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Phase Transitions in the Information Distortion
NIPS 2003 workshop on Information Theory and Learning:
The Bottleneck and Distortion Approach
December 13, 2003
Albert E. Parker
Department of Mathematical Sciences Center for Computational Biology
Montana State University
Collaborators: Tomas Gedeon, Alex Dimitrov, John Miller, and Zane Aldworth
The Goal:To determine the phase transitions or the bifurcation structure of solutions to clustering
problems of the form
maxqG(q) constrained by D(q)I0
where
is the set of valid conditional probabilities in RNK.
• G and D are sufficiently smooth in .• G and D have symmetry: they are invariant to relabelling of the classes of Z.• The Hessians qG and q D are block diagonal.
X Z
q(Z|X)
K objects N clusters
A similar formulation:Using the method Lagrange multipliers, the goal of determining the bifurcation structure of
solutions of the optimization problem can be rephrased as finding the bifurcation structure of stationary points of the problem
maxq(G(q)+D(q))
where [0,). is the set of valid conditional probabilities in RNK.
• G and D are sufficiently smooth in .• G and D have symmetry: they are invariant to relabelling of the classes of Z.• The Hessian q(G+ D) is block diagonal, and satisfies a set of regularity conditions at
bifurcation: (e.g. the kernel of each block is one dimensional)
X Z
q(Z|X)
K objects N clusters
How: Use the Symmetries
By capitalizing on the symmetries of the cost functions, we have described the bifurcation structure of stationary points to problems of the form
maxqG(q) constrained by D(q)I0
or
maxq(G(q)+D(q))
where [0,). is the set of valid conditional probabilities in RNK.• G and D are sufficiently smooth in .• G and D have symmetry: they are invariant to relabelling of the classes of Z.• The Hessian q(G+ D) is block diagonal, and satisfies a set of regularity conditions at bifurcation: (e.g. the
kernel is one dimensional)
• Rate Distortion Theory (Shannon 1950’s) Minimal Informative Compression
min I(X,Z) constrained by D(X,Z) D0
• Deterministic Annealing (Rose 1990’s) A Clustering Algorithm
max H(Z|X) constrained by D(X,Z) D0
qC,
Examplesoptimizing at a distortion level D(Y,Z) D0
q
• Rate Distortion Theory (Shannon 1950’s) Minimal Informative Compression
max -I(X,Z) constrained by D(X,Z) D0
• Deterministic Annealing (Rose 1998) A Clustering Algorithm
max H(Z|X) constrained by D(X,Z) D0
I(X,Z)=H(Z) – H(Z|X)
qC,
Examplesoptimizing at a distortion level D(Y,Z) D0
q
Y X
Inputs Outputs
Zq(Z|X)
Clustered Outputs
K objects {xi} N objects {zi}L objects {yi}
p(X,Y)
Inputs and Outputs and Clustered Outputs
Y X
Inputs Outputs
Zq(Z|X)
Clustered Outputs
K objects {xi} N objects {zi}L objects {yi}
p(X,Y)
Inputs and Outputs and Clustered Outputs
• Information Bottleneck Method (Tishby, Pereira, Bialek 1999)
min I(X,Z) constrained by DI(X,Z) D0
max –I(X,Z) + I(Y;Z)
• Information Distortion Method (Dimitrov and Miller 2001)
max H(Z|X) constrained by DI(X,Z) D0
max H(Z|X) + I(Y;Z)
q
Two methods which use an information distortion function to cluster
q
q
q
• Information Bottleneck Method (Tishby, Pereira, Bialek 1999)
min I(X,Z) constrained by DI(X,Z) D0
max –I(X,Z) + I(Y;Z)
• Information Distortion Method (Dimitrov and Miller 2001)
max H(Z|X) constrained by DI(X,Z) D0
max H(Z|X) + I(Y;Z)
q
Two methods which use an information distortion function to cluster
q
q
q
The Hessian is always singular … (-I(X,Z)is not strictly concave)
The theory whichfollows does not apply
• Information Bottleneck Method (Tishby, Pereira, Bialek 1999)
min I(X,Z) constrained by DI(X,Z) D0
max –I(X,Z) + I(Y;Z)
• Information Distortion Method (Dimitrov and Miller 2001)
max H(Z|X) constrained by DI(X,Z) D0
max H(Z|X) + I(Y;Z)
q
Two methods which use an information distortion function to cluster
q
q
q
The Hessian is always singular … (I(X,Z)is not strictly concave)
The theory whichfollows does not apply
H(Z|X) is strictly concave)
The theory whichfollows does apply
A basic annealing algorithmto solve
maxq(G(q)+D(q))
Let q0 be the maximizer of maxq G(q), and let 0 =0. For k 0, let (qk , k ) be a solution to maxq G(q) + D(q ). Iterate the following steps until K = max for some K.
1. Perform -step: Let k+1 = k + dk where dk>0
2. The initial guess for qk+1 at k+1 is qk+1(0) = qk + for some small
perturbation .
3. Optimization: solve maxq (G(q) + k+1 D(q)) to get the maximizer qk+1 , using initial guess qk+1
(0) .
Application of the annealing method to the Information Distortion problem maxq (H(Z|X) + I(X;Z))
when p(X,Y) is defined by four gaussian blobs
Y, Inputs
X, Outputs
Y X
K=52 outputsL=52 inputs
p(X,Y) X Zq(Z|X)
K=52 outputs N=4 clustered outputs
X, Outputs
Z, C
lust
ered
Ou
tpu
ts
Evolution of the optimal clustering: Observed Bifurcations for the Four Blob problem:
We just saw the optimal clusterings q* at some *= max . What do the clusterings look like for < max ??
I(Y
,Z)
bits
??????
Why are there only 3 bifurcations observed? In general, are there only N-1 bifurcations?
What kinds of bifurcations do we expect: pitchfork-like, transcritical, saddle-node, or some other type?
How many bifurcating branches are there?
What do the bifurcating branches look like? Are they 1st order phase transitions (subcritical) or 2nd order phase transitions (supercritical) ?
What is the stability of the bifurcating branches? Is there always a bifurcating branch which contains solutions of the optimization problem?
Are there bifurcations after all of the classes have resolved ?
q*
Conceptual Bifurcation Structure
Observed Bifurcations for the 4 Blob Problem
I(Y
,Z)
bits
Recall the Symmetries:
To better understand the bifurcation structure, we capitalize on the symmetries of the function G(q)+D(q)
X Zq(Z|X) : a clustering
K objects {xi} N objects {zi}
class 1
class 3
X Zq(Z|X) : a clustering
K objects {xi} N objects {zi}
class 3
class 1
Recall the Symmetries:
To better understand the bifurcation structure, we capitalize on the symmetries of the function G(q)+D(q)
The symmetry group of all permutations on N symbols
is
SN.
4S
3S3S
3S 3S
2S2S 2S2S2S2S2S2S
1
2S 2S 2S2S
A partial subgroup lattice for SN when N=4.
4S
34,12 24,13
23,14
A partial lattice of the maximal subgroups S2 x S2 of S4
This Group Structure determines the
Bifurcation Structure
Define a Gradient FlowGoal: To determine the bifurcation structure of stationary points of
maxq (G(q) + D(q))
Method: Study the equilibria of the of the flow
• Equilibria of this system (in RNK+K ) are possible solutions of the optimization problem
• The Jacobian q,L(q*,*) is symmetric, and so only bifurcations of equilibria can occur.
• The first equilibrium is q*(0 = 0) 1/N.
• If wT qF(q*,) w < 0 for every wker J, then q*() is a maximizer of .
• The first equilibrium is q*(0 = 0) 1/N.
Yy z
yqq xzqqDqGqq
1)|()()(:),,( ,, L
Symmetry Breaking Bifurcations
q*
4
11
N
q
41 by fixed is SSq N
N
4S
3S3S
3S 3S
2S2S 2S2S2S2S2S2S
1
2S 2S 2S2S
Symmetry Breaking Bifurcations
q*
4
11
N
q
*q
41 by fixed is SSq N
N
31* by fixed is SSq N
4S
3S3S
3S 3S
2S2S 2S2S2S2S2S2S
1
2S 2S 2S2S
Symmetry Breaking Bifurcations
q*
4
11
N
q
*q
41 by fixed is SSq N
N
31* by fixed is SSq N
*q
22* by fixed is SSq N
4S
3S3S
3S 3S
2S2S 2S2S2S2S2S2S
1
2S 2S 2S2S
Symmetry Breaking Bifurcations
q*
4S
3S3S
3S 3S
2S2S 2S2S2S2S2S2S
1
2S 2S 2S2S
Symmetry Breaking Bifurcations
q*
4S
34,12 24,13
23,14
Symmetry Breaking Bifurcations
q*
*q
)34(),12(by fixed is 22* SSq
4S
34,12 24,13
23,14
Existence Theorems for Bifurcating Branches
Given a bifurcation at a point fixed by SN ,
• Equivariant Branching Lemma The Smoller-Wasserman Theorem (Vanderbauwhede and Cicogna 1980-1) (Smoller and Wasserman 1985-6)
• There are N bifurcating branches, each which have symmetry SN-1 .
• There are N!/(2m!n!) bifurcating branches which have symmetry Sm x Sn if N=m+n.
q*
q*Existence Theorems for Bifurcating Branches
Given a bifurcation at a point fixed by SN-1 ,
• Equivariant Branching Lemma The Smoller-Wasserman Theorem (Vanderbauwhede and Cicogna 1980-1) (Smoller and Wasserman 1985-6)
• There are N-1 bifurcating branches, each which have symmetry SN-2 .
• There are (N-1)!/(2m!n!) bifurcating branches which have symmetry Sm x Sn if N-1=m+n.
4S
3S3S
3S 3S
2S2S 2S2S2S2S2S2S
1
2S 2S 2S2S
Group Structure
Observed Bifurcation Structure
Group Structure
q*Observed Bifurcation Structure
4S
3S3S
3S 3S
2S2S 2S2S2S2S2S2S
1
2S 2S 2S2S
The Equivariant Branching Lemma shows that the bifurcation structure contains the branches …
Group Structure
q*Observed Bifurcation Structure
4S
34,12 24,13
23,14
The subgroups {S2x S2} give additional structure …
Group Structure
q*Observed Bifurcation Structure
4S
34,12 24,13
23,14
The subgroups {S2x S2} give additional structure …
q*
Theorem: There are at exactly K bifurcations on the branch (q1/N , ) whenever G(q1/N) is nonsingular
There are K=52bifurcations on the first
branch
Observed Bifurcation Structure
4S
3S3S
3S 3S
0
3
vv
v
v
0
3
vv
v
v
0
3vv
v
v
0
3vv
v
v
2S2S 2S2S2S2S2S2S
1
0
2
0
vv
v
2S 2S 2S2S
0
2
0
vv
v
0
2
0
vv
v
0
0
2
vv
v
0
2
0
vv
v
0
2
0
vv
v
0
0
2
v
v
v
0
20v
v
v
0
0
2
v
v
v
0
0
2
v
v
v
0
0
2
v
v
v
0
02v
v
v
A partial subgroup lattice for S4 and the corresponding bifurcating directions given by the Equivariant Branching Lemma
4S
34,12 24,13
23,14
v
v
v
v
v
v
v
v
v
v
v
v
A partial subgroup lattice for S4 and the corresponding bifurcating directions corresponding to subgroups isomorphic to S2 x S2.
This theory enables us to answer the
questions previously posed …
??????
Why are there only 3 bifurcations observed? In general, are there only N-1 bifurcations?
What kinds of bifurcations do we expect: pitchfork-like, transcritical, saddle-node, or some other type?
How many bifurcating solutions are there?
What do the bifurcating branches look like? Are they subcritical or supercritical ?
What is the stability of the bifurcating branches? Is there always a bifurcating branch which contains solutions of the optimization problem?
Are there bifurcations after all of the classes have resolved ?
q*
Conceptual Bifurcation Structure
Observed Bifurcations for the 4 Blob Problem
Why are there only 3 bifurcations observed? In general, are there only N-1 bifurcations?There are N-1 symmetry breaking bifurcations from SM to SM-1 for M N.
What kinds of bifurcations do we expect: pitchfork-like, transcritical, saddle-node, or some other type?
How many bifurcating solutions are there? There are at least N from the first bifurcation, at least N-1 from the next one, etc.
What do the bifurcating branches look like? They are subcritical or supercritical depending on the sign of the bifurcation discriminator (q*,*,uk) .
What is the stability of the bifurcating branches? Is there always a bifurcating branch which contains solutions of the optimization problem? No.
Are there bifurcations after all of the classes have resolved ? Generically, no.
Conceptual Bifurcation Structure
q*
4S
3S3S
3S 3S
2S2S 2S2S2S2S2S2S
1
2S 2S 2S2S
Continuation techniques numerically illustrate the theory
using the Information Distortion
q*
I(Y
,Z)
bits
Bifurcating branches with symmetry S2 x S2
= <(12),(34)>
q*
I(Y
,Z)
bits
Additional structure!!
I(Y
,Z)
bits
I(Y
,Z)
bits
I(Y
,Z)
bits
A closer look …
q*
I(Y
,Z)
bits
Bifurcation from S4 to S3…
q*
I(Y
,Z)
bits
The bifurcation from S4 to S3 is subcritical …
(the theory predicted this since the bifurcation discriminator (q1/4,*,u)<0 )
I(Y
,Z)
bits
(4) RH(I0) = maxqH(Z|X) constrained by I(Y,Z) I0
(7) maxq(H(Z|X) + I(Y,Z))
What does this mean regardingsolutions ofthe original problems?
I(Y
,Z)
bits
Theorem:• dR/dI0 = -(I0)• d2R/dI0
2 = -d(I0)/dI0
(4) RH(I0) = maxqH(Z|X) constrained by I(Y,Z) I0
(7) maxq(H(Z|X) + I(Y,Z))
Theorem:• dR/dI0 = -(I0)• d2R/dI0
2 = d(I0)/dI0
RH as a function of I0
RH(I0) = maxq H(Z|Y) constrained by I(X;Z) I0
• is not convex and not concave• is a monotonically decreasing, continuous function
RH
Consequences??
• Analogue for the Information Distortion
RH(I0) = maxq H(Z|X) constrained by I(Y;Z) I0
is neither concave nor convex since subcritical bifurcations and saddle nodes exist.
• Rate Distortion Function (from Information Theory)R(D0) = minq I(X;Z) constrained by D(X,Z) D0
is convex if D(Y,Z) is linear in q (Rose, 1994; Cover and Thomas; Grey).
• Relevance Compression Function (for Information Bottleneck)RI(I0) = minq I(X;Z) constrained by I(Y;Z) I0
is convex if N>K+1 (Witsenhausen and Wyner 1975, Bachrach et al 2003)
Analogue for the Information Distortion
RH(I0) = maxq H(Z|X) constrained by I(Y;Z) I0
is neither concave nor convex since subcritical bifurcations and saddle nodes exist.
Relevance Compression Function (for Information Bottleneck)RI(I0) = minq I(X;Z) constrained by I(Y;Z) I0
is convex if N>K+1 (Bachrach et al 2003)
• RI(I0) and RH(I0) are related by I(X;Z) = H(Z) - H(Z|X).
• The Information Bottleneck can not have a subcritical bifurcation when N > K+1. Are there subcritical bifurcations when N<K+1 ?
• Is RH(I0) convex when N>K+1 ? That would mean that the subcritical bifurcations go away when considering the gradient flow in R(K+2)K instead of RNK.
So What??
Application to cricket sensory data
E(Y|Z): stimulusmeans conditioned
on each of the classes
spikepatterns
optimal clustering
Conclusions …
We have a complete theoretical picture of how the clusterings evolve for a class of annealing problems of the form
maxq(G(q)+D(q))
subject to the assumptions stated earlier.
o When clustering to N classes, there are N-1 bifurcations.o In general, there are only pitchfork and saddle-node bifurcations.o We can determine whether pitchfork bifurcations are either subcritical or
supercritical (1st or 2nd order phase transitions)o We know the explicit bifurcating directions
SO WHAT?? There are theoretical consequences … This suggests an algorithm for solving the
annealing problem … (NIPS 2002)