More generic data

[BenConnault]

(In my understanding) the package currently assumes the n sample points constituting the data are given as an (d,n) array, where each sample point is a d-dimensional vector of some sort. It would be great if we could have access to a more generic data presentation (n widgets of WidgetType types). This would work particularly well with #23 and would have applications in tons of contexts. Eg you have sentences and you define distance between sentences, US Senators and you define ideological distances between senators, pictures of cats, Julia packages, etc.

[KristofferC]

The package right now is heavily focused towards computing things on a set of geometric points but as you say, you can compute distances between other objects.

And yes, the correct input should just be a list of things where distance(thing1, thing2) is defined. It might be worth making something like:

immutable Point{N, T}
    coordinates::NTuple{N, T}
end

One can then do:

julia> p = rand(3, 5)
3x5 Array{Float64,2}:
 0.310207  0.783898  0.780439   0.607608  0.0795108
 0.221908  0.53741   0.553835   0.373283  0.522569 
 0.291331  0.210073  0.0570539  0.809761  0.397538 

julia> reinterpret(Point{3, Float64}, p, (5,))
5-element Array{Point{3,Float64},1}:
 Point{3,Float64}((0.31020720319673756,0.22190808492334102,0.2913307988933631))
 Point{3,Float64}((0.783898246452825,0.5374103691511933,0.21007290525123268))  
 Point{3,Float64}((0.7804389420189257,0.5538345595214069,0.0570539417966307))  
 Point{3,Float64}((0.6076080127056103,0.3732832928705745,0.8097605299871604))  
 Point{3,Float64}((0.07951078253622623,0.5225687936560375,0.3975381159479192)) 

To quickly convert between a matrix to a vector of points. This is not type stable but the cost is only paid once by making sure to use a function barrier.

[BenConnault]

Thanks for all these answers. Would you be willing to pay a performance penalty for those cases where the data could in fact be presented in an (d,n) array (probably the overwhelming majority of cases), or are you thinking allowing both interfaces would be the way to go?

[KristofferC]

I wouldn't like keeping both the interfaces, would be too much code duplication and awkward to test everything. I would just convert a Matrix{T} -> Vector{Point{N, T}} the first thing I do with a function barrier. For a sufficient number of input points (not that large) the cost should be insignificant. It can just be documented as well so that if the number of points is low the user has to create the Vector{Point{N,T}} him/herself.

This issue is actually a bit related: #17

[BenConnault]

Ok. I would be a little worried that a performance penalty would show up beyond the initial reinterpret, conceptually whenever slice(data,:,i) would now be data[i], but you certainly know better than I do.

[KristofferC]

data[i] is cheap and does not cause any heap allocation like slicing into an array. In fact it would be faster than slice because slice points to a heap allocated object and must thus also be allocated on the heap.

As an example:

immutable Point{N, T} <: AbstractVector{T}
    coordinates::NTuple{N, T}
end

Base.length{N}(::Point{N}) = N
Base.size{N}(::Point{N}) = (N,)
Base.linearindexing(::Type{Point}) = Base.LinearFast()

function Base.getindex(p::Point, i::Int)
    @inbounds v = p.coordinates[i]
    return v
end

function compare{N, T}(A1::Matrix{T}, A2::Matrix{T},  
                       V1::Vector{Point{N, T}}, V2::Vector{Point{N, T}})
    m = Euclidean()
    @time begin
        for i in 1:length(V1)
            evaluate(m, V1[i], V2[i])
        end
    end

     @time begin
        for i in 1:size(A1, 2)
            evaluate(m, slice(A1, :, i), slice(A2, :, i))
        end
    end
end


A1 = rand(3, 10^6)
A2 = rand(3, 10^6)
V1 = reinterpret(Point{3, Float64}, A1, (10^6,))
V2 = reinterpret(Point{3, Float64}, A2, (10^6,))

compare(A1, A2, V1, V2)

gives:

julia> compare(A1, A2, V1, V2)
  0.012314 seconds
  0.070496 seconds (4.00 M allocations: 152.588 MB, 26.75% gc time)

so using Points is about 7 times faster than slicing.

[BenConnault]

Very cool & illuminating! Thanks.

[BenConnault]

Hi Kristoffer,
In my understanding create_bsphere in hyperspheres.jl assumes the sample space is something like a real vector space to find a center by point by taking an average of points. Of course you can't do that for abstract sample spaces (I guess you can't ever step out of the set of sample points in the abstract). Not sure if this is an implementation detail or a hard limit of the algorithm.

[KristofferC]

Ball trees should work with arbitrary objects which has a defined metric. However, I am no expert no these stuff and I don't know how to compute the bounding hyper sphere nor how to split the points into sub spheres. Right now I am splitting them in the same way as for the KD trees where you sort the objects along a certain dimension but for general objects you don't even have he concept of a dimension.

[BenConnault]

I see, thanks. I guess a refactor of the package towards more generic data would not be super useful without an abstract tree algorithm for metric spaces. Although, in order to separate the workload, I think the interface for generic data could be implemented with a fallback on BruteTree for abstract metric spaces with minimal work, as a start.

Assuming abstract metric space tree algorithms would suffer from a performance penalty (probable), would you be willing to pay that price for BallTrees, or would you prefer two implementations living side by side?

[KristofferC]

I believe we could use multiple dispatch to overload things in such a way that we don't need two completely separate implementations. The knnand inrange functions probably don't need to be touched, only how the tree is built.

[BenConnault]

A couple of things I found out after digging for a little time.

Unfortunately it looks like what we want for abstract metric spaces are "VP trees" and that the algorithm for traversing VP trees is not the same as the one for ball trees. Here is a description:
http://stevehanov.ca/blog/index.php?id=130

As a much quicker hack, I think dispatching a center() function in create_bsphere to either a vector space average (when applicable) or a naive random draw among all points will produce a valid-if-not-that-great search tree.

[KristofferC]

This is better now and you can have a Vector of Widgets. However, each Widget need to be an AbstractVector and have a defined eltype(Type{Widget}) and length(Type{Widget}) defined so perhaps not as general as you would like.

[BenConnault]

Thanks Kristoffer, this looks great as always.

Implementing all the changes we mentioned in this thread proved a bit overwhelming for me, especially since I don't have your command of efficient generic programming. I ended up putting together a basic VP Tree algorithm for my own use. I just pushed some comments to it here.

The data is given as data::Vector{T}. x::T is a point in an abstract metric space: the only assumption is that we can evaluate(distance,x::T,y::T) (x+y is not necessarily defined, etc.).