Markov Entropy Approximation Scheme
David Poulin
Département de PhysiqueUniversité de Sherbrooke
Joint work with: Matt Leifer, Ersen Bilgin, and Matt Hastings
Quantum Computation and Quantum Spin SystemsVienna, August 2009
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 1 / 33
TaskCompute thermal properties of spin systems on lattices.
Origin of the method
Refinement of quantum generalized belief propagation.
ResultAlgorithm to produce lower bounds to free energy by solving finite-sizeconvex optimization problem with linear constraints.
Why is this interesting?
Variational methods (PEPS, MPS, MERA) provide UPPER bounds tofree energy.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 2 / 33
TaskCompute thermal properties of spin systems on lattices.
Origin of the method
Refinement of quantum generalized belief propagation.
ResultAlgorithm to produce lower bounds to free energy by solving finite-sizeconvex optimization problem with linear constraints.
Why is this interesting?
Variational methods (PEPS, MPS, MERA) provide UPPER bounds tofree energy.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 2 / 33
TaskCompute thermal properties of spin systems on lattices.
Origin of the method
Refinement of quantum generalized belief propagation.
ResultAlgorithm to produce lower bounds to free energy by solving finite-sizeconvex optimization problem with linear constraints.
Why is this interesting?
Variational methods (PEPS, MPS, MERA) provide UPPER bounds tofree energy.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 2 / 33
TaskCompute thermal properties of spin systems on lattices.
Origin of the method
Refinement of quantum generalized belief propagation.
ResultAlgorithm to produce lower bounds to free energy by solving finite-sizeconvex optimization problem with linear constraints.
Why is this interesting?
Variational methods (PEPS, MPS, MERA) provide UPPER bounds tofree energy.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 2 / 33
Outline
1 Belief propagation
2 Quantum belief propagation
3 Markov Entropy Approximation Scheme
4 Closing
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 3 / 33
Belief propagation
Outline
1 Belief propagation
2 Quantum belief propagation
3 Markov Entropy Approximation Scheme
4 Closing
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 4 / 33
Belief propagation
Description of the algorithm
PurposeApproximate the solution to statistical inference problems involvinga large number of random variables v (spins) located at thevertices v ∈ V of a graph G = (V ,E).The graph’s edges (u, v) ∈ E encode some king of directdependency relation of the random variables.
Algorithm architectureHighly parallelizable: one processor per vertex.Messages exchanged between processors related by an edge.Outgoing messages at v depend on local "fields" and incommingmessages.Exact when G is a tree and complexity = depth(G).Good heuristic on loopy graphs.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 5 / 33
Belief propagation
Description of the algorithm
PurposeApproximate the solution to statistical inference problems involvinga large number of random variables v (spins) located at thevertices v ∈ V of a graph G = (V ,E).The graph’s edges (u, v) ∈ E encode some king of directdependency relation of the random variables.
Algorithm architectureHighly parallelizable: one processor per vertex.Messages exchanged between processors related by an edge.Outgoing messages at v depend on local "fields" and incommingmessages.Exact when G is a tree and complexity = depth(G).Good heuristic on loopy graphs.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 5 / 33
Belief propagation
Description of the algorithm
PurposeApproximate the solution to statistical inference problems involvinga large number of random variables v (spins) located at thevertices v ∈ V of a graph G = (V ,E).The graph’s edges (u, v) ∈ E encode some king of directdependency relation of the random variables.
Algorithm architectureHighly parallelizable: one processor per vertex.Messages exchanged between processors related by an edge.Outgoing messages at v depend on local "fields" and incommingmessages.Exact when G is a tree and complexity = depth(G).Good heuristic on loopy graphs.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 5 / 33
Belief propagation
Description of the algorithm
PurposeApproximate the solution to statistical inference problems involvinga large number of random variables v (spins) located at thevertices v ∈ V of a graph G = (V ,E).The graph’s edges (u, v) ∈ E encode some king of directdependency relation of the random variables.
Algorithm architectureHighly parallelizable: one processor per vertex.Messages exchanged between processors related by an edge.Outgoing messages at v depend on local "fields" and incommingmessages.Exact when G is a tree and complexity = depth(G).Good heuristic on loopy graphs.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 5 / 33
Belief propagation
Description of the algorithm
PurposeApproximate the solution to statistical inference problems involvinga large number of random variables v (spins) located at thevertices v ∈ V of a graph G = (V ,E).The graph’s edges (u, v) ∈ E encode some king of directdependency relation of the random variables.
Algorithm architectureHighly parallelizable: one processor per vertex.Messages exchanged between processors related by an edge.Outgoing messages at v depend on local "fields" and incommingmessages.Exact when G is a tree and complexity = depth(G).Good heuristic on loopy graphs.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 5 / 33
Belief propagation
Description of the algorithm
PurposeApproximate the solution to statistical inference problems involvinga large number of random variables v (spins) located at thevertices v ∈ V of a graph G = (V ,E).The graph’s edges (u, v) ∈ E encode some king of directdependency relation of the random variables.
Algorithm architectureHighly parallelizable: one processor per vertex.Messages exchanged between processors related by an edge.Outgoing messages at v depend on local "fields" and incommingmessages.Exact when G is a tree and complexity = depth(G).Good heuristic on loopy graphs.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 5 / 33
Belief propagation
Description of the algorithm
PurposeApproximate the solution to statistical inference problems involvinga large number of random variables v (spins) located at thevertices v ∈ V of a graph G = (V ,E).The graph’s edges (u, v) ∈ E encode some king of directdependency relation of the random variables.
Algorithm architectureHighly parallelizable: one processor per vertex.Messages exchanged between processors related by an edge.Outgoing messages at v depend on local "fields" and incommingmessages.Exact when G is a tree and complexity = depth(G).Good heuristic on loopy graphs.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 5 / 33
Belief propagation
Transfer matrix
Consider the 1d classical system with hamiltonianH =
∑i h(vi) +
∑〈ij〉 J(vi , vj).
Its Gibbs distribution is (µ(v) = e−βh(v) and ν(u : v) = e−βJ(u,v))
ρ(v1, v2, . . .) =1Z
e−βH(v1,v2,...)
=1Zµ(v1)ν(v1, v2)µ(v2)ν(v2, v3)µ(v3) . . .
So the reduced state of spin N can be evaluated step by step:
m1→2(v2) =∑v1
µ(v1)ν(v1 : v2)
m2→3(v3) =∑v2
mv1→v2(v2)µ(v2)ν(v2 : v3)
m3→4(v4) =∑v3
mv2→v3(v3)µ(v3)ν(v3 : v4)...
ρ(vN) = mvN−1→vN (vN)µ(vN)
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 6 / 33
Belief propagation
Transfer matrix
Consider the 1d classical system with hamiltonianH =
∑i h(vi) +
∑〈ij〉 J(vi , vj).
Its Gibbs distribution is (µ(v) = e−βh(v) and ν(u : v) = e−βJ(u,v))
ρ(v1, v2, . . .) =1Z
e−βH(v1,v2,...)
=1Zµ(v1)ν(v1, v2)µ(v2)ν(v2, v3)µ(v3) . . .
So the reduced state of spin N can be evaluated step by step:
m1→2(v2) =∑v1
µ(v1)ν(v1 : v2)
m2→3(v3) =∑v2
mv1→v2(v2)µ(v2)ν(v2 : v3)
m3→4(v4) =∑v3
mv2→v3(v3)µ(v3)ν(v3 : v4)...
ρ(vN) = mvN−1→vN (vN)µ(vN)
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 6 / 33
Belief propagation
Transfer matrix
Consider the 1d classical system with hamiltonianH =
∑i h(vi) +
∑〈ij〉 J(vi , vj).
Its Gibbs distribution is (µ(v) = e−βh(v) and ν(u : v) = e−βJ(u,v))
ρ(v1, v2, . . .) =1Z
e−βH(v1,v2,...)
=1Zµ(v1)ν(v1, v2)µ(v2)ν(v2, v3)µ(v3) . . .
So the reduced state of spin N can be evaluated step by step:
m1→2(v2) =∑v1
µ(v1)ν(v1 : v2)
m2→3(v3) =∑v2
mv1→v2(v2)µ(v2)ν(v2 : v3)
m3→4(v4) =∑v3
mv2→v3(v3)µ(v3)ν(v3 : v4)...
ρ(vN) = mvN−1→vN (vN)µ(vN)
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 6 / 33
Belief propagation
Transfer matrix
Consider the 1d classical system with hamiltonianH =
∑i h(vi) +
∑〈ij〉 J(vi , vj).
Its Gibbs distribution is (µ(v) = e−βh(v) and ν(u : v) = e−βJ(u,v))
ρ(v1, v2, . . .) =1Z
e−βH(v1,v2,...)
=1Zµ(v1)ν(v1, v2)µ(v2)ν(v2, v3)µ(v3) . . .
So the reduced state of spin N can be evaluated step by step:
m1→2(v2) =∑v1
µ(v1)ν(v1 : v2)
m2→3(v3) =∑v2
mv1→v2(v2)µ(v2)ν(v2 : v3)
m3→4(v4) =∑v3
mv2→v3(v3)µ(v3)ν(v3 : v4)...
ρ(vN) = mvN−1→vN (vN)µ(vN)
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 6 / 33
Belief propagation
Transfer matrix
Consider the 1d classical system with hamiltonianH =
∑i h(vi) +
∑〈ij〉 J(vi , vj).
Its Gibbs distribution is (µ(v) = e−βh(v) and ν(u : v) = e−βJ(u,v))
ρ(v1, v2, . . .) =1Z
e−βH(v1,v2,...)
=1Zµ(v1)ν(v1, v2)µ(v2)ν(v2, v3)µ(v3) . . .
So the reduced state of spin N can be evaluated step by step:
m1→2(v2) =∑v1
µ(v1)ν(v1 : v2)
m2→3(v3) =∑v2
mv1→v2(v2)µ(v2)ν(v2 : v3)
m3→4(v4) =∑v3
mv2→v3(v3)µ(v3)ν(v3 : v4)...
ρ(vN) = mvN−1→vN (vN)µ(vN)
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 6 / 33
Belief propagation
Transfer matrix
Consider the 1d classical system with hamiltonianH =
∑i h(vi) +
∑〈ij〉 J(vi , vj).
Its Gibbs distribution is (µ(v) = e−βh(v) and ν(u : v) = e−βJ(u,v))
ρ(v1, v2, . . .) =1Z
e−βH(v1,v2,...)
=1Zµ(v1)ν(v1, v2)µ(v2)ν(v2, v3)µ(v3) . . .
So the reduced state of spin N can be evaluated step by step:
m1→2(v2) =∑v1
µ(v1)ν(v1 : v2)
m2→3(v3) =∑v2
mv1→v2(v2)µ(v2)ν(v2 : v3)
m3→4(v4) =∑v3
mv2→v3(v3)µ(v3)ν(v3 : v4)...
ρ(vN) = mvN−1→vN (vN)µ(vN)
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 6 / 33
Belief propagation
Transfer matrix
Consider the 1d classical system with hamiltonianH =
∑i h(vi) +
∑〈ij〉 J(vi , vj).
Its Gibbs distribution is (µ(v) = e−βh(v) and ν(u : v) = e−βJ(u,v))
ρ(v1, v2, . . .) =1Z
e−βH(v1,v2,...)
=1Zµ(v1)ν(v1, v2)µ(v2)ν(v2, v3)µ(v3) . . .
So the reduced state of spin N can be evaluated step by step:
m1→2(v2) =∑v1
µ(v1)ν(v1 : v2)
m2→3(v3) =∑v2
mv1→v2(v2)µ(v2)ν(v2 : v3)
m3→4(v4) =∑v3
mv2→v3(v3)µ(v3)ν(v3 : v4)...
ρ(vN) = mvN−1→vN (vN)µ(vN)
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 6 / 33
Belief propagation
General setting
Graph G = (V ,E).Random variables vi for i ∈ V .Functions µi(vi) for i ∈ V and νij(vi : vj) for (i , j) ∈ E .
You can think of them as exponentiels of local fields µ(v) = e−βh(v)
and couplings ν(u, v) = e−βJ(u,v).
Task (basic case)Evaluate
ρk (vk ) =∑{vi}−vk
∏i∈V
µi(vi)∏ij∈E
νij(vi : vj)
=∑{vi}−vk
e−β(P
i∈V hi (vi )+P
i,j∈E Jij (vi ,vj ))
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 7 / 33
Belief propagation
General setting
Graph G = (V ,E).Random variables vi for i ∈ V .Functions µi(vi) for i ∈ V and νij(vi : vj) for (i , j) ∈ E .
You can think of them as exponentiels of local fields µ(v) = e−βh(v)
and couplings ν(u, v) = e−βJ(u,v).
Task (basic case)Evaluate
ρk (vk ) =∑{vi}−vk
∏i∈V
µi(vi)∏ij∈E
νij(vi : vj)
=∑{vi}−vk
e−β(P
i∈V hi (vi )+P
i,j∈E Jij (vi ,vj ))
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 7 / 33
Belief propagation
General setting
Graph G = (V ,E).Random variables vi for i ∈ V .Functions µi(vi) for i ∈ V and νij(vi : vj) for (i , j) ∈ E .
You can think of them as exponentiels of local fields µ(v) = e−βh(v)
and couplings ν(u, v) = e−βJ(u,v).
Task (basic case)Evaluate
ρk (vk ) =∑{vi}−vk
∏i∈V
µi(vi)∏ij∈E
νij(vi : vj)
=∑{vi}−vk
e−β(P
i∈V hi (vi )+P
i,j∈E Jij (vi ,vj ))
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 7 / 33
Belief propagation
General setting
Graph G = (V ,E).Random variables vi for i ∈ V .Functions µi(vi) for i ∈ V and νij(vi : vj) for (i , j) ∈ E .
You can think of them as exponentiels of local fields µ(v) = e−βh(v)
and couplings ν(u, v) = e−βJ(u,v).
Task (basic case)Evaluate
ρk (vk ) =∑{vi}−vk
∏i∈V
µi(vi)∏ij∈E
νij(vi : vj)
=∑{vi}−vk
e−β(P
i∈V hi (vi )+P
i,j∈E Jij (vi ,vj ))
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 7 / 33
Belief propagation
General setting
Graph G = (V ,E).Random variables vi for i ∈ V .Functions µi(vi) for i ∈ V and νij(vi : vj) for (i , j) ∈ E .
You can think of them as exponentiels of local fields µ(v) = e−βh(v)
and couplings ν(u, v) = e−βJ(u,v).
Task (basic case)Evaluate
ρk (vk ) =∑{vi}−vk
∏i∈V
µi(vi)∏ij∈E
νij(vi : vj)
=∑{vi}−vk
e−β(P
i∈V hi (vi )+P
i,j∈E Jij (vi ,vj ))
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 7 / 33
Belief propagation
Belief propagation algorithm
Algorithm
Initialization mu→v (v) = cte.Iterations mu→v (v) ∝∑u µ(u)ν(u : v)
∏v ′∈n(u)−v mv ′→u(u).
Beliefs b(u) ∝ µ(u)∏
v∈n(u) mv→u(u).b(u, v) ∝ µ(u)µ(v)ν(u : v)
∏w∈n(u)−v mw→u(u)
∏w∈n(v)−u mw→v (v).
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 8 / 33
Belief propagation
Belief propagation algorithm
Algorithm
Initialization mu→v (v) = cte.Iterations mu→v (v) ∝∑u µ(u)ν(u : v)
∏v ′∈n(u)−v mv ′→u(u).
... u v
a
b
g
ma!
u
mg!u
mb!u
µu!(u : v)mu!v
Beliefs b(u) ∝ µ(u)∏
v∈n(u) mv→u(u).b(u, v) ∝ µ(u)µ(v)ν(u : v)
∏w∈n(u)−v mw→u(u)
∏w∈n(v)−u mw→v (v).
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 8 / 33
Belief propagation
Belief propagation algorithm
Algorithm
Initialization mu→v (v) = cte.Iterations mu→v (v) ∝∑u µ(u)ν(u : v)
∏v ′∈n(u)−v mv ′→u(u).
... u v
a
b
g
ma!
u
mg!u
mb!u
µu!(u : v)mu!v
Beliefs b(u) ∝ µ(u)∏
v∈n(u) mv→u(u).b(u, v) ∝ µ(u)µ(v)ν(u : v)
∏w∈n(u)−v mw→u(u)
∏w∈n(v)−u mw→v (v).
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 8 / 33
Belief propagation
Belief propagation algorithm
Algorithm
Initialization mu→v (v) = cte.Iterations mu→v (v) ∝∑u µ(u)ν(u : v)
∏v ′∈n(u)−v mv ′→u(u).
... u v
a
b
g
ma!
u
mg!u
mb!u
µu!(u : v)mu!v
Beliefs b(u) ∝ µ(u)∏
v∈n(u) mv→u(u).b(u, v) ∝ µ(u)µ(v)ν(u : v)
∏w∈n(u)−v mw→u(u)
∏w∈n(v)−u mw→v (v).
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 8 / 33
Belief propagation
Bethe free energy
Let H =∑
v hv +∑〈u,v〉 Juv be a local Hamiltonian on G.
Given a probability P(V ), the Gibbs free energy G = E − TSwhere
E =∑
V
P(V )H(V ) and S = −∑
V
P(V ) log P(V )
The Gibbs distribution P(V ) = 1Z e−βH is the stationary point of G.
E only depends on the two-body probabilities
E =∑〈u,v〉
P(u, v)Juv +∑
u
P(u)hu
So does S if P is the product of local functions on a tree (whichincludes Gibbs distributions)
S = −∑〈u,v〉
P(u, v) log P(u, v)−∑
u
(1− dv )P(u) log P(u)
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 9 / 33
Belief propagation
Bethe free energy
Let H =∑
v hv +∑〈u,v〉 Juv be a local Hamiltonian on G.
Given a probability P(V ), the Gibbs free energy G = E − TSwhere
E =∑
V
P(V )H(V ) and S = −∑
V
P(V ) log P(V )
The Gibbs distribution P(V ) = 1Z e−βH is the stationary point of G.
E only depends on the two-body probabilities
E =∑〈u,v〉
P(u, v)Juv +∑
u
P(u)hu
So does S if P is the product of local functions on a tree (whichincludes Gibbs distributions)
S = −∑〈u,v〉
P(u, v) log P(u, v)−∑
u
(1− dv )P(u) log P(u)
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 9 / 33
Belief propagation
Bethe free energy
Let H =∑
v hv +∑〈u,v〉 Juv be a local Hamiltonian on G.
Given a probability P(V ), the Gibbs free energy G = E − TSwhere
E =∑
V
P(V )H(V ) and S = −∑
V
P(V ) log P(V )
The Gibbs distribution P(V ) = 1Z e−βH is the stationary point of G.
E only depends on the two-body probabilities
E =∑〈u,v〉
P(u, v)Juv +∑
u
P(u)hu
So does S if P is the product of local functions on a tree (whichincludes Gibbs distributions)
S = −∑〈u,v〉
P(u, v) log P(u, v)−∑
u
(1− dv )P(u) log P(u)
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 9 / 33
Belief propagation
Bethe free energy
Let H =∑
v hv +∑〈u,v〉 Juv be a local Hamiltonian on G.
Given a probability P(V ), the Gibbs free energy G = E − TSwhere
E =∑
V
P(V )H(V ) and S = −∑
V
P(V ) log P(V )
The Gibbs distribution P(V ) = 1Z e−βH is the stationary point of G.
E only depends on the two-body probabilities
E =∑〈u,v〉
P(u, v)Juv +∑
u
P(u)hu
So does S if P is the product of local functions on a tree (whichincludes Gibbs distributions)
S = −∑〈u,v〉
P(u, v) log P(u, v)−∑
u
(1− dv )P(u) log P(u)
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 9 / 33
Belief propagation
Bethe free energy
Let H =∑
v hv +∑〈u,v〉 Juv be a local Hamiltonian on G.
Given a probability P(V ), the Gibbs free energy G = E − TSwhere
E =∑
V
P(V )H(V ) and S = −∑
V
P(V ) log P(V )
The Gibbs distribution P(V ) = 1Z e−βH is the stationary point of G.
E only depends on the two-body probabilities
E =∑〈u,v〉
P(u, v)Juv +∑
u
P(u)hu
So does S if P is the product of local functions on a tree (whichincludes Gibbs distributions)
S = −∑〈u,v〉
P(u, v) log P(u, v)−∑
u
(1− dv )P(u) log P(u)
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 9 / 33
Belief propagation
Bethe free energy
The Bethe free energy is extending this expression to arbitrarygraphs
GBethe =∑〈u,v〉
P(u, v)(T log P(u, v)+Juv )+∑
u
(1−du)P(u)(T log P(u)+hu).
This is seemingly easier to handle because it involves onlytwo-body distributions.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 10 / 33
Belief propagation
Bethe free energy
The Bethe free energy is extending this expression to arbitrarygraphs
GBethe =∑〈u,v〉
P(u, v)(T log P(u, v)+Juv )+∑
u
(1−du)P(u)(T log P(u)+hu).
This is seemingly easier to handle because it involves onlytwo-body distributions.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 10 / 33
Belief propagation
Bethe free energy and BP
Theorem (Yedidia, Freeman, and Weiss)
The fixed point of the belief propagation algorithms b(u, v) and b(u)are stationary points of the Bethe free energy.
Use BP to solve Bethe free energy minimization.Exact on tree, complexity = depth(G).Most successful on trees with only large loops:
LDPC and Turbo codes.Spin glasses on Bethe lattices.Random k -SAT assignments.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 11 / 33
Belief propagation
Bethe free energy and BP
Theorem (Yedidia, Freeman, and Weiss)
The fixed point of the belief propagation algorithms b(u, v) and b(u)are stationary points of the Bethe free energy.
Use BP to solve Bethe free energy minimization.Exact on tree, complexity = depth(G).Most successful on trees with only large loops:
LDPC and Turbo codes.Spin glasses on Bethe lattices.Random k -SAT assignments.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 11 / 33
Belief propagation
Bethe free energy and BP
Theorem (Yedidia, Freeman, and Weiss)
The fixed point of the belief propagation algorithms b(u, v) and b(u)are stationary points of the Bethe free energy.
Use BP to solve Bethe free energy minimization.Exact on tree, complexity = depth(G).Most successful on trees with only large loops:
LDPC and Turbo codes.Spin glasses on Bethe lattices.Random k -SAT assignments.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 11 / 33
Belief propagation
Bethe free energy and BP
Theorem (Yedidia, Freeman, and Weiss)
The fixed point of the belief propagation algorithms b(u, v) and b(u)are stationary points of the Bethe free energy.
Use BP to solve Bethe free energy minimization.Exact on tree, complexity = depth(G).Most successful on trees with only large loops:
LDPC and Turbo codes.Spin glasses on Bethe lattices.Random k -SAT assignments.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 11 / 33
Belief propagation
Bethe free energy and BP
Theorem (Yedidia, Freeman, and Weiss)
The fixed point of the belief propagation algorithms b(u, v) and b(u)are stationary points of the Bethe free energy.
Use BP to solve Bethe free energy minimization.Exact on tree, complexity = depth(G).Most successful on trees with only large loops:
LDPC and Turbo codes.Spin glasses on Bethe lattices.Random k -SAT assignments.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 11 / 33
Belief propagation
Generalized belief propagation
When the graph contains small loops, “collapse" them into asingle super-site.Messages exchanged between (possibly overlapping) regions.
If clustered graph is a tree, method is exact.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 12 / 33
Belief propagation
Generalized belief propagation
When the graph contains small loops, “collapse" them into asingle super-site.Messages exchanged between (possibly overlapping) regions.
If clustered graph is a tree, method is exact.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 12 / 33
Belief propagation
Generalized belief propagation
When the graph contains small loops, “collapse" them into asingle super-site.Messages exchanged between (possibly overlapping) regions.
If clustered graph is a tree, method is exact.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 12 / 33
Belief propagation
Generalized belief propagation
When the graph contains small loops, “collapse" them into asingle super-site.Messages exchanged between (possibly overlapping) regions.
If clustered graph is a tree, method is exact.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 12 / 33
Belief propagation
Generalized belief propagation
When the graph contains small loops, “collapse" them into asingle super-site.Messages exchanged between (possibly overlapping) regions.
If clustered graph is a tree, method is exact.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 12 / 33
Belief propagation
Kikuchi free energy
Fix a set of regions on the graph.Approximate the entropy by the sum of the entropy of the regions.Correct for the over counting of the sub-regions.Correct for the sub-sub-regions ...
1 2 3
4 5 6
7 8 9
Corresponds to Bethe’s free energy when regions are nearestneighbor pairs.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 13 / 33
Belief propagation
Kikuchi free energy
Fix a set of regions on the graph.Approximate the entropy by the sum of the entropy of the regions.Correct for the over counting of the sub-regions.Correct for the sub-sub-regions ...
1 2 3
4 5 6
7 8 9
S = S1245 + S2356 + S478 + S5689
Corresponds to Bethe’s free energy when regions are nearestneighbor pairs.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 13 / 33
Belief propagation
Kikuchi free energy
Fix a set of regions on the graph.Approximate the entropy by the sum of the entropy of the regions.Correct for the over counting of the sub-regions.Correct for the sub-sub-regions ...
1 2 3
4 5 6
7 8 9
S = S1245 + S2356 + S478 + S5689
!S4 ! S8 ! S25 ! S56
Corresponds to Bethe’s free energy when regions are nearestneighbor pairs.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 13 / 33
Belief propagation
Kikuchi free energy
Fix a set of regions on the graph.Approximate the entropy by the sum of the entropy of the regions.Correct for the over counting of the sub-regions.Correct for the sub-sub-regions ...
1 2 3
4 5 6
7 8 9
S = S1245 + S2356 + S478 + S5689
!S4 ! S8 ! S25 ! S56
Corresponds to Bethe’s free energy when regions are nearestneighbor pairs.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 13 / 33
Belief propagation
Kikuchi free energy
Fix a set of regions on the graph.Approximate the entropy by the sum of the entropy of the regions.Correct for the over counting of the sub-regions.Correct for the sub-sub-regions ...
1 2 3
4 5 6
7 8 9
S = S1245 + S2356 + S478 + S5689
!S4 ! S8 ! S25 ! S56
Corresponds to Bethe’s free energy when regions are nearestneighbor pairs.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 13 / 33
Belief propagation
Kikuchi free energy and generalized BP
Theorem (Yedidia, Freeman, and Weiss)The fixed point of the generalized belief propagation algorithms arestationary points of the Kikushi free energy computed using the sameregions.a
aTechnical detail: the counting number of each sub-region must be non-positive.
Greatly improves numerical precision for graphs containing smallloops.Can use GBP instead of minimizing Kikuchi free energy.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 14 / 33
Belief propagation
Kikuchi free energy and generalized BP
Theorem (Yedidia, Freeman, and Weiss)The fixed point of the generalized belief propagation algorithms arestationary points of the Kikushi free energy computed using the sameregions.a
aTechnical detail: the counting number of each sub-region must be non-positive.
Greatly improves numerical precision for graphs containing smallloops.Can use GBP instead of minimizing Kikuchi free energy.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 14 / 33
Belief propagation
Kikuchi free energy and generalized BP
Theorem (Yedidia, Freeman, and Weiss)The fixed point of the generalized belief propagation algorithms arestationary points of the Kikushi free energy computed using the sameregions.a
aTechnical detail: the counting number of each sub-region must be non-positive.
Greatly improves numerical precision for graphs containing smallloops.Can use GBP instead of minimizing Kikuchi free energy.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 14 / 33
Quantum belief propagation
Outline
1 Belief propagation
2 Quantum belief propagation
3 Markov Entropy Approximation Scheme
4 Closing
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 15 / 33
Quantum belief propagation
The algorithm
Defining the log-exp product A� B = elog A+log B, the quantum andclassical setting become very similar.
Let G = (V ,E) be a graph with Hamiltonian H =∑
hi +∑
Jij .Defining µi = e−βhi , νij = e−βJij , the Gibbs state is
ρV =1Z
(⊗i∈V
µi
)�( ⊙
(i,j)∈E
νij
)We can execute BP by substituting ordinary products for �.
TheoremIf G is a tree and (G, ρV ) is a quantum Markov random field, then thebeliefs bu converge to the correct marginals ρu = TrV−u{ρV} in a timeproportional to depth(G).
This doesn’t happen very often, but similar conditions forgeneralized BP are more natural.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 16 / 33
Quantum belief propagation
The algorithm
Defining the log-exp product A� B = elog A+log B, the quantum andclassical setting become very similar.
Let G = (V ,E) be a graph with Hamiltonian H =∑
hi +∑
Jij .Defining µi = e−βhi , νij = e−βJij , the Gibbs state is
ρV =1Z
(⊗i∈V
µi
)�( ⊙
(i,j)∈E
νij
)We can execute BP by substituting ordinary products for �.
TheoremIf G is a tree and (G, ρV ) is a quantum Markov random field, then thebeliefs bu converge to the correct marginals ρu = TrV−u{ρV} in a timeproportional to depth(G).
This doesn’t happen very often, but similar conditions forgeneralized BP are more natural.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 16 / 33
Quantum belief propagation
The algorithm
Defining the log-exp product A� B = elog A+log B, the quantum andclassical setting become very similar.
Let G = (V ,E) be a graph with Hamiltonian H =∑
hi +∑
Jij .Defining µi = e−βhi , νij = e−βJij , the Gibbs state is
ρV =1Z
(⊗i∈V
µi
)�( ⊙
(i,j)∈E
νij
)We can execute BP by substituting ordinary products for �.
TheoremIf G is a tree and (G, ρV ) is a quantum Markov random field, then thebeliefs bu converge to the correct marginals ρu = TrV−u{ρV} in a timeproportional to depth(G).
This doesn’t happen very often, but similar conditions forgeneralized BP are more natural.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 16 / 33
Quantum belief propagation
The algorithm
Defining the log-exp product A� B = elog A+log B, the quantum andclassical setting become very similar.
Let G = (V ,E) be a graph with Hamiltonian H =∑
hi +∑
Jij .Defining µi = e−βhi , νij = e−βJij , the Gibbs state is
ρV =1Z
(⊗i∈V
µi
)�( ⊙
(i,j)∈E
νij
)We can execute BP by substituting ordinary products for �.
TheoremIf G is a tree and (G, ρV ) is a quantum Markov random field, then thebeliefs bu converge to the correct marginals ρu = TrV−u{ρV} in a timeproportional to depth(G).
This doesn’t happen very often, but similar conditions forgeneralized BP are more natural.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 16 / 33
Quantum belief propagation
The algorithm
Defining the log-exp product A� B = elog A+log B, the quantum andclassical setting become very similar.
Let G = (V ,E) be a graph with Hamiltonian H =∑
hi +∑
Jij .Defining µi = e−βhi , νij = e−βJij , the Gibbs state is
ρV =1Z
(⊗i∈V
µi
)�( ⊙
(i,j)∈E
νij
)We can execute BP by substituting ordinary products for �.
TheoremIf G is a tree and (G, ρV ) is a quantum Markov random field, then thebeliefs bu converge to the correct marginals ρu = TrV−u{ρV} in a timeproportional to depth(G).
This doesn’t happen very often, but similar conditions forgeneralized BP are more natural.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 16 / 33
Quantum belief propagation
The algorithm
Defining the log-exp product A� B = elog A+log B, the quantum andclassical setting become very similar.
Let G = (V ,E) be a graph with Hamiltonian H =∑
hi +∑
Jij .Defining µi = e−βhi , νij = e−βJij , the Gibbs state is
ρV =1Z
(⊗i∈V
µi
)�( ⊙
(i,j)∈E
νij
)We can execute BP by substituting ordinary products for �.
TheoremIf G is a tree and (G, ρV ) is a quantum Markov random field, then thebeliefs bu converge to the correct marginals ρu = TrV−u{ρV} in a timeproportional to depth(G).
This doesn’t happen very often, but similar conditions forgeneralized BP are more natural.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 16 / 33
Quantum belief propagation
Effective thermal hamiltonian
-5 -4 -3 -2 -1 0 1 2 3 4 5... ...
! =1Z
exp{!"H}H =!!
i="!hi + Ji,i+1
Effective Thermal Hamiltonian =∑∞
i=1 hi + Ji,i+1 +V1 +V2+V3+V4 . . .
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 17 / 33
Quantum belief propagation
Effective thermal hamiltonian
-5 -4 -3 -2 -1 0 1 2 3 4 5... ...
! =1Z
exp{!"H}
Effective Thermal Hamiltonian =∑∞
i=1 hi + Ji,i+1 +V1 +V2+V3+V4 . . .
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 17 / 33
Quantum belief propagation
Effective thermal hamiltonian
-5 -4 -3 -2 -1 0 1 2 3 4 5... ...
!! = Tr"#..."1{!} ! =1Z
exp{!"H}
Effective Thermal Hamiltonian =∑∞
i=1 hi + Ji,i+1 +V1 +V2+V3+V4 . . .
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 17 / 33
Quantum belief propagation
Effective thermal hamiltonian
-5 -4 -3 -2 -1 0 1 2 3 4 5... ...
!! = Tr"#..."1{!} !! =1Z ! exp{!"He!}
Effective Thermal Hamiltonian =∑∞
i=1 hi + Ji,i+1 +V1 +V2+V3+V4 . . .
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 17 / 33
Quantum belief propagation
Effective thermal hamiltonian
-5 -4 -3 -2 -1 0 1 2 3 4 5... ...
!! = Tr"#..."1{!} !! =1Z ! exp{!"He!}
Effective Thermal Hamiltonian =∑∞
i=1 hi + Ji,i+1 +V1 +V2+V3+V4 . . .
10 20 30 40 50 60 70 80 90 10010!12
10!10
10!8
10!6
10!4
10!2
100
!!z 0!
z j"
j
! = 20
! = 1
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 17 / 33
Quantum belief propagation
Effective thermal hamiltonian
-5 -4 -3 -2 -1 0 1 2 3 4 5... ...
!! = Tr"#..."1{!} !! =1Z ! exp{!"He!}
Effective Thermal Hamiltonian =∑∞
i=1 hi + Ji,i+1 +V1 +V2+V3+V4 . . .
10 20 30 40 50 60 70 80 90 10010!12
10!10
10!8
10!6
10!4
10!2
100
!!z 0!
z j"
j
! = 20
! = 1
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 17 / 33
Quantum belief propagation
Effective thermal hamiltonian
-5 -4 -3 -2 -1 0 1 2 3 4 5... ...
!! = Tr"#..."1{!} !! =1Z ! exp{!"He!}
Effective Thermal Hamiltonian =∑∞
i=1 hi + Ji,i+1 +V1 +V2+V3+V4 . . .
10 20 30 40 50 60 70 80 90 10010!12
10!10
10!8
10!6
10!4
10!2
100
!!z 0!
z j"
j
! = 20
! = 1
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 17 / 33
Quantum belief propagation
Effective thermal hamiltonian
-5 -4 -3 -2 -1 0 1 2 3 4 5... ...
!! = Tr"#..."1{!} !! =1Z ! exp{!"He!}
Effective Thermal Hamiltonian =∑∞
i=1 hi + Ji,i+1 +V1 +V2+V3+V4 . . .
10 20 30 40 50 60 70 80 90 10010!12
10!10
10!8
10!6
10!4
10!2
100
!!z 0!
z j"
j
! = 20
! = 1
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 17 / 33
Quantum belief propagation
Effective thermal hamiltonian
-5 -4 -3 -2 -1 0 1 2 3 4 5... ...
!! = Tr"#..."1{!} !! =1Z ! exp{!"He!}
Effective Thermal Hamiltonian =∑∞
i=1 hi + Ji,i+1 +V1 +V2+V3+V4 . . .
10 20 30 40 50 60 70 80 90 10010!12
10!10
10!8
10!6
10!4
10!2
100
!!z 0!
z j"
j
! = 20
! = 1
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 17 / 33
Quantum belief propagation
Effective thermal hamiltonian
-5 -4 -3 -2 -1 0 1 2 3 4 5... ...
!! = Tr"#..."1{!} !! =1Z ! exp{!"He!}
Effective Thermal Hamiltonian =∑∞
i=1 hi + Ji,i+1 +V1 +V2+V3+V4 . . .
10 20 30 40 50 60 70 80 90 10010!12
10!10
10!8
10!6
10!4
10!2
100
!!z 0!
z j"
j
! = 20
! = 1
1 2 3 4 5 610!12
10!10
10!8
10!6
10!4
10!2
100
!Vj! 2
j
! = 1
! = 20
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 17 / 33
Quantum belief propagation
Generalized quantum belief propagation
σ1−4 = e−β(h1+h2+h3+h4+J12+J23+J34)
σ′2−4 = Tr1{σ1−4} h′2−4 = −1β
logσ′2−4
σ2−5 = e−β(h′2−4+h5+J45)
σ′3−5 = Tr2{σ2−5} h′3−5 = −1β
logσ′3−5
σ3−6 = e−β(h′3−5+h6+J56)
...ρN ≈ bN = TrN−3,N−2,N−1{σN−3,N−2,N−1,N}
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 18 / 33
Quantum belief propagation
Generalized quantum belief propagation
σ1−4 = e−β(h1+h2+h3+h4+J12+J23+J34)
σ′2−4 = Tr1{σ1−4} h′2−4 = −1β
logσ′2−4
σ2−5 = e−β(h′2−4+h5+J45)
σ′3−5 = Tr2{σ2−5} h′3−5 = −1β
logσ′3−5
σ3−6 = e−β(h′3−5+h6+J56)
...ρN ≈ bN = TrN−3,N−2,N−1{σN−3,N−2,N−1,N}
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 18 / 33
Quantum belief propagation
Generalized quantum belief propagation
σ1−4 = e−β(h1+h2+h3+h4+J12+J23+J34)
σ′2−4 = Tr1{σ1−4} h′2−4 = −1β
logσ′2−4
σ2−5 = e−β(h′2−4+h5+J45)
σ′3−5 = Tr2{σ2−5} h′3−5 = −1β
logσ′3−5
σ3−6 = e−β(h′3−5+h6+J56)
...ρN ≈ bN = TrN−3,N−2,N−1{σN−3,N−2,N−1,N}
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 18 / 33
Quantum belief propagation
Generalized quantum belief propagation
{!1!4
σ1−4 = e−β(h1+h2+h3+h4+J12+J23+J34)
σ′2−4 = Tr1{σ1−4} h′2−4 = −1β
logσ′2−4
σ2−5 = e−β(h′2−4+h5+J45)
σ′3−5 = Tr2{σ2−5} h′3−5 = −1β
logσ′3−5
σ3−6 = e−β(h′3−5+h6+J56)
...ρN ≈ bN = TrN−3,N−2,N−1{σN−3,N−2,N−1,N}
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 18 / 33
Quantum belief propagation
Generalized quantum belief propagation
{
h!2"4
σ1−4 = e−β(h1+h2+h3+h4+J12+J23+J34)
σ′2−4 = Tr1{σ1−4} h′2−4 = −1β
logσ′2−4
σ2−5 = e−β(h′2−4+h5+J45)
σ′3−5 = Tr2{σ2−5} h′3−5 = −1β
logσ′3−5
σ3−6 = e−β(h′3−5+h6+J56)
...ρN ≈ bN = TrN−3,N−2,N−1{σN−3,N−2,N−1,N}
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 18 / 33
Quantum belief propagation
Generalized quantum belief propagation
{!2!5
σ1−4 = e−β(h1+h2+h3+h4+J12+J23+J34)
σ′2−4 = Tr1{σ1−4} h′2−4 = −1β
logσ′2−4
σ2−5 = e−β(h′2−4+h5+J45)
σ′3−5 = Tr2{σ2−5} h′3−5 = −1β
logσ′3−5
σ3−6 = e−β(h′3−5+h6+J56)
...ρN ≈ bN = TrN−3,N−2,N−1{σN−3,N−2,N−1,N}
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 18 / 33
Quantum belief propagation
Generalized quantum belief propagation
{
h!3"5
σ1−4 = e−β(h1+h2+h3+h4+J12+J23+J34)
σ′2−4 = Tr1{σ1−4} h′2−4 = −1β
logσ′2−4
σ2−5 = e−β(h′2−4+h5+J45)
σ′3−5 = Tr2{σ2−5} h′3−5 = −1β
logσ′3−5
σ3−6 = e−β(h′3−5+h6+J56)
...ρN ≈ bN = TrN−3,N−2,N−1{σN−3,N−2,N−1,N}
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 18 / 33
Quantum belief propagation
Generalized quantum belief propagation
{
!3!6
σ1−4 = e−β(h1+h2+h3+h4+J12+J23+J34)
σ′2−4 = Tr1{σ1−4} h′2−4 = −1β
logσ′2−4
σ2−5 = e−β(h′2−4+h5+J45)
σ′3−5 = Tr2{σ2−5} h′3−5 = −1β
logσ′3−5
σ3−6 = e−β(h′3−5+h6+J56)
...ρN ≈ bN = TrN−3,N−2,N−1{σN−3,N−2,N−1,N}
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 18 / 33
Quantum belief propagation
Generalized quantum belief propagation
{
!N!3,N!2,N!1,N
σ1−4 = e−β(h1+h2+h3+h4+J12+J23+J34)
σ′2−4 = Tr1{σ1−4} h′2−4 = −1β
logσ′2−4
σ2−5 = e−β(h′2−4+h5+J45)
σ′3−5 = Tr2{σ2−5} h′3−5 = −1β
logσ′3−5
σ3−6 = e−β(h′3−5+h6+J56)
...ρN ≈ bN = TrN−3,N−2,N−1{σN−3,N−2,N−1,N}
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 18 / 33
Quantum belief propagation
When does this work?
... ...A B C
This quantum generalized belief propagation is exact when
I(A : C|B) = S(AB) + S(BC)− S(B)− S(ABC) = 0
This is probably never exactly the case, but as B gets larger it seemsto be a good approximation.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 19 / 33
Quantum belief propagation
When does this work?
... ...A B C
This quantum generalized belief propagation is exact when
I(A : C|B) = S(AB) + S(BC)− S(B)− S(ABC) = 0
This is probably never exactly the case, but as B gets larger it seemsto be a good approximation.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 19 / 33
Quantum belief propagation
Results on trees
0 0.5 1 1.5 20
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Bethe ansatzQBP
0.46 0.47 0.48 0.49 0.50.349
0.3492
0.3494
0.3496
0.3498
Spec
ific h
eat
Temperature
Sliding window
Bethe Ansatz
Temperature
Tra
nsvers
e F
ield
Str
ength
Edwards!Anderson Order Parameter on a Cayley Tree with Transverse Ising Hamiltonian
0.2 0.4 0.6 0.8 1 1.2 1.4
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Ising spin glass on Cayley tree
Tran
sver
se fi
eld
Bx
Temperature 1.13
17
Finally, we note that much of the phase diagram is surprisingly stable to variation in Nt.
We have explored various regions of the phase space at Nt = 6, 7, 8, 9, 10, 11. The classical
line (Bt = 0) at all temperatures is completely stable down to Nt = 1 as expected. Perhaps
more surprisingly, moving between Nt = 8 and Nt = 10, qEA is essentially stable below
Bt = 1 down to temperatures ! ! 0.15. Of course, the high field, low temperature part
of the phase transition curve moves downward as the finite discretization asymptote goes
towards the ! = 0 axis. See Figure 6 for the low temperature critical curves estimated using
vertical stripes run at five di!erent temperatures (corresponding to " = 3.5, 4, 4.5, 5, 5.5) at
various Nt.
FIG. 4: (a) Phase diagram at q = 3. The solid phase transition curve has been calculated at
Nt = 10, Nrods = 2500, Niter = 1000Nrods on a fine mesh in the (!, Bt) plane. The vertical dotted
line is the asymptotic critical line for large Bt at Nt = 10 (ie ! = !c/Nt). The points marked x
with error bars indicate Nt " # fits based on Figure 6. The dashed transition curve is a weighted
quadratic fit through the estimated low temperature points and the Nt = 10 points in the range
0.5 < ! < 1. This leads to an estimated Bct = 1.775 ± 0.03. As this fit is clearly heuristic, we
have suggested a much larger range for our estimate of Bct in the Figure. Our phase diagram
clearly disagrees with that of [18], who treat the identical model using a spherical approximation.
The stars and stripes indicate points in the phase space which we have investigated in more detail
below. (b) The average von Neumann entropy of a central spin as a function of (!, Bt).Laumann, Scardicchio, and Sondhi ’07, Bilgin and Poulin.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 20 / 33
Quantum belief propagation
2D anti-ferromagnetic Heisenberg model
Doesn’t work so well in 2D...
Quantum Monte Calrlo: M.S. Makivic and H.-Q. Ding PRB’91.10th-order J/T expansion.Generalized Quantum Belief propagation, window size 7.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 21 / 33
Quantum belief propagation
2D anti-ferromagnetic Heisenberg model
Doesn’t work so well in 2D...
!"" #"" $"" %""
!&$
!&'
!&(
)!*
Quantum Monte Calrlo: M.S. Makivic and H.-Q. Ding PRB’91.10th-order J/T expansion.Generalized Quantum Belief propagation, window size 7.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 21 / 33
Markov Entropy Approximation Scheme
Outline
1 Belief propagation
2 Quantum belief propagation
3 Markov Entropy Approximation Scheme
4 Closing
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 22 / 33
Markov Entropy Approximation Scheme
Entropy is a mess...
F (T ) = minρ{E(ρ)− TS(ρ)} = min
ρ{Tr(ρH) + T · Tr(ρ log ρ)}
Energy can be computed from pieces of ρ:
E(ρ) =∑
i
Tr(ρihi) +∑
ij
Tr(ρijJij)
Hence it is easy to compute, involves small matrices.The same cannot be achieved with entropy.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 23 / 33
Markov Entropy Approximation Scheme
Entropy is a mess...
F (T ) = minρ{E(ρ)− TS(ρ)} = min
ρ{Tr(ρH) + T · Tr(ρ log ρ)}
Energy can be computed from pieces of ρ:
E(ρ) =∑
i
Tr(ρihi) +∑
ij
Tr(ρijJij)
Hence it is easy to compute, involves small matrices.The same cannot be achieved with entropy.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 23 / 33
Markov Entropy Approximation Scheme
Entropy is a mess...
F (T ) = minρ{E(ρ)− TS(ρ)} = min
ρ{Tr(ρH) + T · Tr(ρ log ρ)}
Energy can be computed from pieces of ρ:
E(ρ) =∑
i
Tr(ρihi) +∑
ij
Tr(ρijJij)
Hence it is easy to compute, involves small matrices.The same cannot be achieved with entropy.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 23 / 33
Markov Entropy Approximation Scheme
Entropy is a mess...
F (T ) = minρ{E(ρ)− TS(ρ)} = min
ρ{Tr(ρH) + T · Tr(ρ log ρ)}
Energy can be computed from pieces of ρ:
E(ρ) =∑
i
Tr(ρihi) +∑
ij
Tr(ρijJij)
Hence it is easy to compute, involves small matrices.The same cannot be achieved with entropy.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 23 / 33
Markov Entropy Approximation Scheme
Breaking entropy apart
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 24 / 33
Markov Entropy Approximation Scheme
Breaking entropy apart
1 2 3
N
...
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 24 / 33
Markov Entropy Approximation Scheme
Breaking entropy apart
S = SN
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 24 / 33
Markov Entropy Approximation Scheme
Breaking entropy apart
S = SN !SN!1 + SN!1
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 24 / 33
Markov Entropy Approximation Scheme
Breaking entropy apart
S = SN !SN!1 + SN!1 !SN!2 + SN!2
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 24 / 33
Markov Entropy Approximation Scheme
Breaking entropy apart
S = SN !SN!1 + SN!1 !SN!2 + SN!2
=!
j
Sj ! Sj!1
=!
j
S(j|1, 2, ..., j ! 1)
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 24 / 33
Markov Entropy Approximation Scheme
Breaking entropy apart
S(j|1, 2, ..., j ! 1)
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 24 / 33
Markov Entropy Approximation Scheme
Breaking entropy apart
S(j|1, 2, ..., j ! 1)
! S(j|{1, 2, ..., j " 1} #N (j))
Strong sub-additivity
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 24 / 33
Markov Entropy Approximation Scheme
Bounding the free energy
S∗ =∑
j
S(j |{1,2, ..., j − 1} ∩ N (j)) ≥ S
can be computed from pieces ρP(j) of ρ (P(j) = {1,2, ..., j} ∩ N (j)).Provides approximation for free energy
F (T ) ≥ F ∗(T ) = minρP(j):Consistent
{E({ρP(j)})− TS∗({ρP(j)})
}Consistent means ∃ρ : ρP(j) = TrP(j)(ρ). QMA-complete.Relax constraint to local consistency, i.e.
TrP(j)∩P(k)(ρP(j) − ρP(k)) = 0
F ∗(T ) ≥ F ∗∗(T ) = minρP(j):Local consistent
{E({ρP(j)})− TS∗({ρP(j)})
}Convex optimization with linear constraints.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 25 / 33
Markov Entropy Approximation Scheme
Bounding the free energy
S∗ =∑
j
S(j |{1,2, ..., j − 1} ∩ N (j)) ≥ S
can be computed from pieces ρP(j) of ρ (P(j) = {1,2, ..., j} ∩ N (j)).Provides approximation for free energy
F (T ) ≥ F ∗(T ) = minρP(j):Consistent
{E({ρP(j)})− TS∗({ρP(j)})
}Consistent means ∃ρ : ρP(j) = TrP(j)(ρ). QMA-complete.Relax constraint to local consistency, i.e.
TrP(j)∩P(k)(ρP(j) − ρP(k)) = 0
F ∗(T ) ≥ F ∗∗(T ) = minρP(j):Local consistent
{E({ρP(j)})− TS∗({ρP(j)})
}Convex optimization with linear constraints.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 25 / 33
Markov Entropy Approximation Scheme
Bounding the free energy
S∗ =∑
j
S(j |{1,2, ..., j − 1} ∩ N (j)) ≥ S
can be computed from pieces ρP(j) of ρ (P(j) = {1,2, ..., j} ∩ N (j)).Provides approximation for free energy
F (T ) ≥ F ∗(T ) = minρP(j):Consistent
{E({ρP(j)})− TS∗({ρP(j)})
}Consistent means ∃ρ : ρP(j) = TrP(j)(ρ). QMA-complete.Relax constraint to local consistency, i.e.
TrP(j)∩P(k)(ρP(j) − ρP(k)) = 0
F ∗(T ) ≥ F ∗∗(T ) = minρP(j):Local consistent
{E({ρP(j)})− TS∗({ρP(j)})
}Convex optimization with linear constraints.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 25 / 33
Markov Entropy Approximation Scheme
Bounding the free energy
S∗ =∑
j
S(j |{1,2, ..., j − 1} ∩ N (j)) ≥ S
can be computed from pieces ρP(j) of ρ (P(j) = {1,2, ..., j} ∩ N (j)).Provides approximation for free energy
F (T ) ≥ F ∗(T ) = minρP(j):Consistent
{E({ρP(j)})− TS∗({ρP(j)})
}Consistent means ∃ρ : ρP(j) = TrP(j)(ρ). QMA-complete.Relax constraint to local consistency, i.e.
TrP(j)∩P(k)(ρP(j) − ρP(k)) = 0
F ∗(T ) ≥ F ∗∗(T ) = minρP(j):Local consistent
{E({ρP(j)})− TS∗({ρP(j)})
}Convex optimization with linear constraints.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 25 / 33
Markov Entropy Approximation Scheme
Bounding the free energy
S∗ =∑
j
S(j |{1,2, ..., j − 1} ∩ N (j)) ≥ S
can be computed from pieces ρP(j) of ρ (P(j) = {1,2, ..., j} ∩ N (j)).Provides approximation for free energy
F (T ) ≥ F ∗(T ) = minρP(j):Consistent
{E({ρP(j)})− TS∗({ρP(j)})
}Consistent means ∃ρ : ρP(j) = TrP(j)(ρ). QMA-complete.Relax constraint to local consistency, i.e.
TrP(j)∩P(k)(ρP(j) − ρP(k)) = 0
F ∗(T ) ≥ F ∗∗(T ) = minρP(j):Local consistent
{E({ρP(j)})− TS∗({ρP(j)})
}Convex optimization with linear constraints.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 25 / 33
Markov Entropy Approximation Scheme
Bounding the free energy
S∗ =∑
j
S(j |{1,2, ..., j − 1} ∩ N (j)) ≥ S
can be computed from pieces ρP(j) of ρ (P(j) = {1,2, ..., j} ∩ N (j)).Provides approximation for free energy
F (T ) ≥ F ∗(T ) = minρP(j):Consistent
{E({ρP(j)})− TS∗({ρP(j)})
}Consistent means ∃ρ : ρP(j) = TrP(j)(ρ). QMA-complete.Relax constraint to local consistency, i.e.
TrP(j)∩P(k)(ρP(j) − ρP(k)) = 0
F ∗(T ) ≥ F ∗∗(T ) = minρP(j):Local consistent
{E({ρP(j)})− TS∗({ρP(j)})
}Convex optimization with linear constraints.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 25 / 33
Markov Entropy Approximation Scheme
Bounding the free energy
S∗ =∑
j
S(j |{1,2, ..., j − 1} ∩ N (j)) ≥ S
can be computed from pieces ρP(j) of ρ (P(j) = {1,2, ..., j} ∩ N (j)).Provides approximation for free energy
F (T ) ≥ F ∗(T ) = minρP(j):Consistent
{E({ρP(j)})− TS∗({ρP(j)})
}Consistent means ∃ρ : ρP(j) = TrP(j)(ρ). QMA-complete.Relax constraint to local consistency, i.e.
TrP(j)∩P(k)(ρP(j) − ρP(k)) = 0
F ∗(T ) ≥ F ∗∗(T ) = minρP(j):Local consistent
{E({ρP(j)})− TS∗({ρP(j)})
}Convex optimization with linear constraints.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 25 / 33
Markov Entropy Approximation Scheme
2D anti-ferromagnetic Heisenberg model revisited
! !"# !"$ !"% !"& ' '"# '"$ '"% '"& #!'"(
!'"#(
!'
!!")(
!!"(
!!"#(
*+,,-./,+012-3.45
./,+012-3.45
./,+012-.6789-:;70</7=;>79;</-$6$-
./,+012-3</9,-?7+=<
*+,,-./,+012-@;0A-B,CD,+79E+,-5,+;,F
./,+012-.6789-:;70</7=;>79;</-G6G
!"#$ !"## !"#% !"#&!!"'$
!!"'!(
!!"'!)
!!"'!'
!!"'!*
!!"'!+
!!"'!&
Temperature (1/J)
E0 ! "0.7062...
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 26 / 33
Markov Entropy Approximation Scheme
Improved finite size effects
MEAS requires 8-site diagonalization and matches 16-sitediagonalization.The constraints force the system to behave as if were part of alarger lattice.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 27 / 33
Markov Entropy Approximation Scheme
Improved finite size effects
MEAS requires 8-site diagonalization and matches 16-sitediagonalization.The constraints force the system to behave as if were part of alarger lattice.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 27 / 33
Markov Entropy Approximation Scheme
Improved finite size effects
MEAS requires 8-site diagonalization and matches 16-sitediagonalization.The constraints force the system to behave as if were part of alarger lattice.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 27 / 33
Markov Entropy Approximation Scheme
Improved finite size effects
MEAS requires 8-site diagonalization and matches 16-sitediagonalization.The constraints force the system to behave as if were part of alarger lattice.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 27 / 33
Markov Entropy Approximation Scheme
Improved finite size effects
MEAS requires 8-site diagonalization and matches 16-sitediagonalization.The constraints force the system to behave as if were part of alarger lattice.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 27 / 33
Markov Entropy Approximation Scheme
Positive entropy density constraint
Alternative method to estimate ground energy
Minimize Tr(ρH) on finite lattice with translational constraints on ρ.
Our method is more accurate because it imposes positive entropydensity constraints.The free energy at the the temperature where S∗ goes negative isa rigorous lower bound on E0 in the thermodynamical limit.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 28 / 33
Markov Entropy Approximation Scheme
Positive entropy density constraint
Alternative method to estimate ground energy
Minimize Tr(ρH) on finite lattice with translational constraints on ρ.
Our method is more accurate because it imposes positive entropydensity constraints.The free energy at the the temperature where S∗ goes negative isa rigorous lower bound on E0 in the thermodynamical limit.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 28 / 33
Markov Entropy Approximation Scheme
Positive entropy density constraint
Alternative method to estimate ground energy
Minimize Tr(ρH) on finite lattice with translational constraints on ρ.
Our method is more accurate because it imposes positive entropydensity constraints.The free energy at the the temperature where S∗ goes negative isa rigorous lower bound on E0 in the thermodynamical limit.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 28 / 33
Closing
Outline
1 Belief propagation
2 Quantum belief propagation
3 Markov Entropy Approximation Scheme
4 Closing
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 29 / 33
Closing
Tightening the bound
S∗ > S because it neglects some correlations in the systems:Had we worked with a different patch, we would have neglectedother correlations:pS∗1 + (1− p)S∗2 is a valid upper bound to S.In addition, we can impose cross-constraints that will tighten thebound.
Combine patches that explore multiple length-scales.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 30 / 33
Closing
Tightening the bound
S∗ > S because it neglects some correlations in the systems:Had we worked with a different patch, we would have neglectedother correlations:pS∗1 + (1− p)S∗2 is a valid upper bound to S.In addition, we can impose cross-constraints that will tighten thebound.
Combine patches that explore multiple length-scales.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 30 / 33
Closing
Tightening the bound
S∗ > S because it neglects some correlations in the systems:Had we worked with a different patch, we would have neglectedother correlations:pS∗1 + (1− p)S∗2 is a valid upper bound to S.In addition, we can impose cross-constraints that will tighten thebound.
Combine patches that explore multiple length-scales.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 30 / 33
Closing
Tightening the bound
S∗ > S because it neglects some correlations in the systems:Had we worked with a different patch, we would have neglectedother correlations:pS∗1 + (1− p)S∗2 is a valid upper bound to S.In addition, we can impose cross-constraints that will tighten thebound.
Combine patches that explore multiple length-scales.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 30 / 33
Closing
Tightening the bound
S∗ > S because it neglects some correlations in the systems:Had we worked with a different patch, we would have neglectedother correlations:pS∗1 + (1− p)S∗2 is a valid upper bound to S.In addition, we can impose cross-constraints that will tighten thebound.
Combine patches that explore multiple length-scales.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 30 / 33
Closing
Dual program
Numerical minimization might fail to find the true minimum of F ∗.In that case our result would not be a rigorous lower bound to thesystem’s free energy.Any solution to the dual program would provide a lower bound tothe minimum of F ∗, and hence to the system’s free energy.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 31 / 33
Closing
Dual program
Numerical minimization might fail to find the true minimum of F ∗.In that case our result would not be a rigorous lower bound to thesystem’s free energy.Any solution to the dual program would provide a lower bound tothe minimum of F ∗, and hence to the system’s free energy.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 31 / 33
Closing
Dual program
Numerical minimization might fail to find the true minimum of F ∗.In that case our result would not be a rigorous lower bound to thesystem’s free energy.Any solution to the dual program would provide a lower bound tothe minimum of F ∗, and hence to the system’s free energy.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 31 / 33
Conclusion
Belief propagation is a powerful heuristic to solve statisticalinference problems involving a large number of random variablesin a Markov Random Field.
Fixed points of BP = Stationary point of Bethe free energy.Generalized belief propagation improves the case where thegraph has small loops.
Fixed points of GBP = Stationary point of Kikuchi free energy.
Quantum generalized BP provides reliable estimates because theeffective thermal hamiltonian is short ranged.Markov Entropy Approximation Scheme uses strong sub-additivityto lower bound the system’s free energy.Linearizing the constraints decreases this bound and yields asolvable problem.Numerical results (simple) interpolate nicely between High Tseries and exact diagonalization of larger lattices.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 32 / 33
Conclusion
Belief propagation is a powerful heuristic to solve statisticalinference problems involving a large number of random variablesin a Markov Random Field.
Fixed points of BP = Stationary point of Bethe free energy.Generalized belief propagation improves the case where thegraph has small loops.
Fixed points of GBP = Stationary point of Kikuchi free energy.
Quantum generalized BP provides reliable estimates because theeffective thermal hamiltonian is short ranged.Markov Entropy Approximation Scheme uses strong sub-additivityto lower bound the system’s free energy.Linearizing the constraints decreases this bound and yields asolvable problem.Numerical results (simple) interpolate nicely between High Tseries and exact diagonalization of larger lattices.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 32 / 33
Conclusion
Belief propagation is a powerful heuristic to solve statisticalinference problems involving a large number of random variablesin a Markov Random Field.
Fixed points of BP = Stationary point of Bethe free energy.Generalized belief propagation improves the case where thegraph has small loops.
Fixed points of GBP = Stationary point of Kikuchi free energy.
Quantum generalized BP provides reliable estimates because theeffective thermal hamiltonian is short ranged.Markov Entropy Approximation Scheme uses strong sub-additivityto lower bound the system’s free energy.Linearizing the constraints decreases this bound and yields asolvable problem.Numerical results (simple) interpolate nicely between High Tseries and exact diagonalization of larger lattices.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 32 / 33
Conclusion
Belief propagation is a powerful heuristic to solve statisticalinference problems involving a large number of random variablesin a Markov Random Field.
Fixed points of BP = Stationary point of Bethe free energy.Generalized belief propagation improves the case where thegraph has small loops.
Fixed points of GBP = Stationary point of Kikuchi free energy.
Quantum generalized BP provides reliable estimates because theeffective thermal hamiltonian is short ranged.Markov Entropy Approximation Scheme uses strong sub-additivityto lower bound the system’s free energy.Linearizing the constraints decreases this bound and yields asolvable problem.Numerical results (simple) interpolate nicely between High Tseries and exact diagonalization of larger lattices.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 32 / 33
Conclusion
Belief propagation is a powerful heuristic to solve statisticalinference problems involving a large number of random variablesin a Markov Random Field.
Fixed points of BP = Stationary point of Bethe free energy.Generalized belief propagation improves the case where thegraph has small loops.
Fixed points of GBP = Stationary point of Kikuchi free energy.
Quantum generalized BP provides reliable estimates because theeffective thermal hamiltonian is short ranged.Markov Entropy Approximation Scheme uses strong sub-additivityto lower bound the system’s free energy.Linearizing the constraints decreases this bound and yields asolvable problem.Numerical results (simple) interpolate nicely between High Tseries and exact diagonalization of larger lattices.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 32 / 33
Conclusion
Belief propagation is a powerful heuristic to solve statisticalinference problems involving a large number of random variablesin a Markov Random Field.
Fixed points of BP = Stationary point of Bethe free energy.Generalized belief propagation improves the case where thegraph has small loops.
Fixed points of GBP = Stationary point of Kikuchi free energy.
Quantum generalized BP provides reliable estimates because theeffective thermal hamiltonian is short ranged.Markov Entropy Approximation Scheme uses strong sub-additivityto lower bound the system’s free energy.Linearizing the constraints decreases this bound and yields asolvable problem.Numerical results (simple) interpolate nicely between High Tseries and exact diagonalization of larger lattices.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 32 / 33
Conclusion
Belief propagation is a powerful heuristic to solve statisticalinference problems involving a large number of random variablesin a Markov Random Field.
Fixed points of BP = Stationary point of Bethe free energy.Generalized belief propagation improves the case where thegraph has small loops.
Fixed points of GBP = Stationary point of Kikuchi free energy.
Quantum generalized BP provides reliable estimates because theeffective thermal hamiltonian is short ranged.Markov Entropy Approximation Scheme uses strong sub-additivityto lower bound the system’s free energy.Linearizing the constraints decreases this bound and yields asolvable problem.Numerical results (simple) interpolate nicely between High Tseries and exact diagonalization of larger lattices.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 32 / 33
Conclusion
Belief propagation is a powerful heuristic to solve statisticalinference problems involving a large number of random variablesin a Markov Random Field.
Fixed points of BP = Stationary point of Bethe free energy.Generalized belief propagation improves the case where thegraph has small loops.
Fixed points of GBP = Stationary point of Kikuchi free energy.
Quantum generalized BP provides reliable estimates because theeffective thermal hamiltonian is short ranged.Markov Entropy Approximation Scheme uses strong sub-additivityto lower bound the system’s free energy.Linearizing the constraints decreases this bound and yields asolvable problem.Numerical results (simple) interpolate nicely between High Tseries and exact diagonalization of larger lattices.
David Poulin (Sherbrooke) Markov Entropy Approximation Scheme Vienna’09 32 / 33