arXiv subjects
Publications and source records attributed to Rahul Garg.
In this article, we define classical and non-classical Besov and Triebel--Lizorkin spaces associated with the Grushin operator with or without drift. We establish various characterisations of these spaces, and study their complex interpolation, embedding properties, and fractional Leibniz rules.
We define Sobolev spaces associated with the Grushin operator with or without drift. We study some of their norm equivalences and establish embedding and algebra properties. We also prove the boundedness of the Riesz transforms on these spaces.
We introduce MathConstraint, a hard, adaptive benchmark for evaluating the combinatorial reasoning capabilities of LLMs. We combine constraint satisfaction problems with rigorous solver-based verification and design an adaptive generator to create instances that remain challenging as the LLMs improve in their reasoning capabilities. Unlike existing benchmarks that quickly saturate on fixed datasets or use LLM-as-a-judge for checking solutions,MathConstraint uses parameterized problem types that enable scalable generation of arbitrarily difficult and automatically verifiable instances. We release MathConstraint-Easy ($266$ instances), on which frontier models achieve between $72.6\%$ (gemini-3.1-flash-lite) and $87.6\%$ (gpt-5.5) accuracy, and MathConstraint ($329$ instances) on which the same models drop to between $18.5\%$ (claude-4.6-sonnet) and $66.9\%$ (gpt-5.5) accuracy, demonstrating the resilience of our benchmark generator against rapid progress in LLM reasoning capabilities. We evaluate 12 frontier and open-weight models with and without access to a sandboxed Python environment that includes generic SAT/SMT solvers. Tool access roughly doubles frontier accuracy on MathConstraint (mean $+28$pp; up to $+52$pp for claude-4.6-sonnet). Further, halving the tool-call budget from $8$ to $4$ rounds erases up to $37$ points -- a sensitivity that most single-budget benchmarks miss. We release the generator, dataset, and evaluation harness as a robust environment for studying combinatorial reasoning and tool-use behavior under adversarially-tunable difficulty.
We consider the Grushin operator with drift which is symmetric with respect to a measure having exponential growth. For the corresponding Riesz transforms, we study strong-type $(p, p)$, $1 < p < \infty$, and weak-type $(1, 1)$ boundedness.
We construct operator analogues of Hermite functions which form an orthonormal basis for the Hilbert space $ \mathcal{S}_2$ of Hilbert-Schmidt operators on $ L^2(\R^n).$ We use this orthonormal basis to define Fourier transform on $ \mathcal{S}_2 $ and study some of its basic properties.
We investigate whether video generative models can exhibit visuospatial intelligence, a capability central to human cognition, using only visual data. To this end, we present Video4Spatial, a framework showing that video diffusion models conditioned solely on video-based scene context can perform complex spatial tasks. We validate on two tasks: scene navigation - following camera-pose instructions while remaining consistent with 3D geometry of the scene, and object grounding - which requires semantic localization, instruction following, and planning. Both tasks use video-only inputs, without auxiliary modalities such as depth or poses. With simple yet effective design choices in the framework and data curation, Video4Spatial demonstrates strong spatial understanding from video context: it plans navigation and grounds target objects end-to-end, follows camera-pose instructions while maintaining spatial consistency, and generalizes to long contexts and out-of-domain environments. Taken together, these results advance video generative models toward general visuospatial reasoning.
We introduce a framework that enables both multi-view character consistency and 3D camera control in video diffusion models through a novel customization data pipeline. We train the character consistency component with recorded volumetric capture performances re-rendered with diverse camera trajectories via 4D Gaussian Splatting (4DGS), lighting variability obtained with a video relighting model. We fine-tune state-of-the-art open-source video diffusion models on this data to provide strong multi-view identity preservation, precise camera control, and lighting adaptability. Our framework also supports core capabilities for virtual production, including multi-subject generation using two approaches: joint training and noise blending, the latter enabling efficient composition of independently customized models at inference time; it also achieves scene and real-life video customization as well as control over motion and spatial layout during customization. Extensive experiments show improved video quality, higher personalization accuracy, and enhanced camera control and lighting adaptability, advancing the integration of video generation into virtual production. Our project page is available at: https://eyeline-labs.github.io/Virtually-Being.
Toxicity identification in online multimodal environments remains a challenging task due to the complexity of contextual connections across modalities (e.g., textual and visual). In this paper, we propose a novel framework that integrates Knowledge Distillation (KD) from Large Visual Language Models (LVLMs) and knowledge infusion to enhance the performance of toxicity detection in hateful memes. Our approach extracts sub-knowledge graphs from ConceptNet, a large-scale commonsense Knowledge Graph (KG) to be infused within a compact VLM framework. The relational context between toxic phrases in captions and memes, as well as visual concepts in memes enhance the model's reasoning capabilities. Experimental results from our study on two hate speech benchmark datasets demonstrate superior performance over the state-of-the-art baselines across AU-ROC, F1, and Recall with improvements of 1.1%, 7%, and 35%, respectively. Given the contextual complexity of the toxicity detection task, our approach showcases the significance of learning from both explicit (i.e. KG) as well as implicit (i.e. LVLMs) contextual cues incorporated through a hybrid neurosymbolic approach. This is crucial for real-world applications where accurate and scalable recognition of toxic content is critical for creating safer online environments.
We consider the Riesz transforms of arbitrary order associated with the twisted Laplacian with drift on $\mathbb{C}^n$ and study their strong-type $(p, p)$, $1<p<\infty$, and weak-type $(1, 1)$ boundedness.
The Facial Action Coding System (FACS) for studying facial expressions is manual and requires significant effort and expertise. This paper explores the use of automated techniques to generate Action Units (AUs) for studying facial expressions. We propose an unsupervised approach based on Principal Component Analysis (PCA) and facial keypoint tracking to generate data-driven AUs called PCA AUs using the publicly available DISFA dataset. The PCA AUs comply with the direction of facial muscle movements and are capable of explaining over 92.83 percent of the variance in other public test datasets (BP4D-Spontaneous and CK+), indicating their capability to generalize facial expressions. The PCA AUs are also comparable to a keypoint-based equivalence of FACS AUs in terms of variance explained on the test datasets. In conclusion, our research demonstrates the potential of automated techniques to be an alternative to manual FACS labeling which could lead to efficient real-time analysis of facial expressions in psychology and related fields. To promote further research, we have made code repository publicly available.
The development of existing facial coding systems, such as the Facial Action Coding System (FACS), relied on manual examination of facial expression videos for defining Action Units (AUs). To overcome the labor-intensive nature of this process, we propose the unsupervised learning of an automated facial coding system by leveraging computer-vision-based facial keypoint tracking. In this novel facial coding system called the Data-driven Facial Expression Coding System (DFECS), the AUs are estimated by applying dimensionality reduction to facial keypoint movements from a neutral frame through a proposed Full Face Model (FFM). FFM employs a two-level decomposition using advanced dimensionality reduction techniques such as dictionary learning (DL) and non-negative matrix factorization (NMF). These techniques enhance the interpretability of AUs by introducing constraints such as sparsity and positivity to the encoding matrix. Results show that DFECS AUs estimated from the DISFA dataset can account for an average variance of up to 91.29 percent in test datasets (CK+ and BP4D-Spontaneous) and also surpass the variance explained by keypoint-based equivalents of FACS AUs in these datasets. Additionally, 87.5 percent of DFECS AUs are interpretable, i.e., align with the direction of facial muscle movements. In summary, advancements in automated facial coding systems can accelerate facial expression analysis across diverse fields such as security, healthcare, and entertainment. These advancements offer numerous benefits, including enhanced detection of abnormal behavior, improved pain analysis in healthcare settings, and enriched emotion-driven interactions. To facilitate further research, the code repository of DFECS has been made publicly accessible.
This is a continuation of our work [BBGG23, BBGG22] where we have initiated the study of sparse domination and quantitative weighted estimates for Grushin pseudo-multipliers. In this article, we further extend this analysis to study analogous estimates for a family of operator-valued Fourier pseudo-multipliers associated to Grushin operators $G = - \Delta_{x^{\prime}} - |x^{\prime}|^2 \Delta_{x^{\prime \prime}}$ on $\mathbb{R}^{n_1+n_2}.$
On the twisted Fock spaces $ \mathcal{F}^\lambda(\mathbb{C}^{2n}) $ we consider two types of convolution operators $ S_\varphi $ and $ \widetilde{S}_\varphi $ associated to an element $ \varphi \in \mathcal{F}^\lambda(\mathbb{C}^{2n}) .$ We find a necessary and sufficient condition on $ \varphi $ so that $ S_\varphi $ (resp. $ \widetilde{S}_\varphi $ ) is bounded on $ \mathcal{F}^\lambda(\mathbb{C}^{2n}).$ We show that for any given non constant $ \varphi $ at least one of these two operators is unbounded.
Most video restoration networks are slow, have high computational load, and can't be used for real-time video enhancement. In this work, we design an efficient and fast framework to perform real-time video enhancement for practical use-cases like live video calls and video streams. Our proposed method, called Recurrent Bottleneck Mixer Network (ReBotNet), employs a dual-branch framework. The first branch learns spatio-temporal features by tokenizing the input frames along the spatial and temporal dimensions using a ConvNext-based encoder and processing these abstract tokens using a bottleneck mixer. To further improve temporal consistency, the second branch employs a mixer directly on tokens extracted from individual frames. A common decoder then merges the features form the two branches to predict the enhanced frame. In addition, we propose a recurrent training approach where the last frame's prediction is leveraged to efficiently enhance the current frame while improving temporal consistency. To evaluate our method, we curate two new datasets that emulate real-world video call and streaming scenarios, and show extensive results on multiple datasets where ReBotNet outperforms existing approaches with lower computations, reduced memory requirements, and faster inference time.
Yoga is a globally acclaimed and widely recommended practice for a healthy living. Maintaining correct posture while performing a Yogasana is of utmost importance. In this work, we employ transfer learning from Human Pose Estimation models for extracting 136 key-points spread all over the body to train a Random Forest classifier which is used for estimation of the Yogasanas. The results are evaluated on an in-house collected extensive yoga video database of 51 subjects recorded from 4 different camera angles. We propose a 3 step scheme for evaluating the generalizability of a Yoga classifier by testing it on 1) unseen frames, 2) unseen subjects, and 3) unseen camera angles. We argue that for most of the applications, validation accuracies on unseen subjects and unseen camera angles would be most important. We empirically analyze over three public datasets, the advantage of transfer learning and the possibilities of target leakage. We further demonstrate that the classification accuracies critically depend on the cross validation method employed and can often be misleading. To promote further research, we have made key-points dataset and code publicly available.
Motivated by the recent works [Huan Yu, Quansen Jiu, and Dongsheng Li, 2021] and [Yanping Chen and Zihua Guo, 2021], we study the following extension of Calder\'on-Zygmund type singular integrals $$ T_{\beta}f (x) = p.v. \int_{\mathbb{R}^n} \frac{\Omega(y)}{|y|^{n-\beta}} f(x-y) \, dy, $$ for $0 < \beta < n$, and their commutators. We establish estimates of these singular integrals on Lipschitz spaces, Hardy spaces and Muckenhoupt $A_p$-weighted $L^p$-spaces. We also establish Lebesgue and Hardy space estimates of their commutators. Our estimates are uniform in small $\beta$, and therefore one can pass onto the limits as $\beta \to 0$ to deduce analogous estimates for the classical Calder\'on-Zygmund type singular integrals and their commutators.
In this article, we establish pointwise sparse domination results for Grushin pseudo-multipliers corresponding to various symbol classes, as a continuation of our investigation initiated in [BBGG21]. As a consequence, we deduce quantitative weighted estimates for these pseudo-multipliers.
In this article, we prove sharp quantitative weighted $L^p$-estimates for Grushin pseudo-multipliers satisfying H\"ormander's condition as an application of pointwise domination of Grushin pseudo-multipliers by appropriate sparse operators.