Problem: Compute distance between points with uncertain locations (given by samples, or differing observations, or clusters). For example, if I have the following three “points” in the plane, as indicated by their colors, which is closer, blue to green, or blue to red? It’s not obvious, and there are multiple factors at work: the red points have fewer samples, but we can be more certain about the position;

When NP-hardness pops up on the internet, say because some silly blogger wants to write about video games, it’s often tempting to conclude that the problem being proved NP-hard is actually very hard! “Scientists proved Super Mario is NP-hard? I always knew there was a reason I wasn’t very good at it!” Sorry, these two are unrelated. NP-hardness means hard in a narrow sense this post should hopefully make clear.

Binary search is one of the most basic algorithms I know. Given a sorted list of comparable items and a target item being sought, binary search looks at the middle of the list, and compares it to the target. If the target is larger, we repeat on the smaller half of the list, and vice versa. With each comparison the binary search algorithm cuts the search space in half.

Previously in this series: Linear programming and healthy diets — Part 1 Linear programing and the simplex algorithm Foods of the Father My dad’s an interesting guy. Every so often he picks up a health trend and/or weight loss goal that would make many people’s jaw drop.

Last week I was in Boston for the Geometry of Redistricting workshop. It was an optimistic gathering of over 500 mathematicians, computer scientists, lawyers, policy makers, teachers, and interested people of all stripes. There was a ton of information in the talks and subsequent discussions. I’ll try to distill the main ideas and avenues for research as best I can.

Problem: Express a boolean logic formula using polynomials. I.e., if an input variable $ x$ is set to $ 0$, that is interpreted as false, while $ x=1$ is interpreted as true. The output of the polynomial should be 0 or 1 according to whether the formula is true or false as a whole. Solution: You can do this using a single polynomial.

As a fun side project to distract me from my abysmal progress on my book, I decided to play around with the math genealogy graph! For those who don’t know, since 1996, mathematicians, starting with the labor of Harry Coonce et al, have been managing a database of all mathematicians. More specifically, they’ve been keeping track of who everyone’s thesis advisors and subsequent students were.

This post is a sequel to Formulating the Support Vector Machine Optimization Problem. The Karush-Kuhn-Tucker theorem Generic optimization problems are hard to solve efficiently. However, optimization problems whose objective and constraints have special structure often succumb to analytic simplifications.

The hypothesis and the setup This blog post has an interactive demo (mostly used toward the end of the post). The source for this demo is available in a Github repository. Last time we saw how the inner product of two vectors gives rise to a decision rule: if $ w$ is the normal to a line (or hyperplane) $ L$, the sign of the inner product $ \langle x, w \rangle$ tells you whether $ x$ is on the same side of $ L$ as $ w$.

The standard inner product of two vectors has some nice geometric properties. Given two vectors $ x, y \in \mathbb{R}^n$, where by $ x_i$ I mean the $ i$-th coordinate of $ x$, the standard inner product (which I will interchangeably call the dot product) is defined by the formula $$\displaystyle \langle x, y \rangle = x_1 y_1 + \dots + x_n y_n$$ This formula, simple as it is, produces a lot of interesting geometry.

Problem: Determine if two polynomial expressions represent the same function. Specifically, if $ p(x_1, x_2, \dots, x_n)$ and $ q(x_1, x_2, \dots, x_n)$ are a polynomial with inputs, outputs and coefficients in a field $ F$, where $ |F|$ is sufficiently large, then the problem is to determine if $ p(\mathbf{x}) = q(\mathbf{x})$ for every $ x \in F$, in time polynomial in the number of bits required to write down $ p$ and $ q$.