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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., Comment: Accepted in ECCV 2024 Workshop on Out-of-distribution generalization in computer vision (OOD-CV)

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., Comment: 7 pages

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, following Silverman, to divisibility sequences arising from the split 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 $\sum_{n \leq x}\,\gcd(n,u_n)^\lambda$, 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., Comment: 15 pages, 1 figure

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.

The quantum speed limits (QSLs) set fundamental lower bounds on the time required for a quantum system to evolve from a given initial state to a final state. In this work, we investigate CP violation and the mass hierarchy problem of neutrino oscillations in matter using the QSL time as a key analytical tool. We examine the QSL time for the unitary evolution of two- and three-flavor neutrino states, both in vacuum and in the presence of matter. Two-flavor neutrino oscillations are used as a precursor to their three-flavor counterparts. We further compute the QSL time for neutrino state evolution and entanglement in terms of neutrino survival and oscillation probabilities, which are experimentally measurable quantities in neutrino experiments. A difference in the QSL time between the normal and inverted mass hierarchy scenarios, for neutrino state evolution as well as for entanglement, under the effect of a CP violation phase is observed. Our results are illustrated using energy-varying sets of accelerator neutrino sources from experiments such as T2K, NOvA, and DUNE. Notably, three-flavor neutrino oscillations in constant matter density exhibit faster state evolution across all these neutrino experiments in the normal mass hierarchy scenario. Additionally, we observe fast entanglement growth in DUNE assuming a normal mass hierarchy., Comment: v1: 18 pages, 10 figures. Comments welcome

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 Segmentation), a Semi-Supervised Learning approach for semantic segmentation of defects in EL images that reduces reliance on extensive labeling. PV-S3 is a Deep learning model trained using a few labeled images along with numerous unlabeled images. 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 IoU, 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 UCF-EL dataset (largest dataset available for semantic segmentation of EL images)showing improvement in performance while reducing the annotation costs by 80%.

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., Comment: Additional visualizations (.gif): https://github.com/abskjha/CenterVectorSSL

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., Comment: Accepted at SPIE 2024. arXiv admin note: text overlap with arXiv:2402.00175

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 $x$.

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 $\lambda,$ we have $$\sum_{\substack{p\le x\\p\text{ prime}}} (\log g_{u}(p-1))^{\lambda} \sim P_{u,\lambda}\pi(x),$$ where $P_{u, \lambda}$ is a constant depending on $u$ and $\lambda$ which is expressible as an infinite series. Additionally, we provide estimates for $P_{u,\lambda}$ and $M_{u,\lambda},$ where $M_{u, \lambda}$ 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., Comment: 17 pages; comments welcome

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.

We map neutrinos to qubit and qutrit states of quantum information theory by constructing the Poincar\'e sphere using SU(2) Pauli matrices and SU(3) Gell-Mann matrices, respectively. The construction of the Poincar\'e sphere in the two-qubit system enables us to construct the Bloch matrix, which yields valuable symmetries in the Bloch vector space of two neutrino systems. By identifying neutrinos with qutrits, we calculate the measures of qutrit entanglement for neutrinos. We use SU(3) Gell-Mann matrices tensor products to construct the Poincar\'e sphere of two qutrits neutrino systems. The comparison between the entanglement measures of bipartite qubits and bipartite qutrits in the two neutrino system are shown. The result warrants a study of two qutrits entanglement in the three neutrino system., Comment: This version of the paper was submitted to the journal. 33 pages and 3 figures

Visual question answering is a vision-and-language multimodal task, that aims at predicting answers given samples from the question and image modalities. Most recent methods focus on learning a good joint embedding space of images and questions, either by improving the interaction between these two modalities, or by making it a more discriminant space. However, how informative this joint space is, has not been well explored. In this paper, we propose a novel regularization for VQA models, Constrained Optimization using Barlow's theory (COB), that improves the information content of the joint space by minimizing the redundancy. It reduces the correlation between the learned feature components and thereby disentangles semantic concepts. Our model also aligns the joint space with the answer embedding space, where we consider the answer and image+question as two different `views' of what in essence is the same semantic information. We propose a constrained optimization policy to balance the categorical and redundancy minimization forces. When built on the state-of-the-art GGE model, the resulting model improves VQA accuracy by 1.4% and 4% on the VQA-CP v2 and VQA v2 datasets respectively. The model also exhibits better interpretability.

Fine-tuning pre-trained language models improves the quality of commercial reply suggestion systems, but at the cost of unsustainable training times. Popular training time reduction approaches are resource intensive, thus we explore low-cost model compression techniques like Layer Dropping and Layer Freezing. We demonstrate the efficacy of these techniques in large-data scenarios, enabling the training time reduction for a commercial email reply suggestion system by 42%, without affecting the model relevance or user engagement. We further study the robustness of these techniques to pre-trained model and dataset size ablation, and share several insights and recommendations for commercial applications.

Recently, there has been some interest in values of arithmetical functions on members of special sequences, such as Euler's totient function $\phi$ on factorials, linear recurrences, etc. In this article, we investigate, for given positive integers $a$ and $b$, the least positive integer $c=c(a,b)$ such that the quotient $\phi(c!)/\phi(a!)\phi(b!)$ is an integer. We derive results on the limit of the ratio $c(a,b)/(a+b)$ as $a$ and $b$ tend to infinity. Furthermore, we show that $c(a,b)>a+b$ for all pairs of positive integers $(a,b)$ with an exception of a set of density zero., Comment: 13 pages, 2 figures; to appear in Bulletin of Australian Mathematical Society

Active visual exploration aims to assist an agent with a limited field of view to understand its environment based on partial observations made by choosing the best viewing directions in the scene. Recent methods have tried to address this problem either by using reinforcement learning, which is difficult to train, or by uncertainty maps, which are task-specific and can only be implemented for dense prediction tasks. In this paper, we propose the Glimpse-Attend-and-Explore model which: (a) employs self-attention to guide the visual exploration instead of task-specific uncertainty maps; (b) can be used for both dense and sparse prediction tasks; and (c) uses a contrastive stream to further improve the representations learned. Unlike previous works, we show the application of our model on multiple tasks like reconstruction, segmentation and classification. Our model provides encouraging results while being less dependent on dataset bias in driving the exploration. We further perform an ablation study to investigate the features and attention learned by our model. Finally, we show that our self-attention module learns to attend different regions of the scene by minimizing the loss on the downstream task. Code: https://github.com/soroushseifi/glimpse-attend-explore.

Cilleruelo conjectured that if $f\in\mathbb{Z}[x]$ is an irreducible polynomial of degree $d\ge 2$ then, $\log \operatorname{lcm} \{f(n)\mid n

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 $G(x)=x.$, Comment: Improved results and incorporated referee comments, 19 pages To appear in New York Journal of Mathematics

Two and three flavor oscillating neutrinos are shown to exhibit the properties bipartite and tripartite quantum entanglement. The two and three flavor neutrinos are mapped to qubit states used in quantum information theory. Such quantum bits of the neutrino state can be encoded on a IBMQ computer using quantum computing as a tool. We show the implementation of entanglement in the two neutrino system on the IBM quantum processor., Comment: 8 pages, 6 figures, Prepared for the proceedings of The 5th International Conference on Particle Physics and Astrophysics (ICPPA-2020), Moscow, Russia, 5th-9th October,2020. New version has new references in introduction

The ongoing COVID-19 pandemic calls for a multi-faceted public health response comprising complementary interventions to control the spread of the disease while vaccines and therapies are developed. Many of these interventions need to be informed by epidemic risk predictions given available data, including symptoms, contact patterns, and environmental factors. Here we propose a novel probabilistic formalism based on Individual-Level Models (ILMs) that offers rigorous formulas for the probability of infection of individuals, which can be parameterised via Maximum Likelihood Estimation (MLE) applied on compartmental models defined at the population level. We describe an approach where individual data collected in real-time is integrated with overall case counts to update the a predictor of the susceptibility of infection as a function of individual risk factors., Comment: 16 pages, 5 figures. arXiv admin note: text overlap with arXiv:1908.06822 by other authors