arXiv subjects

Saikat Roy

Publications and source records attributed to Saikat Roy.

At least 19 recordsLinked to original sources

arXiv2026

Accurate representation of the continuum orbital in a Gaussian basis set: application to Auger decay rates

Many processes in chemistry and physics, such as autoionization, photoionization, or electron-impact collisions, involve unbound electrons. Despite their importance, theoretical methods for modeling continuum electrons remain significantly less developed than those for purely bound electronic states. Here we present a novel approach for treating continuum electrons within Gaussian-based quantum chemistry. The method represents continuum orbital in a discrete basis of standard Gaussian-type orbitals (GTOs), where the continuum wave function is obtained by solving the Lippmann-Schwinger equation with proper boundary conditions using an analytical representation of the Green's function in an auxiliary GTO basis set. The resulting continuum orbital is multicentered and multichannel, providing a rigorous description of scattering states. As an initial application of the new methodology, we compute Auger decay rates for several atomic and molecular systems, including Ne, H$_2$O, CO$_2$, and Ar. We demonstrate a consistent improvement of the results with increasing the quality of the continuum representation. We show also that higher-order electron-correlation effects in the bound domain are essential for an accurate treatment of satellite transitions. The present formalism therefore provides a systematically improvable Gaussian-based framework for continuum-electron problems that is fully compatible with standard quantum chemistry methods.

physics.chem-ph↗

arXiv2026

Norming Approximate Orthogonality in Normed Linear Spaces

We introduce and study the notion of \emph{norming approximate orthogonality}, a two-parameter generalization of Birkhoff--James orthogonality in normed linear spaces. For $δ, \varepsilon \in [0,1)$ with $\varepsilon < (1-δ)^2$, we say $x \nperp y$ in $X$ if there exists $f \in X^*$ with $|f(x)| \geq (1-δ)\|f\|\|x\|$ and $|f(y)| \leq \frac{\varepsilon}{1-δ}\|f\|\|y\|$, simultaneously relaxing both the norming condition on $x$ and the vanishing condition on $y$. It is proved that \[ x\nperp y \iff \|x+λy\|\geq (1-δ)\|x\|-\frac{\varepsilon}{1-δ}\|λy\|~\qquad \forall ~\text{scalars}~λ. \] This framework interpolates between two notions of approximate orthogonality in normed linear spaces due to Chmieliński and Dragomir, and recovers three existing notions of orthogonality in extreme cases: exact Birkhoff--James orthogonality at $δ= \varepsilon = 0$, the approximate orthogonality of Chmieliński at $δ= 0$, and the approximate orthogonality of Dragomir at $\varepsilon = 0$. A two-parameter proximity result generalizing Chmlieński's characterization of $\perp_B^\varepsilon$ is established. The forward and converse implications are governed by the distinct thresholds $\frac{\varepsilon}{(1-δ)^2}$ and $\frac{\varepsilon}{1-δ}$, which collapse to $\varepsilon$ of Chmieliński precisely when $δ=0$, and the strictness of this gap is confirmed by counterexamples in $\ell_\infty^2$. A dual formulation of norming approximate orthogonality is established with a complete equivalence in the reflexive case. We apply our results to (vector-valued) continuous function spaces, which extends some earlier results and recovers few operator theoretical results with alternative proofs using measure theoretic techniques.

math.FA↗

arXiv2026

Inclusive Federated Learning Through Compliance-Weighted Noise Allocation in Healthcare AI

Background: Federated learning (FL) enables collaborative training of clinical AI models without centralizing patient data, but adoption is limited by privacy concerns, heterogeneous institutional compliance, and resource disparities; standard differential privacy (DP) applies uniform noise to all clients, penalizing well-compliant or under-resourced institutions. Objective: We introduce a compliance-aware FL framework that adapts DP to institutional compliance, letting lower-compliance sites participate without uniformly penalizing others. Methods: A compliance scoring tool aligned with HIPAA, GDPR, NIST, ISO, and HL7/FHIR maps each client score to a per-step Gaussian noise scale for server-side DP-SGD on a small aggregator dataset. The formal $(ε,δ)$ bound applies to the aggregator dataset under a semi-honest aggregator; client-level DP needs secure aggregation (future work). We evaluate five FL strategies on PneumoniaMNIST and BreastMNIST (16 clients, 50 rounds, five seeds); the cumulative aggregator-dataset $ε$ is about 1434 (Breast) and 513 (Pneumonia) at $δ=10^{-5}$. Results: Including 12 lower-compliance clients (Experiment 1) versus a compliant-only baseline (Experiment 4) changed BreastMNIST accuracy by +4.5 (FedAvg), +6.8 (FedMedian), +5.2 (FedProx), +1.6 (FedYogi), and -4.1 (FedAdam) percentage points (pooled +2.8 pp; not significant at n=5; up to +17 pp per configuration); compliance-weighted allocation matched uniform server-side DP at equal mean noise (+0.1 pp), carrying no utility penalty, and first-round noise cost 1.3 pp (Breast) and 2.5 pp (Pneumonia, FedAvg). Conclusions: Compliance-weighted server-side DP lets lower-compliance institutions join FL without degrading performance, giving auditable per-site noise control at no utility cost; formal guarantees apply to the aggregator dataset, with client-level DP requiring secure aggregation.

cs.LG↗

arXiv2026

Operator Geometry of Hilbert Ball Automorphisms

We consider the operator--theoretic model for the group of biholomorphic automorphisms $Aut(B)$ of the unit ball $B$ of a complex Hilbert space $\H$ by representing each automorphism as a bounded linear operator on the augmented Hilbert space $\H\oplus \mathbb{C}$. Any member of $Aut(B)$ admits a natural block operator matrix representation acting on $\mathcal{H}\oplus\mathbb{C}$. We study the geometry of the subset $M(\mathcal{H})$ of $\mathcal{B}(\mathcal{H}\oplus\mathbb{C})$ consisting of these block operator matrices. It is shown that every element corresponding to a non-rotation automorphism is a smooth point of $\mathcal{B}(\mathcal{H}\oplus\mathbb{C})$. Orthogonality between two such matrices is characterized geometrically by the antipodality of the corresponding Möbius images of a boundary point of the ball. This orthogonality characterization is applied to show that an inner automorphism of $\Aut(B)$ that preserves Birkhoff--James orthogonality in both directions if and only if it is conjugation by a pure rotation, yielding a rigidity result. The normalized block matrices are $J$-unitary under a suitable normalization, where $J = \operatorname{diag}(I_{\mathcal{H}}, -1)$. We show that norm of such block matrices satisfy a submultiplicativity under a certain composition rule other than usual operator multiplication, and induce a metric on certain subsets of $Aut(B)$ which recover the hyperbolic metric on the Hilbert ball. The symmetric structure of Birkhoff--James orthogonality within $M(\mathcal{H})$ is also studied: there are no left-symmetric points, while the only right-symmetric points are pure rotations.

math.FA↗

arXiv2026

Polyhedral norms and smooth Hahn-Banach extension

We find a necessary and sufficient condition for a smooth functional on a subspace to admit a norm-preserving smooth extension to the entire space in polyhedral norms. The characterization is geometric: such an extension exists if and only if the unique absolute norm-attaining point of the smooth functional is an extreme point of both the unit ball of the subspace and that of the ambient space. We show by example that such a result is not true in non-polyhedral norms, even under sufficiently strong hypothesis. Extremity of the norm preserving restrictions of extreme functionals are also discussed.

math.FA↗

arXiv2026

Primus: Enforcing Attention Usage for 3D Medical Image Segmentation

Transformers have achieved remarkable success across multiple fields, yet their impact on 3D medical image segmentation remains limited with convolutional networks still dominating major benchmarks. In this work, (A) we analyze current Transformer-based segmentation models and identify critical shortcomings, particularly their over-reliance on convolutional blocks. Further, we demonstrate that in some architectures, performance is unaffected by the absence of the Transformer, thereby demonstrating their limited effectiveness. To address these challenges, we move away from hybrid architectures and (B) introduce Transformer-centric segmentation architectures, termed Primus and PrimusV2. Primus leverages high-resolution tokens, combined with advances in positional embeddings and block design, to maximally leverage its Transformer blocks, while PrimusV2 expands on this through an iterative patch embedding. Through these adaptations, Primus surpasses current Transformer-based methods and competes with a default nnU-Net while PrimusV2 exceeds it and is on par with the state-of-the-art CNNs such as ResEnc-L and MedNeXt architectures across nine public datasets. In doing so, we introduce the first competitive Transformer-centric model, making Transformers state-of-the-art in 3D medical image segmentation. The code is available here: https://github.com/MIC-DKFZ/nnUNet/blob/master/documentation/primus.md.

cs.CV↗

arXiv2026

Constitutive relations for colloidal gel

The theoretical treatment of depletion gels with central interactions often involves expanding the free energy around a stress-free reference state to derive a constitutive relation between global stress and strain. The premise upon which the previous continuum theories are based, i.e., the stress-free reference state and the affine deformation, both of which do not hold in the context of amorphous gel materials. Gels never reach a true global minimum in the potential energy landscape and contain local regions of significant compressive and tensile stress, interspersed with zero-stress regions. Hence, expansion of free energy around a stressed reference state will produce scalar terms in harmonic expansion, the effects of which are qualitatively different from the terms appearing in the expansion around an unstressed reference state. In this study, we demonstrate the limitations of traditional continuum theories and propose simple constitutive relations that better capture the mechanical response of gel materials. The robustness of the proposed relations is established through large-scale numerical simulations of depletion and frictional gels across a vast parameter space.

cond-mat.soft↗

arXiv2026

A characterization of Banach spaces with numerical index one

We investigate the extremal properties of the unit ball of $L(X)_w^*$, the dual space of bounded linear operators defined on a Banach space $X$ equipped with the numerical radius norm. As an application of the present study, we obtain a geometric characterization of Banach spaces with numerical index one, which extends the well-known McGregor's characterization of finite-dimensional Banach spaces with numerical index one. We also present refinements of several earlier results in this direction, including an explicit description of the extreme points of $B_{L(X)_w^*}$, the unit ball of $L(X)_w^*$, for any finite-dimensional Banach space $X$. This allows us to obtain an independent and elementary proof of McGregor's characterization of finite-dimensional Banach spaces with numerical index one.

math.FA↗

arXiv2025

MedNeXt-v2: Scaling 3D ConvNeXts for Large-Scale Supervised Representation Learning in Medical Image Segmentation

Large-scale supervised pretraining is rapidly reshaping 3D medical image segmentation. However, existing efforts focus primarily on increasing dataset size and overlook the question of whether the backbone network is an effective representation learner at scale. In this work, we address this gap by revisiting ConvNeXt-based architectures for volumetric segmentation and introducing MedNeXt-v2, a compound-scaled 3D ConvNeXt that leverages improved micro-architecture and data scaling to deliver state-of-the-art performance. First, we show that routinely used backbones in large-scale pretraining pipelines are often suboptimal. Subsequently, we use comprehensive backbone benchmarking prior to scaling and demonstrate that stronger from scratch performance reliably predicts stronger downstream performance after pretraining. Guided by these findings, we incorporate a 3D Global Response Normalization module and use depth, width, and context scaling to improve our architecture for effective representation learning. We pretrain MedNeXt-v2 on 18k CT volumes and demonstrate state-of-the-art performance when fine-tuning across six challenging CT and MR benchmarks (144 structures), showing consistent gains over seven publicly released pretrained models. Beyond improvements, our benchmarking of these models also reveals that stronger backbones yield better results on similar data, representation scaling disproportionately benefits pathological segmentation, and that modality-specific pretraining offers negligible benefit once full finetuning is applied. In conclusion, our results establish MedNeXt-v2 as a strong backbone for large-scale supervised representation learning in 3D Medical Image Segmentation. Our code and pretrained models are made available with the official nnUNet repository at: https://www.github.com/MIC-DKFZ/nnUNet

eess.IV↗

arXiv2025

CRONOS: Continuous Time Reconstruction for 4D Medical Longitudinal Series

Forecasting how 3D medical scans evolve over time is important for disease progression, treatment planning, and developmental assessment. Yet existing models either rely on a single prior scan, fixed grid times, or target global labels, which limits voxel-level forecasting under irregular sampling. We present CRONOS, a unified framework for many-to-one prediction from multiple past scans that supports both discrete (grid-based) and continuous (real-valued) timestamps in one model, to the best of our knowledge the first to achieve continuous sequence-to-image forecasting for 3D medical data. CRONOS learns a spatio-temporal velocity field that transports context volumes toward a target volume at an arbitrary time, while operating directly in 3D voxel space. Across three public datasets spanning Cine-MRI, perfusion CT, and longitudinal MRI, CRONOS outperforms other baselines, while remaining computationally competitive. We will release code and evaluation protocols to enable reproducible, multi-dataset benchmarking of multi-context, continuous-time forecasting.

cs.CV↗

arXiv2025

Expectation-Maximization as the Engine of Scalable Medical Intelligence

Large, high-quality, annotated datasets are the foundation of medical AI research, but constructing even a small, moderate-quality, annotated dataset can take years of effort from multidisciplinary teams. Although active learning can prioritize what to annotate, scaling up still requires extensive manual efforts to revise the noisy annotations. We formulate this as a missing-data problem and develop ScaleMAI, a framework that unifies data annotation and model development co-evolution through an Expectation-Maximization (EM) process. In this iterative process, the AI model automatically identifies and corrects the mistakes in annotations (Expectation), while the refined annotated data retrain the model to improve accuracy (Maximization). In addition to the classical EM algorithm, ScaleMAI brings human experts into the loop to review annotations that cannot be adequately addressed by either Expectation or Maximization step (<5%). As a result, ScaleMAI progressively creates an annotated dataset of 47,315 CT scans (4.8x larger than the largest public dataset, PanTS) including 4,163,720 per-voxel annotations for benign/malignant tumors and 88 anatomical structures. ScaleMAI iteratively trains a model that exceeds human expert performance in tumor diagnosis (+7%), and outperforms models developed from smaller, moderate-quality datasets, with statistically significant gains in tumor detection (+10%) and segmentation (+14%) on two prestigious benchmarks.

cs.CV↗

arXiv2025

Temporal Flow Matching for Learning Spatio-Temporal Trajectories in 4D Longitudinal Medical Imaging

Understanding temporal dynamics in medical imaging is crucial for applications such as disease progression modeling, treatment planning and anatomical development tracking. However, most deep learning methods either consider only single temporal contexts, or focus on tasks like classification or regression, limiting their ability for fine-grained spatial predictions. While some approaches have been explored, they are often limited to single timepoints, specific diseases or have other technical restrictions. To address this fundamental gap, we introduce Temporal Flow Matching (TFM), a unified generative trajectory method that (i) aims to learn the underlying temporal distribution, (ii) by design can fall back to a nearest image predictor, i.e. predicting the last context image (LCI), as a special case, and (iii) supports $3D$ volumes, multiple prior scans, and irregular sampling. Extensive benchmarks on three public longitudinal datasets show that TFM consistently surpasses spatio-temporal methods from natural imaging, establishing a new state-of-the-art and robust baseline for $4D$ medical image prediction.

cs.CV↗

arXiv2025

Distance and best approximations in operator norm and trace class norm

We study the best approximation and distance problems in the operator space $\B(\HS)$ and in the space of trace class operators $\LS^1(\B(\HS))$. Formulations of distances are obtained in both cases. The case of finite-dimensional $C^*$-algebras is also considered. The computational advantage of the results is illustrated through examples.

math.FA↗

arXiv2025

A dilation theoretic approach to Banach spaces

For a complex Banach space $\mathbb X$, we prove that $\mathbb X$ is a Hilbert space if and only if every strict contraction $T$ on $\mathbb X$ dilates to an isometry if and only if for every strict contraction $T$ on $\mathbb X$ the function $A_T: \mathbb X \rightarrow [0, \infty]$ defined by $A_T(x)=(\|x\|^2 -\|Tx\|^2)^{\frac{1}{2}}$ gives a norm on $\mathbb X$. We also find several other necessary and sufficient conditions in this thread such that a Banach sapce becomes a Hilbert space. We construct examples of strict contractions on non-Hilbert Banach spaces that do not dilate to isometries. Then we characterize all strict contractions on a non-Hilbert Banach space that dilate to isometries and find explicit isometric dilation for them. We prove several other results including characterizations of complemented subspaces in a Banach space, extension of a Wold isometry to a Banach space unitary and describing norm attainment sets of Banach space operators in terms of dilations.

math.FA↗

arXiv2025

Investigating the Feasibility of Patch-based Inference for Generalized Diffusion Priors in Inverse Problems for Medical Images

Plug-and-play approaches to solving inverse problems such as restoration and super-resolution have recently benefited from Diffusion-based generative priors for natural as well as medical images. However, solutions often use the standard albeit computationally intensive route of training and inferring with the whole image on the diffusion prior. While patch-based approaches to evaluating diffusion priors in plug-and-play methods have received some interest, they remain an open area of study. In this work, we explore the feasibility of the usage of patches for training and inference of a diffusion prior on MRI images. We explore the minor adaptation necessary for artifact avoidance, the performance and the efficiency of memory usage of patch-based methods as well as the adaptability of whole image training to patch-based evaluation - evaluating across multiple plug-and-play methods, tasks and datasets.

eess.IV↗

arXiv2025

LesionLocator: Zero-Shot Universal Tumor Segmentation and Tracking in 3D Whole-Body Imaging

In this work, we present LesionLocator, a framework for zero-shot longitudinal lesion tracking and segmentation in 3D medical imaging, establishing the first end-to-end model capable of 4D tracking with dense spatial prompts. Our model leverages an extensive dataset of 23,262 annotated medical scans, as well as synthesized longitudinal data across diverse lesion types. The diversity and scale of our dataset significantly enhances model generalizability to real-world medical imaging challenges and addresses key limitations in longitudinal data availability. LesionLocator outperforms all existing promptable models in lesion segmentation by nearly 10 dice points, reaching human-level performance, and achieves state-of-the-art results in lesion tracking, with superior lesion retrieval and segmentation accuracy. LesionLocator not only sets a new benchmark in universal promptable lesion segmentation and automated longitudinal lesion tracking but also provides the first open-access solution of its kind, releasing our synthetic 4D dataset and model to the community, empowering future advancements in medical imaging. Code is available at: www.github.com/MIC-DKFZ/LesionLocator

cs.CV↗

arXiv2025

Touchstone Benchmark: Are We on the Right Way for Evaluating AI Algorithms for Medical Segmentation?

How can we test AI performance? This question seems trivial, but it isn't. Standard benchmarks often have problems such as in-distribution and small-size test sets, oversimplified metrics, unfair comparisons, and short-term outcome pressure. As a consequence, good performance on standard benchmarks does not guarantee success in real-world scenarios. To address these problems, we present Touchstone, a large-scale collaborative segmentation benchmark of 9 types of abdominal organs. This benchmark is based on 5,195 training CT scans from 76 hospitals around the world and 5,903 testing CT scans from 11 additional hospitals. This diverse test set enhances the statistical significance of benchmark results and rigorously evaluates AI algorithms across various out-of-distribution scenarios. We invited 14 inventors of 19 AI algorithms to train their algorithms, while our team, as a third party, independently evaluated these algorithms on three test sets. In addition, we also evaluated pre-existing AI frameworks--which, differing from algorithms, are more flexible and can support different algorithms--including MONAI from NVIDIA, nnU-Net from DKFZ, and numerous other open-source frameworks. We are committed to expanding this benchmark to encourage more innovation of AI algorithms for the medical domain.

cs.CV↗

arXiv2025

A unified approach to a family of optimization problems in Banach spaces

Our principal aim is to illustrate that the concept Birkhoff-James orthogonality can be applied effectively to obtain a unified approach to a large family of optimization problems in Banach spaces. We study such optimization problems from the perspective of Birkhoff-James orthogonality in certain suitable Banach spaces. In particular, we demonstrate the duality between the Fermat-Torricelli problem and the Chebyshev center problem which are important particular cases of the least square problem. We revisit the Fermat-Torricelli problem for three and four points and solve it using the same technique. We also investigate the behavior of the Fermat-Torricelli points under the addition or replacement of a new point, and present several new results involving the locations of the Fermat-Torricelli point and the Chebyshev center.

math.FA↗