The quadratic sandwich

If you have ever tried to minimize a function with gradient descent, you probably noticed that some functions are a joy to optimize and others are a nightmare. The difference often boils down to two properties: strong convexity and L-smoothness. These two concepts define a “sandwich” of quadratic bounds around your function that tells you exactly how well-behaved it is. If the sandwich is tight, life is good. If one slice of bread is missing, things get ugly fast.

Think Linear Algebra — Think Linear Algebra

What is an elliptic curve? Informal and formal definition

The informal definition of an elliptic curve is simple. The formal definition is more involved.

Beyond Smoothed Analysis: Analyzing the Simplex Method by the Book

Narrowing the gap between theory and practice is a longstanding goal of the algorithm analysis community. To further progress our understanding of how algorithms work in practice, we propose a new algorithm analysis framework that we call by the book analysis. In contrast to earlier frameworks, by the book analysis not only models an algorithm's input data, but also the algorithm itself. Results from by the book analysis are meant to correspond well with established knowledge of an algorithm's practical behavior, as they are meant to be grounded in observations from implementations, input modeling best practices, and measurements on practical benchmark instances. We apply our framework to the simplex method, an algorithm which is beloved for its excellent performance in practice and notorious for its high running time under worst-case analysis. The simplex method similarly showcased the state of the art framework smoothed analysis (Spielman and Teng, STOC'01). We explain how our framework overcomes several weaknesses of smoothed analysis and we prove that under input scaling assumptions, feasibility tolerances and other design principles used by simplex method implementations, the simplex method indeed attains a polynomial running time.

A Trick for Backpropagation of Linear Transformations

Linear transformations such as sums, matrix products, dot products, Hadamard products, and many more can often be represented using an [einsum](https://numpy.org/doc/stable/reference/generated/numpy.

What Is the Fourier Transform? | Quanta Magazine

Amid the chaos of revolutionary France, one man’s mathematical obsession gave way to a calculation that now underpins much of mathematics and physics. The calculation, called the Fourier transform, decomposes any function into its parts.

Busy Beaver Hunters Reach Numbers That Overwhelm Ordinary Math | Quanta Magazine

The quest to find the longest-running simple computer program has identified a new champion. It’s physically impossible to write out the numbers involved using standard mathematical notation.

Korea TSPs

Grid Computing Shatters World Record for Goldbach Conjecture Verification

I’ve achieved a new world record in verifying the Goldbach Conjecture, a famous unsolved problem in mathematics, by extending the verification up to 4 quadrillion (4×10¹⁸) + 70 trillion (7×10¹³)…

‘Once in a Century’ Proof Settles Math’s Kakeya Conjecture

The deceptively simple Kakeya conjecture has bedeviled mathematicians for 50 years. A new proof of the conjecture in three dimensions illuminates a whole crop of related problems.