Modern deep learning approaches haveshown promising results inmeteorological applications like precipitation nowcasting, synthetic radar generation, front detection and several others. Inorder toeffectively train and validate these complex algorithms, large and diverse datasets containing high-resolution imagery are required. Petabytes of weather data, such as from the Geostationary Environmental SatelliteSystem(GOES)andtheNext-Generation Radar(NEXRAD) system, are available to the public; however, the size and complexity of these datasets isahindrance todeveloping and training deep models.

The first subset (in red) is utilized to evaluate a traditional accuracy-basedlossfunction `a,suchasthecrossentropy. This benchmark is based on a loss function designed to incentivize the trained model to produce the smallest possible conformal prediction sets with the desired coverage (e.g., 90% ifα = 0.1). The hybrid training procedure is similar to Algorithm 1, in the sense that it relies on analogous soft-sorting, soft-ranking, and soft-indexing algorithms toevaluate adifferentiable approximation Wi oftheconformity scoreWi in(8). Above, the second equality follows directly from the fact thatS(x,U;π,t), defined in (A2), is by construction increasing in t, and therefore Y / S(x,U;π,1 α) if and only if min{t [0,1]:Y S(x,U;π,t)}>1 α. The proof consists of showing that`a and`u are separately minimized by ˆπ = π,although only approximately inthelatter case.

However, both L2 and BL have two deficiencies. First, the noise in the annotation process is not considered in a principled way. L2 and BL make an assumption about per-pixel i.i.d.

Electrochemical impedance spectroscopy (EIS) data is typically modeled using an equivalent circuit model (ECM), with parameters obtained by minimizing a loss function via nonlinear least squares fitting. This paper introduces two new loss functions, log-B and log-BW, derived from the Bode representation of EIS. Using a large dataset of generated EIS data, the performance of proposed loss functions was evaluated alongside existing ones in terms of R2 scores, chi-squared, computational efficiency, and the mean absolute percentage error (MAPE) between the predicted component values and the original values. Statistical comparisons revealed that the choice of loss function impacts convergence, computational efficiency, quality of fit, and MAPE. Our analysis showed that X2 loss function (squared sum of residuals with proportional weighting) achieved the highest performance across multiple quality of fit metrics, making it the preferred choice when the quality of fit is the primary goal. On the other hand, log-B offered a slightly lower quality of fit while being approximately 1.4 times faster and producing lower MAPE for most circuit components, making log-B as a strong alternative. This is a critical factor for large-scale least squares fitting in data-driven applications, such as training machine learning models on extensive datasets or iterations.

The appendix is organized as follows. We first introduce the basic definitions and inequalities used throughout the appendices. In Appendix A, we provide more details about the datasets, computational resources, and more experiment results on CIFAR10, CIFAR100 and miniImageNet datasets. In Appendix B, we prove that CE, FL and LS satisfy the contrastive property in Definition 1. In Appendix C, we provide a detailed proof for Theorem 1, showing that the Simplex ETFs are the only global minimizers, as long as the loss function satisfies the Definition 1. Finally, in Appendix D, we present the whole proof for Theorem 2 that the FL function is a locally strict saddle function with no spurious local minimizers existing locally and LS function is a globally strict saddle function with no spurious local minimizers existing globally. The following Lemma extends the standard variational form of the nuclear norm.