Résumé

Asymmetric information distances are used to define asymmetric norms and quasimetrics on the statistical manifold and its dual space of random variables. Quasimetric topology, generated by the Kullback-Leibler (KL) divergence, is considered as the main example, and some of its topological properties are investigated.

Asymmetric Topologies on Statistical Manifolds

Média

Voir la vidéo
YouTube
0:00
unavailable

Métriques

154
4
333.63 Ko
 application/pdf
bitcache://90afebf6829a20dc81758b0c4f56575776201349

Licence

Creative Commons

Sponsors

Organisateurs

Sponsors

logo-lix.png
logorioniledefrance.jpg
isc-pif_logo.png
csdcunitwinlogo.jpg
<resource  xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
                xmlns="http://datacite.org/schema/kernel-4"
                xsi:schemaLocation="http://datacite.org/schema/kernel-4 http://schema.datacite.org/meta/kernel-4/metadata.xsd">
        <identifier identifierType="DOI">10.23723/11784/14261</identifier><creators><creator><creatorName>Roman Belavkin</creatorName></creator></creators><titles>
            <title>Asymmetric Topologies on Statistical Manifolds</title></titles>
        <publisher>SEE</publisher>
        <publicationYear>2015</publicationYear>
        <resourceType resourceTypeGeneral="Text">Text</resourceType><dates>
	    <date dateType="Created">Sat 7 Nov 2015</date>
	    <date dateType="Updated">Wed 31 Aug 2016</date>
            <date dateType="Submitted">Sun 22 Oct 2017</date>
	</dates>
        <alternateIdentifiers>
	    <alternateIdentifier alternateIdentifierType="bitstream">90afebf6829a20dc81758b0c4f56575776201349</alternateIdentifier>
	</alternateIdentifiers>
        <formats>
	    <format>application/pdf</format>
	</formats>
	<version>24602</version>
        <descriptions>
            <description descriptionType="Abstract">
Asymmetric information distances are used to define asymmetric norms and quasimetrics on the statistical manifold and its dual space of random variables. Quasimetric topology, generated by the Kullback-Leibler (KL) divergence, is considered as the main example, and some of its topological properties are investigated.

</description>
        </descriptions>
    </resource>
.

Asymmetric Topologies on Statistical Manifolds Roman V. Belavkin School of Science and Technology Middlesex University, London NW4 4BT, UK GSI2015, October 28, 2015 Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 1 / 16 Sources and Consequences of Asymmetry Method: Symmetric Sandwich Results Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 2 / 16 Sources and Consequences of Asymmetry Sources and Consequences of Asymmetry Method: Symmetric Sandwich Results Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 3 / 16 Sources and Consequences of Asymmetry Asymmetric Information Distances Kullback-Leibler divergence D[p, q] = Eq{ln(p/q)} q Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 4 / 16 Sources and Consequences of Asymmetry Asymmetric Information Distances Kullback-Leibler divergence D[p, q] = Eq{ln(p/q)} D[p1⊗p2, q1⊗q2] = D[p1, q1]+D[p2, q2] q Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 4 / 16 Sources and Consequences of Asymmetry Asymmetric Information Distances Kullback-Leibler divergence D[p, q] = Eq{ln(p/q)} D[p1⊗p2, q1⊗q2] = D[p1, q1]+D[p2, q2] ln : (R+, ×) → (R, +) q Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 4 / 16 Sources and Consequences of Asymmetry Asymmetric Information Distances Kullback-Leibler divergence D[p, q] = Eq{ln(p/q)} D[p1⊗p2, q1⊗q2] = D[p1, q1]+D[p2, q2] ln : (R+, ×) → (R, +) q Asymmetry of the KL-divergence D[p, q] = D[q, p] Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 4 / 16 Sources and Consequences of Asymmetry Asymmetric Information Distances Kullback-Leibler divergence D[p, q] = Eq{ln(p/q)} D[p1⊗p2, q1⊗q2] = D[p1, q1]+D[p2, q2] ln : (R+, ×) → (R, +) q Asymmetry of the KL-divergence D[p, q] = D[q, p] D[q + (p − q), q] = D[q − (p − q), q] Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 4 / 16 Sources and Consequences of Asymmetry Asymmetric Information Distances Kullback-Leibler divergence D[p, q] = Eq{ln(p/q)} D[p1⊗p2, q1⊗q2] = D[p1, q1]+D[p2, q2] ln : (R+, ×) → (R, +) q Asymmetry of the KL-divergence D[p, q] = D[q, p] D[q + (p − q), q] = D[q − (p − q), q] Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 4 / 16 Sources and Consequences of Asymmetry Asymmetric Information Distances Kullback-Leibler divergence D[p, q] = Eq{ln(p/q)} D[p1⊗p2, q1⊗q2] = D[p1, q1]+D[p2, q2] ln : (R+, ×) → (R, +) q Asymmetry of the KL-divergence D[p, q] = D[q, p] D[q + (p − q), q] = D[q − (p − q), q] p − q| = inf{α−1 > 0 : D[q + α(p − q), q] ≤ 1} Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 4 / 16 Sources and Consequences of Asymmetry Asymmetric Information Distances Kullback-Leibler divergence D[p, q] = Eq{ln(p/q)} D[p1⊗p2, q1⊗q2] = D[p1, q1]+D[p2, q2] ln : (R+, ×) → (R, +) q Asymmetry of the KL-divergence D[p, q] = D[q, p] D[q + (p − q), q] = D[q − (p − q), q] p − q| = inf{α−1 > 0 : D[q + α(p − q), q] ≤ 1} sup x {Ep−q{x} : Eq{ex − 1 − x} ≤ 1} Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 4 / 16 Sources and Consequences of Asymmetry Asymmetric Information Distances Kullback-Leibler divergence D[p, q] = Eq{ln(p/q)} D[p1⊗p2, q1⊗q2] = D[p1, q1]+D[p2, q2] ln : (R+, ×) → (R, +) q Asymmetry of the KL-divergence D[p, q] = D[q, p] D[q + (p − q), q] = D[q − (p − q), q] p − q = inf{α−1 > 0 : D[q + α|(p − q)|, q] ≤ 1} sup x {Ep−q{x} : Eq{e|x| − 1 − |x|} ≤ 1} Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 4 / 16 Sources and Consequences of Asymmetry Functional Analysis in Asymmetric Spaces Theorem (e.g. Theorem 1.5 in Fletcher and Lindgren (1982)) Every topological space with a countable base is quasi-pseudometrizable. Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 5 / 16 Sources and Consequences of Asymmetry Functional Analysis in Asymmetric Spaces Theorem (e.g. Theorem 1.5 in Fletcher and Lindgren (1982)) Every topological space with a countable base is quasi-pseudometrizable. An asymmetric seminormed space can be T0, but not T1 (and hence not Hausdorff T2). Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 5 / 16 Sources and Consequences of Asymmetry Functional Analysis in Asymmetric Spaces Theorem (e.g. Theorem 1.5 in Fletcher and Lindgren (1982)) Every topological space with a countable base is quasi-pseudometrizable. An asymmetric seminormed space can be T0, but not T1 (and hence not Hausdorff T2). Dual quasimetrics ρ(x, y) and ρ−1(x, y) = ρ(y, x) induce two different topologies. Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 5 / 16 Sources and Consequences of Asymmetry Functional Analysis in Asymmetric Spaces Theorem (e.g. Theorem 1.5 in Fletcher and Lindgren (1982)) Every topological space with a countable base is quasi-pseudometrizable. An asymmetric seminormed space can be T0, but not T1 (and hence not Hausdorff T2). Dual quasimetrics ρ(x, y) and ρ−1(x, y) = ρ(y, x) induce two different topologies. There are 7 notions of Cauchy sequences: left (right) Cauchy, left (right) K-Cauchy, weakly left (right) K-Cauchy, Cauchy. Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 5 / 16 Sources and Consequences of Asymmetry Functional Analysis in Asymmetric Spaces Theorem (e.g. Theorem 1.5 in Fletcher and Lindgren (1982)) Every topological space with a countable base is quasi-pseudometrizable. An asymmetric seminormed space can be T0, but not T1 (and hence not Hausdorff T2). Dual quasimetrics ρ(x, y) and ρ−1(x, y) = ρ(y, x) induce two different topologies. There are 7 notions of Cauchy sequences: left (right) Cauchy, left (right) K-Cauchy, weakly left (right) K-Cauchy, Cauchy. This gives 14 notions of completeness (with respect to ρ or ρ−1). Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 5 / 16 Sources and Consequences of Asymmetry Functional Analysis in Asymmetric Spaces Theorem (e.g. Theorem 1.5 in Fletcher and Lindgren (1982)) Every topological space with a countable base is quasi-pseudometrizable. An asymmetric seminormed space can be T0, but not T1 (and hence not Hausdorff T2). Dual quasimetrics ρ(x, y) and ρ−1(x, y) = ρ(y, x) induce two different topologies. There are 7 notions of Cauchy sequences: left (right) Cauchy, left (right) K-Cauchy, weakly left (right) K-Cauchy, Cauchy. This gives 14 notions of completeness (with respect to ρ or ρ−1). Compactness is related to outer precompactness or precompactness, which are strictly weaker properties than total boundedness. Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 5 / 16 Sources and Consequences of Asymmetry Functional Analysis in Asymmetric Spaces Theorem (e.g. Theorem 1.5 in Fletcher and Lindgren (1982)) Every topological space with a countable base is quasi-pseudometrizable. An asymmetric seminormed space can be T0, but not T1 (and hence not Hausdorff T2). Dual quasimetrics ρ(x, y) and ρ−1(x, y) = ρ(y, x) induce two different topologies. There are 7 notions of Cauchy sequences: left (right) Cauchy, left (right) K-Cauchy, weakly left (right) K-Cauchy, Cauchy. This gives 14 notions of completeness (with respect to ρ or ρ−1). Compactness is related to outer precompactness or precompactness, which are strictly weaker properties than total boundedness. An asymmetric seminormed space may fail to be a topological vector space, because y → αy can be discontinuous (Borodin, 2001). Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 5 / 16 Sources and Consequences of Asymmetry Functional Analysis in Asymmetric Spaces Theorem (e.g. Theorem 1.5 in Fletcher and Lindgren (1982)) Every topological space with a countable base is quasi-pseudometrizable. An asymmetric seminormed space can be T0, but not T1 (and hence not Hausdorff T2). Dual quasimetrics ρ(x, y) and ρ−1(x, y) = ρ(y, x) induce two different topologies. There are 7 notions of Cauchy sequences: left (right) Cauchy, left (right) K-Cauchy, weakly left (right) K-Cauchy, Cauchy. This gives 14 notions of completeness (with respect to ρ or ρ−1). Compactness is related to outer precompactness or precompactness, which are strictly weaker properties than total boundedness. An asymmetric seminormed space may fail to be a topological vector space, because y → αy can be discontinuous (Borodin, 2001). Practically all other results have to be reconsidered (e.g. Baire category theorem, Alaoglu-Bourbaki, etc). (Cobzas, 2013). Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 5 / 16 Sources and Consequences of Asymmetry Random Variables as the Source of Asymmetry M◦ := {x : x, y ≤ 1, ∀ y ∈ M} M Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 6 / 16 Sources and Consequences of Asymmetry Random Variables as the Source of Asymmetry M◦ := {x : x, y ≤ 1, ∀ y ∈ M} M Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 6 / 16 Sources and Consequences of Asymmetry Random Variables as the Source of Asymmetry M◦ := {x : x, y ≤ 1, ∀ y ∈ M} M Minkowski functional: µM◦ (x) = inf{α > 0 : x/α ∈ M◦ } Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 6 / 16 Sources and Consequences of Asymmetry Random Variables as the Source of Asymmetry M◦ := {x : x, y ≤ 1, ∀ y ∈ M} M Minkowski functional: µM◦ (x) = inf{α > 0 : x/α ∈ M◦ } Support function = sM(x) = sup{ x, y : y ∈ M} Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 6 / 16 Sources and Consequences of Asymmetry Random Variables as the Source of Asymmetry M◦ := {x : x, y ≤ 1, ∀ y ∈ M} M Minkowski functional: µM◦ (x) = inf{α > 0 : x/α ∈ M◦ } Support function = sM(x) = sup{ x, y : y ∈ M} M = {u : D[(1 + u)z, z] ≤ 1} Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 6 / 16 Sources and Consequences of Asymmetry Random Variables as the Source of Asymmetry M◦ := {x : x, y ≤ 1, ∀ y ∈ M} M Minkowski functional: µM◦ (x) = inf{α > 0 : x/α ∈ M◦ } Support function = sM(x) = sup{ x, y : y ∈ M} M = {u : D[(1 + u)z, z] ≤ 1} D = (1 + u) ln(1 + u) − u, z Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 6 / 16 Sources and Consequences of Asymmetry Random Variables as the Source of Asymmetry M◦ := {x : x, y ≤ 1, ∀ y ∈ M} M Minkowski functional: µM◦ (x) = inf{α > 0 : x/α ∈ M◦ } M◦ {y : D∗[x, 0] ≤ 1} Support function = sM(x) = sup{ x, y : y ∈ M} M = {u : D[(1 + u)z, z] ≤ 1} D = (1 + u) ln(1 + u) − u, z Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 6 / 16 Sources and Consequences of Asymmetry Random Variables as the Source of Asymmetry M◦ := {x : x, y ≤ 1, ∀ y ∈ M} M Minkowski functional: µM◦ (x) = inf{α > 0 : x/α ∈ M◦ } M◦ {y : D∗[x, 0] ≤ 1} D∗[x, 0] = ex − 1 − x, z Support function = sM(x) = sup{ x, y : y ∈ M} M = {u : D[(1 + u)z, z] ≤ 1} D = (1 + u) ln(1 + u) − u, z Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 6 / 16 Sources and Consequences of Asymmetry Examples Example (St. Peterbourgh lottery) x = 2n, q = 2−n, n ∈ N. Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 7 / 16 Sources and Consequences of Asymmetry Examples Example (St. Peterbourgh lottery) x = 2n, q = 2−n, n ∈ N. Eq{x} = ∞ n=1(2n/2n) → ∞ Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 7 / 16 Sources and Consequences of Asymmetry Examples Example (St. Peterbourgh lottery) x = 2n, q = 2−n, n ∈ N. Eq{x} = ∞ n=1(2n/2n) → ∞ Ep{x} < ∞ for all biased p = 2−(1+α)n, α > 0. Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 7 / 16 Sources and Consequences of Asymmetry Examples Example (St. Peterbourgh lottery) x = 2n, q = 2−n, n ∈ N. Eq{x} = ∞ n=1(2n/2n) → ∞ Ep{x} < ∞ for all biased p = 2−(1+α)n, α > 0. 2n /∈ dom Eq{ex}, −2n ∈ dom Eq{ex} Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 7 / 16 Sources and Consequences of Asymmetry Examples Example (St. Peterbourgh lottery) x = 2n, q = 2−n, n ∈ N. Eq{x} = ∞ n=1(2n/2n) → ∞ Ep{x} < ∞ for all biased p = 2−(1+α)n, α > 0. 2n /∈ dom Eq{ex}, −2n ∈ dom Eq{ex} 0 /∈ Int(dom Eq{ex}) Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 7 / 16 Sources and Consequences of Asymmetry Examples Example (St. Peterbourgh lottery) x = 2n, q = 2−n, n ∈ N. Eq{x} = ∞ n=1(2n/2n) → ∞ Ep{x} < ∞ for all biased p = 2−(1+α)n, α > 0. 2n /∈ dom Eq{ex}, −2n ∈ dom Eq{ex} 0 /∈ Int(dom Eq{ex}) Example (Error minimization) Minimize x = 1 2 a − b 2 2 subject to DKL[w, q ⊗ p] ≤ λ, a, b ∈ Rn. Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 7 / 16 Sources and Consequences of Asymmetry Examples Example (St. Peterbourgh lottery) x = 2n, q = 2−n, n ∈ N. Eq{x} = ∞ n=1(2n/2n) → ∞ Ep{x} < ∞ for all biased p = 2−(1+α)n, α > 0. 2n /∈ dom Eq{ex}, −2n ∈ dom Eq{ex} 0 /∈ Int(dom Eq{ex}) Example (Error minimization) Minimize x = 1 2 a − b 2 2 subject to DKL[w, q ⊗ p] ≤ λ, a, b ∈ Rn. Ew{x} < ∞ minimized at w ∝ e−βxq ⊗ p. Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 7 / 16 Sources and Consequences of Asymmetry Examples Example (St. Peterbourgh lottery) x = 2n, q = 2−n, n ∈ N. Eq{x} = ∞ n=1(2n/2n) → ∞ Ep{x} < ∞ for all biased p = 2−(1+α)n, α > 0. 2n /∈ dom Eq{ex}, −2n ∈ dom Eq{ex} 0 /∈ Int(dom Eq{ex}) Example (Error minimization) Minimize x = 1 2 a − b 2 2 subject to DKL[w, q ⊗ p] ≤ λ, a, b ∈ Rn. Ew{x} < ∞ minimized at w ∝ e−βxq ⊗ p. Maximization of x has no solution. Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 7 / 16 Sources and Consequences of Asymmetry Examples Example (St. Peterbourgh lottery) x = 2n, q = 2−n, n ∈ N. Eq{x} = ∞ n=1(2n/2n) → ∞ Ep{x} < ∞ for all biased p = 2−(1+α)n, α > 0. 2n /∈ dom Eq{ex}, −2n ∈ dom Eq{ex} 0 /∈ Int(dom Eq{ex}) Example (Error minimization) Minimize x = 1 2 a − b 2 2 subject to DKL[w, q ⊗ p] ≤ λ, a, b ∈ Rn. Ew{x} < ∞ minimized at w ∝ e−βxq ⊗ p. Maximization of x has no solution. 1 2 a − b 2 2 /∈ dom Eq⊗p{ex}, −1 2 a − b 2 2 ∈ dom Eq⊗p{ex} Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 7 / 16 Sources and Consequences of Asymmetry Examples Example (St. Peterbourgh lottery) x = 2n, q = 2−n, n ∈ N. Eq{x} = ∞ n=1(2n/2n) → ∞ Ep{x} < ∞ for all biased p = 2−(1+α)n, α > 0. 2n /∈ dom Eq{ex}, −2n ∈ dom Eq{ex} 0 /∈ Int(dom Eq{ex}) Example (Error minimization) Minimize x = 1 2 a − b 2 2 subject to DKL[w, q ⊗ p] ≤ λ, a, b ∈ Rn. Ew{x} < ∞ minimized at w ∝ e−βxq ⊗ p. Maximization of x has no solution. 1 2 a − b 2 2 /∈ dom Eq⊗p{ex}, −1 2 a − b 2 2 ∈ dom Eq⊗p{ex} 0 /∈ Int(dom Eq⊗p{ex}) Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 7 / 16 Method: Symmetric Sandwich Sources and Consequences of Asymmetry Method: Symmetric Sandwich Results Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 8 / 16 Method: Symmetric Sandwich Method: Symmetric Sandwich s[−A ∩ A] ≤ sA ≤ s[−A ∪ A] Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 9 / 16 Method: Symmetric Sandwich Method: Symmetric Sandwich s[−A ∩ A] ≤ sA ≤ s[−A ∪ A] µco [−A◦ ∪ A◦] ≤ µA◦ ≤ µ[−A◦ ∩ A◦] Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 9 / 16 Method: Symmetric Sandwich Method: Symmetric Sandwich s[−A ∩ A] ≤ sA ≤ s[−A ∪ A] µco [−A◦ ∪ A◦] ≤ µA◦ ≤ µ[−A◦ ∩ A◦] s[−A ∩ A] = s(−A)co ∧ sA = inf{sA(z) + sA(z − y) : z ∈ Y } Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 9 / 16 Method: Symmetric Sandwich Method: Symmetric Sandwich s[−A ∩ A] ≤ sA ≤ s[−A ∪ A] µco [−A◦ ∪ A◦] ≤ µA◦ ≤ µ[−A◦ ∩ A◦] s[−A ∩ A] = s(−A)co ∧ sA = inf{sA(z) + sA(z − y) : z ∈ Y } s[−A ∪ A] = s(−A) ∨ sA Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 9 / 16 Method: Symmetric Sandwich Method: Symmetric Sandwich s[−A ∩ A] ≤ sA ≤ s[−A ∪ A] µco [−A◦ ∪ A◦] ≤ µA◦ ≤ µ[−A◦ ∩ A◦] s[−A ∩ A] = s(−A)co ∧ sA = inf{sA(z) + sA(z − y) : z ∈ Y } s[−A ∪ A] = s(−A) ∨ sA Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 9 / 16 Method: Symmetric Sandwich Method: Symmetric Sandwich s[−A ∩ A] ≤ sA ≤ s[−A ∪ A] µco [−A◦ ∪ A◦] ≤ µA◦ ≤ µ[−A◦ ∩ A◦] s[−A ∩ A] = s(−A)co ∧ sA = inf{sA(z) + sA(z − y) : z ∈ Y } s[−A ∪ A] = s(−A) ∨ sA Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 9 / 16 Method: Symmetric Sandwich Method: Symmetric Sandwich s[−A ∩ A] ≤ sA ≤ s[−A ∪ A] µco [−A◦ ∪ A◦] ≤ µA◦ ≤ µ[−A◦ ∩ A◦] s[−A ∩ A] = s(−A)co ∧ sA = inf{sA(z) + sA(z − y) : z ∈ Y } s[−A ∪ A] = s(−A) ∨ sA µM◦ ≤ µ(−M◦ ) ∨ µM◦ µ(−M)co ∧ µM ≤ µM Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 9 / 16 Method: Symmetric Sandwich Method: Symmetric Sandwich s[−A ∩ A] ≤ sA ≤ s[−A ∪ A] µco [−A◦ ∪ A◦] ≤ µA◦ ≤ µ[−A◦ ∩ A◦] s[−A ∩ A] = s(−A)co ∧ sA = inf{sA(z) + sA(z − y) : z ∈ Y } s[−A ∪ A] = s(−A) ∨ sA µ(−M◦ )co ∧ µM◦ ≤ µM◦ µM ≤ µ(−M) ∨ µM Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 9 / 16 Method: Symmetric Sandwich Lower and upper Luxemburg (Orlicz) norms −2 −1 0 1 2 ϕ∗ (x) = ex − 1 − x −2 −1 0 1 2 ϕ(u) = (1 + u) ln(1 + u) − u Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 10 / 16 Method: Symmetric Sandwich Lower and upper Luxemburg (Orlicz) norms −2 −1 0 1 2 ϕ∗ (x) = ex − 1 − x ϕ∗ +(x) = ϕ∗ (|x|) /∈ ∆2 −2 −1 0 1 2 ϕ(u) = (1 + u) ln(1 + u) − u ϕ+(u) = ϕ(|u|) ∈ ∆2 Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 10 / 16 Method: Symmetric Sandwich Lower and upper Luxemburg (Orlicz) norms −2 −1 0 1 2 ϕ∗ (x) = ex − 1 − x ϕ∗ +(x) = ϕ∗ (|x|) /∈ ∆2 ϕ∗ −(x) = ϕ∗ (−|x|) ∈ ∆2 −2 −1 0 1 2 ϕ(u) = (1 + u) ln(1 + u) − u ϕ+(u) = ϕ(|u|) ∈ ∆2 ϕ−(u) = ϕ(−|u|) /∈ ∆2 Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 10 / 16 Method: Symmetric Sandwich Lower and upper Luxemburg (Orlicz) norms −2 −1 0 1 2 ϕ∗ (x) = ex − 1 − x ϕ∗ +(x) = ϕ∗ (|x|) /∈ ∆2 ϕ∗ −(x) = ϕ∗ (−|x|) ∈ ∆2 x|∗ ϕ = µ{x : ϕ∗ (x), z ≤ 1} −2 −1 0 1 2 ϕ(u) = (1 + u) ln(1 + u) − u ϕ+(u) = ϕ(|u|) ∈ ∆2 ϕ−(u) = ϕ(−|u|) /∈ ∆2 u|ϕ = µ{u : ϕ(u), z ≤ 1} Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 10 / 16 Method: Symmetric Sandwich Lower and upper Luxemburg (Orlicz) norms −2 −1 0 1 2 ϕ∗ (x) = ex − 1 − x ϕ∗ +(x) = ϕ∗ (|x|) /∈ ∆2 ϕ∗ −(x) = ϕ∗ (−|x|) ∈ ∆2 x|∗ ϕ = µ{x : ϕ∗ (x), z ≤ 1} −2 −1 0 1 2 ϕ(u) = (1 + u) ln(1 + u) − u ϕ+(u) = ϕ(|u|) ∈ ∆2 ϕ−(u) = ϕ(−|u|) /∈ ∆2 u|ϕ = µ{u : ϕ(u), z ≤ 1} Proposition · ∗ ϕ+, · ∗ ϕ− are Luxemburg norms and x ∗ ϕ− ≤ x|∗ ϕ ≤ x ∗ ϕ+ · ϕ+, · ϕ− are Luxemburg norms and u ϕ+ ≤ u|ϕ ≤ u ϕ− Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 10 / 16 Method: Symmetric Sandwich Lower and upper Luxemburg (Orlicz) norms −2 −1 0 1 2 ϕ∗ (x) = ex − 1 − x ϕ∗ +(x) = ϕ∗ (|x|) /∈ ∆2 ϕ∗ −(x) = ϕ∗ (−|x|) ∈ ∆2 x|∗ ϕ = µ{x : ϕ∗ (x), z ≤ 1} −2 −1 0 1 2 ϕ(u) = (1 + u) ln(1 + u) − u ϕ+(u) = ϕ(|u|) ∈ ∆2 ϕ−(u) = ϕ(−|u|) /∈ ∆2 u|ϕ = µ{u : ϕ(u), z ≤ 1} Proposition · ∗ ϕ+, · ∗ ϕ− are Luxemburg norms and x ∗ ϕ− ≤ x|∗ ϕ ≤ x ∗ ϕ+ · ϕ+, · ϕ− are Luxemburg norms and u ϕ+ ≤ u|ϕ ≤ u ϕ− Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 10 / 16 Results Sources and Consequences of Asymmetry Method: Symmetric Sandwich Results Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 11 / 16 Results KL Induces Hausdorff (T2) Asymmetric Topology Theorem (Y, · |ϕ) (resp. (X, · |∗ ϕ)) is Hausdorff. Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 12 / 16 Results KL Induces Hausdorff (T2) Asymmetric Topology Theorem (Y, · |ϕ) (resp. (X, · |∗ ϕ)) is Hausdorff. Proof. u ϕ+ ≤ u|ϕ (resp. x ϕ− ≤ x|ϕ) implies (Y, · |ϕ) (resp. (X, · |∗ ϕ)) is finer than normed space (Y, · ϕ+) (resp. (X, · ∗ ϕ−)). Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 12 / 16 Results Separable Subspaces Theorem (Y, · ϕ+) (resp. (X, · ∗ ϕ−)) is a separable Orlicz subspace of (Y, · |ϕ) (resp. (X, · |∗ ϕ)). Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 13 / 16 Results Separable Subspaces Theorem (Y, · ϕ+) (resp. (X, · ∗ ϕ−)) is a separable Orlicz subspace of (Y, · |ϕ) (resp. (X, · |∗ ϕ)). Proof. ϕ+(u) = (1 + |u|) ln(1 + |u|) − |u| ∈ ∆2 (resp. ϕ∗ −(x) = e−|x| − 1 + |x| ∈ ∆2). Note that ϕ− /∈ ∆2 and ϕ∗ + /∈ ∆2. Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 13 / 16 Results Completeness Theorem (Y, · |ϕ) (resp. (X, · |∗ ϕ)) is 1 Bi-Complete: ρs-Cauchy yn ρs → y. Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 14 / 16 Results Completeness Theorem (Y, · |ϕ) (resp. (X, · |∗ ϕ)) is 1 Bi-Complete: ρs-Cauchy yn ρs → y. 2 ρ-sequentially complete: ρs-Cauchy yn ρ → y. Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 14 / 16 Results Completeness Theorem (Y, · |ϕ) (resp. (X, · |∗ ϕ)) is 1 Bi-Complete: ρs-Cauchy yn ρs → y. 2 ρ-sequentially complete: ρs-Cauchy yn ρ → y. 3 Right K-sequentially complete: right K-Cauchy yn ρ → y. Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 14 / 16 Results Completeness Theorem (Y, · |ϕ) (resp. (X, · |∗ ϕ)) is 1 Bi-Complete: ρs-Cauchy yn ρs → y. 2 ρ-sequentially complete: ρs-Cauchy yn ρ → y. 3 Right K-sequentially complete: right K-Cauchy yn ρ → y. Proof. ρs(y, z) = z − y|ϕ ∨ y − z|ϕ ≤ y − z ϕ−, where (Y, · ϕ−) is Banach. Then use theorems of Reilly et al. (1982) and Chen et al. (2007). Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 14 / 16 Results Summary and Further Questions Topologies induced by asymmetric information divergences may not have the same properties as their symmetrized counterparts (e.g. Banach spaces), and therefore many properties have to be re-examined. Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 15 / 16 Results Summary and Further Questions Topologies induced by asymmetric information divergences may not have the same properties as their symmetrized counterparts (e.g. Banach spaces), and therefore many properties have to be re-examined. We have proved that topologies induced by the KL-divergence are: Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 15 / 16 Results Summary and Further Questions Topologies induced by asymmetric information divergences may not have the same properties as their symmetrized counterparts (e.g. Banach spaces), and therefore many properties have to be re-examined. We have proved that topologies induced by the KL-divergence are: Hausdorff. Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 15 / 16 Results Summary and Further Questions Topologies induced by asymmetric information divergences may not have the same properties as their symmetrized counterparts (e.g. Banach spaces), and therefore many properties have to be re-examined. We have proved that topologies induced by the KL-divergence are: Hausdorff. Bi-complete, ρ-sequentially complete and right K-sequentially complete. Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 15 / 16 Results Summary and Further Questions Topologies induced by asymmetric information divergences may not have the same properties as their symmetrized counterparts (e.g. Banach spaces), and therefore many properties have to be re-examined. We have proved that topologies induced by the KL-divergence are: Hausdorff. Bi-complete, ρ-sequentially complete and right K-sequentially complete. Contain a separable Orlicz subspace. Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 15 / 16 Results Summary and Further Questions Topologies induced by asymmetric information divergences may not have the same properties as their symmetrized counterparts (e.g. Banach spaces), and therefore many properties have to be re-examined. We have proved that topologies induced by the KL-divergence are: Hausdorff. Bi-complete, ρ-sequentially complete and right K-sequentially complete. Contain a separable Orlicz subspace. Total boundedness, compactness? Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 15 / 16 Results Summary and Further Questions Topologies induced by asymmetric information divergences may not have the same properties as their symmetrized counterparts (e.g. Banach spaces), and therefore many properties have to be re-examined. We have proved that topologies induced by the KL-divergence are: Hausdorff. Bi-complete, ρ-sequentially complete and right K-sequentially complete. Contain a separable Orlicz subspace. Total boundedness, compactness? Other asymmetric information distances (e.g. Renyi divergence). Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 15 / 16 References Sources and Consequences of Asymmetry Method: Symmetric Sandwich Results Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 16 / 16 Results Borodin, P. A. (2001). The Banach-Mazur theorem for spaces with asymmetric norm. Mathematical Notes, 69(3–4), 298–305. Chen, S.-A., Li, W., Zou, D., & Chen, S.-B. (2007, Aug). Fixed point theorems in quasi-metric spaces. In Machine learning and cybernetics, 2007 international conference on (Vol. 5, p. 2499-2504). IEEE. Cobzas, S. (2013). Functional analysis in asymmetric normed spaces. Birkh¨auser. Fletcher, P., & Lindgren, W. F. (1982). Quasi-uniform spaces (Vol. 77). New York: Marcel Dekker. Reilly, I. L., Subrahmanyam, P. V., & Vamanamurthy, M. K. (1982). Cauchy sequences in quasi-pseudo-metric spaces. Monatshefte f¨ur Mathematik, 93, 127–140. Roman Belavkin (Middlesex University) Asymmetric Topologies October 28, 2015 16 / 16