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The most popular blog posts about math & algorithms on Hacker News. 2,391 posts.
It is a wonder that we have yet to officially write about probability theory on this blog. Probability theory underlies a huge portion of artificial intelligence, machine learning, and statistics, and a number of our future posts will rely on the ideas and terminology we lay out in this post. Our first formal theory…
When my friends hear me say that I’m buying a lottery ticket for a big draw I often get ...
In the previous post we discussed homomorphic encryption in it’s most abstract terms we presented Paillier as a partially homomorphic encryption . Thank you @mortendahlcs for making a clarification about the terms described :
An aggregation is some kind of summary of a set of data. This can be the sum, length, minimum, etc. It is quite common to want to calculate such a summary repeatedly, e.g. “the maximum noise level in dB for the past 30 seconds” for a nuisance detector. In such a case we say there is a sliding window over our data, and…
This article is a starting article in a series that is going to briefly skim through a good engineering textbook on linear algebra, making notes that help yo...
Generative AI tools like ChatGPT and Midjournery are able to replicate (and often exceed) human-like performance on tasks like taking exams, generating text and making art.
We have established that there is a homomorphism called θ from B, the PROP of diagrams, to Mat, the PROP of matrices. To do this, it was enough to say where θ takes the four stars of the story so f…
I'm the kind of person who reads encyclopedias front to back just for fun, and I've been fascinated by mathematical special functions at least since my first encounter with the gamma function as a teenager. My work implementing numerical and symbolic algorithms in mpmath , Arb and other software projects during the…
A hands-on introduction to elliptic curve cryptography. Start with curve geometry, build point addition and scalar multiplication, see why ECDLP is hard, and then use that math in ECDH, ECDSA, and ECIES.
Engineer friends often ask me: Graph Deep Learning sounds great, but are there any big commercial success stories? Is it being deployed in practical applications? Besides the obvious ones–recommendation systems at Pinterest, Alibaba and Twitter–a slightly nuanced success story is the Transformer architecture, which…
I’m thrilled to report that I’m a co-author of the article “ An ultra-stable gold-coordinated protein cage displaying reversible assembly “, which was recently published in Nature . This work is the result of an exciting collaboration between biochemists, physicists, structural biologists, mathematicians, and others…
In causal inference, it is often important to study varying treatment effects: for example, a treatment could be more effective for men than for women, or for healthy than for unhealthy patients. Suppose a study is designed to have 80% power to detect a main effect at a 95% confidence level. Further suppose that…
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.
Lambda Screen allows you to draw images using terms of pure lambda calculus. By using fixed-point combinators, you can even create infinitely detailed fractals.
See also Frequentism and Bayesianism: A Python-driven Primer , a peer-reviewed article partially based on this content.
The Game of Life is based on very simple rules, but artificial neural networks struggle to learn its rules through the classic deep learning training process.
This is a tale of two approaches to regular expression matching. One of them is in widespread use in the standard interpreters for many languages, including Perl. The other is used only in a few places, notably most implementations of awk and grep. The two approaches have wildly different performance characteristics:
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…
A global pandemic, apocalyptic fires, and the possible descent of the US into violent anarchy three days from now can do strange things to the soul. Bertrand Russell—and if he’d done no…
This is the first part of a series of posts on big changes for SymPy with particular focus on speed. What I am going to describe in these posts are:
I have a side project to find the longest line of sight on the planet 1 using a novel Total Viewshed algorithm . At a resolution of 3 arc-seconds (~92m x 92m), the planet contains around 4.5 billion elevation samples 2 . Now obviously we don’t need to calculate the visibility between literally every single one of…
The fun of owning rural property is that knowing where your land actually starts and ends is a guess usually. Sure, you have a parcel description, and maybe even an online service with “official” GIS boundaries (for us here in New Brunswick, this is GeoNB ). However, since it’s in the middle of the forest, the odds of…
At least a familiarity with the lambda calculus, including how it is evaluated. Some base knowledge of programming languages is also assumed.
Originally developed by Aristid Lindenmayer in 1968, an L-System is a formalism to generate self-similar structures. Although, L-Systems are capable of generating famous fractals like the Sierpinski Triangle (Axiom: FXF--FF--FF, F → FF , X → --FXF++FXF++FXF-- , angle = 60 ) or the Gosper Curve (Axiom: FX, X →…
Generative Adversarial Networks refer to a family of generative models that seek to discover the underlying distribution behind a certain data generating process. This distribution is discovered through an adversarial competition between a generator and a discriminator. As we saw in an earlier introductory post on…
These remarks from “THE GENERAL AND LOGICAL THEORY OF AUTOMATA” 1947 are enormously insightful – no surprise. Von Neumann essentially predicts the emergence of the field of analysis of algorithms and algorithmic complexity. Not only does he point out the importance of understanding how many steps it will take to carry…
Note: this post was written for Lean 3; the latest version, Lean 4, is a very different language. Turn back the clock to 2009: a confused physics major newly infatuated with math and computer science, I enrolled in MATH 273: Numbers and Proofs at the University of Calgary. This wasn’t my first encounter with…
How a bit of algorithm engineering and low-level details can alter what seems like a straightforward trade-off on a blackboard.
Singular Value Decomposition (SVD) is a fundamental concept in linear algebra, and it is particularly important in the field of machine learning for tasks such as dimensionality reduction, data compression, and noise reduction.
Memories from Oxford days Mathematical Institute src Peter Neumann, Professor of Mathematics at Queen’s College, Oxford University, passed away two weeks ago. Last Monday would have been his …
