arXiv subjects

Somnath Hazra

Publications and source records attributed to Somnath Hazra.

14 recordsLinked to original sources

arXiv2026

Möb Homogeneous Analytic Hilbert Modules over the Bidisc

An analytic Hilbert module $\mathcal{H}$ over the polynomial ring, consisting of holomorphic functions over the bidisc, is said to be Möb-homogeneous if the corresponding pair of multiplication operators is homogeneous with respect to the diagonal action of the group $\{(φ,φ): φ\in \mbox{Möb}\} \cong \mbox{Möb}$. In this article, we construct three families of mutually unitarily inequivalent Möb-homogeneous analytic Hilbert modules, distinct from the family of weighted Bergman modules over the bidisc. We further show that none of the reproducing kernels in one of these families induces a Kähler--Einstein metric on the bidisc.

math.FA↗

arXiv2026

Complete Nevanlinna-Pick property of $\mathbb K$-Invariant Reproducing Kernels

Let $Ω$ be a Cartan domain and $K = \sum_{\underline s}a_{\underline s}K_{\underline s}$ be a $\mathbb K$-invariant kernel on $Ω$. In this article, we first obtain a necessary condition on $K$ to have the complete Nevanlinna-Pick property in terms of the sequence $\{a_{\underline s}\}_{\underline s}$ with the assumption that each $a_{\underline s}$ is non-zero and $K$ is non-vanishing. This generalizes the well-known Kaluza's Lemma in the context of $\mathbb K$-invariant kernels. The notion of the characteristic function of the classical Sz.-Nagy--Foias Theory is extended to a commuting tuple of $\frac{1}{K}$-contraction where $K$ is an irreducible $\mathbb K$-invariant kernel. An explicit construction of the characteristic function of a $\frac{1}{K}$-contraction is provided. A characterization of a $\mathbb K$-invariant kernel with the complete Nevanlinna-Pick property is obtained via the existence of characteristic functions associated with $\frac{1}{K}$-contractions.

math.FA↗

arXiv2026

AD$^2$: Analysis and Detection of Adversarial Threats in Visual Perception for End-to-End Autonomous Driving Systems

End-to-end autonomous driving systems have achieved significant progress, yet their adversarial robustness remains largely underexplored. In this work, we conduct a closed-loop evaluation of state-of-the-art autonomous driving agents under black-box adversarial threat models in CARLA. Specifically, we consider three representative attack vectors on the visual perception pipeline: (i) a physics-based blur attack induced by acoustic waves, (ii) an electromagnetic interference attack that distorts captured images, and (iii) a digital attack that adds ghost objects as carefully crafted bounded perturbations on images. Our experiments on two advanced agents, Transfuser and Interfuser, reveal severe vulnerabilities to such attacks, with driving scores dropping by up to 99% in the worst case, raising valid safety concerns. To help mitigate such threats, we further propose a lightweight Attack Detection model for Autonomous Driving systems (AD$^2$) based on attention mechanisms that capture spatial-temporal consistency. Comprehensive experiments across multi-camera inputs on CARLA show that our detector achieves superior detection capability and computational efficiency compared to existing approaches.

cs.CV↗

arXiv2025

Incentivizing Safer Actions in Policy Optimization for Constrained Reinforcement Learning

Constrained Reinforcement Learning (RL) aims to maximize the return while adhering to predefined constraint limits, which represent domain-specific safety requirements. In continuous control settings, where learning agents govern system actions, balancing the trade-off between reward maximization and constraint satisfaction remains a significant challenge. Policy optimization methods often exhibit instability near constraint boundaries, resulting in suboptimal training performance. To address this issue, we introduce a novel approach that integrates an adaptive incentive mechanism in addition to the reward structure to stay within the constraint bound before approaching the constraint boundary. Building on this insight, we propose Incrementally Penalized Proximal Policy Optimization (IP3O), a practical algorithm that enforces a progressively increasing penalty to stabilize training dynamics. Through empirical evaluation on benchmark environments, we demonstrate the efficacy of IP3O compared to the performance of state-of-the-art Safe RL algorithms. Furthermore, we provide theoretical guarantees by deriving a bound on the worst-case error of the optimality achieved by our algorithm.

cs.LG↗

arXiv2025

Homogeneous analytic Hilbert modules -- the case of non-transitive action

This work investigates analytic Hilbert modules $\mathcal{H}$, over the polynomial ring, consisting of holomorphic functions on a $G$-space $Ω\subset \mathbb{C}^m$ that are homogeneous under the natural action of the group $G$. In a departure from the past studies of such questions, here we don't assume transitivity of the group action. The primary finding reveals that unitary invariants such as curvature and the reproducing kernel of a homogeneous analytic Hilbert module can be deduced from their values on a fundamental set $Λ$ of the group action. Next, utilizing these techniques, we examine the analytic Hilbert modules associated with the symmetrized bi-disc $\mathbb{G}_2$ and its homogeneity under the automorphism group of $\mathbb{G}_2$. It follows from one of our main theorems that none of the weighted Bergman metrics on the symmetrized bi-disc is Kähler-Einstein.

math.FA↗

arXiv2025

Tackling Uncertainties in Multi-Agent Reinforcement Learning through Integration of Agent Termination Dynamics

Multi-Agent Reinforcement Learning (MARL) has gained significant traction for solving complex real-world tasks, but the inherent stochasticity and uncertainty in these environments pose substantial challenges to efficient and robust policy learning. While Distributional Reinforcement Learning has been successfully applied in single-agent settings to address risk and uncertainty, its application in MARL is substantially limited. In this work, we propose a novel approach that integrates distributional learning with a safety-focused loss function to improve convergence in cooperative MARL tasks. Specifically, we introduce a Barrier Function based loss that leverages safety metrics, identified from inherent faults in the system, into the policy learning process. This additional loss term helps mitigate risks and encourages safer exploration during the early stages of training. We evaluate our method in the StarCraft II micromanagement benchmark, where our approach demonstrates improved convergence and outperforms state-of-the-art baselines in terms of both safety and task completion. Our results suggest that incorporating safety considerations can significantly enhance learning performance in complex, multi-agent environments.

cs.LG↗

arXiv2024

Representations of the Möbius group and pairs of homogeneous operators in the Cowen-Douglas class

Let Möb be the biholomorphic automorphism group of the unit disc of the complex plane, $\mathcal{H}$ be a complex separable Hilbert space and $\mathcal{U}(\mathcal{H})$ be the group of all unitary operators. Suppose $\mathcal{H}$ is a reproducing kernel Hilbert space consisting of holomorphic functions over the poly-disc $\mathbb D^n$ and contains all the polynomials. If $π: \mbox{Möb} \to \mathcal{U}(\mathcal{H})$ is a multiplier representation, then we prove that there exist $λ_1, λ_2, \ldots, λ_n > 0$ such that $π$ is unitarily equivalent to $(\otimes_{i=1}^{n} D_{λ_i}^+)|_{\mbox{Möb}}$, where each $D_{λ_i}^+$ is a holomorphic discrete series representation of Möb. As an application, we prove that if $(T_1, T_2)$ is a Möb - homogeneous pair in the Cowen - Douglas class of rank $1$ over the bi-disc, then each $T_i$ posses an upper triangular form with respect to a decomposition of the Hilbert space. In this upper triangular form of each $T_i$, the diagonal operators are identified. We also prove that if $\mathcal{H}$ consists of symmetric (resp. anti-symmetric) holomorphic functions over $\mathbb D^2$ and contains all the symmetric (resp. anti-symmetric) polynomials, then there exists $λ> 0$ such that $π\cong \oplus_{m = 0}^\infty D^+_{λ+ 4m}$ (resp. $π\cong \oplus_{m=0}^\infty D^+_{λ+ 4m + 2}$).

math.FA↗

arXiv2023

A Family of Homogeneous Operators In The Cowen-Douglas Class Over The Poly-disc

We construct a large family of positive-definite kernels $K: \mathbb{D}^n\times \mathbb{D}^n \to \mbox{M} (r, \mathbb C)$, holomorphic in the first variable and anti-holomorphic in the second, that are quasi-invariant with respect to the subgroup $\mbox{Möb} \times\cdots\times \mbox{Möb}$ ($n$ times) of the bi-holomorphic automorphism group of $\mathbb{D}^n$. The adjoint of the $n$ - tuples of multiplication operators by the co-ordinate functions on the Hilbert spaces $\mathcal H_K$ determined by $K$ is then homogeneous with respect to this subgroup. We show that these $n$ - tuples are irreducible, are in the Cowen-Douglas class $\mathrm B_r(\mathbb D^n)$ and that they are mutually pairwise unitarily inequivalent.

math.FA↗

arXiv2022

Penalizing Proposals using Classifiers for Semi-Supervised Object Detection

Obtaining gold standard annotated data for object detection is often costly, involving human-level effort. Semi-supervised object detection algorithms solve the problem with a small amount of gold-standard labels and a large unlabelled dataset used to generate silver-standard labels. But training on the silver standard labels does not produce good results, because they are machine-generated annotations. In this work, we design a modified loss function to train on large silver standard annotated sets generated by a weak annotator. We include a confidence metric associated with the annotation as an additional term in the loss function, signifying the quality of the annotation. We test the effectiveness of our approach on various test sets and use numerous variations to compare the results with some of the current approaches to object detection. In comparison with the baseline where no confidence metric is used, we achieved a 4% gain in mAP with 25% labeled data and 10% gain in mAP with 50% labeled data by using the proposed confidence metric.

cs.CV↗

arXiv2022

On analytic structure of weighted shifts on generalized directed semi-trees

Inspired by natural classes of examples, we define generalized directed semi-tree and construct weighted shifts on the generalized directed semi-trees. Given an $n$-tuple of directed directed semi-trees with certain properties, we associate an $n$-tuple of multiplication operators on a Hilbert space $\mathscr{H}^2(β)$ of formal power series. Under certain conditions, $\mathscr{H}^2(β)$ turns out to be a reproducing kernel Hilbert space consisting of holomorphic functions on some domain in $\mathbb C^n$ and the $n$-tuple of multiplication operators on $\mathscr{H}^2(β)$ is unitarily equivalent to an $n$-tuple of weighted shifts on the generalized directed semi-trees. Finally, we exhibit two classes of examples of $n$-tuple of operators which can be intrinsically identified as weighted shifts on generalized directed semi-trees.

math.FA↗

arXiv2022

Homogeneous Hermitian Holomorphic Vector Bundles And Operators In The Cowen-Douglas Class Over The Poly-disc

In this article, we obtain two sets of results. The first set of complete results are exclusively for the case of the bi-disc while the second set of results describe in part, which of these carry over to the general case of the poly-disc: * A classification of irreducible hermitian holomorphic vector bundles over $\mathbb{D}^2$, homogeneous with respect to $\mbox{Möb}\times \mbox{Möb}$, is obtained assuming that the associated representations are \textit{multiplicity-free}. Among these the ones that give rise to an operator in the Cowen-Douglas class of $\mathbb{D}^2$ of rank $1,2$ or $3$ is determined. * Any hermitian holomorphic vector bundle of rank $2$ over $\mathbb{D}^n$, homogeneous with respect to the $n$-fold product of the group $\mbox{Möb}$ is shown to be a tensor product of $n-1$ hermitian holomorphic line bundles, each of which is homogeneous with respect to $\mbox{Möb}$ and a hermitian holomorphic vector bundle of rank $2$, homogeneous with respect to $\mbox{Möb}$. * The classification of irreducible homogeneous hermitian holomorphic vector buldles over $\mathbb{D}^2$ of rank $3$ (as well as the corresponding Cowen-Douglas class of operators) is extended to the case of $\mathbb{D}^n$, $n>2$. * It is shown that there is no irreducible $n$ - tuple of operators in the Cowen-Douglas class $\mathrm B_2(\mathbb{D}^n)$ that is homogeneous with respect $\mbox{Aut}(\mathbb{D}^n)$, $n >1$. Also, pairs of operators in $\mathrm B_3(\mathbb{D}^2)$ homogeneous with respect to $\mbox{Aut}(\mathbb{D}^2)$ are produced, while it is shown that no $n$ - tuple of operators in $\mathrm B_3(\mathbb{D}^n)$ is homogeneous with respect to $\mbox{Aut}(\mathbb{D}^n)$, $n > 2$.

math.FA↗

arXiv2020

Semi-Lexical Languages -- A Formal Basis for Unifying Machine Learning and Symbolic Reasoning in Computer Vision

Human vision is able to compensate imperfections in sensory inputs from the real world by reasoning based on prior knowledge about the world. Machine learning has had a significant impact on computer vision due to its inherent ability in handling imprecision, but the absence of a reasoning framework based on domain knowledge limits its ability to interpret complex scenarios. We propose semi-lexical languages as a formal basis for dealing with imperfect tokens provided by the real world. The power of machine learning is used to map the imperfect tokens into the alphabet of the language and symbolic reasoning is used to determine the membership of input in the language. Semi-lexical languages also have bindings that prevent the variations in which a semi-lexical token is interpreted in different parts of the input, thereby leaning on deduction to enhance the quality of recognition of individual tokens. We present case studies that demonstrate the advantage of using such a framework over pure machine learning and pure symbolic methods.

cs.AI↗

arXiv2019

A product formula for homogeneous characteristic functions

A bounded linear operator $T$ on a Hilbert space is said to be homogeneous if $φ(T)$ is unitarily equivalent to $T$ for all $φ$ in the group Möb of bi-holomorphic automorphisms of the unit disc. A projective unitary representation $σ$ of Möb is said to be associated with an operator T if $φ(T)= σ(φ)^\star T σ(φ)$ for all $φ$ in Möb. In this paper, we develop a Möbius equivariant version of the Sz.-Nagy--Foias model theory for completely non-unitary (cnu) contractions. As an application, we prove that if T is a cnu contraction with associated (projective unitary) representation $σ$, then there is a unique projective unitary representation $\hatσ$, extending $σ$, associated with the minimal unitary dilation of $T$. The representation $\hatσ$ is given in terms of $σ$ by the formula $$ \hatσ = (π\otimes D_1^+) \oplus σ\oplus (π_\star \otimes D_1^-), $$ where $D_1^\pm$ are the two Discrete series representations (one holomorphic and the other anti-holomorphic) living on the Hardy space $H^2(\mathbb D)$, and $π, π_\star$ are representations of Möb living on the two defect spaces of $T$ defined explicitly in terms of $σ$. Moreover, a cnu contraction $T$ has an associated representation if and only if its Sz.-Nagy--Foias characteristic function $θ_T$ has the product form $θ_T(z) = π_\star(φ_z)^* θ_T(0) π(φ_z),$ $z\in \mathbb D$, where $φ_z$ is the involution in Möb mapping $z$ to $0.$ We obtain a concrete realization of this product formula %the two representations $π_\star$ and $π$ for a large subclass of homogeneous cnu contractions from the Cowen-Douglas class.

math.FA↗

arXiv2018

Homogeneous $2$-shifts

The classification of homogeneous scalar weighted shifts is known. Recently, Korányi obtained a large class of inequivalent irreducible homogeneous bi-lateral $2$-by-$2$ block shifts. In this paper, we construct two distinct classes of examples not in the list of Korányi. It is then shown that these new examples of irreducible homogeneous bi-lateral $2$-by-$2$ block shifts, together with the ones found earlier by Korányi, account for every unitarily inequivalent irreducible homogeneous bi-lateral $2$-by-$2$ block shift.

math.FA↗