At the root of the fast transform is the simple fact that

    ax + bx = (a+b)x

The right hand side has fewer arithmetic operations. It's about finding common factors and pushing parentheses in. Because of the inherent symmetry of the FT expression there are lots of opportunities for this optimization.

Efficient decoding of LDPC codes also use the same idea. LDPCs were quite a revolution (pun intended) in coding/information theory.

On the other hand, something completely random, few days ago I found out that Tukey (then a Prof) and Feynman (then a student) along with other students were so enamored and intrigued by flexagons that they had set up an informal committee to understand them. Unfortunately their technical report never got published because the war intervened.

Strangely, it does not find a mention in Surely You're Joking.

▲ rigtorp 4 days ago | parent | next [-]

How is belief propagation used for decoding LDPC codes related to FFT?

▲ srean 4 days ago | parent [-]

At the core both derive their optimization from the distributive property. If the expression graph has symmetry, you get more optimization out of it.

https://www.cs.ubc.ca/~murphyk/Teaching/Papers/GDL.pdf

Check out the first paragraph

    THE humble distributive
    law, in its simplest form
    states that...this leads
    to a large family of fast
    algorithms, including 
    Viterbi’s algorithm and 
    the fast Fourier
    transform (FFT).

Two extremely influential papers appeared back to back in transactions information theory. This is one of them.

The other is

https://vision.unipv.it/IA2/Factor graphs and the sum-product algorithm.pdf

Both are absolute gems of papers. The editor made sure that both appear in the same volume.

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rigtorp 4 days ago | parent | next [-]

Interesting, of course many computations can be expressed as a graph. In the case of the bipartite graph we perform belief propagation on to decode LDPC where is the optimization from the distributive property? The parity matrix would typically be constructed so that there's few subexpression to factor out, to maximize the error correcting properties.

I agree both FFT and belief propagation can be expressed as message passing algorithms.

	▲	srean 4 days ago \| parent [-]
		It shows up in pushing in the parenthesis and pulling common terms out in the expression that is a sum (over all possible assignments) of products of terms. Doing the summation the naive way will be exponential in the number of variables. The goal is to this in an efficient way exploiting the distributive property and symmetry if any, much like in the FFT case. This can be done efficiently, for example, when the graph is a tree. (Even if it isn't, one can pretend as if it is. Surprisingly that often works very well but that's a different topic entirely) Read the paper it's not difficult to follow.

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kqbx 4 days ago | parent | prev | next [-]

The second link is broken on HN because it contains a space. Here's a clickable version: https://vision.unipv.it/IA2/Factor%20graphs%20and%20the%20su...

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adamnemecek 4 days ago | parent | prev [-]

There’s a whole subfield called generalized distributive law https://en.wikipedia.org/wiki/Generalized_distributive_law

▲ ajross 4 days ago | parent | prev [-]

> At the root of the fast transform is the simple fact that

Actually... no? That's a constant factor optimization; the second expression has 75% the operations of the first. The FFT is algorithmically faster. It's O(N·log2(N)) in the number of samples instead of O(N²).

That property doesn't come from factorization per se, but from the fact that the factorization can be applied recursively by creatively ordering the terms.

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srean 4 days ago | parent | next [-]

It's the symmetry that gives recursive opportunities to apply the optimization. It's the same optimization folded over and over again. Butterfly diagrams are great for understanding this. https://news.ycombinator.com/item?id=45291978 has pointers to more in depth exploration of the idea.

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emil-lp 4 days ago | parent | prev [-]

Well, actually ... Summation is linear time, multiplication is superlinear (eg n log n in number of digits).

Meaning that this takes k summations and one multiplication rather than k multiplications and k summations.

... Where k is the number of terms.

	▲	ajross 4 days ago \| parent [-]
		"Digits" are constant in an FFT (or rather ignored, really, precision is out of scope of the algorithm definition). Obviously in practice these are implemented as (pairs of, for a complex FFT, though real-valued DCTs are much more common) machine words in practice, and modern multipliers and adders pipeline at one per cycle.