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0d639745f1
This mostly changes package name and code, but also fixes a couple of name clashes with the new package names
253 lines
8.2 KiB
Go
253 lines
8.2 KiB
Go
// Copyright ©2016 The gonum Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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package distmv
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import (
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"math"
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"gonum.org/v1/gonum/floats"
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"gonum.org/v1/gonum/mat"
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"gonum.org/v1/gonum/stat"
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)
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// Bhattacharyya is a type for computing the Bhattacharyya distance between
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// probability distributions.
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//
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// The Battachara distance is defined as
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// D_B = -ln(BC(l,r))
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// BC = \int_x (p(x)q(x))^(1/2) dx
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// Where BC is known as the Bhattacharyya coefficient.
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// The Bhattacharyya distance is related to the Hellinger distance by
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// H = sqrt(1-BC)
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// For more information, see
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// https://en.wikipedia.org/wiki/Bhattacharyya_distance
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type Bhattacharyya struct{}
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// DistNormal computes the Bhattacharyya distance between normal distributions l and r.
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// The dimensions of the input distributions must match or DistNormal will panic.
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//
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// For Normal distributions, the Bhattacharyya distance is
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// Σ = (Σ_l + Σ_r)/2
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// D_B = (1/8)*(μ_l - μ_r)^T*Σ^-1*(μ_l - μ_r) + (1/2)*ln(det(Σ)/(det(Σ_l)*det(Σ_r))^(1/2))
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func (Bhattacharyya) DistNormal(l, r *Normal) float64 {
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dim := l.Dim()
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if dim != r.Dim() {
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panic(badSizeMismatch)
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}
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var sigma mat.SymDense
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sigma.AddSym(&l.sigma, &r.sigma)
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sigma.ScaleSym(0.5, &sigma)
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var chol mat.Cholesky
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chol.Factorize(&sigma)
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mahalanobis := stat.Mahalanobis(mat.NewVector(dim, l.mu), mat.NewVector(dim, r.mu), &chol)
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mahalanobisSq := mahalanobis * mahalanobis
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dl := l.chol.LogDet()
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dr := r.chol.LogDet()
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ds := chol.LogDet()
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return 0.125*mahalanobisSq + 0.5*ds - 0.25*dl - 0.25*dr
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}
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// DistUniform computes the Bhattacharyya distance between uniform distributions l and r.
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// The dimensions of the input distributions must match or DistUniform will panic.
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func (Bhattacharyya) DistUniform(l, r *Uniform) float64 {
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if len(l.bounds) != len(r.bounds) {
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panic(badSizeMismatch)
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}
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// BC = \int \sqrt(p(x)q(x)), which for uniform distributions is a constant
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// over the volume where both distributions have positive probability.
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// Compute the overlap and the value of sqrt(p(x)q(x)). The entropy is the
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// negative log probability of the distribution (use instead of LogProb so
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// it is not necessary to construct an x value).
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//
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// BC = volume * sqrt(p(x)q(x))
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// logBC = log(volume) + 0.5*(logP + logQ)
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// D_B = -logBC
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return -unifLogVolOverlap(l.bounds, r.bounds) + 0.5*(l.Entropy()+r.Entropy())
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}
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// unifLogVolOverlap computes the log of the volume of the hyper-rectangle where
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// both uniform distributions have positive probability.
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func unifLogVolOverlap(b1, b2 []Bound) float64 {
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var logVolOverlap float64
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for dim, v1 := range b1 {
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v2 := b2[dim]
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// If the surfaces don't overlap, then the volume is 0
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if v1.Max <= v2.Min || v2.Max <= v1.Min {
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return math.Inf(-1)
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}
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vol := math.Min(v1.Max, v2.Max) - math.Max(v1.Min, v2.Min)
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logVolOverlap += math.Log(vol)
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}
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return logVolOverlap
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}
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// CrossEntropy is a type for computing the cross-entropy between probability
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// distributions.
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//
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// The cross-entropy is defined as
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// - \int_x l(x) log(r(x)) dx = KL(l || r) + H(l)
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// where KL is the Kullback-Leibler divergence and H is the entropy.
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// For more information, see
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// https://en.wikipedia.org/wiki/Cross_entropy
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type CrossEntropy struct{}
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// DistNormal returns the cross-entropy between normal distributions l and r.
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// The dimensions of the input distributions must match or DistNormal will panic.
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func (CrossEntropy) DistNormal(l, r *Normal) float64 {
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if l.Dim() != r.Dim() {
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panic(badSizeMismatch)
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}
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kl := KullbackLeibler{}.DistNormal(l, r)
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return kl + l.Entropy()
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}
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// Hellinger is a type for computing the Hellinger distance between probability
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// distributions.
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//
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// The Hellinger distance is defined as
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// H^2(l,r) = 1/2 * int_x (\sqrt(l(x)) - \sqrt(r(x)))^2 dx
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// and is bounded between 0 and 1.
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// The Hellinger distance is related to the Bhattacharyya distance by
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// H^2 = 1 - exp(-Db)
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// For more information, see
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// https://en.wikipedia.org/wiki/Hellinger_distance
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type Hellinger struct{}
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// DistNormal returns the Hellinger distance between normal distributions l and r.
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// The dimensions of the input distributions must match or DistNormal will panic.
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//
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// See the documentation of Bhattacharyya.DistNormal for the formula for Normal
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// distributions.
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func (Hellinger) DistNormal(l, r *Normal) float64 {
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if l.Dim() != r.Dim() {
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panic(badSizeMismatch)
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}
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db := Bhattacharyya{}.DistNormal(l, r)
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bc := math.Exp(-db)
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return math.Sqrt(1 - bc)
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}
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// KullbackLiebler is a type for computing the Kullback-Leibler divergence from l to r.
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// The dimensions of the input distributions must match or the function will panic.
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//
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// The Kullback-Liebler divergence is defined as
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// D_KL(l || r ) = \int_x p(x) log(p(x)/q(x)) dx
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// Note that the Kullback-Liebler divergence is not symmetric with respect to
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// the order of the input arguments.
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type KullbackLeibler struct{}
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// DistNormal returns the KullbackLeibler distance between normal distributions l and r.
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// The dimensions of the input distributions must match or DistNormal will panic.
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//
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// For two normal distributions, the KL divergence is computed as
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// D_KL(l || r) = 0.5*[ln(|Σ_r|) - ln(|Σ_l|) + (μ_l - μ_r)^T*Σ_r^-1*(μ_l - μ_r) + tr(Σ_r^-1*Σ_l)-d]
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func (KullbackLeibler) DistNormal(l, r *Normal) float64 {
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dim := l.Dim()
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if dim != r.Dim() {
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panic(badSizeMismatch)
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}
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mahalanobis := stat.Mahalanobis(mat.NewVector(dim, l.mu), mat.NewVector(dim, r.mu), &r.chol)
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mahalanobisSq := mahalanobis * mahalanobis
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// TODO(btracey): Optimize where there is a SolveCholeskySym
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// TODO(btracey): There may be a more efficient way to just compute the trace
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// Compute tr(Σ_r^-1*Σ_l) using the fact that Σ_l = U^T * U
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var u mat.TriDense
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u.UFromCholesky(&l.chol)
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var m mat.Dense
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err := m.SolveCholesky(&r.chol, u.T())
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if err != nil {
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return math.NaN()
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}
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m.Mul(&m, &u)
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tr := mat.Trace(&m)
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return r.logSqrtDet - l.logSqrtDet + 0.5*(mahalanobisSq+tr-float64(l.dim))
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}
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// DistUniform returns the KullbackLeibler distance between uniform distributions
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// l and r. The dimensions of the input distributions must match or DistUniform
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// will panic.
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func (KullbackLeibler) DistUniform(l, r *Uniform) float64 {
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bl := l.Bounds(nil)
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br := r.Bounds(nil)
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if len(bl) != len(br) {
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panic(badSizeMismatch)
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}
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// The KL is ∞ if l is not completely contained within r, because then
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// r(x) is zero when l(x) is non-zero for some x.
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contained := true
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for i, v := range bl {
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if v.Min < br[i].Min || br[i].Max < v.Max {
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contained = false
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break
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}
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}
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if !contained {
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return math.Inf(1)
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}
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// The KL divergence is finite.
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//
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// KL defines 0*ln(0) = 0, so there is no contribution to KL where l(x) = 0.
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// Inside the region, l(x) and r(x) are constant (uniform distribution), and
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// this constant is integrated over l(x), which integrates out to one.
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// The entropy is -log(p(x)).
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logPx := -l.Entropy()
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logQx := -r.Entropy()
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return logPx - logQx
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}
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// Wasserstein is a type for computing the Wasserstein distance between two
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// probability distributions.
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//
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// The Wasserstein distance is defined as
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// W(l,r) := inf 𝔼(||X-Y||_2^2)^1/2
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// For more information, see
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// https://en.wikipedia.org/wiki/Wasserstein_metric
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type Wasserstein struct{}
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// DistNormal returns the Wasserstein distance between normal distributions l and r.
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// The dimensions of the input distributions must match or DistNormal will panic.
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//
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// The Wasserstein distance for Normal distributions is
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// d^2 = ||m_l - m_r||_2^2 + Tr(Σ_l + Σ_r - 2(Σ_l^(1/2)*Σ_r*Σ_l^(1/2))^(1/2))
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// For more information, see
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// http://djalil.chafai.net/blog/2010/04/30/wasserstein-distance-between-two-gaussians/
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func (Wasserstein) DistNormal(l, r *Normal) float64 {
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dim := l.Dim()
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if dim != r.Dim() {
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panic(badSizeMismatch)
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}
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d := floats.Distance(l.mu, r.mu, 2)
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d = d * d
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// Compute Σ_l^(1/2)
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var ssl mat.SymDense
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ssl.PowPSD(&l.sigma, 0.5)
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// Compute Σ_l^(1/2)*Σ_r*Σ_l^(1/2)
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var mean mat.Dense
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mean.Mul(&ssl, &r.sigma)
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mean.Mul(&mean, &ssl)
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// Reinterpret as symdense, and take Σ^(1/2)
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meanSym := mat.NewSymDense(dim, mean.RawMatrix().Data)
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ssl.PowPSD(meanSym, 0.5)
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tr := mat.Trace(&r.sigma)
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tl := mat.Trace(&l.sigma)
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tm := mat.Trace(&ssl)
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return d + tl + tr - 2*tm
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}
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