Image Labeling by Assignment

Abstract

We introduce a novel geometric approach to the image labeling problem. Abstracting from specific labeling applications, a general objective function is defined on a manifold of stochastic matrices, whose elements assign prior data that are given in any metric space, to observed image measurements. The corresponding Riemannian gradient flow entails a set of replicator equations, one for each data point, that are spatially coupled by geometric averaging on the manifold. Starting from uniform assignments at the barycenter as natural initialization, the flow terminates at some global maximum, each of which corresponds to an image labeling that uniquely assigns the prior data. Our geometric variational approach constitutes a smooth non-convex inner approximation of the general image labeling problem, implemented with sparse interior-point numerics in terms of parallel multiplicative updates that converge efficiently.

Appendices

Appendix 1: Basic Notation

For $n \in {\mathbb {N}}$, we set $[n] = \{1,2,\ldots ,n\}$. ${\mathbbm {1}}= (1,1,\ldots ,1)^{\top }$ denotes the vector with all components equal to 1, whose dimension can either be inferred from the context or is indicated by a subscript, e.g., ${\mathbbm {1}}_{n}$. Vectors $v^{1}, v^{2},\ldots$ are indexed by lowercase letters and superscripts, whereas subscripts $v_{i},\, i \in [n]$, index vector components. $e^{1},\ldots ,e^{n}$ denotes the canonical orthonormal basis of $\mathbb {R}^{n}$.

We assume data to be indexed by a graph ${\mathscr {G}}=({\mathscr {V}},{\mathscr {E}})$ with nodes $i \in {\mathscr {V}}=[m]$ and associated locations $x^{i} \in \mathbb {R}^{d}$, and with edges ${\mathscr {E}}$. A regular grid graph and $d=2$ is the canonical example. But ${\mathscr {G}}$ may also be irregular due to some preprocessing like forming super-pixels, for instance, or correspond to 3D images or videos ($d=3$). For simplicity, we call i location although this actually is $x^{i}$.

If $A \in \mathbb {R}^{m \times n}$, then the row and column vectors are denoted by $A_{i} \in \mathbb {R}^{n},\, i \in [m]$ and $A^{j} \in \mathbb {R}^{m},\, j \in [n]$, respectively, and the entries by $A_{ij}$. This notation of row vectors $A_{i}$ is the only exception from our rule of indexing vectors stated above.

The componentwise application of functions $f :\mathbb {R}\rightarrow \mathbb {R}$ to a vector is simply denoted by f(v), e.g., 

$$\begin{aligned} \forall v \in \mathbb {R}^{n},\qquad \sqrt{v}&:= (\sqrt{v_{1}},\ldots ,\sqrt{v_{n}})^{\top }, \end{aligned}$$

(6.1a)

$$\begin{aligned} \exp (v)&:= \big (e^{v_{1}},\ldots ,e^{v_{n}}\big )^{\top } \quad \text {etc.} \end{aligned}$$

(6.1b)

Likewise, binary relations between vectors apply componentwise, e.g., $u \ge v \;\Leftrightarrow \; u_{i} \ge v_{i},\; i \in [n]$, and binary componentwise operations are simply written in terms of the vectors. For example,

$$\begin{aligned} p q := (\ldots , p_{i} q_{i},\ldots )^{\top },\qquad \frac{p}{q} := \Big (\ldots ,\frac{p_{i}}{q_{i}},\ldots \Big )^{\top }, \end{aligned}$$

(6.2)

where the latter operation is only applied to strictly positive vectors $q > 0$. The support ${{\mathrm{supp}}}(p) = \{p_{i} \ne 0 :i \in {{\mathrm{supp}}}(p)\} \subset [n]$ of a vector $p \in \mathbb {R}^{n}$ is the index set of all non-nonvanishing components of p.

$\langle x, y \rangle$ denotes the standard Euclidean inner product and $\Vert x\Vert = \langle x, x \rangle ^{1/2}$ the corresponding norm. Other $\ell _{p}$-norms, $1 \le p \ne 2 \le \infty$, are indicated by a corresponding subscript, $\Vert x\Vert _{p} = \big (\sum _{i \in [d]} |x_{i}|^{p}\big )^{1/p},$ except for the case $\Vert x\Vert = \Vert x\Vert _{2}$. For matrices $A, B \in \mathbb {R}^{m \times n}$, the canonical inner product is $\langle A, B \rangle = \hbox {tr}(A^{\top } B)$ with the corresponding Frobenius norm $\Vert A\Vert = \langle A, A \rangle ^{1/2}$. ${{\mathrm{Diag}}}(v) \in \mathbb {R}^{n \times n},\, v \in \mathbb {R}^{n}$, is the diagonal matrix with the vector v on its diagonal.

Other basic sets and their notation are

• the positive orthant

  $$\begin{aligned} \mathbb {R}_{+}^{n} = \{ p \in \mathbb {R}^{n} :p \ge 0 \}, \end{aligned}$$

  (6.3)

• the set of strictly positive vectors

  $$\begin{aligned} \mathbb {R}_{++}^{n} = \{p \in \mathbb {R}^{n} :p > 0\}, \end{aligned}$$

  (6.4)

• the ball of radius r centered at p

  $$\begin{aligned} {\mathbb {B}}_{r}(p) = \{p \in \mathbb {R}^{n} :\Vert p\Vert \le r\}, \end{aligned}$$

  (6.5)

• the unit sphere

  $$\begin{aligned} {\mathbb {S}}^{n-1} = \{p \in \mathbb {R}^{n} :\Vert p\Vert =1\}, \end{aligned}$$

  (6.6)

• the probability simplex

  $$\begin{aligned} \varDelta _{n-1} = \{p \in \mathbb {R}_{+}^{n} :\langle {\mathbbm {1}}, p \rangle = 1 \} \end{aligned}$$

  (6.7)

• and its relative interior

  $$\begin{aligned} {\mathscr {S}}&= \mathring{\varDelta }_{n-1} = \varDelta _{n-1} \cap \mathbb {R}_{++}^{n}, \end{aligned}$$

  (6.8a)
  $$\begin{aligned} {\mathscr {S}}_{n}&= {\mathscr {S}}\;\text {with concrete value of} n (e.g.,~{\mathscr {S}}_{3}), \end{aligned}$$

  (6.8b)

• closure (not regarded as manifold)

  $$\begin{aligned} \overline{{\mathscr {S}}} = \varDelta _{n-1}, \end{aligned}$$

  (6.9)

• the sphere with radius 2

  $$\begin{aligned} {\mathscr {N}} = 2 {\mathbb {S}}^{n-1}, \end{aligned}$$

  (6.10)

• the assignment manifold

  $$\begin{aligned} {\mathscr {W}} = {\mathscr {S}} \times \cdots \times {\mathscr {S}}, \quad (\text {m times}) \end{aligned}$$

  (6.11)

• and its closure (not regarded as manifold)

  $$\begin{aligned} \overline{{\mathscr {W}}} = \overline{{\mathscr {S}}} \times \cdots \times \overline{{\mathscr {S}}}, \quad (\text {m times}). \end{aligned}$$

  (6.12)

For a discrete distribution $p \in \varDelta _{n-1}$ and a finite set $S=\{s^{1},\ldots ,s^{n}\}$ vectors, we denote by

$$\begin{aligned} {\mathbb {E}}_{p}[S] := \sum _{i \in [n]} p_{i} s^{i} \end{aligned}$$

(6.13)

the mean of S with respect to p.

Let ${\mathscr {M}}$ be a any differentiable manifold. Then $T_{p}{\mathscr {M}}$ denotes the tangent space at base point $p \in {\mathscr {M}}$ and $T{\mathscr {M}}$ the total space of the tangent bundle of ${\mathscr {M}}$. If $F :{\mathscr {M}} \rightarrow {\mathscr {N}}$ is a smooth mapping between differentiable manifold ${\mathscr {M}}$ and ${\mathscr {N}}$, then the differential of F at $p \in {\mathscr {M}}$ is denoted by

$$\begin{aligned} DF(p) :T_{p}{\mathscr {M}} \rightarrow T_{F(p)}{\mathscr {N}},\qquad DF(p) :v \mapsto DF(p)[v]. \end{aligned}$$

(6.14)

If $F :\mathbb {R}^{m} \rightarrow \mathbb {R}^{n}$, then $DF(p) \in \mathbb {R}^{n \times m}$ is the Jacobian matrix at p, and the application DF(p)[v] to a vector $v \in \mathbb {R}^{m}$ means matrix-vector multiplication. We then also write DF(p)v. If $F = F(p,q)$, then $D_{p}F(p,q)$ and $D_{q}F(p,q)$ are the Jacobians of the functions $F(\cdot ,q)$ and $F(p,\cdot )$, respectively.

The gradient of a differentiable function $f :\mathbb {R}^{n} \rightarrow \mathbb {R}$ is denoted by $\nabla f(x) = \big (\partial _{1} f(x),\ldots ,\partial _{n} f(x)\big )^{\top }$, whereas the Riemannian gradient of a function $f :{\mathscr {M}} \rightarrow \mathbb {R}$ defined on Riemannian manifold ${\mathscr {M}}$ is denoted by $\nabla _{{\mathscr {M}}} f$. Eq. (2.5) recalls the formal definition.

The exponential mapping [21, Def. 1.4.3]

$$\begin{aligned} {{\mathrm{Exp}}}_{p} :T_{p}{\mathscr {M}}&\rightarrow {\mathscr {M}}, \quad v \mapsto {{\mathrm{Exp}}}_{p}(v) = \gamma _{v}(1), \end{aligned}$$

(6.15a)

$$\begin{aligned} \gamma _{v}(0)&=p,\; \dot{\gamma }_{v}(0)=\frac{\hbox {d}}{\hbox {d}t}\gamma _{v}(t)\big |_{t=0} = v, \end{aligned}$$

(6.15b)

maps the tangent vector v to the point $\gamma _{v}(1) \in {\mathscr {M}}$, uniquely defined by the geodesic curve $\gamma _{v}(t)$ emanating at p in direction v. $\gamma _{v}(t)$ is the shortest path on ${\mathscr {M}}$ between the points $p, q \in {\mathscr {M}}$ that $\gamma _{v}$ connects. This minimal length equals the Riemannian distance $d_{{\mathscr {M}}}(p,q)$ induced by the Riemannian metric, denoted by

$$\begin{aligned} \langle u, v \rangle _{p}, \end{aligned}$$

(6.16)

i.e., the inner product on the tangent spaces $T_{p}{\mathscr {M}},\,p \in {\mathscr {M}}$, that smoothly varies with p. Existence and uniqueness of geodesics will not be an issue for the manifolds ${\mathscr {M}}$ considered in this paper.

Remark 8

The exponential mapping ${{\mathrm{Exp}}}_{p}$ should not be confused with

• the exponential function $e^{v}$ used, e.g., in (6.1);

• the mapping $\exp _{p} :T_{p}{\mathscr {S}} \rightarrow {\mathscr {S}}$ defined by Eq. (3.8a).

The abbreviations “l.h.s.” and “r.h.s.” mean left-hand side and right-hand side of some equation, respectively. We abbreviate with respect to by “wrt.”

Appendix 2: Proofs and Further Details

1.1 Proofs of Section 2

Proof

(of Lemma 1) Let $p \in {\mathscr {S}}$ and $v \in T_{p}{\mathscr {S}}$. We have

$$\begin{aligned} D\psi (p) = {{\mathrm{Diag}}}(p)^{-1/2} \end{aligned}$$

(7.1)

and $\big \langle \psi (p), D\psi (p)[v] \big \rangle = \langle 2 \sqrt{p}, \frac{v}{\sqrt{p}} \rangle = 2 \langle {\mathbbm {1}}, v \rangle = 0$, that is, $D\psi (p)[v] \in T_{\psi (p)}{\mathscr {N}}$. Furthermore,

$$\begin{aligned} \big \langle D\psi (p)[u], D\psi (p)[v] \big \rangle = \big \langle u/\sqrt{p}, v/\sqrt{p} \rangle \overset{(2.1)}{=} \langle u, v \rangle _{p}, \end{aligned}$$

(7.2)

i.e., the Riemannian metric is preserved and hence also the length L(s) of curves $s(t) \in {\mathscr {N}},\, t \in [a,b]$: Put $\gamma (t) = \psi ^{-1}\big (s(t)\big ) = \frac{1}{4} s^{2}(t) \in {\mathscr {S}},\, t \in [a,b]$. Then $\dot{\gamma }(t)=\frac{1}{2} s(t) \dot{s}(t) = \frac{1}{2} \psi \big (\gamma (t)\big ) \dot{s}(t) = \sqrt{\gamma (t)} \dot{s}(t)$ and

$$\begin{aligned} L(s)&= \int _{a}^{b} \Vert \dot{s}(t)\Vert \mathrm{d}t = \int _{a}^{b} \bigg \langle \frac{\dot{\gamma }(t)}{\sqrt{\gamma (t)}}, \frac{\dot{\gamma }(t)}{\sqrt{\gamma (t)}} \bigg \rangle ^{1/2} \mathrm{d}t \end{aligned}$$

(7.3a)

$$\begin{aligned}&\overset{(2.1)}{=} \int _{a}^{b} \Vert \dot{\gamma }(t)\Vert _{\gamma (t)} \mathrm{d}t = L(\gamma ). \end{aligned}$$

(7.3b)

$\square$

Proof

(of Prop. 1) Setting $g:{\mathscr {N}} \rightarrow \mathbb {R}$, $q \mapsto g(s) := f\big (\psi ^{-1}(s)\big )$ with $s = \psi (p) = 2 \sqrt{p}$ from (2.3), we have

$$\begin{aligned} \nabla _{{\mathscr {N}}} g(s) = \bigg (I - \frac{s}{\Vert s\Vert } \frac{s^{\top }}{\Vert s\Vert }\bigg ) \nabla g(s), \end{aligned}$$

(7.4)

because the 2-sphere ${\mathscr {N}}=2{\mathbb {S}}^{n-1}$ is an embedded submanifold, and hence the Riemannian gradient equals the orthogonal projection of the Euclidean gradient onto the tangent space. Pulling back the vector field $\nabla _{{\mathscr {N}}} g$ by $\psi$ using

$$\begin{aligned} \nabla g(s) = \nabla f\big (\psi ^{-1}(s)\big ) = \nabla f\Big (\frac{1}{4} s^{2}\Big ) = \frac{1}{2} s \big (\nabla f(p)\big ), \end{aligned}$$

(7.5)

we get with (7.1), (7.4) and $\Vert s\Vert =2$ and hence $s/\Vert s\Vert = \frac{1}{2} \psi (p)=\sqrt{p}$

$$\begin{aligned} \nabla f_{{\mathscr {S}}}(p)&= \big (D\psi (p)\big )^{-1}\big (\nabla _{{\mathscr {N}}} g(\psi (p))\big ) \end{aligned}$$

(7.6a)

$$\begin{aligned}&= {{\mathrm{Diag}}}(\sqrt{p}) \Big (\big (I - \sqrt{p} \sqrt{p}^{\top }\big ) \sqrt{p} \big (\nabla f(p)\big )\Big ) \end{aligned}$$

(7.6b)

$$\begin{aligned}&= p \big (\nabla f(p)\big ) - \langle p, \nabla f(p) \rangle p, \end{aligned}$$

(7.6c)

which equals (2.6). We finally check that $\nabla f_{{\mathscr {S}}}(p)$ satisfies (2.5) (with ${\mathscr {S}}$ in place of ${\mathscr {M}}$). Using (2.1), we have

$$\begin{aligned} \langle \nabla f_{{\mathscr {S}}}(p), v \rangle _{p}&= \Big \langle \sqrt{p} \big (\nabla f(p)\big ) - \langle p, \nabla f(p) \rangle \sqrt{p}, \frac{v}{\sqrt{p}} \Big \rangle \end{aligned}$$

(7.7a)

$$\begin{aligned}&= \langle \nabla f(p), v \rangle - \langle p, \nabla f(p) \rangle \langle {\mathbbm {1}}, v \rangle \end{aligned}$$

(7.7b)

$$\begin{aligned}&\overset{(2.2)}{=} \langle \nabla f(p), v \rangle ,\quad \forall v \in T_{p}{\mathscr {S}}. \end{aligned}$$

(7.7c)

$\square$

Proof

(of Prop. 2) The geodesic on the 2-sphere emanating at $s(0) \in {\mathscr {N}}$ in direction $w=\dot{s}(0) \in T_{s(0)}{\mathscr {N}}$ is given by

$$\begin{aligned} s(t) = s(0) \cos \Big (\frac{\Vert w\Vert }{2} t\Big ) + 2 \frac{w}{\Vert w\Vert } \sin \Big (\frac{\Vert w\Vert }{2} t\Big ). \end{aligned}$$

(7.8)

Setting $s(0)=\psi (p)$ and $w = D\psi (p)[v]=v/\sqrt{p}$, the geodesic emanating at $p=\gamma _{v}(0)$ in direction v is given by $\psi ^{-1}\big (s(t)\big )$ due to Lemma 1, which results in (2.7a) after elementary computations. $\square$

1.2 Proofs of Section 3 and Further Details

Proof

(of Prop. 3) We have $p = \exp _{p}(0)$ and

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t} \exp _{p}(u t)&= \frac{\langle p, e^{u t} \rangle p e^{u t} u - p e^{u t} \langle p, e^{u t} u \rangle }{\langle p, e^{u t} \rangle ^{2}} \end{aligned}$$

(7.9a)

$$\begin{aligned}&= p(t) u - \langle p(t), u \rangle p(t), \end{aligned}$$

(7.9b)

which confirms (3.10), is equal to (3.9) at $t=0$ and hence yields the first expression of (3.11). The second expression of (3.11) follows from a Taylor expansion of (2.7a)

$$\begin{aligned} \gamma _{v}(t) \approx p + v t + \frac{1}{4}\big (v_{p}^{2}-\Vert v_{p}\Vert ^{2} p\big ) t^{2},\qquad v_{p} = \frac{v}{\sqrt{p}}. \end{aligned}$$

(7.10)

$\square$

Proof

(of Lemma 4) By construction, $S(W) \in {\mathscr {W}}$, that is, $S_{i}(W) \in {\mathscr {S}},\; i \in [m]$. Consequently,

$$\begin{aligned} 0 \le J(W)=\sum _{i \in [m]} \langle S_{i}(W), W_{i} \rangle \le \sum _{i \in [m]} \Vert S_{i}(W)\Vert \Vert W_{i}\Vert < m. \end{aligned}$$

(7.11)

The upper bound corresponds to matrices $\overline{W}^{*} \in \overline{{\mathscr {W}}}$ and $S(\overline{W}^{*})$ where for each $i \in [m]$, both $\overline{W}^{*}_{i}$ and $S_{i}(\overline{W}^{*})$ equal the same unit vector $e^{k_{i}}$ for some $k_{i} \in [m]$. $\square$

Proof

(Explicit form of (3.27)) The matrices $T^{ij}(W) = \frac{\partial }{\partial W_{ij}} S(W)$ are implicitly given through the optimality condition (2.9) that each vector $S_{k}(W),\, k \in [m]$, defined by (3.13) has to satisfy

$$\begin{aligned} S_{k}(W)&= {\mathrm {mean}}_{{\mathscr {S}}}\{L_{r}(W_{r})\}_{r \in \tilde{{\mathscr {N}}}_{{\mathscr {E}}}(k)} \end{aligned}$$

(7.12a)

$$\begin{aligned} \qquad \Leftrightarrow \qquad 0&= \sum _{r \in \tilde{{\mathscr {N}}}_{{\mathscr {E}}}(k)} {{\mathrm{Exp}}}_{S_{k}(W)}^{-1}\big (L_{r}(W_{r})\big ). \end{aligned}$$

(7.12b)

Writing

$$\begin{aligned} \phi \big (S_{k}(W),L_{r}(W_{r})\big ) := {{\mathrm{Exp}}}_{S_{k}(W)}^{-1}\big (L_{r}(W_{r})\big ), \end{aligned}$$

(7.13)

while temporarily dropping below W as argument to simplify the notation, and using the indicator function $\delta _{{\mathrm {P}}} = 1$ if the predicate ${\mathrm {P}}={\mathrm {true}}$ and $\delta _{{\mathrm {P}}} = 0$ otherwise, we differentiate the optimality condition on the r.h.s. of (7.12),

$$\begin{aligned} 0&= \frac{\partial }{\partial W_{ij}} \sum _{r \in \tilde{{\mathscr {N}}}_{{\mathscr {E}}}(k)} \phi \big (S_{k}(W),L_{r}(W_{r})\big ) \end{aligned}$$

(7.14a)

$$\begin{aligned}&= \sum _{r \in \tilde{{\mathscr {N}}}_{{\mathscr {E}}}(k)} \Big ( D_{S_{k}}\phi (S_{k},L_{r}) \Big [\frac{\partial }{\partial W_{ij}} S_{k}(W)\Big ] \end{aligned}$$

(7.14b)

$$\begin{aligned}&\quad + \delta _{i=r} D_{L_{r}}\phi (S_{k},L_{r}) \Big [\frac{\partial }{\partial W_{rj}} L_{r}(W_{r})\Big ] \Big ) \end{aligned}$$

(7.14c)

$$\begin{aligned}&= \Big (\sum _{r \in \tilde{{\mathscr {N}}}_{{\mathscr {E}}}(k)} D_{S_{k}}\phi (S_{k},L_{r}) \Big ) \Big (\frac{\partial }{\partial W_{ij}} S_{k}(W)\Big ) \end{aligned}$$

(7.14d)

$$\begin{aligned}&\quad + \delta _{i \in \tilde{{\mathscr {N}}}_{{\mathscr {E}}}(k)} D_{L_{i}}\phi (S_{k},L_{i})\Big (\frac{\partial }{\partial W_{ij}} L_{i}(W_{i}) \Big ) \end{aligned}$$

(7.14e)

$$\begin{aligned}&=: H^{k}(W) \Big (\frac{\partial }{\partial W_{ij}} S_{k}(W)\Big ) + h^{k,ij}(W). \end{aligned}$$

(7.14f)

Since the vectors $\phi (S_{k},L_{r})$ given by (7.13) are the negative Riemannian gradients of the (locally) strictly convex objectives (2.8) defining the means $S_{k}$ [21, Thm. 4.6.1], the regularity of the matrices $H^{k}(W)$ follows. Thus, using (7.14f) and defining the matrices

$$\begin{aligned}&T^{ij}(W) \in \mathbb {R}^{m \times n},\quad T^{ij}_{kl}(W) := \frac{\partial }{\partial S_{kl}(W)}{W_{ij}},\nonumber \\&\quad i,k \in [m],\; j,l \in [n], \end{aligned}$$

(7.15)

results in (3.27). The explicit form of this expression results from computing and inserting into (7.14f) the corresponding Jacobians $D_{p}\phi (p,q)$ and $D_{q}\phi (p,q)$ of

$$\begin{aligned} \phi (p,q)&={{\mathrm{Exp}}}_{p}^{-1}(q) \end{aligned}$$

(7.16a)

$$\begin{aligned}&= \frac{d_{{\mathscr {S}}}(p,q)}{\sqrt{1-\langle \sqrt{p},\sqrt{q}\rangle ^{2}}}\big (\sqrt{p q}-\langle \sqrt{p},\sqrt{q}\rangle p\big ), \end{aligned}$$

(7.16b)

and

$$\begin{aligned} \frac{\partial }{\partial W_{ij}} L_{i}(W_{i})&= \frac{e^{-U_{ij}}}{\langle W_{i}, e^{-U_{i}} \rangle } \big (e^{j} - L_{i}(W_{i})\big ). \end{aligned}$$

(7.16c)

The term (7.16b) results from mapping back the corresponding vector from the 2-sphere ${\mathscr {N}}$,

$$\begin{aligned} {{\mathrm{Exp}}}_{p}^{-1}(q) = -\big (D\psi (p)\big )^{-1}\Big (\frac{1}{2}\nabla _{{\mathscr {N}}}d_{{\mathscr {N}}}^{2}\big (\psi (p),\psi (q)\big )\Big ), \end{aligned}$$

(7.17)

where $\psi$ is the sphere map (2.3) and $d_{{\mathscr {N}}}$ is the geodesic distance on ${\mathscr {N}}$. The term (7.16c) results from directly evaluating (3.12). $\square$

Proof

(of Lemma 5) We first compute $\exp _{p}^{-1}$. Suppose

$$\begin{aligned} q = \exp _{p}(u) = \frac{p e^{u}}{\langle p, e^{u} \rangle }, \qquad p,q \in {\mathscr {S}},\quad u \in \mathbb {R}^{n}. \end{aligned}$$

(7.18)

Then

$$\begin{aligned} \log (q)&= \log (p)+u-\log (\langle p, e^{u} \rangle ) {\mathbbm {1}}, \end{aligned}$$

(7.19a)

$$\begin{aligned} \log (\langle p, e^{u} \rangle )&= \frac{1}{n} \langle {\mathbbm {1}}, \log (p)-\log (q) \rangle , \end{aligned}$$

(7.19b)

and

$$\begin{aligned} u = \exp _{p}^{-1}(q) = \left( I-\frac{1}{n} {\mathbbm {1}}{\mathbbm {1}}^{\top }\right) \big (\log (q)-\log (p)\big ). \end{aligned}$$

(7.20)

Thus, in view of (3.9), we approximate

$$\begin{aligned}&{{\mathrm{Exp}}}_{p}^{-1}(q) \approx v \nonumber \\&\quad = \big ({{\mathrm{Diag}}}(p)-p p^{\top }\big ) u \end{aligned}$$

(7.21a)

$$\begin{aligned}&\quad = \left( {{\mathrm{Diag}}}(p)-\frac{1}{n} p {\mathbbm {1}}^{\top } - p p^{\top } + \frac{1}{n} p {\mathbbm {1}}^{\top }\right) \log \Big (\frac{q}{p}\Big ) \end{aligned}$$

(7.21b)

$$\begin{aligned}&\quad = \big ({{\mathrm{Diag}}}(p)-p p^{\top }\big ) \log \Big (\frac{q}{p}\Big ). \end{aligned}$$

(7.21c)

Applying this to the point set ${\mathscr {P}}$, i.e., setting

$$\begin{aligned} v^{i} = \big ({{\mathrm{Diag}}}(p)-p p^{\top }\big ) \log \frac{p^{i}}{p}, \qquad i \in [N], \end{aligned}$$

(7.22)

step (3) of (3.31) yields

$$\begin{aligned} v&:= \frac{1}{N} \sum _{i \in [N]} v^{i} = \frac{1}{N} \big ({{\mathrm{Diag}}}(p)-p p^{\top }\big )\nonumber \\&\quad \Big (\sum _{i \in [N]} \log (p^{i}) - N \log (p)\Big ) \end{aligned}$$

(7.23a)

$$\begin{aligned}&= \big ({{\mathrm{Diag}}}(p)-p p^{\top }\big ) \log \bigg ( \frac{1}{p} \Big (\prod _{i \in [N]} p^{i}\Big )^{\frac{1}{N}} \bigg ) \end{aligned}$$

(7.23b)

$$\begin{aligned}&= \big ({{\mathrm{Diag}}}(p)-p p^{\top }\big ) \log \Big (\frac{{\mathrm {mean}}_{g}({\mathscr {P}})}{p}\Big ) \end{aligned}$$

(7.23c)

$$\begin{aligned}&=: \big ({{\mathrm{Diag}}}(p)-p p^{\top }\big ) u. \end{aligned}$$

(7.23d)

Finally, approximating step (4) of (3.31) results in view of Prop. 3 in the update of p

$$\begin{aligned} \exp _{p}(u) = \frac{p e^{u}}{\langle p, e^{u} \rangle } = \frac{{\mathrm {mean}}_{g}({\mathscr {P}})}{\langle {\mathbbm {1}}, {\mathrm {mean}}_{g}({\mathscr {P}}) \rangle }. \end{aligned}$$

(7.24)

$\square$