Abstract--We study the expressivity of one-dimensional (1D) ReLU deep neural networks through the lens of their linear regions. We also propose a function-adaptive notion of sparsity that compares the expected regions used by the network to the minimal number needed to approximate a target within a fixed tolerance. Deep Neural Networks (DNNs) with Rectified Linear Unit (ReLU) activation functions are piecewise-linear functions whose expressive power can be studied via the number of linear regions that they create [1]-[3]. However, achieving such approximations for complicated functions typically demands substantial computational resources. The Lottery Ticket Hypothesis states that we can often remove many connections while maintaining similar performance, motivating the study of sparse DNNs [7].

We study the energy-optimal shortest path problem for electric vehicles (EVs) in large-scale road networks, where recuperated energy along downhill segments introduces negative energy costs. While traditional point-to-point pathfinding algorithms for EVs assume a known initial energy level, many real-world scenarios involving uncertainty in available energy require planning optimal paths for all possible initial energy levels, a task known as energy-optimal profile search. Existing solutions typically rely on specialized profile-merging procedures within a label-correcting framework that results in searching over complex profiles. In this paper, we propose a simple yet effective label-setting approach based on multi-objective A* search, which employs a novel profile dominance rule to avoid generating and handling complex profiles. We develop four variants of our method and evaluate them on real-world road networks enriched with realistic energy consumption data. Experimental results demonstrate that our energy profile A* search achieves performance comparable to energy-optimal A* with a known initial energy level.

Interval-valued outputs arise naturally in fields such as computational biology and survival analysis. In the latter setting, one is interested in predicting the time until some adverse event, such as death, occurs.

Can we incorporate discrete optimization algorithms within modern machine learning models?

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In this paper, we explore two fundamental first-order algorithms in convex optimization, namely, gradient descent (GD) and proximal gradient method (ProxGD). Our focus is on making these algorithms entirely adaptive by leveraging local curvature information of smooth functions.

Algorithms for clustering points in metric spaces is a long-studied area of research.

In this setting, the unknown regression parameter is allowed to vary in time.