The problem of interpolating a rigid body motion is to find a spatial trajectory between a prescribed initial and terminal pose. Two variants of this interpolation problem are addressed. The first is to find a solution that satisfies initial conditions on the k-1 derivatives of the rigid body twist. This is called the kth-order initial value trajectory interpolation problem (k-IV-TIP). The second is to find a solution that satisfies conditions on the rigid body twist and its k-1 derivatives at the initial and terminal pose. This is called the kth-order boundary value trajectory interpolation problem (k-BV-TIP). Solutions to the k-IV-TIP for k=1,...,4, i.e. the initial twist and up to the 4th time derivative are prescribed. Further, a solution to the 1-IV-TBP is presented, i.e. the initial and terminal twist are prescribed. The latter is a novel cubic interpolation between two spatial configurations with given initial and terminal twist. This interpolation is automatically identical to the minimum acceleration curve when the twists are set to zero. The general approach to derive higher-order solutions is presented. Numerical results are shown for two examples.

We present a scalable Gaussian Process (GP) method that can fit and predict full derivative observations called DSoftKI. It extends SoftKI, a method that approximates a kernel via softmax interpolation from learned interpolation point locations, to the setting with derivatives. DSoftKI enhances SoftKI's interpolation scheme to incorporate the directional orientation of interpolation points relative to the data. This enables the construction of a scalable approximate kernel, including its first and second-order derivatives, through interpolation. We evaluate DSoftKI on a synthetic function benchmark and high-dimensional molecular force field prediction (100-1000 dimensions), demonstrating that DSoftKI is accurate and can scale to larger datasets with full derivative observations than previously possible.

Structured Kernel Interpolation (SKI) (Wilson et al. 2015) helps scale Gaussian Processes (GPs) by approximating the kernel matrix via interpolation at inducing points, achieving linear computational complexity. However, it lacks rigorous theoretical error analysis. This paper bridges the gap: we prove error bounds for the SKI Gram matrix and examine the error's effect on hyperparameter estimation and posterior inference. We further provide a practical guide to selecting the number of inducing points under convolutional cubic interpolation: they should grow as $n^{d/3}$ for error control. Crucially, we identify two dimensionality regimes governing the trade-off between SKI Gram matrix spectral norm error and computational complexity. For $d \leq 3$, any error tolerance can achieve linear time for sufficiently large sample size. For $d > 3$, the error must increase with sample size to maintain linear time. Our analysis provides key insights into SKI's scalability-accuracy trade-offs, establishing precise conditions for achieving linear-time GP inference with controlled approximation error.

The fast and accurate estimation of planetary mass-loss rates is critical for planet population and evolution modelling. We use machine learning (ML) for fast interpolation across an existing large grid of hydrodynamic upper atmosphere models, providing mass-loss rates for any planet inside the grid boundaries with superior accuracy compared to previously published interpolation schemes. We consider an already available grid comprising about 11000 hydrodynamic upper atmosphere models for training and generate an additional grid of about 250 models for testing purposes. We develop the ML interpolation scheme (dubbed "atmospheric Mass Loss INquiry frameworK"; MLink) using a Dense Neural Network, further comparing the results with what was obtained employing classical approaches (e.g. linear interpolation and radial basis function-based regression). Finally, we study the impact of the different interpolation schemes on the evolution of a small sample of carefully selected synthetic planets. MLink provides high-quality interpolation across the entire parameter space by significantly reducing both the number of points with large interpolation errors and the maximum interpolation error compared to previously available schemes. For most cases, evolutionary tracks computed employing MLink and classical schemes lead to comparable planetary parameters at Gyr-timescales. However, particularly for planets close to the top edge of the radius gap, the difference between the predicted planetary radii at a given age of tracks obtained employing MLink and classical interpolation schemes can exceed the typical observational uncertainties. Machine learning can be successfully used to estimate atmospheric mass-loss rates from model grids paving the way to explore future larger and more complex grids of models computed accounting for more physical processes.

Multiple sclerosis is a disease that affects the brain and spinal cord, it can lead to severe disability and has no known cure. The majority of prior work in machine learning for multiple sclerosis has been centered around using Magnetic Resonance Imaging scans or laboratory tests; these modalities are both expensive to acquire and can be unreliable. In a recent paper it was shown that disease progression can be predicted effectively using performance outcome measures and demographic data. In our work we build on this to investigate the modeling side, using continuous time models to predict progression. We benchmark four continuous time models using a publicly available multiple sclerosis dataset. We find that the best continuous model is often able to outperform the best benchmarked discrete time model. We also carry out an extensive ablation to discover the sources of performance gains, we find that standardizing existing features leads to a larger performance increase than interpolating missing features.

There has been a surge in Explainable-AI (XAI) methods that provide insights into the workings of Deep Neural Network (DNN) models. Integrated Gradients (IG) is a popular XAI algorithm that attributes relevance scores to input features commensurate with their contribution to the model's output. However, it requires multiple forward \& backward passes through the model. Thus, compared to a single forward-pass inference, there is a significant computational overhead to generate the explanation which hinders real-time XAI. This work addresses the aforementioned issue by accelerating IG with a hardware-aware algorithm optimization. We propose a novel non-uniform interpolation scheme to compute the IG attribution scores which replaces the baseline uniform interpolation. Our algorithm significantly reduces the total interpolation steps required without adversely impacting convergence. Experiments on the ImageNet dataset using a pre-trained InceptionV3 model demonstrate \textit{2.6-3.6}$\times$ performance speedup on GPU systems for iso-convergence. This includes the minimal \textit{0.2-3.2}\% latency overhead introduced by the pre-processing stage of computing the non-uniform interpolation step-sizes.

Textbook wisdom advocates for smooth function fits and implies that interpolation of noisy data should lead to poor generalization. A related heuristic is that fitting parameters should be fewer than measurements (Occam's Razor). Surprisingly, contemporary machine learning (ML) approaches, cf. deep nets (DNNs), generalize well despite interpolating noisy data. This may be understood via Statistically Consistent Interpolation (SCI), i.e. data interpolation techniques that generalize optimally for big data. In this article we elucidate SCI using the weighted interpolating nearest neighbors (wiNN) algorithm, which adds singular weight functions to kNN (k-nearest neighbors). This shows that data interpolation can be a valid ML strategy for big data. SCI clarifies the relation between two ways of modeling natural phenomena: the rationalist approach (strong priors) of theoretical physics with few parameters and the empiricist (weak priors) approach of modern ML with more parameters than data. SCI shows that the purely empirical approach can successfully predict. However data interpolation does not provide theoretical insights, and the training data requirements may be prohibitive. Complex animal brains are between these extremes, with many parameters, but modest training data, and with prior structure encoded in species-specific mesoscale circuitry. Thus, modern ML provides a distinct epistemological approach different both from physical theories and animal brains.

With the advance in user-friendly and powerful video editing tools, anyone can easily manipulate videos without leaving prominent visual traces. Frame-rate up-conversion (FRUC), a representative temporal-domain operation, increases the motion continuity of videos with a lower frame-rate and is used by malicious counterfeiters in video tampering such as generating fake frame-rate video without improving the quality or mixing temporally spliced videos. FRUC is based on frame interpolation schemes and subtle artifacts that remain in interpolated frames are often difficult to distinguish. Hence, detecting such forgery traces is a critical issue in video forensics. This paper proposes a frame-rate conversion detection network (FCDNet) that learns forensic features caused by FRUC in an end-to-end fashion. The proposed network uses a stack of consecutive frames as the input and effectively learns interpolation artifacts using network blocks to learn spatiotemporal features. This study is the first attempt to apply a neural network to the detection of FRUC. Moreover, it can cover the following three types of frame interpolation schemes: nearest neighbor interpolation, bilinear interpolation, and motion-compensated interpolation. In contrast to existing methods that exploit all frames to verify integrity, the proposed approach achieves a high detection speed because it observes only six frames to test its authenticity. Extensive experiments were conducted with conventional forensic methods and neural networks for video forensic tasks to validate our research. The proposed network achieved state-of-the-art performance in terms of detecting the interpolated artifacts of FRUC. The experimental results also demonstrate that our trained model is robust for an unseen dataset, unlearned frame-rate, and unlearned quality factor.

Application of discrete-time survival methods for continuous-time survival prediction is considered. For this purpose, a scheme for discretization of continuous-time data is proposed by considering the quantiles of the estimated event-time distribution, and, for smaller data sets, it is found to be preferable over the commonly used equidistant scheme. Furthermore, two interpolation schemes for continuous-time survival estimates are explored, both of which are shown to yield improved performance compared to the discrete-time estimates. The survival methods considered are based on the likelihood for right-censored survival data, and parameterize either the probability mass function (PMF) or the discrete-time hazard rate, both with neural networks. Through simulations and study of real-world data, the hazard rate parametrization is found to perform slightly better than the parametrization of the PMF. Inspired by these investigations, a continuous-time method is proposed by assuming that the continuous-time hazard rate is piecewise constant. The method, named PC-Hazard, is found to be highly competitive with the aforementioned methods in addition to other methods for survival prediction found in the literature.

In this paper, we investigate the application of data mining to existing techniques for quality control/anomaly detection on weather sensor observations. Specifically we adapt the popular Barnes Spatial interpolation method to use time-series distance rather than spatial distance to develop an online algorithm that uses readings from similar stations based on current and historical observations for interpolation and we demonstrate that this new algorithm exhibits less model error than the Barnes Spatial interpolation-based method. We focus on interpolation, which is a basis for this popular quality control method and other related methods, and examine a dataset of over 233 million temperature observations from California and surrounding areas. Our approach shows improved performance as indicated by mean squared error reduced by approximately one half for predicted values versus reported values.