Unitary.jl

This package implements a differentiable parametrization of a group of unitary matrices as described in paper Sum-Product-Transform Networks: Exploiting Symmetries using Invertible Transformations, Tomas Pevny, Vasek Smidl, Martin Trapp, Ondrej Polacek, Tomas Oberhuber, 2020 https://arxiv.org/abs/2005.01297

The actual "Dense" node implementing f(x) = σ.(W * x .+ b), where W is in svd form has moved to https://github.com/pevnak/SumProductTransform.jl to keep this simple. Since in the paper, we have experimented with different ways, how to efficiently implement Dense matrices featuring efficient inversion and calculation of determinant, the repository contains a little bit more.

• Givens - representation of a unitary matrix using Givens rotations
• UnitaryHouseholder - representation of a unitary matrix using Householder reflections, an approach common in Machine Learning
• LU - representation of a matrix using LU decomposition
• LDU - representation of a matrix using LDU decomposition

The usage is simple:

using Unitary, Flux, BenchmarkTools
using Unitary: Givens, lowup

x = randn(Float32, 50, 100)
xx = randn(Float32, 100, 50)

a = Givens(50)
@btime a * x;		
#  224.097 μs (4 allocations: 20.00 KiB)
@btime xx * a;	
#  79.517 μs (4 allocations: 20.00 KiB)

ps = Flux.params(a)
@btime gradient(() -> sum(a * x), ps);	# 890.323 μs (58 allocations: 71.52 KiB)
# 891.481 μs (60 allocations: 72.42 KiB)
@btime gradient(() -> sum(xx * a), ps);	# 473.158 μs (58 allocations: 71.52 KiB)
@ 468.794 μs (60 allocations: 72.42 KiB)

a = Givens(50)
@btime a * x;
# 646.874 μs (10154 allocations: 2.37 MiB)

@btime xx * a;
#  726.198 μs (10204 allocations: 2.39 MiB)

@btime gradient(() -> sum(a * x), ps);  
#  103.869 ms (44538 allocations: 179.60 MiB)

@btime gradient(() -> sum(xx * a), ps);
#  105.061 ms (44688 allocations: 179.67 MiB)

Matrices support only multiplication, because that is what they have been designed for, but you can always convert them to normal matrices using Matrix (but this is not at the moment differentiable).