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annotate src/LazyTensors/lazy_tensor_operations.jl @ 474:3ad327378b2d feature/outer_product

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Simplify implementation of \otimes
author Jonatan Werpers <jonatan@werpers.com>
date Mon, 26 Oct 2020 23:01:47 +0100
parents 0c3decc04649
children 2c0e76d5832d
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1 """
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2 LazyTensorMappingApplication{T,R,D} <: LazyArray{T,R}
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3
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4 Struct for lazy application of a TensorMapping. Created using `*`.
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5
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6 Allows the result of a `TensorMapping` applied to a vector to be treated as an `AbstractArray`.
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7 With a mapping `m` and a vector `v` the LazyTensorMappingApplication object can be created by `m*v`.
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8 The actual result will be calcualted when indexing into `m*v`.
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9 """
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10 struct LazyTensorMappingApplication{T,R,D, TM<:TensorMapping{T,R,D}, AA<:AbstractArray{T,D}} <: LazyArray{T,R}
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11 t::TM
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12 o::AA
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13 end
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14 # TODO: Do boundschecking on creation!
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15 export LazyTensorMappingApplication
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16
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17 # TODO: Go through and remove unneccerary type parameters on functions
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19 Base.getindex(ta::LazyTensorMappingApplication{T,R,D}, I::Vararg{Index,R}) where {T,R,D} = apply(ta.t, ta.o, I...)
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20 Base.getindex(ta::LazyTensorMappingApplication{T,R,D}, I::Vararg{Int,R}) where {T,R,D} = apply(ta.t, ta.o, Index{Unknown}.(I)...)
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21 Base.size(ta::LazyTensorMappingApplication) = range_size(ta.t)
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22 # TODO: What else is needed to implement the AbstractArray interface?
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23
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24 Base.:*(a::TensorMapping, v::AbstractArray) = LazyTensorMappingApplication(a,v)
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25 Base.:*(a::TensorMapping, b::TensorMapping) = throw(MethodError(Base.:*,(a,b)))
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26 Base.:*(a::TensorMapping, args::Union{TensorMapping, AbstractArray}...) = foldr(*,(a,args...))
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28 # # We need the associativity to be a→b→c = a→(b→c), which is the case for '→'
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29 # # Should we overload some other infix binary opesrator?
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30 # →(tm::TensorMapping{T,R,D}, o::AbstractArray{T,D}) where {T,R,D} = LazyTensorMappingApplication(tm,o)
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31 # TODO: We need to be really careful about good error messages.
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32 # For example what happens if you try to multiply LazyTensorMappingApplication with a TensorMapping(wrong order)?
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33
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34 """
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35 LazyTensorMappingTranspose{T,R,D} <: TensorMapping{T,D,R}
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36
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37 Struct for lazy transpose of a TensorMapping.
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38
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39 If a mapping implements the the `apply_transpose` method this allows working with
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40 the transpose of mapping `m` by using `m'`. `m'` will work as a regular TensorMapping lazily calling
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41 the appropriate methods of `m`.
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42 """
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43 struct LazyTensorMappingTranspose{T,R,D, TM<:TensorMapping{T,R,D}} <: TensorMapping{T,D,R}
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44 tm::TM
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45 end
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46 export LazyTensorMappingTranspose
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47
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48 # # TBD: Should this be implemented on a type by type basis or through a trait to provide earlier errors?
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49 # Jonatan 2020-09-25: Is the problem that you can take the transpose of any TensorMapping even if it doesn't implement `apply_transpose`?
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50 Base.adjoint(tm::TensorMapping) = LazyTensorMappingTranspose(tm)
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51 Base.adjoint(tmt::LazyTensorMappingTranspose) = tmt.tm
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52
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53 apply(tmt::LazyTensorMappingTranspose{T,R,D}, v::AbstractArray{T,R}, I::Vararg{Index,D}) where {T,R,D} = apply_transpose(tmt.tm, v, I...)
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54 apply_transpose(tmt::LazyTensorMappingTranspose{T,R,D}, v::AbstractArray{T,D}, I::Vararg{Index,R}) where {T,R,D} = apply(tmt.tm, v, I...)
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55
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56 range_size(tmt::LazyTensorMappingTranspose) = domain_size(tmt.tm)
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57 domain_size(tmt::LazyTensorMappingTranspose) = range_size(tmt.tm)
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59
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60 struct LazyTensorMappingBinaryOperation{Op,T,R,D,T1<:TensorMapping{T,R,D},T2<:TensorMapping{T,R,D}} <: TensorMapping{T,D,R}
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61 tm1::T1
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62 tm2::T2
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64 @inline function LazyTensorMappingBinaryOperation{Op,T,R,D}(tm1::T1,tm2::T2) where {Op,T,R,D, T1<:TensorMapping{T,R,D},T2<:TensorMapping{T,R,D}}
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65 return new{Op,T,R,D,T1,T2}(tm1,tm2)
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66 end
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67 end
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68 # TODO: Boundschecking in constructor.
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70 apply(tmBinOp::LazyTensorMappingBinaryOperation{:+,T,R,D}, v::AbstractArray{T,D}, I::Vararg{Index,R}) where {T,R,D} = apply(tmBinOp.tm1, v, I...) + apply(tmBinOp.tm2, v, I...)
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71 apply(tmBinOp::LazyTensorMappingBinaryOperation{:-,T,R,D}, v::AbstractArray{T,D}, I::Vararg{Index,R}) where {T,R,D} = apply(tmBinOp.tm1, v, I...) - apply(tmBinOp.tm2, v, I...)
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72
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73 range_size(tmBinOp::LazyTensorMappingBinaryOperation{Op,T,R,D}) where {Op,T,R,D} = range_size(tmBinOp.tm1)
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74 domain_size(tmBinOp::LazyTensorMappingBinaryOperation{Op,T,R,D}) where {Op,T,R,D} = domain_size(tmBinOp.tm1)
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75
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76 Base.:+(tm1::TensorMapping{T,R,D}, tm2::TensorMapping{T,R,D}) where {T,R,D} = LazyTensorMappingBinaryOperation{:+,T,R,D}(tm1,tm2)
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77 Base.:-(tm1::TensorMapping{T,R,D}, tm2::TensorMapping{T,R,D}) where {T,R,D} = LazyTensorMappingBinaryOperation{:-,T,R,D}(tm1,tm2)
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79 """
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80 TensorMappingComposition{T,R,K,D}
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81
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82 Lazily compose two TensorMappings, so that they can be handled as a single TensorMapping.
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83 """
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84 struct TensorMappingComposition{T,R,K,D, TM1<:TensorMapping{T,R,K}, TM2<:TensorMapping{T,K,D}} <: TensorMapping{T,R,D}
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85 t1::TM1
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86 t2::TM2
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87
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88 @inline function TensorMappingComposition(t1::TensorMapping{T,R,K}, t2::TensorMapping{T,K,D}) where {T,R,K,D}
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89 @boundscheck if domain_size(t1) != range_size(t2)
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90 throw(DimensionMismatch("the first argument has domain size $(domain_size(t1)) while the second has range size $(range_size(t2)) "))
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91 end
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92 return new{T,R,K,D, typeof(t1), typeof(t2)}(t1,t2)
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93 end
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94 # Add check for matching sizes as a boundscheck
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95 end
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96 export TensorMappingComposition
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97
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98 range_size(tm::TensorMappingComposition) = range_size(tm.t1)
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99 domain_size(tm::TensorMappingComposition) = domain_size(tm.t2)
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100
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101 function apply(c::TensorMappingComposition{T,R,K,D}, v::AbstractArray{T,D}, I::Vararg{S,R} where S) where {T,R,K,D}
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102 apply(c.t1, c.t2*v, I...)
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103 end
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104
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105 function apply_transpose(c::TensorMappingComposition{T,R,K,D}, v::AbstractArray{T,R}, I::Vararg{S,D} where S) where {T,R,K,D}
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106 apply_transpose(c.t2, c.t1'*v, I...)
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107 end
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108
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109 Base.@propagate_inbounds Base.:∘(s::TensorMapping, t::TensorMapping) = TensorMappingComposition(s,t)
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110
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111 """
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112 LazyLinearMap{T,R,D,...}(A, range_indicies, domain_indicies)
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113
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114 TensorMapping defined by the AbstractArray A. `range_indicies` and `domain_indicies` define which indicies of A should
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115 be considerd the range and domain of the TensorMapping. Each set of indices must be ordered in ascending order.
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116
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117 For instance, if A is a m x n matrix, and range_size = (1,), domain_size = (2,), then the LazyLinearMap performs the
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118 standard matrix-vector product on vectors of size n.
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119 """
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120 struct LazyLinearMap{T,R,D, RD, AA<:AbstractArray{T,RD}} <: TensorMapping{T,R,D}
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121 A::AA
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122 range_indicies::NTuple{R,Int}
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123 domain_indicies::NTuple{D,Int}
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124
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125 function LazyLinearMap(A::AA, range_indicies::NTuple{R,Int}, domain_indicies::NTuple{D,Int}) where {T,R,D, RD, AA<:AbstractArray{T,RD}}
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126 if !issorted(range_indicies) || !issorted(domain_indicies)
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127 throw(DomainError("range_indicies and domain_indicies must be sorted in ascending order"))
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128 end
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129
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130 return new{T,R,D,RD,AA}(A,range_indicies,domain_indicies)
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131 end
377
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132 end
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133 export LazyLinearMap
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134
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135 range_size(llm::LazyLinearMap) = size(llm.A)[[llm.range_indicies...]]
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136 domain_size(llm::LazyLinearMap) = size(llm.A)[[llm.domain_indicies...]]
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137
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138 function apply(llm::LazyLinearMap{T,R,D}, v::AbstractArray{T,D}, I::Vararg{Index,R}) where {T,R,D}
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139 view_index = ntuple(i->:,ndims(llm.A))
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140 for i ∈ 1:R
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141 view_index = Base.setindex(view_index, Int(I[i]), llm.range_indicies[i])
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142 end
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143 A_view = @view llm.A[view_index...]
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144 return sum(A_view.*v)
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145 end
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146
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147 function apply_transpose(llm::LazyLinearMap{T,R,D}, v::AbstractArray{T,R}, I::Vararg{Index,D}) where {T,R,D}
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148 apply(LazyLinearMap(llm.A, llm.domain_indicies, llm.range_indicies), v, I...)
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149 end
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150
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151
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152 """
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153 IdentityMapping{T,D} <: TensorMapping{T,D,D}
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154
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155 The lazy identity TensorMapping for a given size. Usefull for building up higher dimensional tensor mappings from lower
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156 dimensional ones through outer products. Also used in the Implementation for InflatedTensorMapping.
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157 """
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158 struct IdentityMapping{T,D} <: TensorMapping{T,D,D}
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159 size::NTuple{D,Int}
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160 end
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161 export IdentityMapping
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162
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163 IdentityMapping{T}(size::NTuple{D,Int}) where {T,D} = IdentityMapping{T,D}(size)
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164 IdentityMapping{T}(size::Vararg{Int,D}) where {T,D} = IdentityMapping{T,D}(size)
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165 IdentityMapping(size::Vararg{Int,D}) where D = IdentityMapping{Float64,D}(size)
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166
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167 range_size(tmi::IdentityMapping) = tmi.size
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168 domain_size(tmi::IdentityMapping) = tmi.size
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169
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170 apply(tmi::IdentityMapping{T,D}, v::AbstractArray{T,D}, I::Vararg{Any,D}) where {T,D} = v[I...]
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171 apply_transpose(tmi::IdentityMapping{T,D}, v::AbstractArray{T,D}, I::Vararg{Any,D}) where {T,D} = v[I...]
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172
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173
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174 """
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175 InflatedTensorMapping{T,R,D} <: TensorMapping{T,R,D}
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176
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177 An inflated `TensorMapping` with dimensions added before and afer its actual dimensions.
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178 """
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179 struct InflatedTensorMapping{T,R,D,D_before,R_middle,D_middle,D_after, TM<:TensorMapping{T,R_middle,D_middle}} <: TensorMapping{T,R,D}
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180 before::IdentityMapping{T,D_before}
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181 tm::TM
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182 after::IdentityMapping{T,D_after}
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183
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184 function InflatedTensorMapping(before, tm::TensorMapping{T}, after) where T
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185 R_before = range_dim(before)
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186 R_middle = range_dim(tm)
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187 R_after = range_dim(after)
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188 R = R_before+R_middle+R_after
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189
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190 D_before = domain_dim(before)
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191 D_middle = domain_dim(tm)
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192 D_after = domain_dim(after)
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193 D = D_before+D_middle+D_after
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194 return new{T,R,D,D_before,R_middle,D_middle,D_after, typeof(tm)}(before, tm, after)
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195 end
446
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196 end
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197 export InflatedTensorMapping
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198 """
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199 InflatedTensorMapping(before, tm, after)
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200 InflatedTensorMapping(before,tm)
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201 InflatedTensorMapping(tm,after)
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202
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203 The outer product of `before`, `tm` and `after`, where `before` and `after` are `IdentityMapping`s.
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204
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205 If one of `before` or `after` is left out, a 0-dimensional `IdentityMapping` is used as the default value.
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206 """
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207 InflatedTensorMapping(::IdentityMapping, ::TensorMapping, ::IdentityMapping)
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208 InflatedTensorMapping(before::IdentityMapping, tm::TensorMapping{T}) where T = InflatedTensorMapping(before,tm,IdentityMapping{T}())
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209 InflatedTensorMapping(tm::TensorMapping{T}, after::IdentityMapping) where T = InflatedTensorMapping(IdentityMapping{T}(),tm,after)
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210 # Resolve ambiguity between the two previous methods
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211 InflatedTensorMapping(I1::IdentityMapping{T}, I2::IdentityMapping{T}) where T = InflatedTensorMapping(I1,I2,IdentityMapping{T}())
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212
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213 # TODO: Implement syntax and constructors for products of different combinations of InflatedTensorMapping and IdentityMapping
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214
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215 # TODO: Implement some pretty printing in terms of ⊗. E.g InflatedTensorMapping(I(3),B,I(2)) -> I(3)⊗B⊗I(2)
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216
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217 function range_size(itm::InflatedTensorMapping)
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218 return flatten_tuple(
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219 range_size(itm.before),
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220 range_size(itm.tm),
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221 range_size(itm.after),
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222 )
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223 end
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224
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225 function domain_size(itm::InflatedTensorMapping)
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226 return flatten_tuple(
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227 domain_size(itm.before),
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228 domain_size(itm.tm),
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229 domain_size(itm.after),
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230 )
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231 end
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232
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233 function apply(itm::InflatedTensorMapping{T,R,D}, v::AbstractArray{T,D}, I::Vararg{Any,R}) where {T,R,D}
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234 view_index, inner_index = split_index(itm, I...)
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235
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236 v_inner = view(v, view_index...)
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237 return apply(itm.tm, v_inner, inner_index...)
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238 end
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239
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240
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241 """
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242 split_index(...)
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243
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244 Splits the multi-index into two parts. One part for the view that the inner TensorMapping acts on, and one part for indexing the result
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245 Eg.
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246 ```
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247 (1,2,3,4) -> (1,:,:,4), (2,3)
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248 ```
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249 """
452
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250 function split_index(itm::InflatedTensorMapping{T,R,D}, I::Vararg{Any,R}) where {T,R,D}
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251 I_before = slice_tuple(I, Val(1), Val(range_dim(itm.before)))
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252 I_after = slice_tuple(I, Val(R-range_dim(itm.after)+1), Val(R))
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253
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254 view_index = (I_before..., ntuple((i)->:,domain_dim(itm.tm))..., I_after...)
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255 inner_index = slice_tuple(I, Val(range_dim(itm.before)+1), Val(R-range_dim(itm.after)))
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256
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257 return (view_index, inner_index)
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258 end
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259
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260 # TODO: Can this be replaced by something more elegant while still being type stable? 2020-10-21
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261 # See:
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262 # https://github.com/JuliaLang/julia/issues/34884
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263 # https://github.com/JuliaLang/julia/issues/30386
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264 """
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265 slice_tuple(t, Val(l), Val(u))
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266
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267 Get a slice of a tuple in a type stable way.
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268 Equivalent to t[l:u] but type stable.
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269 """
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270 function slice_tuple(t,::Val{L},::Val{U}) where {L,U}
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271 return ntuple(i->t[i+L-1], U-L+1)
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272 end
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273
457
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274 """
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275 flatten_tuple(t)
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276
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277 Takes a nested tuple and flattens the whole structure
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278 """
446
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279 flatten_tuple(t::NTuple{N, Number} where N) = t
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280 flatten_tuple(t::Tuple) = ((flatten_tuple.(t)...)...,) # simplify?
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281 flatten_tuple(ts::Vararg) = flatten_tuple(ts)
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282
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283
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284 """
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285 LazyOuterProduct(tms...)
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286
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287 Creates a `TensorComposition` for the outerproduct of `tms...`.
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288 This is done by separating the outer product into regular products of outer products involving only identity mappings and one non-identity mapping.
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289
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290 First let
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291 ```math
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292 A = A_{I,J}
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293 B = B_{M,N}
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294 C = C_{P,Q}
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295 ```
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296
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297 where ``I``, ``M``, ``P`` are multi-indexes for the ranges of ``A``, ``B``, ``C``, and ``J``, ``N``, ``Q`` are multi-indexes of the domains.
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298
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299 We use ``⊗`` to denote the outer product
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300 ```math
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301 (A⊗B)_{IM,JN} = A_{I,J}B_{M,N}
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302 ```
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303
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304 We note that
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305 ```math
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306 A⊗B⊗C = (A⊗B⊗C)_{IMP,JNQ} = A_{I,J}B_{M,N}C_{P,Q}
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307 ```
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308 And that
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309 ```math
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310 A⊗B⊗C = (A⊗I_{|M|}⊗I_{|P|})(I_{|J|}⊗B⊗I_{|P|})(I_{|J|}⊗I_{|N|}⊗C)
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311 ```
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312 where |.| of a multi-index is a vector of sizes for each dimension. ``I_v`` denotes the identity tensor of size ``v[i]`` in each direction
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313 To apply ``A⊗B⊗C`` we evaluate
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314
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315 (A⊗B⊗C)v = [(A⊗I_{|M|}⊗I_{|P|}) [(I_{|J|}⊗B⊗I_{|P|}) [(I_{|J|}⊗I_{|N|}⊗C)v]]]
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316 """
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317 function LazyOuterProduct end
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318 export LazyOuterProduct
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319
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320 function LazyOuterProduct(tm1::TensorMapping{T}, tm2::TensorMapping{T}) where T
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321 itm1 = InflatedTensorMapping(tm1, IdentityMapping{T}(range_size(tm2)))
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322 itm2 = InflatedTensorMapping(IdentityMapping{T}(domain_size(tm1)),tm2)
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323
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324 return itm1∘itm2
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325 end
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326
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327 LazyOuterProduct(tms::Vararg{TensorMapping}) = foldl(LazyOuterProduct, tms)
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328
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329 ⊗(a::TensorMapping, b::TensorMapping) = LazyOuterProduct(a,b)
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330 export ⊗
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331
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332 # TODO: Can we implement compositions and kroneckers of LazyIdentities to just return new LazyIdentities?