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SeaHash: Explained
I explain how SeaHash works.
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Interactive Numerical Optimization Tutorial
Numerical Optimization is one of the central techniques in Machine Learning. For many problems it is hard to figure out the best solution directly, but it is relatively easy to set up a loss function that measures how good a solution is - and then minimize the parameters of that function to find the solution.
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How should mathematics be taught to non-mathematicians? (2012)
Michael Gove, the UK’s Secretary of State for Education, has expressed a wish to see almost all school pupils studying mathematics in one form or another up to the age of 18. An obvious quest…
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Black Hole Tech?
In celebration of the detection of gravitational waves, Stephen Wolfram looks forward and discusses what technology black holes could make possible.
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Zero Knowledge Proofs: An illustrated primer (2014)
One of the best things about modern cryptography is the beautiful terminology. You could start any number of punk bands (or Tumblrs) named after cryptography terms like ‘hard-core predicate&#…
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Don’t invert that matrix (2010)
There is hardly ever a good reason to invert a matrix. What do you do if you need to solve Ax = b where A is an n x n matrix? Isn't the solution A−1 b? Yes, theoretically. But that doesn't mean you need to actually find A−1. Solving the equation Ax = b is
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Four pitfalls of hill climbing
An animated look at local maxima, novelty effects, differentiation, emergent maxima.
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Space Emerging from Quantum Mechanics
The other day I was amused to find a quote from Einstein , in 1936, about how hard it would be to quantize gravity: “like an attempt to breathe in empty space.” Eight decades later, I think we can still agree that it’s hard.
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Singular Value Decomposition Part 2: Theorem, Proof, Algorithm
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. This post will be theorem, proof, algorithm, data. The data set we test on is a thousand-story CNN news data set. All of the data, code, and…
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Towards deep symbolic reinforcement learning
Towards deep symbolic reinforcement learning Garnelo et al, 2016 Every now and then I read a paper that makes a really strong connection with me, one where I can’t stop thinking about the imp…
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“American Scientist” Understands Nothing about the Traveling Salesman Problem
I like to think there are some solid foundations to my life. I will be able to do the Monday New York Times crossword and I will not be able to do the Thursday version. My dental hygienist will not be satisfied with the amount of flossing I do. I will get my favorite spaghetti meal served on my birthday.
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Prime After Prime
Are the prime numbers sprinkled randomly along the number line, like windblown seeds? Of course not: Primality is not a matter of chance but a product of simple arithmetic. A number is prime if and only if no smaller positive integer other than 1 divides it evenly.
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Understanding gradient descent
Gradient descent is a standard tool for optimizing complex functions iteratively within a computer program. Its goal is: given some arbitrary function, find a minumum. For some small subset of functions - those that are convex - there's just a single minumum which also happens to be global. For most realistic…
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Markov Chain Monte Carlo Sampling
📍 This article has moved. Read the current version at This is the third part in a short series of blog posts about quantum Monte Carlo (QMC). The series is derived from an introductory lecture I ga…
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A Guide to Bayesian Statistics
A guide for getting started with Bayesian Statistics! Start your way with Bayes' Theorem and end up building your own Bayesian Hypothesis test!
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The Math Inside the US Highway System
At first glance, the US highway seems like a jumble of roads: However, the numbers follow a grid, with rules nobody told me (image source, click to enlarge):
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What is the roundest country?
We can define roundness in many ways. For example, as you may know, the circle is the shape that given a fixed perimeter maximizes the area. This definition has many problems. One of the problems is that countries generally have chaotic perimeters (also known as borders), so they tend to be much longer than they seem…
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Introduction to Mean Shift Algorithm
Its been quite some time since I wrote a Data Mining post . Today, I intend to post on Mean Shift – a really cool but not very well known algorithm. The basic idea is quite simple but the results a…
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Inspecting C's qsort Through Animation
The C standard library includes a qsort() function for sorting arbitrary buffers given a comparator function. The name comes from its original Unix implementation, “quicker sort,” a variation of the well-known quicksort algorithm. The C standard doesn’t specify an algorithm, except to say that it may be unstable (C99…
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Dijkstra's Algorithm
I was at SCNA the other day, and someone approached me about TDD and Dijkstra’s algorithm. He wondered if you could find a sequence of tests that would lead you to the algorithm. This sounded like a fun little exercise, so I decided to give it a try here.
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Spikey Spheres
A word of warning ... I'm presenting here something called an analogy. It's not perfect, it has problems, it's not exact, but there are ways in which it can aid your thinking. I'm giving a way of visualising high-dimensional spheres, a way that has its limitations, but which can help avoid making mistakes based on…
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What is Symbolic Computation?
I've been reading a lot of Lisp books lately, some more advanced than others. All of the introductory books I've seen cover the idea of symbolic computation, but most of them breeze past it with just a few pages about "symbols" and "quoting". For many programmers coming from non-Lisp languages these are new, foreign…
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Perlin noise
I used Perlin noise for the fog effect and title screen in Under Construction. I tweeted about my efforts to speed it up, and several people replied either confused about how Perlin noise works or not clear on what it actually is. I admit I only (somewhat) understand Perlin noise in the first place because I’ve…