Such distance matrices are commonly computed in software packages and have applications to learning image manifolds, handwriting recognition, and multi-dimensional unfolding, among other things. In an attempt to reduce their description size, we study low rank approximation of such matrices. Our main result is to show that for any underlying distance metric $d$, it is possible to achieve an additive error low rank approximation in sublinear time. We note that it is provably impossible to achieve such a guarantee in sublinear time for arbitrary matrices $\AA$, and our proof exploits special properties of distance matrices. We develop a recursive algorithm based on additive projection-cost preserving sampling.

The Boston startup uses AI to translate and verify legacy software for defense contractors, arguing modernization can't come at the cost of new bugs. Code Metal, a Boston-based startup that uses AI to write code and translate it into other programming languages, just closed a $125 million Series B funding round from new and existing investors. The news comes just a few months after the startup raised $36 million in series A financing led by Accel. Code Metal is part of a new wave of startups aiming to modernize the tech industry by using AI to generate code and translate it across programming languages. One of the questions that persists about AI-assisted code, though, is whether the output is any good--and what the consequences might be if it's not.

The need to analyze graphs is ubiquitous across various fields, from social networks tobiological research and recommendation systems.