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Split conformal prediction techniques are applied to regression problems with circular responses by introducing a suitable conformity score, leading to prediction sets with adaptive arc length and finite-sample coverage guarantees for any circular predictive model under exchangeable data. Leveraging the high performance of existing predictive models designed for linear responses, we analyze a general projection procedure that converts any linear response regression model into one suitable for circular responses. When random forests serve as basis models in this projection procedure, we harness the out-of-bag dynamics to eliminate the necessity for a separate calibration sample in the construction of prediction sets. For synthetic and real datasets the resulting projected random forests model produces more efficient out-of-bag conformal prediction sets, with shorter median arc length, when compared to the split conformal prediction sets generated by two existing alternative models., Comment: 7 pages; 4 figures

Statistical inference with finite-sample validity for the value function of a given policy in Markov decision processes (MDPs) is crucial for ensuring the reliability of reinforcement learning. Temporal Difference (TD) learning, arguably the most widely used algorithm for policy evaluation, serves as a natural framework for this purpose.In this paper, we study the consistency properties of TD learning with Polyak-Ruppert averaging and linear function approximation, and obtain three significant improvements over existing results. First, we derive a novel sharp high-dimensional probability convergence guarantee that depends explicitly on the asymptotic variance and holds under weak conditions. We further establish refined high-dimensional Berry-Esseen bounds over the class of convex sets that guarantee faster rates than those in the literature. Finally, we propose a plug-in estimator for the asymptotic covariance matrix, designed for efficient online computation. These results enable the construction of confidence regions and simultaneous confidence intervals for the linear parameters of the value function, with guaranteed finite-sample coverage. We demonstrate the applicability of our theoretical findings through numerical experiments.

Diffusion models have emerged as a powerful tool for image generation and denoising. Typically, generative models learn a trajectory between the starting noise distribution and the target data distribution. Recently Liu et al. (2023b) designed a novel alternative generative model Rectified Flow (RF), which aims to learn straight flow trajectories from noise to data using a sequence of convex optimization problems with close ties to optimal transport. If the trajectory is curved, one must use many Euler discretization steps or novel strategies, such as exponential integrators, to achieve a satisfactory generation quality. In contrast, RF has been shown to theoretically straighten the trajectory through successive rectifications, reducing the number of function evaluations (NFEs) while sampling. It has also been shown empirically that RF may improve the straightness in two rectifications if one can solve the underlying optimization problem within a sufficiently small error. In this paper, we make two key theoretical contributions: 1) we provide the first theoretical analysis of the Wasserstein distance between the sampling distribution of RF and the target distribution. Our error rate is characterized by the number of discretization steps and a new formulation of straightness stronger than that in the original work. 2) In line with the previous empirical findings, we show that, for a rectified flow from a Gaussian to a mixture of two Gaussians, two rectifications are sufficient to achieve a straight flow. Additionally, we also present empirical results on both simulated and real datasets to validate our theoretical findings.

The formation, movement and gluing of clusters can be described through a system of non local balance laws. Here, the well posedness of this system is obtained, as well as various stability estimates. Remarkably, qualitative properties of the solutions are proved, providing information on stationary solutions and on the propagation speed. In some cases, fragmentation leads to clusters developing independently. Moreover, these equations may serve as an encryption/decryption tool. This poses new analytical problems and asks for improved numerical methods., Comment: 28 pages, 7 figures

The generation of entangled photons from radiative cascades has enabled milestone experiments in quantum information science with several applications in photonic quantum technologies. Significant efforts are being devoted to pushing the performances of near-deterministic entangled-photon sources based on single quantum emitters often embedded in photonic cavities, so to boost the flux of photon pairs. The general postulate is that the emitter generates photons in a nearly maximally entangled state of polarization, ready for application purposes. Here, we demonstrate that this assumption is unjustified. We show that in radiative cascades there exists an interplay between photon polarization and emission wavevector, strongly affecting quantum correlations when emitters are embedded in micro-cavities. We discuss how the polarization entanglement of photon pairs from a biexciton-exciton cascade in quantum dots strongly depends on their propagation wavevector, and it can even vanish for large emission angles. Our experimental results, backed by theoretical modelling, yield a brand-new understanding of cascaded emission for various quantum emitters. In addition, our model provides quantitative guidelines for designing optical microcavities that retain both a high degree of entanglement and collection efficiency, moving the community one step further towards an ideal source of entangled photons for quantum technologies.

Bacterial communities are pivotal to maintaining ecological function and preserving the rich tapestry of biological diversity. The rapid development of environmental sequencing technologies, such as metagenomics, has revolutionized our capacity to probe such diversity. However, despite these advances, a theoretical understanding connecting empirical data with ecosystem modelling, in particular in the framework of disordered systems akin to spin glasses, is still in its infancy. Here, we present a comprehensive framework using theories of disordered systems to decode microbiome data, which offers insight into the ecological forces that shape macroecological states. By employing the quenched disordered generalized Lotka-Volterra model, we analyze species abundance data in healthy and diseased human gut microbiomes. Results reveal the emergence of two distinct patterns of species-interaction networks, elucidating the pathways through which dysbiosis may drive microbiome instability. Interaction patterns thus provide a window into the systemic shifts accompanying the transition from health to disease, offering a new perspective on the dynamics of the microbial community. Our findings suggest the potential of disordered systems theory to characterize microbiomes by capturing the essence of ecological interactions and their consequences on stability and functioning, leveraging our understanding of the linkages of dysbiosis and microbial dynamics., Comment: 10 pages, 4 figures

The softmax gating function is arguably the most popular choice in mixture of experts modeling. Despite its widespread use in practice, softmax gating may lead to unnecessary competition among experts, potentially causing the undesirable phenomenon of representation collapse due to its inherent structure. In response, the sigmoid gating function has been recently proposed as an alternative and has been demonstrated empirically to achieve superior performance. However, a rigorous examination of the sigmoid gating function is lacking in current literature. In this paper, we verify theoretically that sigmoid gating, in fact, enjoys a higher sample efficiency than softmax gating for the statistical task of expert estimation. Towards that goal, we consider a regression framework in which the unknown regression function is modeled as a mixture of experts, and study the rates of convergence of the least squares estimator in the over-specified case in which the number of experts fitted is larger than the true value. We show that two gating regimes naturally arise and, in each of them, we formulate identifiability conditions for the expert functions and derive the corresponding convergence rates. In both cases, we find that experts formulated as feed-forward networks with commonly used activation such as $\mathrm{ReLU}$ and $\mathrm{GELU}$ enjoy faster convergence rates under sigmoid gating than softmax gating. Furthermore, given the same choice of experts, we demonstrate that the sigmoid gating function requires a smaller sample size than its softmax counterpart to attain the same error of expert estimation and, therefore, is more sample efficient., Comment: 31 pages, 2 figures

We prove that cycles, wheels and block graphs have sequentially Cohen-Macaulay binomial edge ideals. Moreover, we provide a construction of new families of sequentially Cohen-Macaulay graphs by cones., Comment: In this version we have made a substantial correction of the Introduction. We also improved the overall notation. Moreover, to improve the readability and comprehension, we added Example 2.4, 3.2, 4.3, 4.4 and Remark 4.2

The purpose of this clinical study is to test broad spectral range (635-1060 nm) time domain diffuse optical spectroscopy in monitoring the response of breast cancer patients to neoadjuvant chemotherapy (NAC). The broadband operation allows us to fully analyze tissue composition in terms of hemoglobin, water, lipids and collagen concentration, which has never been systematically studied until now during the course of therapy. Patients are subjected to multiple breast optical imaging sessions, each one performed at different stages of NAC, both on tumor-bearing and contralateral healthy breast. We correlate the optical results with conventional imaging techniques and pathological response. Preliminary outcomes on data of 10 patients show an average significant reduction in the concentrations of oxy-hemoglobin (-53%, p = 0.0020), collagen (-36%, p = 0.0039) and water (-15%, p = 0.0195), and increase in lipids (+39%, p = 0.0137) from baseline to the end of therapy in the tumor-bearing breast of patients who responded to therapy at least partially. With respect to scattering, the scattering amplitude, a, increases slightly by the end of the therapy compared to the baseline, while the scattering slope, b, shows no significant change. Some change in concentrations of constituents was noticed also in the contralateral healthy breast, even though it was significant only for oxy-hemoglobin concentration. Collagen seems to be the only component distinguishing between complete (p = 0.0423) and partial responders (p = 0.1012) by the end of 2-3 weeks after the start of therapy, suggesting that it could serve as a potential biomarker to measure NAC effectiveness early during treatment. These initial results are consistent with those of standard medical modalities and highlight the sensitivity of the technique to changes that occur in breast composition during NAC.

Metapopulation models have been instrumental in demonstrating the ecological impact of landscape structure on the survival of a focal species in complex environments. However, extensions to multiple species with arbitrary dispersal networks often rely on phenomenological assumptions limiting their scope. Here, we develop a multilayer network model of competitive dispersing metacommunities to investigate how spatially structured environments impact species coexistence and ecosystem stability. We show that homogeneous environments always lead to monodominance unless all species' fitness parameters are in an exact trade-off. However, this precise fine-tuning does not guarantee coexistence in generic heterogeneous environments. By introducing general spatial disorder in the model, we solve it exactly in the mean-field limit, finding that stable coexistence becomes possible in the presence of strong disorder. Crucially, coexistence is supported by the spontaneous localization of species through the emergence of ecological niches. Our results remain qualitatively valid in arbitrary dispersal networks, where topological features can improve species coexistence. Finally, we employ our model to study how correlated disorder promotes spatial ecological patterns in realistic terrestrial and riverine landscapes. Our work provides a novel framework to understand how landscape structure enables coexistence in metacommunities by acting as the substrate for ecological interactions., Comment: 11 pages, 4 figures

Consider the inverse design problem for a scalar conservation law, i.e., the problem of finding initial data evolving into a given profile at a given time. The solution we present below takes into account localizations both in the final interval where the profile is assigned and in the initial interval where the datum is sought, as well as additional a priori constraints on the datum's range provided by the model. These results are motivated and can be applied to data assimilation procedures in traffic modeling and accidents localization.

Some species of salmon and sea turtle are famously good at finding their birth place to reproduce after having travelled vast expanses of ocean. In contrast, imagine now a species (maybe ancestral to the salmon or turtle) which has to find its birth place to reproduce but has no navigation skills and relies on chance alone. Would such an imaginary species survive? According to our (very simple) model it would survive if and only if the probability that a given individual find its birth place is strictly larger than 1/2.

Mixture of experts (MoE) model is a statistical machine learning design that aggregates multiple expert networks using a softmax gating function in order to form a more intricate and expressive model. Despite being commonly used in several applications owing to their scalability, the mathematical and statistical properties of MoE models are complex and difficult to analyze. As a result, previous theoretical works have primarily focused on probabilistic MoE models by imposing the impractical assumption that the data are generated from a Gaussian MoE model. In this work, we investigate the performance of the least squares estimators (LSE) under a deterministic MoE model where the data are sampled according to a regression model, a setting that has remained largely unexplored. We establish a condition called strong identifiability to characterize the convergence behavior of various types of expert functions. We demonstrate that the rates for estimating strongly identifiable experts, namely the widely used feed-forward networks with activation functions $\mathrm{sigmoid}(\cdot)$ and $\tanh(\cdot)$, are substantially faster than those of polynomial experts, which we show to exhibit a surprising slow estimation rate. Our findings have important practical implications for expert selection., Comment: Accepted to ICML 2024, 29 pages, 2 figures, 2 tables

VR devices have recently been actively promoted as tools for knowledge workers and prior work has demonstrated that VR can support some knowledge worker tasks. However, only a few studies have explored the effects of prolonged use of VR such as a study observing 16 participant working in VR and a physical environment for one work-week each and reporting mainly on subjective feedback. As a nuanced understanding of participants' behavior in VR and how it evolves over time is still missing, we report on the results from an analysis of 559 hours of video material obtained in this prior study. Among other findings, we report that (1) the frequency of actions related to adjusting the headset reduced by 46% and the frequency of actions related to supporting the headset reduced by 42% over the five days; (2) the HMD was removed 31% less frequently over the five days but for 41% longer periods; (3) wearing an HMD is disruptive to normal patterns of eating and drinking, but not to social interactions, such as talking. The combined findings in this work demonstrate the value of long-term studies of deployed VR systems and can be used to inform the design of better, more ergonomic VR systems as tools for knowledge workers.

Neural Network movement controllers promise a variety of advantages over conventional control methods, however, they are not widely adopted due to their inability to produce reliably precise movements. This research explores a bilateral neural network architecture as a control system for motor tasks. We aimed to achieve hemispheric specialisation similar to what is observed in humans across different tasks; the dominant system (usually the right hand, left hemisphere) excels at tasks involving coordination and efficiency of movement, and the non-dominant system performs better at tasks requiring positional stability. Specialisation was achieved by training the hemispheres with different loss functions tailored to the expected behaviour of the respective hemispheres. We compared bilateral models with and without specialised hemispheres, with and without inter-hemispheric connectivity (representing the biological Corpus Callosum), and unilateral models with and without specialisation. The models were trained and tested on two tasks common in the human motor control literature: the random reach task, suited to the dominant system, a model with better coordination, and the hold position task, suited to the non-dominant system, a model with more stable movement. Each system outperformed the non-preferred system in its preferred task. For both tasks, a bilateral model outperformed the non-preferred hand and was as good or better than the preferred hand. The results suggest that the hemispheres could collaborate on tasks or work independently to their strengths. This study provides ideas for how a biologically inspired bilateral architecture could be exploited for industrial motor control., Comment: ACAIN 2024

The performance of the proposed MATHUSLA detector as an instrument for studying the physics of cosmic rays by measuring extensive air showers is presented. The MATHUSLA detector is designed to observe and study the decay of long-lived particles produced at the pp interaction point of the CMS detector at CERN during the HL-LHC data-taking period. The proposed MATHUSLA detector will be composed of many layers of long scintillating bars that cannot measure more than one hit per bar and correctly report the hit coordinate in case of multiple hits. This study shows that adding a layer of RPC detectors with both analogue and digital readout significantly enhances the capabilities of MATHUSLA to measure the local densities and arrival times of charged particles at the front of air showers. We discuss open issues in cosmic-ray physics that the proposed MATHUSLA detector with an additional layer of RPC detectors could address and conclude by comparing with other air-shower facilities that measure cosmic rays in the PeV energy range., Comment: 64 pages, 58 figures

Do religions codify ecological principles? This paper explores theoretically and empirically the role religious beliefs play in shaping environmental interactions. We study African Traditional Religions (ATR) which place forests within a sacred sphere. We build a model of non-market interactions of the mean-field type where the actions of agents with heterogeneous religious beliefs continuously affect the spatial density of forest cover. The equilibrium extraction policy shows how individual beliefs and their distribution among the population can be a key driver of forest conservation. The model also characterizes the role of resource scarcity in both individual and population extraction decisions. We test the model predictions empirically relying on the unique case of Benin, where ATR adherence is freely reported. Using an instrumental variable strategy that exploits the variation in proximity to the Benin-Nigerian border, we find that a 1 standard deviation increase in ATR adherence has a 0.4 standard deviation positive impact on forest cover change. We study the impact of historically belonging to the ancient Kingdom of Dahomey, birthplace of the Vodun religion. Using the original boundaries as a spatial discontinuity, we find positive evidence of Dahomey affiliation on contemporary forest change. Lastly, we compare observed forest cover to counterfactual outcomes by simulating the absence of ATR beliefs across the population.

The prevailing statistical approach to analyzing persistence diagrams is concerned with filtering out topological noise. In this paper, we adopt a different viewpoint and aim at estimating the actual distribution of a random persistence diagram, which captures both topological signal and noise. To that effect, Chazel and Divol (2019) proved that, under general conditions, the expected value of a random persistence diagram is a measure admitting a Lebesgue density, called the persistence intensity function. In this paper, we are concerned with estimating the persistence intensity function and a novel, normalized version of it -- called the persistence density function. We present a class of kernel-based estimators based on an i.i.d. sample of persistence diagrams and derive estimation rates in the supremum norm. As a direct corollary, we obtain uniform consistency rates for estimating linear representations of persistence diagrams, including Betti numbers and persistence surfaces. Interestingly, the persistence density function delivers stronger statistical guarantees.

Solid-state quantum emitters embedded in circular Bragg resonators are attractive due to their ability to emit quantum states of light with high brightness and low multi-photon probability. As for any emitter-microcavity system, fabrication imperfections limit the spatial and spectral overlap of the emitter with the cavity mode, thus limiting their coupling strength. Here, we show that an initial spectral mismatch can be corrected after device fabrication by repeated wet chemical etching steps. We demonstrate ~16 nm wavelength tuning for optical modes in AlGaAs resonators on oxide, leading to a 4-fold Purcell enhancement of the emission of single embedded GaAs quantum dots. Numerical calculations reproduce the observations and suggest that the achievable performance of the resonator is only marginally affected in the explored tuning range. We expect the method to be applicable also to circular Bragg resonators based on other material platforms, thus increasing the device yield of cavity-enhanced solid-state quantum emitters.

We prove the well posedness of a class of non linear and non local mixed hyperbolic-parabolic systems in bounded domains, with Dirichlet boundary conditions. In view of control problems, stability estimates on the dependence of solutions on data and parameters are also provided. These equations appear in models devoted to population dynamics or to epidemiology, for instance.

VR has recently been promoted as a tool for knowledge workers and studies have shown that it has the potential to improve knowledge work. However, studies on its prolonged use have been scarce. A prior study compared working in VR for one week to working in a physical environment, focusing on performance measures and subjective feedback. However, a nuanced understanding and comparison of participants' behavior in VR and the physical environment is still missing. To this end, we analyzed video material made available from this previously conducted experiment, carried out over a working week, and present our findings on comparing the behavior of participants while working in VR and in a physical environment.

We present polarized |S|=0.99$\pm$0.01, and unpolarized |S|=0.03$\pm$0.01 emission from a single emitter embedded in a single, cylindrically symmetric device design. We show that the polarization stems from a position offset of the single emitter with respect to the cavity center, which breaks the cylindrical symmetry, and a position-dependent coupling to the frequency degenerate eigenmodes of the resonator structure. The experimental results are interpreted by using numerical simulations and by experimental mapping of the polarization-resolved far-field emission patterns. Our findings can be generalized to any nanophotonic structure where two orthogonal eigenmodes are not fully spatially overlapping., Comment: 15 pages, 5 figures

We classify path polyominoes which are level and pseudo-Gorenstein. Moreover, we compute all level and pseudo-Gorenstein simple thin polyominoes with rank less than or equal to 10. We also compute the regularity of the pseudo-Gorenstein simple thin polyominoes in relation to their rank.

We introduce the following model for the evolution of a population. At every discrete time $j\geq 0$ exactly one individual is introduced in the population and is assigned a death probability $c_j$ sampled from $C$, a fixed probability distribution. We think of $c_j$ as a genetic marker of this individual. At every time $n\geq 1$ every individual in the population dies or not independently of each other with its corresponding death probability $c_j$. We show that the population size goes to infinity if and only if $E(1/C)=\infty$. This is in sharp contrast with the model with constant $c$ and with the model in random environment (same random $c_n$ for all individuals at time $n$). Both of these models are always positive recurrent. Thus, genetic diversity does seem to help survival! We also study the point process associated with our model. We show that the limit point process has an accumulation point near 0 for the $c'$s. For certain $C$ distributions, including the uniform, the limit process properly rescaled is also shown to converge to a non-homogeneous Poisson process.