Fourier Holographically Reduced Representations

src/HyperdimensionalComputing.jl

@@ -9,13 +9,15 @@ export AbstractHV,
     GradedBipolarHV,
     RealHV,
     GradedHV,
-    TernaryHV
+    TernaryHV,
+    FHRR
 
 include("representations.jl")
 
 include("operations.jl")
 export bundle,
     bind,
+    unbind,
     shift!,
     shift,
     ρ,

src/encoding.jl

@@ -493,25 +493,25 @@ end
 
 """
 	level(v::HV, n::Int) where {HV <: AbstractHV}
-	level(HV::Type{<:AbstractHV}, n::Int; dims::Int = 10_000)
+	level(HV::Type{<:AbstractHV}, m::Int; n::Int = 10_000)
 
 Creates a set of level correlated hypervectors, where the first and last hypervectors are quasi-orthogonal.
 
 # Arguments
 - `v::HV`: Base hypervector
-- `n::Int`: Number of levels (alternatively, provide a vector to be encoded)
+- `m::Int`: Number of levels (alternatively, provide a vector to be encoded)
 """
-function level(v::HV, n::Int) where {HV <: AbstractHV}
+function level(v::HV, m::Int) where {HV <: AbstractHV}
     hvs = [v]
-    p = 2 / n
-    while length(hvs) < n
+    p = 2 / m
+    while length(hvs) < m
         u = last(hvs)
         push!(hvs, perturbate(u, p))
     end
     return hvs
 end
 
-level(HV::Type{<:AbstractHV}, n::Int; dims::Int = 10_000) = level(HV(dims), n)
+level(HV::Type{<:AbstractHV}, m::Int; n::Int = 10_000) = level(HV(n), m)
 
 level(HVv, vals::AbstractVector) = level(HVv, length(vals))
 level(HVv, vals::UnitRange) = level(HVv, length(vals))
@@ -591,12 +591,37 @@ end
 
 decodelevel(hvlevels::AbstractVector{<:AbstractHV}, a::Number, b::Number) = decodelevel(hvlevels, range(a, b, length(hvlevels)))
 
-decodelevel(HV, numvalues; testbound = false) = decodelevel(level(HV, length(numvalues)), numvalues)
+decodelevel(HV, numvalues; testbound = false) = decodelevel(level(HV, length(numvalues)), numvalues; testbound)
 
 """
     convertlevel(hvlevels, numvals..., kwargs...)
+    convertlevel(HV::AbstractHV, numvals..., kwargs...)
 
 Creates the `encoder` and `decoder` for a level incoding in one step. See `encodelevel`
 and `decodelevel` for their respective documentations.
 """
 convertlevel(hvlevels, numvals...; kwargs...) = encodelevel(hvlevels, numvals...; kwargs...), decodelevel(hvlevels, numvals..., kwargs...)
+
+convertlevel(hv::AbstractHV, numvals...; kwargs...) = encodelevel(hv, numvals...; kwargs...), decodelevel(hv, numvals..., kwargs...)
+
+
+# levels using FHRR
+
+function level(v::FHRR, m::Int; β = 1 / m)
+    return [v^(x * β) for x in 1:m]
+end
+
+function level(v::FHRR, vals::Union{AbstractVector{<:Number}, UnitRange}; β = 1 / (maximum(vals) - minimum(vals)))
+    return [v^(x * β) for x in vals]
+end
+
+function encodelevel(v::FHRR, vals = (0, 1); β = 1 / (maximum(vals) - minimum(vals)))
+    return x -> v^(β * x)
+end
+
+function decodelevel(v::FHRR, vals = (0, 1); β = 1 / (maximum(vals) - minimum(vals)))
+    return u -> @.(real(log(u.v) / log(v.v) / β)) |> mean
+end
+
+
+convertlevel(v::FHRR, vals = (0, 1); kwargs...) = encodelevel(v, vals; kwargs...), decodelevel(v, vals; kwargs...)

src/inference.jl

@@ -20,6 +20,7 @@ similarity(u::GradedBipolarHV, v::GradedBipolarHV) = sim_cos(u, v)
 similarity(u::RealHV, v::RealHV) = sim_cos(u, v)
 similarity(u::BinaryHV, v::BinaryHV) = sim_jacc(u, v)
 similarity(u::GradedHV, v::GradedHV) = sim_jacc(u, v)
+similarity(u::FHRR, v::FHRR) = real(dot(u.v, v.v)) / length(u)
 
 """
     similarity(u::AbstractVector, v::AbstractVector; method::Symbol)

src/operations.jl

@@ -89,8 +89,8 @@ end
     return r
 end
 
-# AGGREGATION
-# -----------
+# BUNDLE
+# ------
 
 # binary and bipolar: use majority
 function bundle(hvr::Union{BinaryHV, BipolarHV}, hdvs, r)
@@ -143,6 +143,14 @@ function bundle(::GradedBipolarHV, hdvs, r)
     return GradedBipolarHV(r)
 end
 
+function bundle(::FHRR, hdvs, r)
+    for hv in hdvs
+        r .+= hv.v
+    end
+    r ./= abs.(r)
+    return FHRR(r)
+end
+
 function bundle(hdvs; kwargs...)
     hv = first(hdvs)
     r = empty_vector(hv)
@@ -153,17 +161,34 @@ Base.:+(hv1::HV, hv2::HV) where {HV <: AbstractHV} = bundle((hv1, hv2))
 
 # BINDING
 # -------
-
+Base.bind(hv1::HV, hv2::HV) where {HV <: AbstractHV} = HV(hv1.v .* hv2.v)  # default
 Base.bind(hv1::BinaryHV, hv2::BinaryHV) = BinaryHV(hv1.v .⊻ hv2.v)
 Base.bind(hv1::BipolarHV, hv2::BipolarHV) = BipolarHV(hv1.v .⊻ hv2.v)
 Base.bind(hv1::TernaryHV, hv2::TernaryHV) = TernaryHV(hv1.v .* hv2.v)
 Base.bind(hv1::RealHV, hv2::RealHV) = RealHV(hv1.v .* hv2.v)
 Base.bind(hv1::GradedHV, hv2::GradedHV) = GradedHV(fuzzy_xor.(hv1.v, hv2.v))
 Base.bind(hv1::GradedBipolarHV, hv2::GradedBipolarHV) = GradedBipolarHV(fuzzy_xor_bipol.(hv1.v, hv2.v))
+Base.bind(hv1::FHRR, hv2::FHRR) = FHRR(hv1.v .* hv2.v)
 Base.:*(hv1::HV, hv2::HV) where {HV <: AbstractHV} = bind(hv1, hv2)
 Base.bind(hvs::AbstractVector{HV}) where {HV <: AbstractHV} = prod(hvs)
 
 
+"""
+    unbind(hv1::HV, hv2::HV)
+
+Unbinds `hv2` from `hv1`. For many types of hypervectors, the binding operator is
+idempotent, i.e., `u * v * v == u`.
+
+Aliases with `\`.
+"""
+unbind(hv1::HV, hv2::HV) where {HV <: AbstractHV} = bind(hv1, hv2)
+
+unbind(hv1::RealHV, hv2::RealHV) where {HV <: AbstractHV} = RealHV(hv1.v ./ hv2.v)
+unbind(hv1::FHRR, hv2::FHRR) where {HV <: AbstractHV} = FHRR(hv1.v ./ hv2.v)
+
+Base.:/(hv1::HV, hv2::HV) where {HV <: AbstractHV} = unbind(hv1, hv2)
+
+
 # SHIFTING
 # --------
 
@@ -286,3 +311,8 @@ end
 perturbate!(hv, args...) = perturbate!(vectype(hv), hv, args...)
 
 perturbate(hv::AbstractHV, args...; kwargs...) = perturbate!(copy(hv), args...; kwargs...)
+
+# OTHER
+# -----
+
+Base.:^(hv::FHRR, x::Number) = FHRR(hv.v .^ x)

src/types.jl

@@ -18,12 +18,10 @@ Every hypervector HV has the following basic functionality
 TODO: 
 - [ ] SparseHV
 - [ ] support for different types
-- [ ] complex HDC
 =#
 
 abstract type AbstractHV{T} <: AbstractVector{T} end
 
-#Base.collect(hv::AbstractHV) = hv.v
 Base.sum(hv::AbstractHV) = sum(hv.v)
 Base.size(hv::AbstractHV) = size(hv.v)
 Base.getindex(hv::AbstractHV, i) = hv.v[i]
@@ -106,7 +104,6 @@ end
 
 RealHV(n::Integer = 10_000, distr::Distribution = eldist(RealHV)) = RealHV(rand(distr, n))
 
-
 Base.similar(hv::RealHV) = RealHV(length(hv), eldist(RealHV))
 
 function normalize!(hv::RealHV)
@@ -167,6 +164,30 @@ eldist(::Type{<:GradedBipolarHV}) = 2eldist(GradedHV) - 1
 Base.similar(hv::GradedBipolarHV) = GradedBipolarHV(length(hv))
 LinearAlgebra.normalize!(hv::GradedBipolarHV) = clamp!(hv.v, -1, 1)
 
+# Fourier Holographically Reduced Represenetations
+# ------------------------------------------------
+
+struct FHRR{T <: Complex} <: AbstractHV{T}
+    v::Vector{T}
+end
+
+#Base.eltype(::FHRR{T}) where {T} = Complex{T}
+
+FHRR(n::Int = 10_000) = FHRR(exp.(2π * im .* rand(n)))
+FHRR(T::Type, n::Int = 10_000) = FHRR(exp.(2π * im .* rand(T, n)))
+
+Base.similar(hv::FHRR{<:Complex{R}}) where {R} = FHRR(exp.(2π * im .* rand(R, length(hv))))
+
+"""
+    LinearAlgebra.normalize!(hv::FHRR)
+
+A Fourier Holographically Reduced Represenetation is normalized by
+setting the norm of each complex element to 1.
+"""
+function LinearAlgebra.normalize!(hv::FHRR)
+    hv.v ./= abs.(hv.v)
+    return hv
+end
 
 # TRAITS
 # ------

test/encoding.jl

@@ -63,4 +63,22 @@
         x = decoder(hv)
         @test 1 ≤ x ≤ 2
     end
+
+    @testset "FHRR numbers" begin
+
+        v = FHRR()
+
+        numvals = 0:0.1:10
+
+        encoder, decoder = convertlevel(v, numvals)
+
+        x, y, z = 2, 5, 10
+
+        hx, hy, hz = encoder.((x, y, z))
+
+        @test hx isa FHRR
+        @test similarity(hx, hy) > similarity(hx, hz)
+
+        @test decoder(hx) < decoder(hy) < decoder(hz)
+    end
 end

test/operations.jl

@@ -69,4 +69,21 @@ using LinearAlgebra, Random
             end
         end
     end
+    @testset "FHRR" begin
+        hv1 = FHRR(n)
+        hv2 = FHRR(n)
+
+        @test bundle([hv1, hv2]) isa FHRR
+        @test hv1 + hv2 isa FHRR
+        @test bind([hv1, hv2]) isa FHRR
+        @test norm(bind([hv1, hv2])) ≈ sqrt(n)
+
+        @test shift(hv1, 2) isa FHRR
+
+        @test similarity(hv1, hv2) < 0.5
+        @test similarity(hv2, hv2) ≈ 1
+
+        @test norm(hv1^3) ≈ sqrt(n)
+    end
+
 end

test/types.jl

@@ -97,4 +97,15 @@ using Distributions, LinearAlgebra
         @test norm(hdv) ≈ norm(hdv.v)
         normalize!(hdv)
     end
+
+    @testset "FHRR" begin
+        hdv = FHRR(n)
+        @test length(hdv) == n
+        @test eltype(hdv) <: Complex
+        @test hdv[2] isa Complex
+
+        @test sum(hdv) ≈ sum(hdv.v)
+        @test norm(hdv) ≈ norm(hdv.v)
+
+    end
 end