计算多元正态密度的一部分的有效方法

时间：2018-03-26 21:41:00

标签： julia

我有兴趣计算数量：

(\boldsymbol{x_i}-\boldsymbol{\mu})^T \boldsymbol{W} (\boldsymbol{x_i}-\boldsymbol{\mu})

其中 x_i 是1xD向量（维度D的N个数据中的一个），μ是DxK矩阵， W 是K DxD矩阵列表。

这应该导致1XK向量。我以下列方式尝试所有N和K：

res = zeros(N,K);
for i in 1:N
    for k in 1:K
        res[i,k] = (x_matrix[i,:]-mus_matrix[:,k])'*
                   w_matrix[k]*(x_matrix[i,:]-mus_matrix[:,k])

如果我尝试对其进行矢量化，请使用以下内容：

 res = zeros(N,K);
for i in 1:N
        res[i,:] = (x_matrix[i,:].-mus_matrix)'.*w_matrix.*(x_matrix[i,:].-mus_matrix)

我收到以下错误：

ERROR: DimensionMismatch("arrays could not be broadcast to a common size")
Stacktrace:
 [1] _bcs1(::Base.OneTo{Int64}, ::Base.OneTo{Int64}) at ./broadcast.jl:70
 [2] _bcs at ./broadcast.jl:63 [inlined]
 [3] broadcast_shape(::Tuple{Base.OneTo{Int64},Base.OneTo{Int64}}, ::Tuple{Base.OneTo{Int64}}, ::Tuple{Base.OneTo{Int64},Base.OneTo{Int64}}, ::Vararg{Tuple{Base.OneTo{Int64},Base.OneTo{Int64}},N} where N) at ./broadcast.jl:57 (repeats 3 times)
 [4] broadcast_indices(::Array{Float64,2}, ::Array{Any,1}, ::Array{Float64,1}, ::Vararg{Any,N} where N) at ./broadcast.jl:53
 [5] broadcast_c(::Function, ::Type{Array}, ::Array{Float64,2}, ::Array{Any,1}, ::Vararg{Any,N} where N) at ./broadcast.jl:311
 [6] broadcast(::Function, ::Array{Float64,2}, ::Array{Any,1}, ::Array{Float64,1}, ::Vararg{Any,N} where N) at ./broadcast.jl:434

以下是一个例子：

julia> N = 5
5

julia> D=2
2

julia> K = 4
4

julia> W=[]
0-element Array{Any,1}

julia> x = rand(N,D)
5×2 Array{Float64,2}:
 0.576477  0.9575  
 0.184454  0.660436
 0.470267  0.729649
 0.648879  0.782561
 0.626453  0.111332

julia> mu = rand(K,D)
4×2 Array{Float64,2}:
 0.989281  0.00126782
 0.659106  0.66136   
 0.50843   0.289442  
 0.327962  0.523229  

julia> for i in 1:K
           push!(W,rand(D,D))
       end

然后运行

julia> (x_matrix[i,:]-mus_matrix[:,k])'*
                               w_matrix[k]*(x_matrix[i,:]-mus_matrix[:,k])
34649.850360744866

但是第二个代码

julia> (x_matrix[i,:].-mus_matrix)'.*w_matrix.*(x_matrix[i,:].-mus_matrix)
ERROR: DimensionMismatch("arrays could not be broadcast to a common size")
Stacktrace:
 [1] _bcs1(::Base.OneTo{Int64}, ::Base.OneTo{Int64}) at ./broadcast.jl:70
 [2] _bcs at ./broadcast.jl:63 [inlined]
 [3] broadcast_shape(::Tuple{Base.OneTo{Int64},Base.OneTo{Int64}}, ::Tuple{Base.OneTo{Int64}}, ::Tuple{Base.OneTo{Int64},Base.OneTo{Int64}}, ::Vararg{Tuple{Base.OneTo{Int64},Base.OneTo{Int64}},N} where N) at ./broadcast.jl:57 (repeats 3 times)
 [4] broadcast_indices(::Array{Float64,2}, ::Array{Any,1}, ::Array{Float64,1}, ::Vararg{Any,N} where N) at ./broadcast.jl:53
 [5] broadcast_c(::Function, ::Type{Array}, ::Array{Float64,2}, ::Array{Any,1}, ::Vararg{Any,N} where N) at ./broadcast.jl:311
 [6] broadcast(::Function, ::Array{Float64,2}, ::Array{Any,1}, ::Array{Float64,1}, ::Vararg{Any,N} where N) at ./broadcast.jl:434

1 个答案:

答案 0 :(得分：1)

TL / DR：下面的优化变体，但Einsum看起来更好，恕我直言。

看起来像是使用Einstein summation notation的情况。在朱莉娅，Einsum.jl可以做到这一点：

julia> N = 5
5

julia> D = 3
3

julia> K = 10
10

julia> x = rand(N, D)
5×3 Array{Float64,2}:
 0.587436  0.210529  0.261725
 0.527269  0.457477  0.482939
 0.52726   0.411209  0.138872
 0.89107   0.464789  0.758392
 0.885267  0.931014  0.672959

julia> μ = rand(D, K)
3×10 Array{Float64,2}:
 0.280792   0.265066   0.81437   0.503377  0.0717916  …  0.275872  0.609961   0.0820088  0.0042564
 0.0177643  0.0959438  0.563948  0.332433  0.088527      0.691971  0.0296638  0.604488   0.956057 
 0.668128   0.444816   0.74203   0.518232  0.48689       0.465067  0.117469   0.729514   0.109973 

julia> W = rand(K, D, D)
10×3×3 Array{Float64,3}:
[:, :, 1] =
 0.320861   0.662103  0.219234
 0.780944   0.769377  0.566203
 0.466207   0.428527  0.330901
 0.15534    0.035435  0.346737
 0.810676   0.328116  0.469505
 0.676575   0.668204  0.285334
 0.455551   0.211295  0.85295 
 0.229995   0.741487  0.783361
 0.0937583  0.401419  0.47032 
 0.956335   0.434213  0.967791

[:, :, 2] =
 0.275903   0.130298   0.184485
 0.941648   0.940107   0.439454
 0.425292   0.252654   0.797115
 0.0203406  0.594075   0.484809
 0.164309   0.941597   0.455314
 0.73628    0.109502   0.920664
 0.906305   0.177235   0.540193
 0.360038   0.0486971  0.20626 
 0.914357   0.699901   0.295872
 0.284143   0.659117   0.291479

[:, :, 3] =
 0.138311   0.921371  0.353719
 0.345247   0.70865   0.246736
 0.361364   0.636543  0.343837
 0.752149   0.581561  0.346399
 0.705888   0.24765   0.703952
 0.992327   0.369668  0.109407
 0.341624   0.223715  0.970667
 0.762169   0.94248   0.917569
 0.0367128  0.589345  0.121106
 0.826602   0.692111  0.229499

julia> using Einsum

julia> @einsum r[n,k] := (x[n,i] - μ[i,k]) * W[k,i,j] * (x[n,j] - μ[j,k])

julia> r
5×10 Array{Float64,2}:
  0.0176889  0.087092   0.522184    0.0417967   …  -0.0430999   0.041266   -0.0596579  0.432076
  0.0521066  0.364059   0.181515    0.00434307     -0.0248712   0.226976   -0.0686294  0.437169
 -0.0472136  0.127803   0.458812    0.0119074       0.0391649  -0.0190299  -0.0585371  0.264379
  0.468634   1.16498   -0.00263205  0.192809        0.273537    1.13787    -0.0653081  1.41321 
  0.749655   2.20266    0.0205068   0.420249        0.573358    1.42499     0.441232   1.67574

哪个@macroexpand基本上是以下循环（加上准备和边界检查）：

begin  
    local k 
    for k = 1:size(μ, 2) 
        begin  
            local n 
            for n = 1:size(x, 1) 
                begin  
                    local s = zero(T) 
                    begin  
                        local j 
                        for j = 1:size(W, 3) 
                            begin  
                                local i 
                                for i = 1:size(x, 2) 
                                    s += (x[n, i] - μ[i, k]) * W[k, i, j] * (x[n, j] - μ[j, k])
                                end
                            end
                        end
                    end 
                    r[n, k] = s
                end
            end
        end
    end
end

现在，为了找到更高效的东西，我使用BenchmarkTools.jl比较了几个变体。您可以在我的笔记本电脑here上查看完整的代码和结果。它表明Einsum变体实际上已经比原来更好了：

# Original: 
#   memory estimate:  1017.73 MiB
#   allocs estimate:  3429967
#   median time:      361.982 ms (15.94% GC)

# Einsum: 
#   memory estimate:  2.64 MiB
#   allocs estimate:  76
#   median time:      127.536 ms (0.00% GC)

到目前为止，效率最高且分配最少的变体如下，需要x = x'和W = permutedims(W, [2, 3, 1])（假设您可以轻松更改您的表示形式）：

function test_optimized!(res, x, μ, W)
    z = zero(eltype(x))

    for k = 1:size(μ, 1) 
        for n = 1:size(x, 1)
            res[n, k] = z

            for i = 1:size(W, 1)
                for j = 1:size(W, 2)
                    @inbounds res[n, k] += (x[i, n] - μ[i, k]) * W[i, j, k] * (x[j, n] - μ[j, k])
                end
            end
        end
    end
end

function test_optimized(x, μ, W)
    res = zeros(N, K)
    test_optimized!(res, x, μ, W)
    res
end

这将我们带到了

#   memory estimate:  2.63 MiB
#   allocs estimate:  2
#   median time:      521.215 μs (0.00% GC)

它使用了一些可以找到的“技巧”in the docs：在一个单独的方法中填充预分配的矩阵，按列主要顺序访问步幅，并使用@inbounds（虽然这只会改进大约一微秒的东西。）

还有TensorOperations.jl，我认为它会更加智能化，但它失败了：

julia> @tensor r[n,k] := (x[n,i] - μ[i,k]) * W[k,i,j] * (x[n,j] - μ[j,k])
ERROR: TensorOperations.IndexError{String}("invalid index specification: (:n, :i) to (:i, :k)")
Stacktrace:
 [1] add_indices(::Tuple{Symbol,Symbol}, ::Tuple{Symbol,Symbol}) at /home/philipp/.julia/v0.6/TensorOperations/src/implementation/indices.jl:22
 [2] + at /home/philipp/.julia/v0.6/TensorOperations/src/indexnotation/sum.jl:40 [inlined]
 [3] -(::TensorOperations.IndexedObject{(:n, :i),:N,Array{Float64,2},Int64}, ::TensorOperations.IndexedObject{(:i, :k),:N,Array{Float64,2},Int64}) at /home/philipp/.julia/v0.6/TensorOperations/src/indexnotation/sum.jl:44

我认为这是故意的，与效率有关，请参阅this issue。

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