arXiv subjects

Somjit Roy

Publications and source records attributed to Somjit Roy.

5 recordsLinked to original sources

arXiv2026

VaSST: Variational Inference for Symbolic Regression using Soft Symbolic Trees

Symbolic regression (SR) has gained recent traction in AI-driven scientific discovery for learning closed-form physical laws. Yet existing methods are dominated by heuristic search or data-intensive approaches that often assume low-noise regimes and lack principled uncertainty quantification, while fully probabilistic SR formulations remain scarce. We introduce a scalable probabilistic framework for SR, VaSST, based on variational inference. VaSST uses soft symbolic trees, a continuous relaxation of symbolic expression trees in which discrete operator and feature assignments are replaced by probability distributions over allowable components. This transforms combinatorial symbolic search through an astronomically large expression space into efficient gradient-based optimization while preserving a coherent probabilistic interpretation. The learned soft representations induce posterior distributions over symbolic structures, enabling uncertainty quantification across plausible symbolic forms through posterior-aware symbolic model selection. On simulated experiments and the Feynman Symbolic Regression Database, VaSST achieves strong structural recovery and predictive accuracy compared to state-of-the-art competing SR methods.

stat.ME↗

arXiv2026

Frequentist Regret Analysis of Gaussian Process Thompson Sampling via Fractional Posteriors

We study Gaussian Process Thompson Sampling (GP-TS) for sequential decision-making over compact, continuous action spaces and provide a frequentist regret analysis based on fractional Gaussian process posteriors, without relying on domain discretization as in prior work. We show that the variance inflation commonly assumed in existing analyses of GP-TS can be interpreted as Thompson Sampling with respect to a fractional posterior with tempering parameter $\alpha \in (0,1)$. We derive a kernel-agnostic regret bound expressed in terms of the information gain parameter $\gamma_t$ and the posterior contraction rate $\epsilon_t$, and identify conditions on the Gaussian process prior under which $\epsilon_t$ can be controlled. As special cases of our general bound, we recover regret of order $\tilde{\mathcal{O}}(T^{\frac{1}{2}})$ for the squared exponential kernel, $\tilde{\mathcal{O}}(T^{\frac{2\nu+3d}{2(2\nu+d)}} )$ for the Mat\'ern-$\nu$ kernel, and a bound of order $\tilde{\mathcal{O}}(T^{\frac{2\nu+3d}{2(2\nu+d)}})$ for the rational quadratic kernel. Overall, our analysis provides a unified and discretization-free regret framework for GP-TS that applies broadly across kernel classes.

math.ST↗

arXiv2025

Probabilistic Symbolic Regression for Equation Discovery via Operator-induced and Regularized Symbolic Forests

Symbolic regression has emerged as a powerful tool for artificial intelligence-driven scientific discovery by learning interpretable analytical expressions that reveal governing relationships directly from data. Existing methods, however, often rely on heuristic search, struggle to balance predictive accuracy with expression complexity in noisy settings, and offer limited characterization of symbolic uncertainty. Probabilistic approaches that address these challenges in a unified manner remain underexplored. We introduce a probabilistic symbolic regression framework that represents mathematical expressions as ensembles of symbolic trees. A regularizing prior over tree topology controls expression complexity, while an Occam's window-based posterior summary captures uncertainty across multiple plausible symbolic models. Given the limited existing theoretical treatment of symbolic regression, we develop posterior concentration guarantees when symbolic expressions approximate the underlying relationship arbitrarily well, with a near-parametric rate when an exact finite formula exists. Additionally, we establish a sharp oracle concentration result under symbolic misspecification. Comparisons of our proposed framework with state-of-the-art competitors demonstrate superior predictive accuracy, optimal symbolic complexity, and stable structural recovery when learning benchmark scientific equations, together with the identification of scientifically interpretable descriptor formulas in a challenging materials discovery application.

stat.ME↗

arXiv2025

A Generalized Tangent Approximation based Variational Inference Framework for Strongly Super-Gaussian Likelihoods

Variational inference, as an alternative to Markov chain Monte Carlo sampling, has played a transformative role in enabling scalable computation for complex Bayesian models. Nevertheless, existing approaches often depend on either rigid model-specific formulations or stochastic black-box optimization routines. Tangent approximation is a principled class of structured variational methods that exploits the geometry of the underlying probability model. However, its utility has largely been confined to logistic regression and related modeling regimes. In this article, we propose a novel variational framework based on tangent transformation for a broad class of probability models characterized by strongly super-Gaussian likelihoods. Our method leverages convex duality to construct tangent minorants of the log-likelihood, thereby inducing conjugacy with Gaussian priors over model parameters in an otherwise intractable setup. Under mild assumptions on the data-generating mechanism, we establish algorithmic convergence guarantees, a contribution that stands in contrast to the limited theoretical assurances typically available for black-box variational methods. Additionally, we derive near-minimax optimal bounds for the variational risk. Superior performance of our proposed methodology is illustrated on simulated and real-data scenarios that challenge state-of-the-art variational algorithms in terms of scalability and their ability to consistently capture complex underlying data structure.

stat.ME↗

arXiv2024

Almost Perfect Mutually Unbiased Bases that are Sparse

In dimension $d$, Mutually Unbiased Bases (MUBs) are a collection of orthonormal bases over $\mathbb{C}^d$ such that for any two vectors $v_1, v_2$ belonging to different bases, the scalar product $|\braket{v_1|v_2}| = \frac{1}{\sqrt{d}}$. The upper bound on the number of such bases is $d+1$. Constructions to achieve this bound are known when $d$ is some power of prime. The situation is more restrictive in other cases and also when we consider the results over real rather than complex. Thus, certain relaxations of this model are considered in literature and consequently Approximate MUBs (AMUB) are studied. This enables one to construct potentially large number of such objects for $\mathbb{C}^d$ as well as in $\mathbb{R}^d$. In this regard, we propose the concept of Almost Perfect MUBs (APMUB), where we restrict the absolute value of inner product $|\braket{v_1|v_2}|$ to be two-valued, one being 0 and the other $ \leq \frac{1+\mathcal{O}(d^{-λ})}{\sqrt{d}}$, such that $λ> 0$ and the numerator $1 + \mathcal{O}(d^{-λ}) \leq 2$. Each such vector constructed, has an important feature that large number of its components are zero and the non-zero components are of equal magnitude. Our techniques are based on combinatorial structures related to RBDs. We show that for several composite dimensions $d$, one can construct $\mathcal{O}(\sqrt{d})$ many APMUBs, in which cases the number of MUBs are significantly small. To be specific, this result works for $d$ of the form $(q-e)(q+f), \ q, e, f \in \mathbb{N}$, with the conditions $0 \leq f \leq e$ for constant $e, f$ and $q$ some power of prime. We also show that such APMUBs provide sets of Bi-angular vectors which are $\mathcal{O}(d^{\frac{3}{2}})$ in numbers, having high angular distances among them. Finally, as the MUBs are equivalent to a set of Hadamard matrices, we show that the APMUBs are so with the set of Weighing matrices.

cs.DM↗