In epidemiological research, modeling the cumulative effects of time-dependent exposures on survival outcomes presents a challenge due to their intricate temporal dynamics. Conventional spline-based statistical methods, though effective, require repeated data transformation for each spline parameter tuning, with survival analysis computations relying on the entire dataset, posing difficulties for large datasets. Meanwhile, existing neural network-based survival analysis methods focus on accuracy but often overlook the interpretability of cumulative exposure patterns. To bridge this gap, we introduce CENNSurv, a novel deep learning approach that captures dynamic risk relationships from time-dependent data. Evaluated on two diverse real-world datasets, CENNSurv revealed a multi-year lagged association between chronic environmental exposure and a critical survival outcome, as well as a critical short-term behavioral shift prior to subscription lapse. This demonstrates CENNSurv's ability to model complex temporal patterns with improved scalability. CENNSurv provides researchers studying cumulative effects a practical tool with interpretable insights.

Real-world datasets often contain missing or corrupted values. Completing multidimensional tensor-structured data with missing entries is essential for numerous applications. Smoothness-constrained low-rank factorization models have shown superior performance with reduced computational costs. While effective at capturing global and long-range correlations, these models struggle to reproduce short-scale, high-frequency variations in the data. In this paper, we introduce the Generalized Least Squares Kernelized Tensor Factorization (GLSKF) framework for tensor completion. GLSKF integrates smoothness-constrained low-rank factorization with a locally correlated residual process; the resulting additive structure can effectively characterize both global dependencies and local variations. In particular, we define the covariance norm to enforce the smoothness of factor matrices in the global low-rank factorization, and use structured covariance/kernel functions to model the local processes. For model estimation, we develop an alternating least squares (ALS) procedure with closed-form solutions for each subproblem. To efficiently handle missing data, GLSKF utilizes projection matrices that preserve the Kronecker structure of covariances, facilitating fast computations through conjugate gradient (CG) and preconditioned conjugate gradient (PCG) algorithms. The proposed framework is evaluated on four real-world datasets across diverse tasks: traffic speed imputation, color image inpainting, video completion, and MRI image reconstruction. Experimental results confirm that GLSKF delivers superior effectiveness and scalability, establishing it as a robust solution for multidimensional tensor completion.

The joint optimization of Neural Radiance Fields (NeRF) and camera trajectories has been widely applied in SLAM tasks due to its superior dense mapping quality and consistency. NeRF-based SLAM learns camera poses using constraints by implicit map representation. A widely observed phenomenon that results from the constraints of this form is jerky and physically unrealistic estimated camera motion, which in turn affects the map quality. To address this deficiency of current NeRF-based SLAM, we propose in this paper TS-SLAM (TS for Trajectory Smoothness). It introduces smoothness constraints on camera trajectories by representing them with uniform cubic B-splines with continuous acceleration that guarantees smooth camera motion. Benefiting from the differentiability and local control properties of B-splines, TS-SLAM can incrementally learn the control points end-to-end using a sliding window paradigm. Additionally, we regularize camera trajectories by exploiting the dynamics prior to further smooth trajectories. Experimental results demonstrate that TS-SLAM achieves superior trajectory accuracy and improves mapping quality versus NeRF-based SLAM that does not employ the above smoothness constraints.

Regression on medical image sequences can capture temporal image pattern changes and predict images at missing or future time points. However, existing geodesic regression methods limit their regression performance by a strong underlying assumption of linear dynamics, while diffusion-based methods have high computational costs and lack constraints to preserve image topology. In this paper, we propose an optimization-based new framework called NODER, which leverages neural ordinary differential equations to capture complex underlying dynamics and reduces its high computational cost of handling high-dimensional image volumes by introducing the latent space. We compare our NODER with two recent regression methods, and the experimental results on ADNI and ACDC datasets demonstrate that our method achieves the state-of-the-art performance in 3D image regression. Our model needs only a couple of images in a sequence for prediction, which is practical, especially for clinical situations where extremely limited image time series are available for analysis.

Steepest descent algorithms, which are commonly used in deep learning, use the gradient as the descent direction, either as-is or after a direction shift using preconditioning. In many scenarios calculating the gradient is numerically hard due to complex or non-differentiable cost functions, specifically next to singular points. In this work we focus on the derivation of the Total Variation semi-norm commonly used in unsupervised cost functions. Specifically, we derive a differentiable proxy to the hard L1 smoothness constraint in a novel iterative scheme which we refer to as Cost Unrolling. Producing more accurate gradients during training, our method enables finer predictions of a given DNN model through improved convergence, without modifying its architecture or increasing computational complexity. We demonstrate our method in the unsupervised optical flow task. Replacing the L1 smoothness constraint with our unrolled cost during the training of a well known baseline, we report improved results on both MPI Sintel and KITTI 2015 unsupervised optical flow benchmarks. Particularly, we report EPE reduced by up to 15.82% on occluded pixels, where the smoothness constraint is dominant, enabling the detection of much sharper motion edges.

Diffusion models have been popular for point cloud generation tasks. Existing works utilize the forward diffusion process to convert the original point distribution into a noise distribution and then learn the reverse diffusion process to recover the point distribution from the noise distribution. However, the reverse diffusion process can produce samples with non-smooth points on the surface because of the ignorance of the point cloud geometric properties. We propose alleviating the problem by incorporating the local smoothness constraint into the diffusion framework for point cloud generation. Experiments demonstrate the proposed model can generate realistic shapes and smoother point clouds, outperforming multiple state-of-the-art methods.

Department of Mathematics, Princeton University, Princeton, NJ 08544, USA (Dated: May 9, 2023) Whether there exist finite time blow-up solutions for the 2-D Boussinesq and the 3-D Euler equations are of fundamental importance to the field of fluid mechanics. We develop a new numerical framework, employing physics-informed neural networks (PINNs), that discover, for the first time, a smooth self-similar blow-up profile for both equations. The solution itself could form the basis of a future computer-assisted proof of blow-up for both equations. In addition, we demonstrate PINNs could be successfully applied to find unstable self-similar solutions to fluid equations by constructing the first example of an unstable self-similar solution to the Córdoba-Córdoba-Fontelos equation. We show that our numerical framework is both robust and adaptable to various other equations. A celebrated open question in fluids is whether or not cylindrical boundary is intrinsically linked to the same from smooth initial data the 3-D Euler equations may problem for the 2-D Bousinessq equations (cf. The mechanism for blowup self-similar blow-up was proven in the groundbreaking for the two equations is believed to be identical.

Overparameterized neural networks enjoy great representation power on complex data, and more importantly yield sufficiently smooth output, which is crucial to their generalization and robustness. Most existing function approximation theories suggest that with sufficiently many parameters, neural networks can well approximate certain classes of functions in terms of the function value. The neural network themselves, however, can be highly nonsmooth. To bridge this gap, we take convolutional residual networks (ConvResNets) as an example, and prove that large ConvResNets can not only approximate a target function in terms of function value, but also exhibit sufficient first-order smoothness. Moreover, we extend our theory to approximating functions supported on a low-dimensional manifold. Our theory partially justifies the benefits of using deep and wide networks in practice. Numerical experiments on adversarial robust image classification are provided to support our theory.

How sensitive should machine learning models be to input changes? We tackle the question of model smoothness and show that it is a useful inductive bias which aids generalization, adversarial robustness, generative modeling and reinforcement learning. We explore current methods of imposing smoothness constraints and observe they lack the flexibility to adapt to new tasks, they don't account for data modalities, they interact with losses, architectures and optimization in ways not yet fully understood. We conclude that new advances in the field are hinging on finding ways to incorporate data, tasks and learning into our definitions of smoothness.