Kantorovich optimal transport problem and Shannons optimal channel - - PowerPoint PPT Presentation

Kantorovich optimal transport problem and Shannon’s

• ptimal channel problem

Roman V. Belavkin

School of Science and Technology Middlesex University, London NW4 4BT, UK

13 June 2016

In honor of Shun-ichi Amari

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 1 / 30

Optimal transportation problems (OTPs) Information and entropy Optimal channel problem (OCP) Geometry of information divergence and optimization Dynamical OTP: Optimization of evolution

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 2 / 30

Optimal transportation problems (OTPs)

Optimal transportation problems (OTPs) Information and entropy Optimal channel problem (OCP) Geometry of information divergence and optimization Dynamical OTP: Optimization of evolution

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 3 / 30

Optimal transportation problems (OTPs)

Kantorovich’s OTP

Optimal Transportation Problem (Kantorovich, 1939, 1942; Vasershtein, 1969; Dobrushin, 1970)

Kc(q, p) ∶= inf { ∫X×Y c(x, y) dw ∶ 휋Xw = q, 휋Yw = p } where c ∶ X × Y → ℝ is a cost function (e.g. a metric). Γ(q, p) ∶= {w ∈ (X ⊗ Y) ∶ 휋Xw = q, 휋Yw = p}

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 4 / 30

Optimal transportation problems (OTPs)

Kantorovich’s OTP

Optimal Transportation Problem (Kantorovich, 1939, 1942; Vasershtein, 1969; Dobrushin, 1970)

Kc(q, p) ∶= inf { ∫X×Y c(x, y) dw ∶ 휋Xw = q, 휋Yw = p } where c ∶ X × Y → ℝ is a cost function (e.g. a metric). Γ(q, p) ∶= {w ∈ (X ⊗ Y) ∶ 휋Xw = q, 휋Yw = p} q p (X) ↔ (X ⊗ Y)

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 4 / 30

Optimal transportation problems (OTPs)

Kantorovich’s OTP

Optimal Transportation Problem (Kantorovich, 1939, 1942; Vasershtein, 1969; Dobrushin, 1970)

Kc(q, p) ∶= inf { ∫X×Y c(x, y) dw ∶ 휋Xw = q, 휋Yw = p } where c ∶ X × Y → ℝ is a cost function (e.g. a metric). Γ(q, p) ∶= {w ∈ (X ⊗ Y) ∶ 휋Xw = q, 휋Yw = p} q p (X) ↔ (X ⊗ Y) 훿21 훿22 훿12 훿11

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 4 / 30

Optimal transportation problems (OTPs)

Optimal Transport Plan

Optimal Transportation Problem

Kc(q, p) ∶= inf { ∫X×Y c(x, y) dw ∶ 휋Xw = q, 휋Yw = p } 훿21 훿22 훿12 훿11

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 5 / 30

Optimal transportation problems (OTPs)

Optimal Transport Plan

Optimal Transportation Problem

Kc(q, p) ∶= inf { ∫X×Y c(x, y) dw ∶ 휋Xw = q, 휋Yw = p }

Optimal Transport Plan

Linear operator (Markov morphism) T ∶ (X) → (Y): q ↦ Tq = p = ∫X p(⋅ ∣ x) dq 훿21 훿22 훿12 훿11

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 5 / 30

Optimal transportation problems (OTPs)

Optimal Transport Plan

Optimal Transportation Problem

Kc(q, p) ∶= inf { ∫X×Y c(x, y) dw ∶ 휋Xw = q, 휋Yw = p }

Optimal Transport Plan

Linear operator (Markov morphism) T ∶ (X) → (Y): q ↦ Tq = p = ∫X p(⋅ ∣ x) dq p(⋅ ∣ x) — Markov transition kernel 훿21 훿22 훿12 훿11

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 5 / 30

Optimal transportation problems (OTPs)

Optimal Transport Plan

Optimal Transportation Problem

Kc(q, p) ∶= inf { ∫X×Y c(x, y) dw ∶ 휋Xw = q, 휋Yw = p }

Optimal Transport Plan

Linear operator (Markov morphism) T ∶ (X) → (Y): q ↦ Tq = p = ∫X p(⋅ ∣ x) dq p(⋅ ∣ x) — Markov transition kernel T is determined by w ∈ (X ⊗ Y): w = p(⋅ ∣ x) ⊗ q 훿21 훿22 훿12 훿11

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 5 / 30

Optimal transportation problems (OTPs)

Monge OTP

Optimal Transportation Problem (Monge, 1781)

Kc(q, p) ∶= inf { ∫X c(x, f(x)) dq ∶ f ∶ p = q◦f −1 } where p = q◦f −1 is push-forward under measurable mapping f ∶ X → Y: p(E) = q◦f −1(E) = q{x ∶ f(x) ∈ E}

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 6 / 30

Optimal transportation problems (OTPs)

Monge OTP

Optimal Transportation Problem (Monge, 1781)

Kc(q, p) ∶= inf { ∫X c(x, f(x)) dq ∶ f ∶ p = q◦f −1 } where p = q◦f −1 is push-forward under measurable mapping f ∶ X → Y: p(E) = q◦f −1(E) = q{x ∶ f(x) ∈ E}

Optimal Transport

p(⋅ ∣ x) has the form: 훿f(x)(E) = { 1 if f(x) ∈ E

• therwise

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 6 / 30

Optimal transportation problems (OTPs)

Monge OTP

Optimal Transportation Problem (Monge, 1781)

Kc(q, p) ∶= inf { ∫X c(x, f(x)) dq ∶ f ∶ p = q◦f −1 } where p = q◦f −1 is push-forward under measurable mapping f ∶ X → Y: p(E) = q◦f −1(E) = q{x ∶ f(x) ∈ E}

Optimal Transport

p(⋅ ∣ x) has the form: 훿f(x)(E) = { 1 if f(x) ∈ E

• therwise

wf ∈ 휕(X ⊗ Y): wf (X, Y ⧵ f(X)) = 0 훿21 훿22 훿12 훿11

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 6 / 30

Information and entropy

Optimal transportation problems (OTPs) Information and entropy Optimal channel problem (OCP) Geometry of information divergence and optimization Dynamical OTP: Optimization of evolution

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 7 / 30

Information and entropy

Shannon’s Information and Entropy

KL-divergence (Kullback & Leibler, 1951)

DKL(p, q) = ∫ [ln p − ln q] dp

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 8 / 30

Information and entropy

Shannon’s Information and Entropy

KL-divergence (Kullback & Leibler, 1951)

DKL(p, q) = ∫ [ln p − ln q] dp 훿21 훿22 훿12 훿11

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 8 / 30

Information and entropy

Shannon’s Information and Entropy

KL-divergence (Kullback & Leibler, 1951)

DKL(p, q) = ∫ [ln p − ln q] dp

Shannon’s information (Shannon, 1948)

For w ∈ Γ(q, p) ⊂ (X ⊗ Y): Iw{x, y} ∶= DKL(w, q ⊗ p) = H(q) − H(q(x ∣ y)) = H(p) − H(p(y ∣ x)) 훿21 훿22 훿12 훿11

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 8 / 30

Information and entropy

Shannon’s Information and Entropy

KL-divergence (Kullback & Leibler, 1951)

DKL(p, q) = ∫ [ln p − ln q] dp

Shannon’s information (Shannon, 1948)

For w ∈ Γ(q, p) ⊂ (X ⊗ Y): Iw{x, y} ∶= DKL(w, q ⊗ p) = H(q) − H(q(x ∣ y)) = H(p) − H(p(y ∣ x))

Entropy H(p) = − ∫ ln p dp

H(p) ∶= sup

w∶휋Yw=p

Iw{x, y} = Iw{y, y} 훿21 훿22 훿12 훿11

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 8 / 30

Information and entropy

w

D(w,q⊗p)

• q ⊗ q D(p,q)
• D(w,q⊗q)
• q ⊗ p

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 9 / 30

Information and entropy

Theorem (Shannon-Pythagorean)

w ∈ (X ⊗ Y), 휋Xw = q, 휋Yw = p DKL(w, q ⊗ q) = DKL(w, q ⊗ p) + DKL(p, q) (Belavkin, 2013a) w

D(w,q⊗p)

• q ⊗ q D(p,q)
• D(w,q⊗q)
• q ⊗ p

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 9 / 30

Information and entropy

Theorem (Shannon-Pythagorean)

w ∈ (X ⊗ Y), 휋Xw = q, 휋Yw = p DKL(w, q ⊗ q) = DKL(w, q ⊗ p) + DKL(p, q) (Belavkin, 2013a) w

D(w,q⊗p)

• q ⊗ q D(p,q)
• D(w,q⊗q)
• q ⊗ p

Proof.

D(w, q⊗q) = D(w, q ⊗ p) ⏟⏞⏞⏞⏞⏟⏞⏞⏞⏞⏟

Iw{x,y}

+ D(q ⊗ p, q ⊗ q) ⏟⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏟

D(p,q)

− ⟨ln q ⊗ p − ln q ⊗ q, q ⊗ p − w⟩ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ln q ⊗ p − ln q ⊗ q = 1X ⊗ (ln p − ln q)

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 9 / 30

Information and entropy

Theorem (Shannon-Pythagorean)

w ∈ (X ⊗ Y), 휋Xw = q, 휋Yw = p DKL(w, q ⊗ q) = DKL(w, q ⊗ p) + DKL(p, q) (Belavkin, 2013a) w

D(w,q⊗p)

• q ⊗ q D(p,q)
• D(w,q⊗q)
• q ⊗ p

Proof.

D(w, q⊗q) = D(w, q ⊗ p) ⏟⏞⏞⏞⏞⏟⏞⏞⏞⏞⏟

Iw{x,y}

+ D(q ⊗ p, q ⊗ q) ⏟⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏟

D(p,q)

− ⟨ln q ⊗ p − ln q ⊗ q, q ⊗ p − w⟩ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ln q ⊗ p − ln q ⊗ q = 1X ⊗ (ln p − ln q)

Cross-Information (Belavkin, 2013a)

DKL(w, q ⊗ q) = −⟨ln q, p⟩ ⏟⏞⏞⏟⏞⏞⏟

Cross-entropy

− [H(p) − DKL(w, q ⊗ p)] ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

H(p(y∣x))

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 9 / 30

Optimal channel problem (OCP)

Optimal transportation problems (OTPs) Information and entropy Optimal channel problem (OCP) Geometry of information divergence and optimization Dynamical OTP: Optimization of evolution

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 10 / 30

Optimal channel problem (OCP)

Shannon’s OCP

Optimal Channel Problem (Shannon, 1948)

Sc(q, 휆) ∶= inf { ∫X×Y c(x, y) dw ∶ 휋Xw = q, Iw{x, y} ≤ 휆 }

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 11 / 30

Optimal channel problem (OCP)

Shannon’s OCP

Optimal Channel Problem (Shannon, 1948)

Sc(q, 휆) ∶= inf { ∫X×Y c(x, y) dw ∶ 휋Xw = q, Iw{x, y} ≤ 휆 } 훿21 훿22 훿12 훿11

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 11 / 30

Optimal channel problem (OCP)

Shannon’s OCP

Optimal Channel Problem (Shannon, 1948)

Sc(q, 휆) ∶= inf { ∫X×Y c(x, y) dw ∶ 휋Xw = q, Iw{x, y} ≤ 휆 }

Exponential family solutions

Optimal T ∶ (X) → (Y) is defined by w = e−훽c−ln Z q⊗p , 훽−1 = −dSc(q, 휆)∕d휆 훿21 훿22 훿12 훿11

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 11 / 30

Optimal channel problem (OCP)

Shannon’s OCP

Optimal Channel Problem (Shannon, 1948)

Sc(q, 휆) ∶= inf { ∫X×Y c(x, y) dw ∶ 휋Xw = q, Iw{x, y} ≤ 휆 }

Exponential family solutions

Optimal T ∶ (X) → (Y) is defined by w = e−훽c−ln Z q⊗p , 훽−1 = −dSc(q, 휆)∕d휆 Observe that w ∉ 휕(X ⊗ Y), unless q ⊗ p ∈ 휕(X ⊗ Y) or 훽 → ∞. 훿21 훿22 훿12 훿11

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 11 / 30

Optimal channel problem (OCP)

Shannon’s OCP

Optimal Channel Problem (Shannon, 1948)

Sc(q, 휆) ∶= inf { ∫X×Y c(x, y) dw ∶ 휋Xw = q, Iw{x, y} ≤ 휆 }

Exponential family solutions

Optimal T ∶ (X) → (Y) is defined by w = e−훽c−ln Z q⊗p , 훽−1 = −dSc(q, 휆)∕d휆 Observe that w ∉ 휕(X ⊗ Y), unless q ⊗ p ∈ 휕(X ⊗ Y) or 훽 → ∞. 훿21 훿22 훿12 훿11

Value of Information (Stratonovich, 1965)

V(휆) ∶= Sc(q, 0) − Sc(q, 휆) = sup{피w{u} ∶ Iw{x, y} ≤ 휆}

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 11 / 30

Optimal channel problem (OCP)

Relation to Kantorovich OTP

Optimal Channel Problem

Sc(q, 휆) ∶= inf { ∫X×Y c(x, y) dw ∶ 휋Xw = q, Iw{x, y} ≤ 휆 }

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 12 / 30

Optimal channel problem (OCP)

Relation to Kantorovich OTP

Optimal Channel Problem

Sc(q, 휆) ∶= inf { ∫X×Y c(x, y) dw ∶ 휋Xw = q, Iw{x, y} ≤ 휆 }

Optimal Transportation Problem

Kc(q, p) ∶= inf { ∫X×Y c(x, y) dw ∶ 휋Xw = q, 휋Yw = p }

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 12 / 30

Optimal channel problem (OCP)

Relation to Kantorovich OTP

Optimal Channel Problem

Sc(q, 휆) ∶= inf { ∫X×Y c(x, y) dw ∶ 휋Xw = q, Iw{x, y} ≤ 휆 }

Optimal Transportation Problem

Kc(q, p) ∶= inf { ∫X×Y c(x, y) dw ∶ 휋Xw = q, 휋Yw = p } q and p have entropies H(q) and H(p) and 0 ≤ Iw{x, y} ≤ min[H(q), H(p)]

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 12 / 30

Optimal channel problem (OCP)

Relation to Kantorovich OTP

Optimal Channel Problem

Sc(q, 휆) ∶= inf { ∫X×Y c(x, y) dw ∶ 휋Xw = q, Iw{x, y} ≤ 휆 }

Optimal Transportation Problem

Kc(q, p) ∶= inf { ∫X×Y c(x, y) dw ∶ 휋Xw = q, 휋Yw = p } q and p have entropies H(q) and H(p) and 0 ≤ Iw{x, y} ≤ min[H(q), H(p)] Kc(q, p) has implicit constraint Iw{x, y} ≤ 휆 = min[H(q), H(p)] and Sc(q, 휆) ≤ Kc(q, p)

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 12 / 30

Optimal channel problem (OCP)

Inverse Optimal Values

Inverse of the OCP Value

S−1

c (q, 휐) ∶= inf

{ Iw{x, y} ∶ 휋Xw = q, ∫ c dw ≤ 휐 }

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 13 / 30

Optimal channel problem (OCP)

Inverse Optimal Values

Inverse of the OCP Value

S−1

c (q, 휐) ∶= inf

{ Iw{x, y} ∶ 휋Xw = q, ∫ c dw ≤ 휐 }

Inverse of the OTP Value

K−1

c (q, p, 휐) ∶= inf

{ Iw{x, y} ∶ 휋Xw = q, 휋Yw = p, ∫ c dw ≤ 휐 }

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 13 / 30

Optimal channel problem (OCP)

Inverse Optimal Values

Inverse of the OCP Value

S−1

c (q, 휐) ∶= inf

{ Iw{x, y} ∶ 휋Xw = q, ∫ c dw ≤ 휐 }

Inverse of the OTP Value

K−1

c (q, p, 휐) ∶= inf

{ Iw{x, y} ∶ 휋Xw = q, 휋Yw = p, ∫ c dw ≤ 휐 } These inverse values represent the smallest amount of Shannon’s information required to achieve expected cost ∫ c dw = 휐.

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 13 / 30

Optimal channel problem (OCP)

Inverse Optimal Values

Inverse of the OCP Value

S−1

c (q, 휐) ∶= inf

{ Iw{x, y} ∶ 휋Xw = q, ∫ c dw ≤ 휐 }

Inverse of the OTP Value

K−1

c (q, p, 휐) ∶= inf

{ Iw{x, y} ∶ 휋Xw = q, 휋Yw = p, ∫ c dw ≤ 휐 } These inverse values represent the smallest amount of Shannon’s information required to achieve expected cost ∫ c dw = 휐. If 휐 = Kc(q, p), then S−1

c (q, 휐) ≤ K−1 c (q, p, 휐)

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 13 / 30

Geometry of information divergence and optimization

Optimal transportation problems (OTPs) Information and entropy Optimal channel problem (OCP) Geometry of information divergence and optimization Dynamical OTP: Optimization of evolution

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 14 / 30

Geometry of information divergence and optimization

Problems on Conditional Extremum

피p{u} = ⟨u, p⟩ expected utility 휔3 휔2 q 피p{f} ≥ 휐 피p{ln(p∕q)} ≤ 휆

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 15 / 30

Geometry of information divergence and optimization

Problems on Conditional Extremum

피p{u} = ⟨u, p⟩ expected utility 휐u(휆) = −S−u(q, 휆) utility of information 휆: 휐u(휆) ∶= sup{⟨u, p⟩ ∶ F(p, q) ≤ 휆} 휔3 휔2 q 피p{f} ≥ 휐 피p{ln(p∕q)} ≤ 휆

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 15 / 30

Geometry of information divergence and optimization

Problems on Conditional Extremum

피p{u} = ⟨u, p⟩ expected utility 휐u(휆) = −S−u(q, 휆) utility of information 휆: 휐u(휆) ∶= sup{⟨u, p⟩ ∶ F(p, q) ≤ 휆} 휆u(휐) = 휐−1

u (휐) information of

utility 휐: 휆u(휐) ∶= inf{F(p, q) ∶ ⟨u, p⟩ ≥ 휐} 휔3 휔2 q 피p{f} ≥ 휐 피p{ln(p∕q)} ≤ 휆

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 15 / 30

Geometry of information divergence and optimization

Problems on Conditional Extremum

피p{u} = ⟨u, p⟩ expected utility 휐u(휆) = −S−u(q, 휆) utility of information 휆: 휐u(휆) ∶= sup{⟨u, p⟩ ∶ F(p, q) ≤ 휆} 휆u(휐) = 휐−1

u (휐) information of

utility 휐: 휆u(휐) ∶= inf{F(p, q) ∶ ⟨u, p⟩ ≥ 휐} p(훽) optimal solutions: p(훽) ∈ 휕F∗(훽u, q) , F(p(훽), q) = 휆 휔3 휔2 q p훽 피p{f} ≥ 휐 피p{ln(p∕q)} ≤ 휆

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 15 / 30

Geometry of information divergence and optimization

Problems on Conditional Extremum

피p{u} = ⟨u, p⟩ expected utility 휐u(휆) = −S−u(q, 휆) utility of information 휆: 휐u(휆) ∶= sup{⟨u, p⟩ ∶ F(p, q) ≤ 휆} 휆u(휐) = 휐−1

u (휐) information of

utility 휐: 휆u(휐) ∶= inf{F(p, q) ∶ ⟨u, p⟩ ≥ 휐} p(훽) optimal solutions: p(훽) ∈ 휕F∗(훽u, q) , F(p(훽), q) = 휆 휔3 휔2 q p훽 피p{f} ≥ 휐 피p{ln(p∕q)} ≤ 휆

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 15 / 30

Geometry of information divergence and optimization

Problems on Conditional Extremum

피p{u} = ⟨u, p⟩ expected utility 휐u(휆) = −S−u(q, 휆) utility of information 휆: 휐u(휆) ∶= sup{⟨u, p⟩ ∶ F(p, q) ≤ 휆} 휆u(휐) = 휐−1

u (휐) information of

utility 휐: 휆u(휐) ∶= inf{F(p, q) ∶ ⟨u, p⟩ ≥ 휐} p(훽) optimal solutions: p(훽) ∈ 휕F∗(훽u, q) , F(p(훽), q) = 휆 휔3 휔2 q p훽 피p{f} ≥ 휐 ‖p − q‖1 ≤ 휆 피p{w} ≥ 휐

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 15 / 30

Geometry of information divergence and optimization

General Solution

Lagrange function for 휐u(휆) ∶= sup{⟨u, p⟩ ∶ F(p, q) ≤ 휆} (휆u(휐) ∶= inf{F(p, q) ∶ ⟨u, p⟩ ≥ 휐}): L(p, 훽−1) = ⟨u, p⟩ + 훽−1[휆 − F(p, q)] ( L(p, 훽) = F(p, q) + 훽[휐 − ⟨u, p⟩] )

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 16 / 30

Geometry of information divergence and optimization

General Solution

Lagrange function for 휐u(휆) ∶= sup{⟨u, p⟩ ∶ F(p, q) ≤ 휆} (휆u(휐) ∶= inf{F(p, q) ∶ ⟨u, p⟩ ≥ 휐}): L(p, 훽−1) = ⟨u, p⟩ + 훽−1[휆 − F(p, q)] ( L(p, 훽) = F(p, q) + 훽[휐 − ⟨u, p⟩] ) Necessary and sufficient conditions 휕L ∋ 0: 휕pL(p, 훽−1) = {훽u} − 휕pF(p, q) ∋ 0 휕훽−1L(p, 훽−1) = 휆 − F(p, q) = 0

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 16 / 30

Geometry of information divergence and optimization

General Solution

Lagrange function for 휐u(휆) ∶= sup{⟨u, p⟩ ∶ F(p, q) ≤ 휆} (휆u(휐) ∶= inf{F(p, q) ∶ ⟨u, p⟩ ≥ 휐}): L(p, 훽−1) = ⟨u, p⟩ + 훽−1[휆 − F(p, q)] ( L(p, 훽) = F(p, q) + 훽[휐 − ⟨u, p⟩] ) Necessary and sufficient conditions 휕L ∋ 0: 휕pL(p, 훽−1) = {훽u} − 휕pF(p, q) ∋ 0 휕훽−1L(p, 훽−1) = 휆 − F(p, q) = 0 Optimal solutions are subgradients of F∗(u, q) = sup{⟨u, p⟩ − F(p, q)}: p(훽) ∈ 휕F∗(훽u) , F(p, q) = 휆 ( ⟨u, p(훽)⟩ = 휐 )

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 16 / 30

Geometry of information divergence and optimization

Example: Exponential Solution

For DKL(p, q) = ⟨ln(p∕q), p⟩ − ⟨1, p − q⟩: L(p, 훽−1) = ⟨u, p⟩ + 훽−1[휆 − ⟨ln(p∕q), p⟩ + ⟨1, p − q⟩]

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 17 / 30

Geometry of information divergence and optimization

Example: Exponential Solution

For DKL(p, q) = ⟨ln(p∕q), p⟩ − ⟨1, p − q⟩: L(p, 훽−1) = ⟨u, p⟩ + 훽−1[휆 − ⟨ln(p∕q), p⟩ + ⟨1, p − q⟩] Necessary and sufficient conditions ∇L(p, 훽−1) = 0: ∇pL(p, 훽−1) = u − 훽−1 ln(p∕q) = 0 휕훽−1L(p, 훽−1) = 휆 − DKL(p, q) = 0

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 17 / 30

Geometry of information divergence and optimization

Example: Exponential Solution

For DKL(p, q) = ⟨ln(p∕q), p⟩ − ⟨1, p − q⟩: L(p, 훽−1) = ⟨u, p⟩ + 훽−1[휆 − ⟨ln(p∕q), p⟩ + ⟨1, p − q⟩] Necessary and sufficient conditions ∇L(p, 훽−1) = 0: ∇pL(p, 훽−1) = u − 훽−1 ln(p∕q) = 0 휕훽−1L(p, 훽−1) = 휆 − DKL(p, q) = 0 Optimal solutions are gradients of D∗

KL(u, q) = ln⟨eu, q⟩:

p(훽) = e훽 u−ln Z(훽u) q , DKL(p(훽), q) = 휆

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 17 / 30

Geometry of information divergence and optimization

Solution to Shannon’s OCP

The solution for Iw{x, y} = DKL(w, q ⊗ p) ≤ 휆: w(훽) = e훽u−ln Z(훽u) q ⊗ p 훿21 훿22 훿12 훿11

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 18 / 30

Geometry of information divergence and optimization

Solution to Shannon’s OCP

The solution for Iw{x, y} = DKL(w, q ⊗ p) ≤ 휆: w(훽) = e훽u−ln Z(훽u) q ⊗ p w ∉ 휕(X ⊗ Y). 훿21 훿22 훿12 훿11

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 18 / 30

Geometry of information divergence and optimization

Solution to Shannon’s OCP

The solution for Iw{x, y} = DKL(w, q ⊗ p) ≤ 휆: w(훽) = e훽u−ln Z(훽u) q ⊗ p w ∉ 휕(X ⊗ Y). T ∶ (X) → (Y) cannot have kernel 훿f(x)(⋅). 훿21 훿22 훿12 훿11

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 18 / 30

Geometry of information divergence and optimization

Solution to Shannon’s OCP

The solution for Iw{x, y} = DKL(w, q ⊗ p) ≤ 휆: w(훽) = e훽u−ln Z(훽u) q ⊗ p w ∉ 휕(X ⊗ Y). T ∶ (X) → (Y) cannot have kernel 훿f(x)(⋅). The dual is strictly convex: D∗

KL(u, q ⊗ p) = ln ∫ eu q ⊗ p

훿21 훿22 훿12 훿11

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 18 / 30

Geometry of information divergence and optimization

Solution to Kantorovich’s OTP

Γ(q, p) is convex: 휋X((1 − t)w1 + tw2) = (1 − t)q + tq = q 훿21 훿22 훿12 훿11

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 19 / 30

Geometry of information divergence and optimization

Solution to Kantorovich’s OTP

Γ(q, p) is convex: 휋X((1 − t)w1 + tw2) = (1 − t)q + tq = q There exists closed convex functional F: Γ(q, p) = {w ∶ F(w, q ⊗ p) ≤ 1} 훿21 훿22 훿12 훿11

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 19 / 30

Geometry of information divergence and optimization

Solution to Kantorovich’s OTP

Γ(q, p) is convex: 휋X((1 − t)w1 + tw2) = (1 − t)q + tq = q There exists closed convex functional F: Γ(q, p) = {w ∶ F(w, q ⊗ p) ≤ 1} Then the solution to OTP is: w(훽) ∈ 휕F∗(−훽c, q ⊗ p) 훿21 훿22 훿12 훿11

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 19 / 30

Geometry of information divergence and optimization

Solution to Kantorovich’s OTP

Γ(q, p) is convex: 휋X((1 − t)w1 + tw2) = (1 − t)q + tq = q There exists closed convex functional F: Γ(q, p) = {w ∶ F(w, q ⊗ p) ≤ 1} Then the solution to OTP is: w(훽) ∈ 휕F∗(−훽c, q ⊗ p) 훿21 훿22 훿12 훿11

Monge-Amper equation

q = p◦∇휑|∇2휑| where 휑 ∶ X → ℝ ∪ {∞} is convex, and ∇휑 ∶ X → Y is such that p = q◦(∇휑)−1 (McCann, 1995; Villani, 2009).

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 19 / 30

Geometry of information divergence and optimization

Strict Inequalities

Theorem (Belavkin, 2013b)

Let {w(훽)}u be a family of w(훽) ∈ (X ⊗ Y) maximizing 피w{u} on sets {w ∶ F(w) ≤ 휆} , ∀ 휆 = F(w)

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 20 / 30

Geometry of information divergence and optimization

Strict Inequalities

Theorem (Belavkin, 2013b)

Let {w(훽)}u be a family of w(훽) ∈ (X ⊗ Y) maximizing 피w{u} on sets {w ∶ F(w) ≤ 휆} , ∀ 휆 = F(w) F ∶ (X ⊗ Y) → ℝ ∪ {∞} closed convex and minimized at q ⊗ p ∈ 휕F∗(0) ⊂ Int((X ⊗ Y))

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 20 / 30

Geometry of information divergence and optimization

Strict Inequalities

Theorem (Belavkin, 2013b)

Let {w(훽)}u be a family of w(훽) ∈ (X ⊗ Y) maximizing 피w{u} on sets {w ∶ F(w) ≤ 휆} , ∀ 휆 = F(w) F ∶ (X ⊗ Y) → ℝ ∪ {∞} closed convex and minimized at q ⊗ p ∈ 휕F∗(0) ⊂ Int((X ⊗ Y)) If F∗ is strictly convex, then

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 20 / 30

Geometry of information divergence and optimization

Strict Inequalities

Theorem (Belavkin, 2013b)

Let {w(훽)}u be a family of w(훽) ∈ (X ⊗ Y) maximizing 피w{u} on sets {w ∶ F(w) ≤ 휆} , ∀ 휆 = F(w) F ∶ (X ⊗ Y) → ℝ ∪ {∞} closed convex and minimized at q ⊗ p ∈ 휕F∗(0) ⊂ Int((X ⊗ Y)) If F∗ is strictly convex, then

1

w(훽) ∈ 휕(X ⊗ Y) iff 휆 ≥ sup F (i.e. 훽 → ∞).

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 20 / 30

Geometry of information divergence and optimization

Strict Inequalities

Theorem (Belavkin, 2013b)

Let {w(훽)}u be a family of w(훽) ∈ (X ⊗ Y) maximizing 피w{u} on sets {w ∶ F(w) ≤ 휆} , ∀ 휆 = F(w) F ∶ (X ⊗ Y) → ℝ ∪ {∞} closed convex and minimized at q ⊗ p ∈ 휕F∗(0) ⊂ Int((X ⊗ Y)) If F∗ is strictly convex, then

1

w(훽) ∈ 휕(X ⊗ Y) iff 휆 ≥ sup F (i.e. 훽 → ∞).

2

For any v ∈ 휕(X ⊗ Y) with F(v) = F(w(훽)) = 휆 피v{u} < 피w(훽){u}

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 20 / 30

Geometry of information divergence and optimization

Strict Inequalities

Theorem (Belavkin, 2013b)

Let {w(훽)}u be a family of w(훽) ∈ (X ⊗ Y) maximizing 피w{u} on sets {w ∶ F(w) ≤ 휆} , ∀ 휆 = F(w) F ∶ (X ⊗ Y) → ℝ ∪ {∞} closed convex and minimized at q ⊗ p ∈ 휕F∗(0) ⊂ Int((X ⊗ Y)) If F∗ is strictly convex, then

1

w(훽) ∈ 휕(X ⊗ Y) iff 휆 ≥ sup F (i.e. 훽 → ∞).

2

For any v ∈ 휕(X ⊗ Y) with F(v) = F(w(훽)) = 휆 피v{u} < 피w(훽){u}

3

For any v ∈ 휕(X ⊗ Y) with 피v{u} = 피w(훽){u} = 휐 F(v) > F(w(훽))

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 20 / 30

Geometry of information divergence and optimization

Strict Bounds for Monge OTP

Corollary

Let wf ∈ Γ(q, p) be a solution to Monge OTP Kc(q, p).

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 21 / 30

Geometry of information divergence and optimization

Strict Bounds for Monge OTP

Corollary

Let wf ∈ Γ(q, p) be a solution to Monge OTP Kc(q, p). Let w(훽) is a solution to Shannon’s OCP Sc(q, 휆).

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 21 / 30

Geometry of information divergence and optimization

Strict Bounds for Monge OTP

Corollary

Let wf ∈ Γ(q, p) be a solution to Monge OTP Kc(q, p). Let w(훽) is a solution to Shannon’s OCP Sc(q, 휆). If wf and w(훽) have equal Iwf {x, y} = Iw(훽){x, y} = 휆 < sup Iw{x, y}, then Kc(q, p) > Sc(q, 휆) > 0

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 21 / 30

Geometry of information divergence and optimization

Strict Bounds for Monge OTP

Corollary

Let wf ∈ Γ(q, p) be a solution to Monge OTP Kc(q, p). Let w(훽) is a solution to Shannon’s OCP Sc(q, 휆). If wf and w(훽) have equal Iwf {x, y} = Iw(훽){x, y} = 휆 < sup Iw{x, y}, then Kc(q, p) > Sc(q, 휆) > 0 If wf and w(훽) achieve equal values Kc(q, p) = Sc(q, 휆) = 휐 > 0, then K−1

c (q, p, 휐) > S−1 c (q, 휐)

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 21 / 30

Geometry of information divergence and optimization

Optimal Transport and the Expected Utility Principle

Let (X × Y, ≲) be a set with preference relation (total pre-order).

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 22 / 30

Geometry of information divergence and optimization

Optimal Transport and the Expected Utility Principle

Let (X × Y, ≲) be a set with preference relation (total pre-order). The Neumann and Morgenstern (1944) EU principle states that for any v, w ∈ (X ⊗ Y): v ≲ w ⟺ 피v{u} ≤ 피w{u} ∃ u ∶ X × Y → ℝ

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 22 / 30

Geometry of information divergence and optimization

Optimal Transport and the Expected Utility Principle

Let (X × Y, ≲) be a set with preference relation (total pre-order). The Neumann and Morgenstern (1944) EU principle states that for any v, w ∈ (X ⊗ Y): v ≲ w ⟺ 피v{u} ≤ 피w{u} ∃ u ∶ X × Y → ℝ Based on linear and Archimedian axioms: v ≲ w ⟺ 휆v ≲ 휆w , ∀ 휆 > 0 (1) v ≲ w ⟺ v + r ≲ w + r , ∀ r ∈ L (2) nv ≲ w , ∀ n ∈ ℕ ⇒ v ≲ 0 (3)

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 22 / 30

Geometry of information divergence and optimization

Optimal Transport and the Expected Utility Principle

Let (X × Y, ≲) be a set with preference relation (total pre-order). The Neumann and Morgenstern (1944) EU principle states that for any v, w ∈ (X ⊗ Y): v ≲ w ⟺ 피v{u} ≤ 피w{u} ∃ u ∶ X × Y → ℝ Based on linear and Archimedian axioms: v ≲ w ⟺ 휆v ≲ 휆w , ∀ 휆 > 0 (1) v ≲ w ⟺ v + r ≲ w + r , ∀ r ∈ L (2) nv ≲ w , ∀ n ∈ ℕ ⇒ v ≲ 0 (3)

Theorem

Pre-ordered linear space (L, ≲) satisfies (1), (2) and (3) if and only if (L, ≲) has a utility representation by a closed linear functional u ∶ L → ℝ.

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 22 / 30

Dynamical OTP: Optimization of evolution

Optimal transportation problems (OTPs) Information and entropy Optimal channel problem (OCP) Geometry of information divergence and optimization Dynamical OTP: Optimization of evolution

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 23 / 30

Dynamical OTP: Optimization of evolution

Dynamical Problems

q(t) ↦ Tq(t) = q(t + 1) q

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 24 / 30

Dynamical OTP: Optimization of evolution

Dynamical Problems

q(t) ↦ Tq(t) = q(t + 1) q

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 24 / 30

Dynamical OTP: Optimization of evolution

Dynamical Problems

q(t) ↦ Tq(t) = q(t + 1) q

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 24 / 30

Dynamical OTP: Optimization of evolution

Dynamical Problems

q(t) ↦ Tq(t) = q(t + 1) q

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 24 / 30

Dynamical OTP: Optimization of evolution

Dynamical Problems

q(t) ↦ Tq(t) = q(t + 1) q

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 24 / 30

Dynamical OTP: Optimization of evolution

Dynamical Problems

q(t) ↦ Tq(t) = q(t + 1) q

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 24 / 30

Dynamical OTP: Optimization of evolution

Dynamical Problems

q(t) ↦ Tq(t) = q(t + 1) q

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 24 / 30

Dynamical OTP: Optimization of evolution

Dynamical Problems

q(t) ↦ Tq(t) = q(t + 1) q

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 24 / 30

Dynamical OTP: Optimization of evolution

Dynamical Problems

q(t) ↦ Tq(t) = q(t + 1) q

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 24 / 30

Dynamical OTP: Optimization of evolution

Dynamical Problems

q(t) ↦ Tq(t) = q(t + 1) q

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 24 / 30

Dynamical OTP: Optimization of evolution

Dynamical Problems

q(t) ↦ Tq(t) = q(t + 1) q(t) ↦ Tsq(t) = q(t + s) q

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 24 / 30

Dynamical OTP: Optimization of evolution

Dynamical Problems

q(t) ↦ Tq(t) = q(t + 1) q(t) ↦ Tsq(t) = q(t + s) q(t) → 훿⊤ q

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 24 / 30

Dynamical OTP: Optimization of evolution

Dynamical Problems

q(t) ↦ Tq(t) = q(t + 1) q(t) ↦ Tsq(t) = q(t + s) q(t) → 훿⊤ Expected utility: 피q(t){u} = ∫X u dq(t) q

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 24 / 30

Dynamical OTP: Optimization of evolution

Dynamical Problems

q(t) ↦ Tq(t) = q(t + 1) q(t) ↦ Tsq(t) = q(t + s) q(t) → 훿⊤ Expected utility: 피q(t){u} = ∫X u dq(t) q

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 24 / 30

Dynamical OTP: Optimization of evolution

Dynamical Problems

q(t) ↦ Tq(t) = q(t + 1) q(t) ↦ Tsq(t) = q(t + s) q(t) → 훿⊤ Expected utility: 피q(t){u} = ∫X u dq(t) q

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 24 / 30

Dynamical OTP: Optimization of evolution

Dynamical Problems

q(t) ↦ Tq(t) = q(t + 1) q(t) ↦ Tsq(t) = q(t + s) q(t) → 훿⊤ Expected utility: 피q(t){u} = ∫X u dq(t) q

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 24 / 30

Dynamical OTP: Optimization of evolution

Dynamical Problems

q(t) ↦ Tq(t) = q(t + 1) q(t) ↦ Tsq(t) = q(t + s) q(t) → 훿⊤ Expected utility: 피q(t){u} = ∫X u dq(t) q

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 24 / 30

Dynamical OTP: Optimization of evolution

Dynamical Problems

q(t) ↦ Tq(t) = q(t + 1) q(t) ↦ Tsq(t) = q(t + s) q(t) → 훿⊤ Expected utility: 피q(t){u} = ∫X u dq(t) q

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 24 / 30

Dynamical OTP: Optimization of evolution

Dynamical Problems

q(t) ↦ Tq(t) = q(t + 1) q(t) ↦ Tsq(t) = q(t + s) q(t) → 훿⊤ Expected utility: 피q(t){u} = ∫X u dq(t) q

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 24 / 30

Dynamical OTP: Optimization of evolution

Dynamical Problems

q(t) ↦ Tq(t) = q(t + 1) q(t) ↦ Tsq(t) = q(t + s) q(t) → 훿⊤ Expected utility: 피q(t){u} = ∫X u dq(t) q

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 24 / 30

Dynamical OTP: Optimization of evolution

Dynamical Problems

q(t) ↦ Tq(t) = q(t + 1) q(t) ↦ Tsq(t) = q(t + s) q(t) → 훿⊤ Expected utility: 피q(t){u} = ∫X u dq(t) q

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 24 / 30

Dynamical OTP: Optimization of evolution

Dynamical Problems

q(t) ↦ Tq(t) = q(t + 1) q(t) ↦ Tsq(t) = q(t + s) q(t) → 훿⊤ Expected utility: 피q(t){u} = ∫X u dq(t) q

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 24 / 30

Dynamical OTP: Optimization of evolution

Dynamical Problems

q(t) ↦ Tq(t) = q(t + 1) q(t) ↦ Tsq(t) = q(t + s) q(t) → 훿⊤ Expected utility: 피q(t){u} = ∫X u dq(t) Why not directly q ↦ Tq = 훿⊤? q

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 24 / 30

Dynamical OTP: Optimization of evolution

Example: Search in Hamming space {1, … , 훼}l

Find ⊤ ∈ {1, … , 훼}l.

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 25 / 30

Dynamical OTP: Optimization of evolution

Example: Search in Hamming space {1, … , 훼}l

Find ⊤ ∈ {1, … , 훼}l. Hamming metric dH(x, y) = ‖y − x‖H = l − ∑l

i=1 훿xiyi

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 25 / 30

Dynamical OTP: Optimization of evolution

Example: Search in Hamming space {1, … , 훼}l

Find ⊤ ∈ {1, … , 훼}l. Hamming metric dH(x, y) = ‖y − x‖H = l − ∑l

i=1 훿xiyi

|{1, … , 훼}l| = 훼l

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 25 / 30

Dynamical OTP: Optimization of evolution

Example: Search in Hamming space {1, … , 훼}l

Find ⊤ ∈ {1, … , 훼}l. Hamming metric dH(x, y) = ‖y − x‖H = l − ∑l

i=1 훿xiyi

|{1, … , 훼}l| = 훼l To find ⊤ in one step we need l log2 훼 bits.

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 25 / 30

Dynamical OTP: Optimization of evolution

Example: Search in Hamming space {1, … , 훼}l

Find ⊤ ∈ {1, … , 훼}l. Hamming metric dH(x, y) = ‖y − x‖H = l − ∑l

i=1 훿xiyi

|{1, … , 훼}l| = 훼l To find ⊤ in one step we need l log2 훼 bits. dH(⊤, x) communicates no more than log2(l + 1) bits.

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 25 / 30

Dynamical OTP: Optimization of evolution

Example: Search in Hamming space {1, … , 훼}l

Find ⊤ ∈ {1, … , 훼}l. Hamming metric dH(x, y) = ‖y − x‖H = l − ∑l

i=1 훿xiyi

|{1, … , 훼}l| = 훼l To find ⊤ in one step we need l log2 훼 bits. dH(⊤, x) communicates no more than log2(l + 1) bits. For l ≥ 2 log2(l + 1) < l log2 훼

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 25 / 30

Dynamical OTP: Optimization of evolution

Optimal Control of Mutation Rate

Optimal transition kernels are P훽(y ∣ x) = e−훽 ‖y−x‖H [1 + (훼 − 1)e−훽]l =

l

∏

i=1

e−훽 (1−훿xiyi) 1 + (훼 − 1)e−훽

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 26 / 30

Dynamical OTP: Optimization of evolution

Optimal Control of Mutation Rate

Optimal transition kernels are P훽(y ∣ x) = e−훽 ‖y−x‖H [1 + (훼 − 1)e−훽]l =

l

∏

i=1

e−훽 (1−훿xiyi) 1 + (훼 − 1)e−훽 훽 = ln (휇−1 − 1) + ln(훼 − 1), where 휇 = 휐∕l is the mutation rate, defined by the constraint 피{‖y − x‖H} ≤ 휐.

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 26 / 30

Dynamical OTP: Optimization of evolution

Optimal Control of Mutation Rate

Optimal transition kernels are P훽(y ∣ x) = e−훽 ‖y−x‖H [1 + (훼 − 1)e−훽]l =

l

∏

i=1

e−훽 (1−훿xiyi) 1 + (훼 − 1)e−훽 훽 = ln (휇−1 − 1) + ln(훼 − 1), where 휇 = 휐∕l is the mutation rate, defined by the constraint 피{‖y − x‖H} ≤ 휐. Optimal randomization corresponds to independent substitution of letters with probability 휇 (mutation rate): P휇(r ∣ n) = ( l r ) 휇(n)r(1 − 휇(n))l−r

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 26 / 30

Dynamical OTP: Optimization of evolution

Optimal Control of Mutation Rate

Optimal transition kernels are P훽(y ∣ x) = e−훽 ‖y−x‖H [1 + (훼 − 1)e−훽]l =

l

∏

i=1

e−훽 (1−훿xiyi) 1 + (훼 − 1)e−훽 훽 = ln (휇−1 − 1) + ln(훼 − 1), where 휇 = 휐∕l is the mutation rate, defined by the constraint 피{‖y − x‖H} ≤ 휐. Optimal randomization corresponds to independent substitution of letters with probability 휇 (mutation rate): P휇(r ∣ n) = ( l r ) 휇(n)r(1 − 휇(n))l−r Which 휇(n)?

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 26 / 30

Dynamical OTP: Optimization of evolution

Optimal Control of Mutation Rate

Optimal transition kernels are P훽(y ∣ x) = e−훽 ‖y−x‖H [1 + (훼 − 1)e−훽]l =

l

∏

i=1

e−훽 (1−훿xiyi) 1 + (훼 − 1)e−훽 훽 = ln (휇−1 − 1) + ln(훼 − 1), where 휇 = 휐∕l is the mutation rate, defined by the constraint 피{‖y − x‖H} ≤ 휐. Optimal randomization corresponds to independent substitution of letters with probability 휇 (mutation rate): P휇(r ∣ n) = ( l r ) 휇(n)r(1 − 휇(n))l−r Which 휇(n)? The CDF method: 휇(n) =

n−1

∑

m=0

( l m ) (훼 − 1)m 훼l

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 26 / 30

Dynamical OTP: Optimization of evolution

Evolution of Fitness in Information

1 2 3 4 5 6 7 5 10 15 20 Distance to optimum n = d(⊤, a) Information (bits) Constant 1∕l Step Linear n∕l max휇 P휇(m < n ∣ n) P0(m < n ∣ n) 휙∗(n; 휆)

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 27 / 30

Dynamical OTP: Optimization of evolution

Mutation Rate Control in E. coli

Used strains of Escherichia coli K-12 MG1665

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 28 / 30

Dynamical OTP: Optimization of evolution

Mutation Rate Control in E. coli

Used strains of Escherichia coli K-12 MG1665 Fluctuation test using media 50휇g∕ml of Rifamipicin

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 28 / 30

Dynamical OTP: Optimization of evolution

Mutation Rate Control in E. coli

Used strains of Escherichia coli K-12 MG1665 Fluctuation test using media 50휇g∕ml of Rifamipicin Estimated mutation rates 휇 in E.coli strains grown in Davis minimal medium with different amount of glucose.

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 28 / 30

Dynamical OTP: Optimization of evolution

Experimental Results (Krašovec et al., 2014)

1 2 3 4 2 4 6 8 10 12 Mutation rate (10−9 / gen) Population density (108 c.f.u./ml) Strong relationship between 휇 and density of cells (p < .0001).

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 29 / 30

Dynamical OTP: Optimization of evolution

Experimental Results (Krašovec et al., 2014)

1 2 3 4 2 4 6 8 10 12 Mutation rate (10−9 / gen) Population density (108 c.f.u./ml) Strong relationship between 휇 and density of cells (p < .0001). No such relationship in the luxS quorum sensing mutant (p = .0234).

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 29 / 30

Dynamical OTP: Optimization of evolution

Experimental Results (Krašovec et al., 2014)

1 2 3 4 2 4 6 8 10 12 Mutation rate (10−9 / gen) Population density (108 c.f.u./ml) Strong relationship between 휇 and density of cells (p < .0001). No such relationship in the luxS quorum sensing mutant (p = .0234).

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 29 / 30

Dynamical OTP: Optimization of evolution

Experimental Results (Krašovec et al., 2014)

1 2 3 4 2 4 6 8 10 12 Mutation rate (10−9 / gen) Population density (108 c.f.u./ml) Strong relationship between 휇 and density of cells (p < .0001). No such relationship in the luxS quorum sensing mutant (p = .0234).

Krašovec, R., Belavkin, R., Aston, J., Channon, A., Aston, E., Rash, B., Kadirvel, M.,

• Forbes. S., Knight, C. G. (2014, April). Mutation-rate-plasticity in rifampicin resistance

depends on Escherichia coli cell-cell interactions. Nature Communications, Vol. 5 (3742).

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 29 / 30

References

Optimal transportation problems (OTPs) Information and entropy Optimal channel problem (OCP) Geometry of information divergence and optimization Dynamical OTP: Optimization of evolution

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 30 / 30

References

Belavkin, R. V. (2013a). Law of cosines and Shannon-Pythagorean theorem for quantum information. In F. Nielsen & F. Barbaresco (Eds.), Geometric science of information (Vol. 8085, pp. 369–376). Heidelberg: Springer. Belavkin, R. V. (2013b). Optimal measures and Markov transition kernels. Journal of Global Optimization, 55, 387–416. Dobrushin, R. L. (1970). Prescribing a system of random variables by conditional distributions. Theory Probab. Appl., 15(3), 458âĂŞ-486. Kantorovich, L. V. (1939). Mathematical methods in the organization and planning of production (Tech. Rep.). Leningrad Univ. (English translation: Management Science, 6, 4 (1960), 363âĂŞ422) Kantorovich, L. V. (1942). On translocation of masses. USSR AS Doklady, 37(7–8), 227–229. ((in Russian). English translation: J. Math. Sci., 133, 4 (2006), 1381âĂŞ1382) Krašovec, R., Belavkin, R. V., Aston, J. A. D., Channon, A., Aston, E., Rash, B. M., et al. (2014, April). Mutation rate plasticity in rifampicin resistance depends on escherichia coli cell-cell interactions. Nature Communications, 5(3742).

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 30 / 30

References

Kullback, S., & Leibler, R. A. (1951). On information and sufficiency. The Annals of Mathematical Statistics, 22(1), 79–86. McCann, R. J. (1995). Existence and uniqueness of monotone measure-preserving maps. Duke Math. J., 80(2), 309âĂŞ-323. Monge, G. (1781). Mémoire sur la théorie des déblais et de remblais. Paris: Histoire de l’AcadÂťemie Royale des Sciences avec les Mémoires de Mathématique & de Physique. Neumann, J. von, & Morgenstern, O. (1944). Theory of games and economic behavior. Princeton, NJ: Princeton University Press. Shannon, C. E. (1948, July and October). A mathematical theory of

• communication. Bell System Technical Journal, 27, 379–423 and

623–656. Stratonovich, R. L. (1965). On value of information. Izvestiya of USSR Academy of Sciences, Technical Cybernetics, 5, 3–12. (In Russian) Vasershtein, L. N. (1969). Markov processes over denumerable products of spaces describing large system of automata. Problems Inform. Transmission, 5(3), 47âĂŞ-52. Villani, C. (2009). Optimal transport. old and new (Vol. 338).

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 30 / 30

Dynamical OTP: Optimization of evolution

Springer-Verlag.

Roman Belavkin (Middlesex University) Kantorovich’s and Shannon’s optimization problems 13 June 2016 30 / 30