arXiv subjects

Abhishek Jha

Publications and source records attributed to Abhishek Jha.

At least 19 recordsLinked to original sources

arXiv2026

The Poisson Tail Conjecture for primes in short intervals

In 1976, Gallagher showed that, conditional on the Hardy--Littlewood conjectures, the number of primes below $x$ in a randomly chosen short interval of length $λ\log x$ asymptotically follows a Poisson distribution with mean $λ$. Correspondingly, the normalized gaps between consecutive primes follow an exponential distribution, provided that the scaling parameter $λ$ is fixed. We investigate the validity and limitations of the associated folklore Poisson Tail Conjecture as $λ$ is allowed to grow. For slowly growing $λ$, and conditional on a strong variant of the Hardy--Littlewood conjectures, we establish asymptotics showing that the local counting statistics agree with these predictions. Furthermore, we identify the scale of the phase transition and establish the breakdown of these distributions for larger $λ$. The proof relies on a novel combination of extremal interval sieve estimates and concentration inequalities from probability.

math.NT↗

arXiv2026

Forgetting, plasticity, and co-observation: a third facet of continual learning

Efficient continual learning remains a fundamental challenge for deep neural networks. While catastrophic forgetting and loss of plasticity are widely considered the primary obstacles to overcome, we show that these two issues cannot fully explain the performance gap between naive sequential training and offline joint training. In this paper, we highlight data co-observation as a distinct factor influencing continual learning performance. By decoupling the constraints of separate data access from stability and plasticity, we systematically investigate the representational benefits gained by observing training data together. Empirically, we demonstrate a consistent performance difference between joint and separate training across both supervised and self-supervised paradigms in generic data-incremental "chunking" scenarios, whilst mitigating forgetting and controlling for plasticity. Our findings indicate that simultaneous observation of training data (co-observation) yields benefits to the learner's generalization that extend well beyond mere knowledge retention, and that this effect does not require a specific continual distribution shift. Furthermore, we contextualize prominent continual learning mechanisms through this lens: while distillation-based approaches act only as effective knowledge retention mechanisms, our results suggest that the empirical success of memory replay goes beyond the mitigation of forgetting, actively reintroducing the benefits of data co-observation into the learning process.

cs.LG↗

arXiv2026

The moments of split greatest common divisors

Sequences of the form $(\gcd(u_n,v_n))_{n \in \mathbb N}$, with $(u_n)_n$, $(v_n)_n$ sums of $S$-units, have been considered by several authors. The study of $\gcd(n,u_n)$ corresponds, after Silverman, to divisibility sequences arising from the algebraic group $\mathbb G_{\mathrm{a}} \times \mathbb G_{\mathrm{m}}$; in this case, Sanna determined all asymptotic moments of the arithmetic function $\log\,\gcd (n,u_n)$ when $(u_n)_n$ is a Lucas sequence. Here, we characterize the asymptotic behavior of the moments themselves $\sum_{n \leq x}\,\gcd(n,u_n)^λ$, thus solving the moment problem for $\mathbb G_{\mathrm{a}} \times \mathbb G_{\mathrm{m}}$. We give both unconditional and conditional results, the latter only relying on standard conjectures in analytic number theory.

math.NT↗

arXiv2026

Self-Supervised Learning with a Multi-Task Latent Space Objective

We propose a multi-task formulation of self-predictive Siamese SSL in which each spatial transformation defines a distinct latent-space alignment task, solved by a dedicated predictor over a shared encoder. This perspective directly explains a long-standing failure of multi-crop training in self-predictive methods such as BYOL, SimSiam, and MoCo v3: a shared predictor is forced to solve heterogeneous alignment tasks simultaneously, leading to unstable optimization. Assigning one predictor per view type resolves this interference, unlocking linear evaluation gains of 3.8-4\% across frameworks. This perspective also suggests a principled way to enrich pre-training by introducing additional spatial transformations as complementary tasks. We demonstrate this by introducing asymmetric cutout views, in which a masked online view is aligned with a complete target, forming a semantic inpainting objective. The resulting framework is stable, backbone-agnostic, and consistently improves the performance of ResNet and ViT models on ImageNet and COCO.

cs.CV↗

arXiv2025

A Study of Anatomical Priors for Deep Learning-Based Segmentation of Pheochromocytoma in Abdominal CT

Accurate segmentation of pheochromocytoma (PCC) in abdominal CT scans is essential for tumor burden estimation, prognosis, and treatment planning. It may also help infer genetic clusters, reducing reliance on expensive testing. This study systematically evaluates anatomical priors to identify configurations that improve deep learning-based PCC segmentation. We employed the nnU-Net framework to evaluate eleven annotation strategies for accurate 3D segmentation of pheochromocytoma, introducing a set of novel multi-class schemes based on organ-specific anatomical priors. These priors were derived from adjacent organs commonly surrounding adrenal tumors (e.g., liver, spleen, kidney, aorta, adrenal gland, and pancreas), and were compared against a broad body-region prior used in previous work. The framework was trained and tested on 105 contrast-enhanced CT scans from 91 patients at the NIH Clinical Center. Performance was measured using Dice Similarity Coefficient (DSC), Normalized Surface Distance (NSD), and instance-wise F1 score. Among all strategies, the Tumor + Kidney + Aorta (TKA) annotation achieved the highest segmentation accuracy, significantly outperforming the previously used Tumor + Body (TB) annotation across DSC (p = 0.0097), NSD (p = 0.0110), and F1 score (25.84% improvement at an IoU threshold of 0.5), measured on a 70-30 train-test split. The TKA model also showed superior tumor burden quantification (R^2 = 0.968) and strong segmentation across all genetic subtypes. In five-fold cross-validation, TKA consistently outperformed TB across IoU thresholds (0.1 to 0.5), reinforcing its robustness and generalizability. These findings highlight the value of incorporating relevant anatomical context into deep learning models to achieve precise PCC segmentation, offering a valuable tool to support clinical assessment and longitudinal disease monitoring in PCC patients.

eess.IV↗

arXiv2025

Advancing Automatic Photovoltaic Defect Detection using Semi-Supervised Semantic Segmentation of Electroluminescence Images

Photovoltaic (PV) systems allow us to tap into all abundant solar energy, however they require regular maintenance for high efficiency and to prevent degradation. Traditional manual health check, using Electroluminescence (EL) imaging, is expensive and logistically challenging which makes automated defect detection essential. Current automation approaches require extensive manual expert labeling, which is time-consuming, expensive, and prone to errors. We propose PV-S3 (Photovoltaic-Semi-supervised Semantic Segmentation), a Semi-Supervised Learning approach for semantic segmentation of defects in EL images that reduces reliance on extensive labeling. PV-S3 is an artificial intelligence (AI) model trained using a few labeled images along with numerous unlabeled images. We introduce a novel Semi Cross-Entropy loss function to deal with class imbalance. We evaluate PV-S3 on multiple datasets and demonstrate its effectiveness and adaptability. With merely 20% labeled samples, we achieve an absolute improvement of 9.7% in mean Intersection-over-Union (mIoU), 13.5% in Precision, 29.15% in Recall, and 20.42% in F1-Score over prior state-of-the-art supervised method (which uses 100% labeled samples) on University of Central Florida-Electroluminescence (UCF-EL) dataset (largest dataset available for semantic segmentation of EL images) showing improvement in performance while reducing the annotation costs by 80%. For more details, visit our GitHub repository: https://github.com/abj247/PV-S3.

eess.IV↗

arXiv2025

Smallest totient in a residue class

We obtain a totient analogue for Linnik's theorem in arithmetic progressions. Specifically, for any coprime pair of positive integers $(m,a)$ such that $m$ is odd, there exists $n\le m^{2+o(1)}$ such that $φ(n)\equiv a\,\mathrm{mod}\,{m}$.

math.NT↗

arXiv2025

Unsupervised Parameter Efficient Source-free Post-pretraining

Following the success in NLP, the best vision models are now in the billion parameter ranges. Adapting these large models to a target distribution has become computationally and economically prohibitive. Addressing this challenge, we introduce UpStep, an Unsupervised Parameter-efficient Source-free post-pretraining approach, designed to efficiently adapt a base model from a source domain to a target domain: i) we design a self-supervised training scheme to adapt a pretrained model on an unlabeled target domain in a setting where source domain data is unavailable. Such source-free setting comes with the risk of catastrophic forgetting, hence, ii) we propose center vector regularization (CVR), a set of auxiliary operations that minimize catastrophic forgetting and additionally reduces the computational cost by skipping backpropagation in 50\% of the training iterations. Finally iii) we perform this adaptation process in a parameter-efficient way by adapting the pretrained model through low-rank adaptation methods, resulting in a fraction of parameters to optimize. We utilize various general backbone architectures, both supervised and unsupervised, trained on Imagenet as our base model and adapt them to a diverse set of eight target domains demonstrating the adaptability and generalizability of our proposed approach.

cs.CV↗

arXiv2024

Maximally Separated Active Learning

Active Learning aims to optimize performance while minimizing annotation costs by selecting the most informative samples from an unlabelled pool. Traditional uncertainty sampling often leads to sampling bias by choosing similar uncertain samples. We propose an active learning method that utilizes fixed equiangular hyperspherical points as class prototypes, ensuring consistent inter-class separation and robust feature representations. Our approach introduces Maximally Separated Active Learning (MSAL) for uncertainty sampling and a combined strategy (MSAL-D) for incorporating diversity. This method eliminates the need for costly clustering steps, while maintaining diversity through hyperspherical uniformity. We demonstrate strong performance over existing active learning techniques across five benchmark datasets, highlighting the method's effectiveness and integration ease. The code is available on GitHub.

cs.LG↗

arXiv2024

Analysis of Spatial augmentation in Self-supervised models in the purview of training and test distributions

In this paper, we present an empirical study of typical spatial augmentation techniques used in self-supervised representation learning methods (both contrastive and non-contrastive), namely random crop and cutout. Our contributions are: (a) we dissociate random cropping into two separate augmentations, overlap and patch, and provide a detailed analysis on the effect of area of overlap and patch size to the accuracy on down stream tasks. (b) We offer an insight into why cutout augmentation does not learn good representation, as reported in earlier literature. Finally, based on these analysis, (c) we propose a distance-based margin to the invariance loss for learning scene-centric representations for the downstream task on object-centric distribution, showing that as simple as a margin proportional to the pixel distance between the two spatial views in the scence-centric images can improve the learned representation. Our study furthers the understanding of the spatial augmentations, and the effect of the domain-gap between the training augmentations and the test distribution.

cs.CV↗

arXiv2024

Decentralized Safe and Scalable Multi-Agent Control under Limited Actuation

To deploy safe and agile robots in cluttered environments, there is a need to develop fully decentralized controllers that guarantee safety, respect actuation limits, prevent deadlocks, and scale to thousands of agents. Current approaches fall short of meeting all these goals: optimization-based methods ensure safety but lack scalability, while learning-based methods scale but do not guarantee safety. We propose a novel algorithm to achieve safe and scalable control for multiple agents under limited actuation. Specifically, our approach includes: $(i)$ learning a decentralized neural Integral Control Barrier function (neural ICBF) for scalable, input-constrained control, $(ii)$ embedding a lightweight decentralized Model Predictive Control-based Integral Control Barrier Function (MPC-ICBF) into the neural network policy to ensure safety while maintaining scalability, and $(iii)$ introducing a novel method to minimize deadlocks based on gradient-based optimization techniques from machine learning to address local minima in deadlocks. Our numerical simulations show that this approach outperforms state-of-the-art multi-agent control algorithms in terms of safety, input constraint satisfaction, and minimizing deadlocks. Additionally, we demonstrate strong generalization across scenarios with varying agent counts, scaling up to 1000 agents.

cs.RO↗

arXiv2024

Strategic Pseudo-Goal Perturbation for Deadlock-Free Multi-Agent Navigation in Social Mini-Games

This work introduces a Strategic Pseudo-Goal Perturbation (SPGP) technique, a novel approach to resolve deadlock situations in multi-agent navigation scenarios. Leveraging the robust framework of Safety Barrier Certificates, our method integrates a strategic perturbation mechanism that guides agents through social mini-games where deadlock and collision occur frequently. The method adopts a strategic calculation process where agents, upon encountering a deadlock select a pseudo goal within a predefined radius around the current position to resolve the deadlock among agents. The calculation is based on controlled strategic algorithm, ensuring that deviation towards pseudo-goal is both purposeful and effective in resolution of deadlock. Once the agent reaches the pseudo goal, it resumes the path towards the original goal, thereby enhancing navigational efficiency and safety. Experimental results demonstrates SPGP's efficacy in reducing deadlock instances and improving overall system throughput in variety of multi-agent navigation scenarios.

cs.MA↗

arXiv2024

The Common Stability Mechanism behind most Self-Supervised Learning Approaches

Last couple of years have witnessed a tremendous progress in self-supervised learning (SSL), the success of which can be attributed to the introduction of useful inductive biases in the learning process to learn meaningful visual representations while avoiding collapse. These inductive biases and constraints manifest themselves in the form of different optimization formulations in the SSL techniques, e.g. by utilizing negative examples in a contrastive formulation, or exponential moving average and predictor in BYOL and SimSiam. In this paper, we provide a framework to explain the stability mechanism of these different SSL techniques: i) we discuss the working mechanism of contrastive techniques like SimCLR, non-contrastive techniques like BYOL, SWAV, SimSiam, Barlow Twins, and DINO; ii) we provide an argument that despite different formulations these methods implicitly optimize a similar objective function, i.e. minimizing the magnitude of the expected representation over all data samples, or the mean of the data distribution, while maximizing the magnitude of the expected representation of individual samples over different data augmentations; iii) we provide mathematical and empirical evidence to support our framework. We formulate different hypotheses and test them using the Imagenet100 dataset.

cs.CV↗

arXiv2024

Weakly Supervised Detection of Pheochromocytomas and Paragangliomas in CT

Pheochromocytomas and Paragangliomas (PPGLs) are rare adrenal and extra-adrenal tumors which have the potential to metastasize. For the management of patients with PPGLs, CT is the preferred modality of choice for precise localization and estimation of their progression. However, due to the myriad variations in size, morphology, and appearance of the tumors in different anatomical regions, radiologists are posed with the challenge of accurate detection of PPGLs. Since clinicians also need to routinely measure their size and track their changes over time across patient visits, manual demarcation of PPGLs is quite a time-consuming and cumbersome process. To ameliorate the manual effort spent for this task, we propose an automated method to detect PPGLs in CT studies via a proxy segmentation task. As only weak annotations for PPGLs in the form of prospectively marked 2D bounding boxes on an axial slice were available, we extended these 2D boxes into weak 3D annotations and trained a 3D full-resolution nnUNet model to directly segment PPGLs. We evaluated our approach on a dataset consisting of chest-abdomen-pelvis CTs of 255 patients with confirmed PPGLs. We obtained a precision of 70% and sensitivity of 64.1% with our proposed approach when tested on 53 CT studies. Our findings highlight the promising nature of detecting PPGLs via segmentation, and furthers the state-of-the-art in this exciting yet challenging area of rare cancer management.

eess.IV↗

arXiv2022

On terms in a dynamical divisibility sequence having a fixed G.C.D with their indices

Let $F$ and $G$ be integer polynomials where $F$ has degree at least $2$. Define the sequence $(a_n)$ by $a_n=F(a_{n-1})$ for all $n\ge 1$ and $a_0=0.$ Let $\mathscr{B}_{F,\,G,\,k}$ be the set of all positive integers $n$ such that $k\mid \gcd(G(n),a_n)$ and if $p\mid \gcd(G(n),a_n)$ for some $p$, then $p\mid k.$ Let $\mathscr{A}_{F,\,G,\,k}$ be the subset of $\mathscr{B}_{F,\,G,\,k}$ such that $\mathscr{A}_{F,\,G,\,k}=\{n\ge 1 : \gcd(G(n),a_n)=k\}$. In this article, we prove that the asymptotic density of $\mathscr{A}_{F,\,G,\,k}$ and $\mathscr{B}_{F,\,G,\,k}$ exists for a class of $(F,G)$ and also compute the explicit density of $\mathscr{A}_{F,\,G,\,k}$ and $\mathscr{B}_{F,\,G,\,k}$ for $G(x)=x.$

math.NT↗

arXiv2022

On the greatest common divisor of $n$ and the $n$th Fibonacci number, II

Let $\mathcal{A}$ be the set of all integers of the form $\gcd(n, F_n)$, where $n$ is a positive integer and $F_n$ denotes the $n$th Fibonacci number. Leonetti and Sanna proved that $\mathcal{A}$ has natural density equal to zero, and asked for a more precise upper bound. We prove that \begin{equation*} \#\big(\mathcal{A} \cap [1, x]\big) \ll \frac{x \log \log \log x}{\log \log x} \end{equation*} for all sufficiently large $x$.

math.NT↗

arXiv2022

The Distribution of G.C.D.s of Shifted Primes and Lucas Sequences

Let $(u_n)_{n \ge 0}$ be a nondegenerate Lucas sequence and $g_u(n)$ be the arithmetic function defined by $\gcd(n, u_n).$ Recent studies have investigated the distributional characteristics of $g_u$. Numerous results have been proven based on the two extreme values $1$ and $n$ of $g_{u}(n)$. Sanna investigated the average behaviour of $g_{u}$ and found asymptotic formulas for the moments of $\log g_{u}$. In a related direction, Jha and Sanna investigated properties of $g_{u}$ at shifted primes. In light of these results, we prove that for each positive integer $λ,$ we have $$\sum_{\substack{p\le x\\p\text{ prime}}} (\log g_{u}(p-1))^λ \sim P_{u,λ}π(x),$$ where $P_{u, λ}$ is a constant depending on $u$ and $λ$ which is expressible as an infinite series. Additionally, we provide estimates for $P_{u,λ}$ and $M_{u,λ},$ where $M_{u, λ}$ is the constant for an analogous sum obtained by Sanna [J. Number Theory 191 (2018), 305-315]. As an application of our results, we prove upper bounds on the count $\#\{p\le x : g_{u}(p-1)>y\}$ and also establish the existence of infinitely many runs of $m$ consecutive primes $p$ in bounded intervals such that $g_{u}(p-1)>y$ based on a breakthrough of Zhang, Maynard, Tao, et al. on small gaps between primes. Exploring further in this direction, it turns out that for Lucas sequences with nonunit discriminant, we have $\max\{g_{u}(n) : n \le x\} \gg x$. As an analogue, we obtain that that $\max\{g_u(p-1) : p \le x\} \gg x^{0.4736}$ unconditionally, while $\max\{g_u(p-1): p \le x\} \gg x^{1 - o(1)}$ under the hypothesis of Montgomery's or Chowla's conjecture.

math.NT↗

arXiv2022

Greatest common divisors of shifted primes and Fibonacci numbers

Let $(F_n)$ be the sequence of Fibonacci numbers and, for each positive integer $k$, let $\mathcal{P}_k$ be the set of primes $p$ such that $\gcd(p - 1, F_{p - 1}) = k$. We prove that the relative density $\text{r}(\mathcal{P}_k)$ of $\mathcal{P}_k$ exists, and we give a formula for $\text{r}(\mathcal{P}_k)$ in terms of an absolutely convergent series. Furthermore, we give an effective criterion to establish if a given $k$ satisfies $\text{r}(\mathcal{P}_k) > 0$, and we provide upper and lower bounds for the counting function of the set of such $k$'s. As an application of our results, we give a new proof of a lower bound for the counting function of the set of integers of the form $\gcd(n, F_n)$, for some positive integer $n$. Our proof is more elementary than the previous one given by Leonetti and Sanna, which relies on a result of Cubre and Rouse.

math.NT↗