Distribution-free, Risk-controlling Prediction Sets

While improving prediction accuracy has been the focus of machine learning in recent years, this alone does not suffice for reliable decision-making. Deploying learning systems in consequential settings also requires calibrating and communicating the uncertainty of predictions. To convey instance-wise uncertainty for prediction tasks, we show how to generate set-valued predictions from a black-box predictor that controls the expected loss on future test points at a user-specified level. Our approach provides explicit finite-sample guarantees for any dataset by using a holdout set to calibrate the size of the prediction sets. This framework enables simple, distribution-free, rigorous error control for many tasks, and we demonstrate it in five large-scale machine learning problems: (1) classification problems where some mistakes are more costly than others; (2) multi-label classification, where each observation has multiple associated labels; (3) classification problems where the labels have a hierarchical structure; (4) image segmentation, where we wish to predict a set of pixels containing an object of interest; and (5) protein structure prediction. Last, we discuss extensions to uncertainty quantification for ranking, metric learning, and distributionally robust learning.

Asymptotic approximation of central binomial coefficients with rigorous error bounds

We show that a well-known asymptotic series for the logarithm of the central binomial coefficient is strictly enveloping in the sense of Pólya and Szegö, so the error incurred in truncating the series is of the same sign as the next term, and is bounded in magnitude by that term. We consider closely related asymptotic series for Binet's function, for \(\ln\Gamma(z+\frac12)\), and for the Riemann-Siegel theta function, and make some historical remarks.

Effects of spinning on residual velocity of ogive-nosed projectile undergoing ordnance velocity impact

In the present article, finite element analysis (FEA) based simulation on the study of the impact of projectiles having ogive nose shape has been made using ANSYS explicit dynamics. The effects of spinning on the residual velocity of ogive nosed projectile undergoing ordnance velocity impact have been presented. The variations of residual velocity due to different projectile materials and target plate thickness have been evaluated when the projectile is impacted by translational and spinning velocity. The target plates and ogive nosed projectile of a given material are discretized, and a rigorous error and convergence study has been made. Subsequently, the residual velocity of the considered model is evaluated by numerical techniques based on FEA. The results with the optimized meshed model are compared with the analytical results using the penetration theory and found that the results are well compared. Parametric study of the residual velocity has been made with varied ogive nosed materials and target plate thickness when the ogive nosed projectile undergoing ordnance velocity impact. Based on the numerical results, it has been found that the ogive nose projectile having tungsten alloy material is more effective undergoing ordnance velocity impact compared to steel 4340 material. For a given target plate thickness, spinning velocity, and impact velocity, the residual velocity is about 3 percent higher for the projectile made up of tungsten alloy compared to the steel 4340. The effects of the target plate thickness on the residual velocity of the ogive nose projectile do not seem to have much significant effects. It may be due to the simple reason that the ratio of the target plate thickness to projectile diameter is remaining within the intermediate range, i.e. within 1 and 10.

Collocation of Next-Generation Operators for Computing the Basic Reproduction Number of Structured Populations

We contribute a full analysis of theoretical and numerical aspects of the collocation approach recently proposed by some of the authors to compute the basic reproduction number of structured population dynamics as spectral radius of certain infinite-dimensional operators. On the one hand, we prove under mild regularity assumptions on the models coefficients that the concerned operators are compact, so that the problem can be properly recast as an eigenvalue problem thus allowing for numerical discretization. On the other hand, we prove through detailed and rigorous error and convergence analyses that the method performs the expected spectral accuracy. Several numerical tests validate the proposed analysis by highlighting diverse peculiarities of the investigated approach.

Progress towards a rigorous error propagation for total least-squares estimates

After several attempts at a formal derivation of the dispersion matrix for Total Least-Squares (TLS) estimates within an Errors-In-Variables (EIV) Model, here a refined approach is presented that makes rigorous use of the nonlinear normal equations, though assuming a Kronecker product structure for both observational dispersion matrices at this point. In this way, iterative linearization of a model (that can be established as being equivalent to the original EIV-Model) is avoided, which might be preferred since such techniques are based on the last iteration step only and, therefore, produce dispersion matrices for the estimated parameters that are generally too optimistic. Here, the error propagation is based on the (linearized total differential of the) exact nonlinear normal equations, which should lead to more trustworthy measures of precision.

Improving Plane Fitting Accuracy with Rigorous Error Models of Structured Light-Based RGB-D Sensors

Plane fitting is a fundamental operation for point cloud data processing. Most existing methods for point cloud plane fitting have been developed based on high-quality Lidar data giving equal weight to the point cloud data. In recent years, using low-quality RGB-Depth (RGB-D) sensors to generate 3D models has attracted much attention. However, with low-quality point cloud data, equal weight plane fitting methods are not optimal as the range errors of RGB-D sensors are distance-related. In this paper, we developed an accurate plane fitting method for a structured light (SL)-based RGB-D sensor. First, we derived an error model of a point cloud dataset from the SL-based RGB-D sensor through error propagation from the raw measurement to the point coordinates. A new cost function based on minimizing the radial distances with the derived rigorous error model was then proposed for the random sample consensus (RANSAC)-based plane fitting method. The experimental results demonstrated that our method is robust and practical for different operating ranges and different working conditions. In the experiments, for the operating ranges from 1.23 meters to 4.31 meters, the mean plane angle errors were about one degree, and the mean plane distance errors were less than six centimeters. When the dataset is of a large-depth-measurement scale, the proposed method can significantly improve the plane fitting accuracy, with a plane angle error of 0.5 degrees and a mean distance error of 4.7 cm, compared to 3.8 degrees and 16.8 cm, respectively, from the conventional un-weighted RANSAC method. The experimental results also demonstrate that the proposed method is applicable for different types of SL-based RGB-D sensor. The rigorous error model of the SL-based RGB-D sensor is essential for many applications such as in outlier detection and data authorization. Meanwhile, the precise plane fitting method developed in our research will benefit algorithms based on high-accuracy plane features such as depth calibration, 3D feature-based simultaneous localization and mapping (SLAM), and the generation of indoor building information models (BIMs).

On assessing the accuracy of defect free energy computations

We develop a rigorous error analysis for coarse-graining of defect-formation free energy. For a one-dimensional constrained atomistic system, we establish the thermodynamic limit of the defect-formation free energy and obtain explicitly the rate of convergence. We then construct a sequence of coarse-grained energies with the same rate but significantly reduced computational cost. We illustrate our analytical results through explicit computations for the case of harmonic potentials and through numerical simulations.

Dynamical ages of the young local associations with Gaia

Context. The young local associations (YLAs) constitute an excellent sample for the study of a variety of astrophysical topics, especially the star formation process in low-density environments. Data from the Gaia mission allows us to undertake studies of the YLAs with unprecedented accuracy. Aims. We determine the dynamical age and place of birth of a set of associations in a uniform and dynamically consistent manner. There are nine YLAs in our sample ϵ Chamaeleontis, TW Hydrae, β Pictoris, Octans, Tucana-Horologium, Columba, Carina, Argus, and AB Doradus. Methods. We designed a method for deriving the dynamical age of the YLAs based on the orbital integration. The method involves a strategy to account for the effect of observational errors. We tested the method using mock YLAs. Finally, we applied it to our set of nine YLAs with astrometry from the first Gaia data release and complementary on-ground radial velocities from the literature. Results. Our orbital analysis yields a first estimate of the dynamical age of 33−0+9 Myr, 1313−0+7 Myr, and 55−0+23 Myr for ϵ Chamaeleontis, β Pictoris, and Tucana-Horologium, respectively. For four other associations (Octans, Columba, Carina, and Argus), we provide a lower limit for the dynamical age. Our rigorous error treatment indicates that TW Hydrae and AB Doradus deserve further study. Conclusions. The dynamical ages that we obtain are compatible spectroscopic and isochrone fitting ages obtained elsewhere. From the orbital analysis, we suggest a scenario for these YLAs where there were two episodes of star formation: one ~40 Myr ago in the first quadrant that gave birth to ϵ Chamaeleontis, TW Hydrae, and β Pictoris, and another 5−15 Myr ago close to the Sun that formed Tucana-Horologium, Columba, and Carina. Future Gaia data will provide the necessary accuracy to improve the present results, especially for the controversial age determinations, and additional evidence for the proposed scenario once a complete census of YLAs and better membership can be obtained.