Discretized conformal prediction for efficient distribution-free inference

In regression problems where there is no known true underlying model, conformal prediction methods enable prediction intervals to be constructed without any assumptions on the distribution of the underlying data, except that the training and test data are assumed to be exchangeable. However, these methods bear a heavy computational cost-and, to be carried out exactly, the regression algorithm would need to be fitted infinitely many times. In practice, the conformal prediction method is run by simply considering only a finite grid of finely spaced values for the response variable. This paper develops discretized conformal prediction algorithms that are guaranteed to cover the target value with the desired probability, and that offer a tradeoff between computational cost and prediction accuracy

Multivariate Convex Regression at Scale

We present new large-scale algorithms for fitting a subgradient regularized multivariate convex regression function to $n$ samples in $d$ dimensions -- a key problem in shape constrained nonparametric regression with widespread applications in statistics, engineering and the applied sciences. The infinite-dimensional learning task can be expressed via a convex quadratic program (QP) with $O(nd)$ decision variables and $O(n^2)$ constraints. While instances with $n$ in the lower thousands can be addressed with current algorithms within reasonable runtimes, solving larger problems (e.g., $n\approx 10^4$ or $10^5$) is computationally challenging. To this end, we present an active set type algorithm on the dual QP. For computational scalability, we perform approximate optimization of the reduced sub-problems; and propose randomized augmentation rules for expanding the active set. Although the dual is not strongly convex, we present a novel linear convergence rate of our algorithm on the dual. We demonstrate that our framework can approximately solve instances of the convex regression problem with $n=10^5$ and $d=10$ within minutes; and offers significant computational gains compared to earlier approaches

Triangular Bézier sub-surfaces on a triangular Bézier surface

This paper considers the problem of computing the Bézier representation for a triangular sub-patch on a triangular Bézier surface. The triangular sub-patch is defined as a composition of the triangular surface and a domain surface that is also a triangular Bézier patch. Based on de Casteljau recursions and shifting operators, previous methods express the control points of the triangular sub-patch as linear combinations of the construction points that are constructed from the control points of the triangular Bézier surface. The construction points contain too many redundancies. This paper derives a simple explicit formula that computes the composite triangular sub-patch in terms of the blossoming points that correspond to distinct construction points and then an efficient algorithm is presented to calculate the control points of the sub-patch

Lidocaine, an anesthetic drug, protects Neuro2A cells against cadmium toxicity

Purpose: To investigate the neuroprotective effect of lidocaine in Neuro2A cells Methods: Differentiated N2a cells were used in this study. Cell viability and neuroprotection were assessed using dimethylthiazol-2-yl)-2,5-diphenyltetrazolium bromide (MTT) and trypan blue assays, while Bax/Bcl-2 expression was assayed by western blotting. Mitochondrial membrane potential, reactive oxygen species and calcium levels were measured using flow cytometry. Results: Lidocaine protected differentiated N2a cells against cadmium-induced toxicity, and also attenuated cadmium toxicity-induced changes in mitochondrial membrane potential (MMP), reactive oxygen species (ROS) and calcium (Ca2+) levels. Furthermore, Bax/Bcl-2 ratio, which was disrupted by cadmium, and cadmium-induced apoptosis, were reversed by lidocaine. Conclusion: Lidocaine protects differentiated N2a cells against cadmium-induced toxicity by reversing apoptosis. Thus, lidocaine is a potential neuroprotective agent