In this task, you are given the abstract of a research paper. Your task is to generate a summary of this abstract. Your summary should not be very short, but it's better if it's not more than 30 words.
--------
Question: Sometimes SRS (Stereotactic Radio Surgery) requires using sphere packing on a Region of Interest (ROI) such as cancer to determine a treatment plan.   We have developed a sphere packing algorithm which packs non-intersecting spheres inside the ROI.   The region of interest in our case are those voxels which are identified as cancer tissues.   In this paper, we analyze the rotational invariant properties of  our sphere-packing algorithm which is based on distance transformations. Epsilon-Rotation invariant means the ability to arbitrary rotate the 3D ROI while keeping the volume properties remaining (almost) same within some limit of epsilon. The applied rotations produce spherical packing which remains highly correlated as we analyze the geometrically properties of sphere packing before and after the rotation of the volume data for the ROI. Our novel sphere packing algorithm has high degree of rotation invariance within the range of +/- epsilon. Our method used a shape descriptor derived from the values of the disjoint set of spheres form the distance-based sphere packing algorithm to extract the invariant descriptor from the ROI. We demonstrated by implementing these ideas using Slicer3D platform available for our research.   The data is based on sing MRI Stereotactic images. We presented several performance results on different benchmarks data of over 30 patients in Slicer3D platform.

Answer: Packing region of Interest (ROI) such as cancerous regions identified in 3D Volume Data, Packing spheres inside the ROI, rotating the ROI , measures of difference in sphere packing before and after the rotation.


Question: We present local ensembles, a method for detecting extrapolation at test time in a pre-trained model. We focus on underdetermination as a key component of extrapolation: we aim to detect when many possible predictions are consistent with the training data and model class. Our method uses local second-order information to approximate the variance of predictions across an ensemble of models from the same class. We compute this approximation by estimating the norm of the component of a test point's gradient that aligns with the low-curvature directions of the Hessian, and provide a tractable method for estimating this quantity. Experimentally, we show that our method is capable of detecting when a pre-trained model is extrapolating on test data, with applications to out-of-distribution detection, detecting spurious correlates, and active learning.

Answer: We present local ensembles, a method for detecting extrapolation in trained models, which approximates the variance of an ensemble using local-second order information.


Question: Overfitting is an ubiquitous problem in neural network training and usually mitigated using a holdout data set.
 Here we challenge this rationale and investigate criteria for overfitting without using a holdout data set.
 Specifically, we train a model for a fixed number of epochs multiple times with varying fractions of randomized labels and for a range of regularization strengths. 
 A properly trained model should not be able to attain an accuracy greater than the fraction of properly labeled data points. Otherwise the model overfits. 
 We introduce two criteria for detecting overfitting and one to detect underfitting. We analyze early stopping, the regularization factor, and network depth.
 In safety critical applications we are interested in models and parameter settings which perform well and are not likely to overfit. The methods of this paper allow characterizing and identifying such models.

Answer:
We introduce and analyze several criteria for detecting overfitting.