Math ∩ Programming

Last time we discussed the setup for the silent duel problem: two players taking actions in $ [0,1]$, player 1 gets $ n$ chances to act, player 2 gets $ m$, and each knows their probability of success when they act.

Two men start running at each other with loaded pistols, ready to shoot!

For the last four years I’ve been working on a book for programmers who want to learn mathematics.

Mathematics students often hear about the classic “blue-eyed islanders” puzzle early in their career.

Over at Math3ma, Tai-Danae Bradley shared the following puzzle, which she also featured in a fantastic (spoiler-free) YouTube video.

Every now and then I hear some ridiculous things about the equals symbol.

Tai-Danae Bradley is one of the hosts of PBS Infinite Series, a delightful series of vignettes into fun parts of math.

Problem: Compute distance between points with uncertain locations (given by samples, or differing observations, or clusters).

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!

Binary search is one of the most basic algorithms I know.

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.

Last week I was in Boston for the Geometry of Redistricting workshop.

Problem: Express a boolean logic formula using polynomials.

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!

This post is a sequel to Formulating the Support Vector Machine Optimization Problem.

The hypothesis and the setup This blog post has an interactive demo (mostly used toward the end of the post).

The standard inner product of two vectors has some nice geometric properties.

Problem: Determine if two polynomial expressions represent the same function.

Problem: You have a catalog of items with discrete ratings (thumbs up/thumbs down, or 5-star ratings, etc.), and you want to display them in the “right” order.

papad Hard to believe Sanjeev Arora and his coauthors consider it “a basic tool [that should be] taught to all algorithms students together with divide-and-conquer, dynamic programming, and random sampling.”

For fixed integers $ r > 0$, and odd $ g$, a Moore graph is an $ r$-regular graph of girth $ g$ which has the minimum number of vertices $ n$ among all such graphs with the same regularity and girth.

This is a guest post by my friend and colleague Samantha Davies.

I wrote a guest post for my friend Samantha Davies’s blog, With High Probability.

The next Monday, when the fathers were all back at work, we kids were playing in a field.

Last time, we saw a specific zero-knowledge proof for graph isomorphism.

Problem: Design a random number generator that is computationally indistinguishable from a truly random number generator.

In this post we’ll get a strong taste for zero knowledge proofs by exploring the graph isomorphism problem in detail.

I’m just going to jump right into the definitions and rigor, so if you haven’t read the previous post motivating the singular value decomposition, go back and do that first.

The singular value decomposition (SVD) of a matrix is a fundamental tool in computer science, data analysis, and statistics.

Variations on a theme Back in 2014 I wrote a post called How to Conquer Tensorphobia that should end up on Math $ \cap$ Programming’s “greatest hits” album.

Data is abundant, data is big, and big is a problem.

So far in this series we’ve seen a lot of motivation and defined basic ideas of what a quantum circuit is.

Problem: Estimate the number of distinct items in a data stream that is too large to fit in memory.

Here’s a bit of folklore I often hear (and retell) that’s somewhere between a joke and deep wisdom: if you’re doing a software interview that involves some algorithms problem that seems hard, your best bet is to use hash tables.

Math and computer science are full of inequalities, but there is one that shows up more often in my work than any other.

Update 2017-01-09: Laci claims to have found a workaround to the previously posted error, and the claim is again quasipolynoimal time!

I was recently an invited speaker in a series of STEM talks at Moraine Valley Community College.

If you haven’t read the first post on fairness, I suggest you go back and read it because it motivates why we’re talking about fairness for algorithms in the first place.

There’s a well-understood phenomenon in machine learning called overfitting.

In this post we’ll implement Reed-Solomon error-correcting codes and use them to play with codes.

It’s about time we got back to computational topology.

In 2014 the White House commissioned a 90-day study that culminated in a report (pdf) on the state of “big data” and related technologies.

A while back we featured a post about why learning mathematics can be hard for programmers, and I claimed a major issue was not understanding the basic methods of proof (the lingua franca between intuition and rigorous mathematics).

When addressing the question of what it means for an algorithm to learn, one can imagine many different models, and there are quite a few.

A while back Peter Norvig posted a wonderful pair of articles about regex golf.

I have a little secret: I don’t like the terminology, notation, and style of writing in statistics.

Last time we defined the Hamming code.

Problem: Given a massive data stream of $ n$ values in $ \{ 1, 2, \dots, m \}$ and the guarantee that one value occurs more than $ n/2$ times in the stream, determine exactly which value does so.

Or how to detect and correct errors Last time we made a quick tour through the main theorems of Claude Shannon, which essentially solved the following two problems about communicating over a digital channel.

There are two basic problems in information theory that are very easy to explain.