EU at far lower parameter and memory cost by applying learned rank-1 perturbations to a shared base network. We show that BatchEnsemble not only underperforms Deep Ensembles but closely tracks a single model baseline in terms of accuracy, calibration and out-of-distribution (OOD) detection on CIFAR10/10C/SVHN. A controlled study on MNIST finds members are near-identical in function and parameter space, indicating limited capacity to realize distinct predictive modes. Thus, BatchEnsemble behaves more like a single model than a true ensemble.

cs.LG↗

arXiv2026

The selection function of the Gaia DR3 open cluster census

Open clusters are among the most useful and widespread tracers of Galactic structure. The completeness of the Galactic open cluster census, however, remains poorly understood. For the first time ever, we establish the selection function of an entire open cluster census, publishing our results as an open-source Python package for use by the community. Our work is valid for the Hunt & Reffert catalogue of clusters in Gaia DR3. We developed and open-sourced our cluster simulator from our first work. Then, we performed 80,590 injection and retrievals of simulated open clusters to test the Hunt & Reffert catalogue's sensitivity. We fit a logistic model of cluster detectability that depends only on a cluster's number of stars, median parallax error, Gaia data density, and a user-specified significance threshold. We find that our simple model accurately predicts cluster detectability, with a 94.53\% accuracy on our training data that is comparable to a machine-learning based model with orders of magnitude more parameters. Our model itself offers numerous insights on why certain clusters are detected. We briefly use our model to show that cluster detectability depends on non-intuitive parameters, such as a cluster's proper motion, and we show that even a modest 25 km/s boost to a cluster's orbital speed can result in an almost 3$\times$ higher detection probability, depending on its position. In addition, we publish our raw cluster injection and retrievals and cluster memberships, which could be used for a number of other science cases -- such as estimating cluster membership incompleteness. Using our results, selection effect-corrected studies are now possible with the open cluster census. Our work will enable a number of brand new types of study, such as detailed comparisons between the Milky Way's cluster census and recent extragalactic cluster samples.

astro-ph.GA↗

arXiv2025

Stellar masses and mass ratios for Gaia open cluster members

Context: Unresolved binaries in star clusters can bias stellar and cluster mass estimates, making their proper treatment essential for studying cluster dynamics and evolution. Aims: We aim to develop a fast and robust framework for jointly deriving stellar masses and multiplicity statistics of member stars, together with optimal cluster parameters. Methods: We use Gaia DR3 parallaxes together with multi-band photometry of open cluster (OC) members to infer stellar masses and binary mass-ratios through simulation-based inference (SBI), while iteratively fitting the cluster parameters. The validation of our SBI framework on simulated clusters demonstrates that the inclusion of infrared photometry significantly improves the detection of low mass-ratio binaries. The minimum mass-ratio threshold for reliably identifying unresolved binaries depends on cluster properties and the available photometry, but typically lies below $q=0.5$. Results: Applying our method to 42 well-populated OCs, we derive a catalogue of stellar masses and mass-ratios for 27201 stars, achieving typical uncertainties of 0.08 in $q$ and $0.01\,\mathrm{M}_\odot$ in the primary stellar mass. We analyse the archetype OCs M67 and NGC 2360 in detail, including mass segregation and mass-ratio distribution among other characteristics, while deriving multiplicity fractions for the rest of the sample. We find evidence that the high mass-ratio ($q\geq 0.6$) binary fraction shows a strong correlation with the age and a weak anti-correlation with the cluster metallicity. Furthermore, the variation of the binary fraction with stellar mass in OCs shows strong accordance with the observed dependence for field stars heavier than $\gtrsim0.6\,\mathrm{M}_\odot$. Conclusions: Our work paves a path for future population-level investigations of multiplicity statistics and precision stellar masses in extended samples of OCs.

astro-ph.GA↗

arXiv2025

Probably Approximately Global Robustness Certification

We propose and investigate probabilistic guarantees for the adversarial robustness of classification algorithms. While traditional formal verification approaches for robustness are intractable and sampling-based approaches do not provide formal guarantees, our approach is able to efficiently certify a probabilistic relaxation of robustness. The key idea is to sample an $ε$-net and invoke a local robustness oracle on the sample. Remarkably, the size of the sample needed to achieve probably approximately global robustness guarantees is independent of the input dimensionality, the number of classes, and the learning algorithm itself. Our approach can, therefore, be applied even to large neural networks that are beyond the scope of traditional formal verification. Experiments empirically confirm that it characterizes robustness better than state-of-the-art sampling-based approaches and scales better than formal methods.

cs.LG↗

arXiv2025

On Local Limits of Sparse Random Graphs: Color Convergence and the Refined Configuration Model

Local convergence has emerged as a fundamental tool for analyzing sparse random graph models. We introduce a new notion of local convergence, color convergence, based on the Weisfeiler-Leman algorithm. Color convergence fully characterizes the class of random graphs that are well-behaved in the limit for message-passing graph neural networks. Building on this, we propose the Refined Configuration Model (RCM), a random graph model that generalizes the configuration model. The RCM is universal with respect to local convergence among locally tree-like random graph models, including Erdős-Rényi, stochastic block and configuration models. Finally, this framework enables a complete characterization of the random trees that arise as local limits of such graphs.

cs.DM↗

arXiv2025

Prime Implicant Explanations for Reaction Feasibility Prediction

Machine learning models that predict the feasibility of chemical reactions have become central to automated synthesis planning. Despite their predictive success, these models often lack transparency and interpretability. We introduce a novel formulation of prime implicant explanations--also known as minimally sufficient reasons--tailored to this domain, and propose an algorithm for computing such explanations in small-scale reaction prediction tasks. Preliminary experiments demonstrate that our notion of prime implicant explanations conservatively captures the ground truth explanations. That is, such explanations often contain redundant bonds and atoms but consistently capture the molecular attributes that are essential for predicting reaction feasibility.

cs.LG↗

arXiv2025

Beyond Topological Self-Explainable GNNs: A Formal Explainability Perspective

Self-Explainable Graph Neural Networks (SE-GNNs) are popular explainable-by-design GNNs, but their explanations' properties and limitations are not well understood. Our first contribution fills this gap by formalizing the explanations extracted by some popular SE-GNNs, referred to as Minimal Explanations (MEs), and comparing them to established notions of explanations, namely Prime Implicant (PI) and faithful explanations. Our analysis reveals that MEs match PI explanations for a restricted but significant family of tasks. In general, however, they can be less informative than PI explanations and are surprisingly misaligned with widely accepted notions of faithfulness. Although faithful and PI explanations are informative, they are intractable to find and we show that they can be prohibitively large. Given these observations, a natural choice is to augment SE-GNNs with alternative modalities of explanations taking care of SE-GNNs' limitations. To this end, we propose Dual-Channel GNNs that integrate a white-box rule extractor and a standard SE-GNN, adaptively combining both channels. Our experiments show that even a simple instantiation of Dual-Channel GNNs can recover succinct rules and perform on par or better than widely used SE-GNNs.

cs.LG↗

arXiv2025

Lifted Inference beyond First-Order Logic

Weighted First Order Model Counting (WFOMC) is fundamental to probabilistic inference in statistical relational learning models. As WFOMC is known to be intractable in general ($\#$P-complete), logical fragments that admit polynomial time WFOMC are of significant interest. Such fragments are called domain liftable. Recent works have shown that the two-variable fragment of first order logic extended with counting quantifiers ($\mathrm{C^2}$) is domain-liftable. However, many properties of real-world data, like acyclicity in citation networks and connectivity in social networks, cannot be modeled in $\mathrm{C^2}$, or first order logic in general. In this work, we expand the domain liftability of $\mathrm{C^2}$ with multiple such properties. We show that any $\mathrm{C^2}$ sentence remains domain liftable when one of its relations is restricted to represent a directed acyclic graph, a connected graph, a tree (resp. a directed tree) or a forest (resp. a directed forest). All our results rely on a novel and general methodology of "counting by splitting". Besides their application to probabilistic inference, our results provide a general framework for counting combinatorial structures. We expand a vast array of previous results in discrete mathematics literature on directed acyclic graphs, phylogenetic networks, etc.

cs.AI↗

arXiv2024

Logical Distillation of Graph Neural Networks

We present a logic based interpretable model for learning on graphs and an algorithm to distill this model from a Graph Neural Network (GNN). Recent results have shown connections between the expressivity of GNNs and the two-variable fragment of first-order logic with counting quantifiers (C2). We introduce a decision-tree based model which leverages an extension of C2 to distill interpretable logical classifiers from GNNs. We test our approach on multiple GNN architectures. The distilled models are interpretable, succinct, and attain similar accuracy to the underlying GNN. Furthermore, when the ground truth is expressible in C2, our approach outperforms the GNN.

cs.LG↗

arXiv2024

Simple and Effective Transfer Learning for Neuro-Symbolic Integration

Deep Learning (DL) techniques have achieved remarkable successes in recent years. However, their ability to generalize and execute reasoning tasks remains a challenge. A potential solution to this issue is Neuro-Symbolic Integration (NeSy), where neural approaches are combined with symbolic reasoning. Most of these methods exploit a neural network to map perceptions to symbols and a logical reasoner to predict the output of the downstream task. These methods exhibit superior generalization capacity compared to fully neural architectures. However, they suffer from several issues, including slow convergence, learning difficulties with complex perception tasks, and convergence to local minima. This paper proposes a simple yet effective method to ameliorate these problems. The key idea involves pretraining a neural model on the downstream task. Then, a NeSy model is trained on the same task via transfer learning, where the weights of the perceptual part are injected from the pretrained network. The key observation of our work is that the neural network fails to generalize only at the level of the symbolic part while being perfectly capable of learning the mapping from perceptions to symbols. We have tested our training strategy on various SOTA NeSy methods and datasets, demonstrating consistent improvements in the aforementioned problems.

cs.LG↗

arXiv2024

Understanding Domain-Size Generalization in Markov Logic Networks

We study the generalization behavior of Markov Logic Networks (MLNs) across relational structures of different sizes. Multiple works have noticed that MLNs learned on a given domain generalize poorly across domains of different sizes. This behavior emerges from a lack of internal consistency within an MLN when used across different domain sizes. In this paper, we quantify this inconsistency and bound it in terms of the variance of the MLN parameters. The parameter variance also bounds the KL divergence between an MLN's marginal distributions taken from different domain sizes. We use these bounds to show that maximizing the data log-likelihood while simultaneously minimizing the parameter variance corresponds to two natural notions of generalization across domain sizes. Our theoretical results apply to Exponential Random Graphs and other Markov network based relational models. Finally, we observe that solutions known to decrease the variance of the MLN parameters, like regularization and Domain-Size Aware MLNs, increase the internal consistency of the MLNs. We empirically verify our results on four different datasets, with different methods to control parameter variance, showing that controlling parameter variance leads to better generalization.

cs.AI↗

arXiv2024

Towards Counting Markov Equivalence Classes with Logical Constraints

We initiate the study of counting Markov Equivalence Classes (MEC) under logical constraints. MECs are equivalence classes of Directed Acyclic Graphs (DAGs) that encode the same conditional independence structure among the random variables of a DAG model. Observational data can only allow to infer a DAG model up to Markov Equivalence. However, Markov equivalent DAGs can represent different causal structures, potentially super-exponentially many. Hence, understanding MECs combinatorially is critical to understanding the complexity of causal inference. In this paper, we focus on analysing MECs of size one, with logical constraints on the graph topology. We provide a polynomial-time algorithm (w.r.t. the number of nodes) for enumerating essential DAGs (the only members of an MEC of size one) with arbitrary logical constraints expressed in first-order logic with two variables and counting quantifiers (C^2). Our work brings together recent developments in tractable first-order model counting and combinatorics of MECs.

cs.LO↗

arXiv2024

On the Metallicity Gradients in the Galactic Disk using Open Clusters

We study the metallicity distribution and evolution in the Galactic disk based on the largest sample of open star clusters in the Galaxy. From the catalogue of 1879 open clusters in the range of Galactocentric distance (R_GC) from 4 to 20 kpc, we investigate the variation of metallicity in the Galactic disk as functions of R_GC, vertical distance (Z), and ages of the clusters. In the direction perpendicular to the Galactic plane, variation in metallicity is found to follow a stepped linear relation. We estimate a vertical metallicity gradient d[Fe/H]/dZ of -0.545+/-0.046 dex/kpc for |Z| < 0.487 kpc, and -0.075+/-0.093 dex/kpc for 0.487 < |Z| < 1.8 kpc. On average, metallicity variations above and below the Galactic plane are found to change at similar rates. The change in metallicity in the radial direction is also found to follow a two-function linear relation. We obtain a radial metallicity gradient d[Fe/H]/d[R_GC] of -0.070+/-0.002 dex/kpc for 4.0<R_GC<12.8 kpc, and -0.005+/-0.018 dex/kpc for 12.8< R_GC < 20.5 kpc which clearly shows a strong variation in the metallicity gradient when moving from the inner to the outer Galactic disk. Age-metallicity relation (AMR) is found to follow a steeper negative slope of -0.031+/-0.006 dex/Gyr for clusters older than 240 Myr, however, there is some hint of positive metallicity age gradient for younger clusters.

astro-ph.GA↗

arXiv2023

Revisiting Galactic disk and spiral arms using open clusters

We use the largest open clusters catalogue in the post-Gaia era to provide the observational view of the Galactic disk. By compiling the physical parameters like age, distance, and kinematic information, we investigate the spatial distribution of the open clusters and revisit the spiral arms and other asymmetries in the Galactic disk. Using the young open clusters as a tracer of spiral arms, we map the spiral structure of the Galaxy and found that most of the clusters start migrating away from the spiral arms in about 10-20 Myr and fill the inter-arm regions as their age progress. Using the 3D kinematic information of 371 open star clusters, we derived different individual pattern speeds for spiral arms closely following the Milky Way rotation curve, hence favouring the transient nature of Milky Way spiral arms. The pattern rotation speeds of each spiral arm suggest that the spiral arms have not accelerated in the last 80 Myrs. Based on the distribution of open clusters younger than 700 Myr above or below the Galactic plane, we found a Solar offset of $z_\odot = 17.0\pm0.9$ pc north of the Galactic plane and estimated the scale height $z_h = 91.7 \pm 1.9$ pc from the Galactic plane.

astro-ph.GA↗

arXiv2023

Weighted First Order Model Counting with Directed Acyclic Graph Axioms

Statistical Relational Learning (SRL) integrates First-Order Logic (FOL) and probability theory for learning and inference over relational data. Probabilistic inference and learning in many SRL models can be reduced to Weighted First Order Model Counting (WFOMC). However, WFOMC is known to be intractable ($\mathrm{\#P_1-}$ complete). Hence, logical fragments that admit polynomial time WFOMC are of significant interest. Such fragments are called domain liftable. Recent line of works have shown the two-variable fragment of FOL, extended with counting quantifiers ($\mathrm{C^2}$) to be domain-liftable. However, many properties of real-world data can not be modelled in $\mathrm{C^2}$. In fact many ubiquitous properties of real-world data are inexressible in FOL. Acyclicity is one such property, found in citation networks, genealogy data, temporal data e.t.c. In this paper we aim to address this problem by investigating the domain liftability of directed acyclicity constraints. We show that the fragment $\mathrm{C^2}$ with a Directed Acyclic Graph (DAG) axiom, i.e., a predicate in the language is axiomatized to represent a DAG, is domain-liftable. We present a method based on principle of inclusion-exclusion for WFOMC of $\mathrm{C^2}$ formulas extended with DAG axioms.

cs.AI↗

arXiv2023

Deep Symbolic Learning: Discovering Symbols and Rules from Perceptions

Neuro-Symbolic (NeSy) integration combines symbolic reasoning with Neural Networks (NNs) for tasks requiring perception and reasoning. Most NeSy systems rely on continuous relaxation of logical knowledge, and no discrete decisions are made within the model pipeline. Furthermore, these methods assume that the symbolic rules are given. In this paper, we propose Deep Symbolic Learning (DSL), a NeSy system that learns NeSy-functions, i.e., the composition of a (set of) perception functions which map continuous data to discrete symbols, and a symbolic function over the set of symbols. DSL learns simultaneously the perception and symbolic functions while being trained only on their composition (NeSy-function). The key novelty of DSL is that it can create internal (interpretable) symbolic representations and map them to perception inputs within a differentiable NN learning pipeline. The created symbols are automatically selected to generate symbolic functions that best explain the data. We provide experimental analysis to substantiate the efficacy of DSL in simultaneously learning perception and symbolic functions.

cs.LG↗

arXiv2022

On Projectivity in Markov Logic Networks

Markov Logic Networks (MLNs) define a probability distribution on relational structures over varying domain sizes. Many works have noticed that MLNs, like many other relational models, do not admit consistent marginal inference over varying domain sizes. Furthermore, MLNs learnt on a certain domain do not generalize to new domains of varied sizes. In recent works, connections have emerged between domain size dependence, lifted inference and learning from sub-sampled domains. The central idea to these works is the notion of projectivity. The probability distributions ascribed by projective models render the marginal probabilities of sub-structures independent of the domain cardinality. Hence, projective models admit efficient marginal inference, removing any dependence on the domain size. Furthermore, projective models potentially allow efficient and consistent parameter learning from sub-sampled domains. In this paper, we characterize the necessary and sufficient conditions for a two-variable MLN to be projective. We then isolate a special model in this class of MLNs, namely Relational Block Model (RBM). We show that, in terms of data likelihood maximization, RBM is the best possible projective MLN in the two-variable fragment. Finally, we show that RBMs also admit consistent parameter learning over sub-sampled domains.

cs.AI↗