The simple essence of automatic differentiation
Abstract Automatic differentiation (AD) in reverse mode (RAD) is a central component of deep learning and other uses of large-scale optimization. Commonly used RAD algorithms such as backpropagation, however, are complex and stateful, hindering deep understanding, improvement, and parallel execution. This paper develops a simple, generalized AD algorithm calculated from a simple, natural specifica
Beautiful differentiation
March 2009 Appeared in ICFP 2009 Abstract Automatic differentiation (AD) is a precise, efficient, and convenient method for computing derivatives of functions. Its forward-mode implementation can be quite simple even when extended to compute all of the higher-order derivatives as well. The higher-dimensional case has also been tackled, though with extra complexity. This paper develops an implement
From Haskell to Hardware via CCCs
From Haskell to Hardware via CCCs Conal Elliott April, 2015 Conal Elliott From Haskell to Hardware via CCCs April, 2015 1 / 70 Tabula Founded by Steve Teig about 11 years ago. Newly defunct. :( Post-FPGA hardware architecture, Spacetime: 3D for shorter paths. Implemented by rapid reconfiguration (2GHz). Dense active logic. Program in a non-sequential language: Haskell. Compiler developed openly an
Push-pull functional reactive programming
AbstractFunctional reactive programming (FRP) has simple and powerful semantics, but has resisted efficient implementation. In particular, most past implementations have used demand-driven sampling, which accommodates FRP’s continuous time semantics and fits well with the nature of functional programming. Consequently, values are wastefully recomputed even when inputs don’t change, and reaction la
Conal Elliott » “Everything is a function” in Haskell?
Inspirations & experiments, mainly about denotative/functional programming in Haskell There a belief about Haskell that keeps popping up in chat rooms and mailing lists — one that I’ve been puzzling over for a while. One expression of the belief is “everything is a function” in Haskell. Of course, there are all of these non-functions that need to be accounted for, including integers, booleans, tup
Conal Elliott » Circuits as a bicartesian closed category
My previous few posts have been about cartesian closed categories (CCCs). In From Haskell to hardware via cartesian closed categories, I gave a brief motivation: typed lambda expressions and the CCC vocabulary are equally expressive, but have different strengths: In Haskell, the CCC vocabulary is overloadable and so can be interpreted more flexibly than lambda and application. Lambda expressions a
Conal Elliott » Optimizing CCCs
In the post Overloading lambda, I gave a translation from a typed lambda calculus into the vocabulary of cartesian closed categories (CCCs). This simple translation leads to unnecessarily complex expressions. For instance, the simple lambda term, “λ ds → (λ (a,b) → (b,a)) ds”, translated to a rather complicated CCC term: apply ∘ (curry (apply ∘ (apply ∘ (const (,) △ (id ∘ exr) ∘ exr) △ (id ∘ exl)
Conal Elliott » From Haskell to hardware via cartesian closed categories
Since fall of last year, I’ve been working at Tabula, a Silicon Valley start-up developing an innovative programmable hardware architecture called “Spacetime”, somewhat similar to an FPGA, but much more flexible and efficient. I met the founder, Steve Teig, at a Bay Area Haskell Hackathon in February of 2011. He described his Spacetime architecture, which is based on the geometry of the same name,
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Compiling to Categories
Conal Elliott » Lazier function definitions by merging partial values
Elegant memoization
Conal Elliott » Elegant memoization with higher-order types
Conal Elliott » Overloading lambda
Folds and unfolds all around us