arXiv subjects

Sayan Mukherjee

Publications and source records attributed to Sayan Mukherjee.

At least 19 recordsLinked to original sources

arXiv2026

Uniform Consistency of Generalized Fréchet Means

Loss-based notions of centre on nonlinear spaces range from the Fréchet 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 Fréchet 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 $ϕ$-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 $ϕ$-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↗

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 $ε$- and $η$-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↗

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

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 $λ_2(G)\cdot 2n/9$ and $λ_n(G)\cdot n/4$, letting $0=λ_1(G)\le \cdots \le λ_n(G)$ be the Laplacian eigenvalues of $G$, and there is an embedding for which the congestion is at most $λ_n(G)\cdot 2n/9$. Beyond these general bounds, we determine the congestion exactly for hypercubes and lattice graphs, and obtain asymptotically tight bounds for random regular graphs and Erdős-Rényi 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↗

arXiv2026

At the Intersection of Deep Sequential Model Framework and State-space Model Framework: Study on Option Pricing

Inference and forecast problems of the nonlinear dynamical system have arisen in a variety of contexts. Reservoir computing and deep sequential models, on the one hand, have demonstrated efficient, robust, and superior performance in modeling simple and chaotic dynamical systems. However, their innate deterministic feature has partially detracted their robustness to noisy system, and their inability to offer uncertainty measurement has also been an insufficiency of the framework. On the other hand, the traditional state-space model framework is robust to noise. It also carries measured uncertainty, forming a just-right complement to the reservoir computing and deep sequential model framework. We propose the unscented reservoir smoother, a model that unifies both deep sequential and state-space models to achieve both frameworks' superiorities. Evaluated in the option pricing setting on top of noisy datasets, URS strikes highly competitive forecasting accuracy, especially those of longer-term, and uncertainty measurement. Further extensions and implications on URS are also discussed to generalize a full integration of both frameworks.

stat.ML↗

arXiv2025

Minimum $Φ$-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 $Φ$-distance estimators. We establish a general theory for the minimum $Φ$-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 $Φ$. 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↗

arXiv2025

Finite Admixture Models: a Bridge with Stochastic Geometry and Choquet Theory

Given a finite admixture model whose components and weights are unknown, let the number of identifiable components be a function of the amount of data sampled from a known distribution on the unit simplex. We use techniques from stochastic convex geometry to find the growth rate of its expected value. In addition, when the components are known but the weights are not, we provide an application of the classic Glivenko-Cantelli's theorem that allows us to retrieve the Choquet measure supported on the identifiable admixture components. In turn, this gives us the identifiable admixture weights. Finally, we propose a novel algorithm that estimates the model capturing the complexity of the data using only the strictly necessary number of components.

math.ST↗

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

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↗

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 $Ω(\log n)$, 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 $\tildeΩ(\sqrt{n})$. 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↗

arXiv2025

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 $ω(\log n/n)$. This indicates that the best privacy budget obtainable for general graphs is $Θ(\log n)$.

cs.CR↗

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

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

Scale Reliant Inference

Many scientific fields, including human gut microbiome science, collect multivariate count data where the sum of the counts is unrelated to the scale of the underlying system being measured (e.g., total microbial load in a subject's colon). This disconnect complicates downstream analyses such as differential analysis in case-control studies. This article is motivated by a novel study of in vitro human gut microbiome models. Popular tools for analyzing these data led to dramatically elevated rates of both false positives and false negatives. To understand those failures, we provide a formal problem statement that frames these challenges of scale in terms of the classical theory of identifiability. We call this the problem of Scale Reliant Inference (SRI). We use this formulation to prove fundamental limits on SRI in terms of criteria such as consistency and type-I error control. We show that the failures of existing methods stem from a fundamental failure to properly quantify uncertainty in the system scale. We demonstrate that a particular type of Bayesian model called a Bayesian Partially Identified Model (PIMs) can correctly quantify uncertainty in SRI. We introduce Scale Simulation Random Variables (SSRVs) as a flexible and efficient approach to specifying and inferring Bayesian PIMs. In the context of both real and simulated data, we find SSRVs drastically decrease type-I and type-II error rates.

stat.ME↗

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

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↗

arXiv2024

Extremal Numbers of Hypergraph Suspensions of Even Cycles

For fixed $k\ge 2$, determining the order of magnitude of the number of edges in an $n$-vertex bipartite graph not containing $C_{2k}$, the cycle of length $2k$, is a long-standing open problem. We consider an extension of this problem to triple systems. In particular, we prove that the maximum number of triples in an $n$-vertex triple system which does not contain a $C_6$ in the link of any vertex, has order of magnitude $n^{7/3}$. Additionally, we construct new families of dense $C_6$-free bipartite graphs with $n$ vertices and $n^{4/3}$ edges in order of magnitude.

math.CO↗

arXiv2024

The Shadow knows: Empirical Distributions of Minimum Spanning Acycles and Persistence Diagrams of Random Complexes

In 1985, Frieze showed that the expected sum of the edge weights of the minimum spanning tree (MST) in the uniformly weighted graph converges to $ζ(3)$. Recently, Hino and Kanazawa extended this result to a uniformly weighted simplicial complex, where the role of the MST is played by its higher-dimensional analog -- the Minimum Spanning Acycle (MSA). Our work goes beyond and describes the histogram of all the weights in this random MST and random MSA. Specifically, we show that their empirical distributions converge to a measure based on a concept called the shadow. The shadow of a graph is the set of all the missing transitive edges, and, for a simplicial complex, it is a related topological generalization. As a corollary, we obtain a similar claim for the death times in the persistence diagram corresponding to the above-weighted complex, a result of interest in applied topology.

math.PR↗