Classification of $N$ points becomes a simultaneous control problem when viewed through the lens of neural ordinary differential equations (neural ODEs), which represent the time-continuous limit of residual networks. For the narrow model, with one neuron per hidden layer, it has been shown that the task can be achieved using $O(N)$ neurons. In this study, we focus on estimating the number of neurons required for efficient cluster-based classification, particularly in the worst-case scenario where points are independently and uniformly distributed in $[0,1]^d$. Our analysis provides a novel method for quantifying the probability of requiring fewer than $O(N)$ neurons, emphasizing the asymptotic behavior as both $d$ and $N$ increase. Additionally, under the sole assumption that the data are in general position, we propose a new constructive algorithm that simultaneously classifies clusters of $d$ points from any initial configuration, effectively reducing the maximal complexity to $O(N/d)$ neurons.

An employer in Spain may not be able to fire a worker caught on a surveillance camera doing something prohibited if the company hasn't informed workers about the video system and its purpose, according to a recent trial court decision. In a case involving an employee fired after a security camera captured him in a parking-lot fight after work hours, a Pamplona labor court ruled that the video evidence was inadmissible under the European Union's General Data Protection Regulation (GDPR) and case law from the European Court of Human Rights (ECHR). "The judgment is of great interest since it is the first ruling by a Spanish court on the validity that can be given to the evidence of video recordings after the publication of the new Spanish Data Protection Law and also an interpretation of the new European Data Protection Regulation," according to a blog post from Manuel Vargas of Barcelona's Marti & Associats law firm. Under Spain's own data-protection law, employers who record a worker doing something illegal are considered to have fulfilled their duty to inform so long as they have posted a sign identifying a video surveillance zone, Vargas wrote. He also noted that recent case law from the Spanish Supreme Court endorses the idea that employers aren't obligated to notify workers that they plan to use video cameras to monitor their activity for possible disciplinary purposes.