arXiv subjects

Sayan Mukherjee

Publications and source records attributed to Sayan Mukherjee.

At least 19 recordsLinked to original sources

arXiv2026

Synthesizing POMDP Policies: Sampling Meets Model-checking via Learning

Partially Observable Markov Decision Processes (POMDPs) are the standard framework for decision-making under uncertainty. While sampling-based methods scale well, they lack formal correctness guarantees, making them unsuitable for safety-critical applications. Conversely, formal synthesis techniques provide correctness-by-construction but often struggle with scalability, as general POMDP synthesis is undecidable. To bridge this gap, we propose a synthesis framework that integrates sampling, automata learning, and model-checking. Inspired by Angluin's $L^*$ algorithm, our approach utilizes sampling as a membership oracle and model-checking as an equivalence oracle. This enables the synthesis of finite-state controllers with formal guarantees, provided the sampling-induced policy is regular. We establish a relative completeness result for this framework. Experimental results from our prototypical implementation demonstrate that this method successfully solves threshold-safety problems that remain challenging for existing formal synthesis tools. We believe our algorithm serves as a valuable component in a portfolio approach to tackling the inherent difficulty of POMDP synthesis problems.

cs.AI↗

arXiv2026

Robust Predictive Uncertainty and Double Descent in Contaminated Bayesian Random Features

We propose a robust Bayesian formulation of random feature (RF) regression that accounts explicitly for prior and likelihood misspecification via Huber-style contamination sets. Starting from the classical equivalence between ridge-regularized RF training and Bayesian inference with Gaussian priors and likelihoods, we replace the single prior and likelihood with $\epsilon$- and $\eta$-contaminated credal sets, respectively, and perform inference using pessimistic generalized Bayesian updating. We derive explicit and tractable bounds for the resulting lower and upper posterior predictive densities. These bounds show that, when contamination is moderate, prior and likelihood ambiguity effectively acts as a direct contamination of the posterior predictive distribution, yielding uncertainty envelopes around the classical Gaussian predictive. We introduce an Imprecise Highest Density Region (IHDR) for robust predictive uncertainty quantification and show that it admits an efficient approximation via an adjusted Gaussian credible interval. We further obtain predictive variance bounds (under a mild truncation approximation for the upper bound) and prove that they preserve the leading-order proportional-growth asymptotics known for RF models. Together, these results establish a robustness theory for Bayesian random features: predictive uncertainty remains computationally tractable, inherits the classical double-descent phase structure, and is improved by explicit worst-case guarantees under bounded prior and likelihood misspecification.

cs.LG↗

arXiv2025

Congestion bounds via Laplacian eigenvalues and their application to tensor networks with arbitrary geometry

Embedding the vertices of arbitrary graphs into trees while minimizing some measure of overlap is an important problem with applications in computer science and physics. In this work, we consider the problem of bijectively embedding the vertices of an $n$-vertex graph $G$ into the \textit{leaves} of an $n$-leaf \textit{rooted binary tree} $\mathcal{T}$. The congestion of such an embedding is given by the largest size of the cut induced by the two components obtained by deleting any vertex of $\mathcal{T}$. We show that for any embedding, the congestion lies between $\lambda_2(G)\cdot 2n/9$ and $\lambda_n(G)\cdot n/4$, letting $0=\lambda_1(G)\le \cdots \le \lambda_n(G)$ be the Laplacian eigenvalues of $G$, and there is an embedding for which the congestion is at most $\lambda_n(G)\cdot 2n/9$. Beyond these general bounds, we determine the congestion exactly for hypercubes and lattice graphs, and obtain asymptotically tight bounds for random regular graphs and Erd\H{o}s-R\'{e}nyi graphs. We further introduce an efficient contraction procedure based on spectral ordering and dynamic programming, which produces low-congestion embeddings in practice. Numerical experiments on structured graphs, random graphs, and tensor network representations of quantum circuits validate our theoretical bounds and demonstrate the effectiveness of the proposed method. These results yield new spectral bounds on the memory and time complexity of exact tensor network contraction in terms of the underlying graph structure.

cs.DS↗

arXiv2025

Learning Event-recording Automata Passively

This paper presents a state-merging algorithm for learning timed languages definable by Event-Recording Automata (ERA) using positive and negative samples in the form of symbolic timed words. Our algorithm, LEAP (Learning Event-recording Automata Passively), constructs a possibly nondeterministic ERA from such samples based on merging techniques. We prove that determining whether two ERA states can be merged while preserving sample consistency is an NP-complete problem, and address this with a practical SMT-based solution. Our implementation demonstrates the algorithm's effectiveness through examples. We also show that every ERA-definable language can be inferred using our algorithm with a suitable sample.

cs.FL↗

arXiv2025

Locally Differentially Private Graph Clustering via the Power Iteration Method

We propose a locally differentially private graph clustering algorithm. Previous works have explored this problem, including approaches that apply spectral clustering to graphs generated via the randomized response algorithm. However, these methods only achieve accurate results when the privacy budget is in $\Omega(\log n)$, which is unsuitable for many practical applications. In response, we present an interactive algorithm based on the power iteration method. Given that the noise introduced by the largest eigenvector constant can be significant, we incorporate a technique to eliminate this constant. As a result, our algorithm attains local differential privacy with a constant privacy budget when the graph is well-clustered and has a minimum degree of $\tilde{\Omega}(\sqrt{n})$. In contrast, while randomized response has been shown to produce accurate results under the same minimum degree condition, it is limited to graphs generated from the stochastic block model. We perform experiments to demonstrate that our method outperforms spectral clustering applied to randomized response results.

cs.DS↗

arXiv2024

Approximately Optimal Search on a Higher-dimensional Sliding Puzzle

Higher-dimensional sliding puzzles are constructed on the vertices of a $d$-dimensional hypercube, where $2^d-l$ vertices are distinctly coloured. Rings with the same colours are initially set randomly on the vertices of the hypercube. The goal of the puzzle is to move each of the $2^d-l$ rings to pre-defined target vertices on the cube. In this setting, the $k$-rule constraint represents a generalisation of edge collision for the movement of colours between vertices, allowing movement only when a hypercube face of dimension $k$ containing a ring is completely free of other rings. Starting from an initial configuration, what is the minimum number of moves needed to make ring colours match the vertex colours? An algorithm that provides us with such a number is called God's algorithm. When such an algorithm exists, it does not have a polynomial time complexity, at least in the case of the 15-puzzle corresponding to $k=1$ in the cubical puzzle. This paper presents a comprehensive computational study of different scenarios of the higher-dimensional puzzle. A benchmark of three computational techniques, an exact algorithm (the A* search) and two approximately optimal search techniques (an evolutionary algorithm (EA) and reinforcement learning (RL)) is presented in this work. The experiments show that all three methods can successfully solve the puzzle of dimension three for different face dimensions and across various difficulty levels. When the dimension increases, the A* search fails, and RL and EA methods can still provide a generally acceptable solution, i.e. a distribution of a number of moves with a median value of less than $30$. Overall, the EA method consistently requires less computational time, while failing in most cases to minimise the number of moves for the puzzle dimensions $d=4$ and $d=5$.

cs.AI↗

arXiv2024

Higher Order Bipartiteness vs Bi-Partitioning in Simplicial Complexes

Bipartite graphs are a fundamental concept in graph theory with diverse applications. A graph is bipartite iff it contains no odd cycles, a characteristic that has many implications in diverse fields ranging from matching problems to the construction of complex networks. Another key identifying feature is their Laplacian spectrum as bipartite graphs achieve the maximum possible eigenvalue of graph Laplacian. However, for modeling higher-order connections in complex systems, hypergraphs and simplicial complexes are required due to the limitations of graphs in representing pairwise interactions. In this article, using simple tools from graph theory, we extend the cycle-based characterization from bipartite graphs to those simplicial complexes that achieve the maximum Hodge Laplacian eigenvalue, known as disorientable simplicial complexes. We show that a $N$-dimensional simplicial complex is disorientable if its down dual graph contains no simple odd cycle of distinct edges and no twisted even cycle of distinct edges. Furthermore, we see that in a $N$-simplicial complex without twisting cycles, the fewer the number of (non-branching) simple odd cycles in its down dual graph, the closer is its maximum eigenvalue to the possible maximum eigenvalue of Hodge Laplacian. Similar to the graph case, the absence of odd cycles plays a crucial role in solving the bi-partitioning problem of simplexes in higher dimensions.

math.CO↗

arXiv2024

Greybox Learning of Languages Recognizable by Event-Recording Automata

In this paper, we revisit the active learning of timed languages recognizable by event-recording automata. Our framework employs a method known as greybox learning, which enables the learning of event-recording automata with a minimal number of control states. This approach avoids learning the region automaton associated with the language, contrasting with existing methods. We have implemented our greybox learning algorithm with various heuristics to maintain low computational complexity. The efficacy of our approach is demonstrated through several examples.

cs.FL↗

arXiv2024

Uniform Consistency of Generalized Fr\'echet Means

Loss-based notions of centre on nonlinear spaces range from the Fr\'echet mean and power means to the geometric median and, in a limiting sense, the Chebyshev centre. To use such summaries statistically, one first needs a law of large numbers that remains valid beyond smooth manifolds and beyond a fixed choice of loss. We study generalized Fr\'echet means on metric spaces with the Heine--Borel property, obtained by replacing squared distance with a convex loss under a mild exponential-growth condition. We prove existence and compactness of the population mean set, establish a sharp diameter bound, obtain almost-sure consistency of empirical $\phi$-means, and derive a uniform strong law over compact classes of losses. The analysis is driven by a deterministic argmin principle together with a Glivenko--Cantelli theorem for monotone classes. For isotropic densities on Riemannian symmetric spaces, we identify the population $\phi$-mean for every strictly increasing loss for which the objective is finite, including bounded robust losses. We also illustrate the framework on spheres and on the polyhedral space of ultrametric phylogenetic trees.

math.ST↗

arXiv2024

Meta-Posterior Consistency for the Bayesian Inference of Metastable System

The vast majority of the literature on learning dynamical systems or stochastic processes from time series has focused on stable or ergodic systems, for both Bayesian and frequentist inference procedures. However, most real-world systems are only metastable, that is, the dynamics appear to be stable on some time scale, but are in fact unstable over longer time scales. Consistency of inference for metastable systems may not be possible, but one can ask about metaconsistency: Do inference procedures converge when observations are taken over a large but finite time interval, but diverge on longer time scales? In this paper we introduce, discuss, and quantify metaconsistency in a Bayesian framework. We discuss how metaconsistency can be exploited to efficiently infer a model for a sub-system of a larger system, where inference on the global behavior may require much more data, or there is no theoretical guarantee as to the asymptotic success of inference procedures. We also discuss the relation between metaconsistency and the spectral properties of the model dynamical system in the case of uniformly ergodic and non-ergodic diffusions.

stat.ML↗

arXiv2024

Pragmatist Intelligence: Where the Principle of Usefulness Can Take ANNs

Artificial neural networks (ANNs) perform extraordinarily on numerous tasks including classification or prediction, e.g., speech processing and image classification. These new functions are based on a computational model that is enabled to select freely all necessary internal model parameters as long as it eventually delivers the functionality it is supposed to exhibit. Here, we review the connection between the model parameter selection in machine learning (ML) algorithms running on ANNs and the epistemological theory of neopragmatism focusing on the theory's utility and anti-representationalist aspects. To understand the consequences of the model parameter selection of an ANN, we suggest using neopragmatist theories whose implications are well studied. Incidentally, neopragmatism's notion of optimization is also based on utility considerations. This means that applying this approach elegantly reveals the inherent connections between optimization in ML, using a numerical method during the learning phase, and optimization in the ethical theory of consequentialism, where it occurs as a maxim of action. We suggest that these connections originate from the way relevance is calculated in ML systems. This could ultimately reveal a tendency for specific actions in ML systems.

cs.AI↗

arXiv2023

Irreducibility of Markov Chains on simplicial complexes, the Spectrum of the Discrete Hodge Laplacian and Homology

Random walks on graphs are a fundamental concept in graph theory and play a crucial role in solving a wide range of theoretical and applied problems in discrete math, probability, theoretical computer science, network science, and machine learning. The connection between Markov chains on graphs and their geometric and topological structures is the main reason why such a wide range of theoretical and practical applications exist. Graph connectedness ensures irreducibility of a Markov chain. The convergence rate to the stationary distribution is determined by the spectrum of the graph Laplacian which is associated with lower bounds on graph curvature. Furthermore, walks on graphs are used to infer structural properties of underlying manifolds in data analysis and manifold learning. However, an important question remains: can similar connections be established between Markov chains on simplicial complexes and the topology, geometry, and spectral properties of complexes? Additionally, can we gain topological, geometric, or analytic information about a manifold by defining appropriate Markov chains on its triangulations? These questions are not only theoretically important but answers to them provide powerful tools for the analysis of complex networks that go beyond the analysis of pairwise interactions. In this paper, we provide an integrated overview of the existing results on random walks on simplicial complexes, using the novel perspective of signed graphs. This perspective sheds light on previously unknown aspects such as irreducibility conditions. We show that while up-walks on higher dimensional simplexes can never be irreducible, the down walks become irreducible if and only if the complex is orientable. We believe that this new integrated perspective can be extended beyond discrete structures and enables exploration of classical problems for triangulable manifolds.

math.SP↗

arXiv2023

Local Differential Privacy-Preserving Spectral Clustering for General Graphs

Spectral clustering is a widely used algorithm to find clusters in networks. Several researchers have studied the stability of spectral clustering under local differential privacy with the additional assumption that the underlying networks are generated from the stochastic block model (SBM). However, we argue that this assumption is too restrictive since social networks do not originate from the SBM. Thus, we delve into an analysis for general graphs in this work. Our primary focus is the edge flipping method -- a common technique for protecting local differential privacy. We show that, when the edges of an $n$-vertex graph satisfying some reasonable well-clustering assumptions are flipped with a probability of $O(\log n/n)$, the clustering outcomes are largely consistent. Empirical tests further corroborate these theoretical findings. Conversely, although clustering outcomes have been stable for non-sparse and well-clustered graphs produced from the SBM, we show that in general, spectral clustering may yield highly erratic results on certain well-clustered graphs when the flipping probability is $\omega(\log n/n)$. This indicates that the best privacy budget obtainable for general graphs is $\Theta(\log n)$.

cs.CR↗

arXiv2023

Exact generalized Tur\'an number for $K_3$ versus suspension of $P_4$

Let $P_4$ denote the path graph on $4$ vertices. The suspension of $P_4$, denoted by $\widehat P_4$, is the graph obtained via adding an extra vertex and joining it to all four vertices of $P_4$. In this note, we demonstrate that for $n\ge 8$, the maximum number of triangles in any $n$-vertex graph not containing $\widehat P_4$ is $\left\lfloor n^2/8\right\rfloor$. Our method uses simple induction along with computer programming to prove a base case of the induction hypothesis.

math.CO↗

arXiv2023

Asymptotics of Bayesian Uncertainty Estimation in Random Features Regression

In this paper we compare and contrast the behavior of the posterior predictive distribution to the risk of the maximum a posteriori estimator for the random features regression model in the overparameterized regime. We will focus on the variance of the posterior predictive distribution (Bayesian model average) and compare its asymptotics to that of the risk of the MAP estimator. In the regime where the model dimensions grow faster than any constant multiple of the number of samples, asymptotic agreement between these two quantities is governed by the phase transition in the signal-to-noise ratio. They also asymptotically agree with each other when the number of samples grow faster than any constant multiple of model dimensions. Numerical simulations illustrate finer distributional properties of the two quantities for finite dimensions. We conjecture they have Gaussian fluctuations and exhibit similar properties as found by previous authors in a Gaussian sequence model, which is of independent theoretical interest.

stat.ML↗

arXiv2023

Bi-Objective Lexicographic Optimization in Markov Decision Processes with Related Objectives

We consider lexicographic bi-objective problems on Markov Decision Processes (MDPs), where we optimize one objective while guaranteeing optimality of another. We propose a two-stage technique for solving such problems when the objectives are related (in a way that we formalize). We instantiate our technique for two natural pairs of objectives: minimizing the (conditional) expected number of steps to a target while guaranteeing the optimal probability of reaching it; and maximizing the (conditional) expected average reward while guaranteeing an optimal probability of staying safe (w.r.t. some safe set of states). For the first combination of objectives, which covers the classical frozen lake environment from reinforcement learning, we also report on experiments performed using a prototype implementation of our algorithm and compare it with what can be obtained from state-of-the-art probabilistic model checkers solving optimal reachability.

cs.GT↗

arXiv2023

Minimum $\Phi$-distance estimators for finite mixing measures

Finite mixture models have long been used across a variety of fields in engineering and sciences. Recently there has been a great deal of interest in quantifying the convergence behavior of the \emph{mixing measure}, a fundamental object that encapsulates all unknown parameters in a mixture distribution. In this paper we propose a general framework for estimating the mixing measure arising in finite mixture models, which we term minimum $\Phi$-distance estimators. We establish a general theory for the minimum $\Phi$-distance estimator, where sharp probability bounds are obtained on the estimation error for the mixing measures in terms of the suprema of the associated empirical processes for a suitably chosen function class $\Phi$. Our framework includes several existing and seemingly distinct estimation methods as special cases using a weakened identifiability condition, but also motivates new estimators. For instance, it extends the minimum Kolmogorov-Smirnov distance estimator to the multivariate setting, and it extends the method of moments to cover a broader family of probability kernels beyond the Gaussian. Moreover, it also includes methods that are applicable to complex (e.g., non-Euclidean) observation domains, using tools from reproducing kernel Hilbert spaces. It will be shown that under general conditions the methods achieve optimal rates of estimation under Wasserstein metrics in either minimax or pointwise sense of convergence; the latter case can be achieved when no upper bound on the finite number of components is given. Also of interest is a sharp inequality that captures the local information geometry for general mixture models precisely in terms of moment differences between mixing measures.

math.ST↗

arXiv2023

Global Optimality of Elman-type RNN in the Mean-Field Regime

We analyze Elman-type Recurrent Reural Networks (RNNs) and their training in the mean-field regime. Specifically, we show convergence of gradient descent training dynamics of the RNN to the corresponding mean-field formulation in the large width limit. We also show that the fixed points of the limiting infinite-width dynamics are globally optimal, under some assumptions on the initialization of the weights. Our results establish optimality for feature-learning with wide RNNs in the mean-field regime

stat.ML↗