conditional expectation manifolds and brain population...
TRANSCRIPT
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Conditional Expectation Manifoldsand
Brain Population Analysis
Samuel Gerber, University of Utah
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Manifold Learning
Some observations on popular algorithms
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Isomap• Approximate geodesic distances by
shortest path in nearest neighbor graph
• Preserve approximate geodesics
•
• Multidimensional scaling
X ∼ Uniform([0,1]d)
P(X ∈ Sd) =πd/2
Γ(d/2+1)2d
limd→∞ P(X ∈ Sd) = 0minx = ∑i, j[δ (yi,y j)−d(xi,x j)]2
var(PY )X = PYyi = ∑N
k=0 P(Ck|xi)(ak +bkxi)ri(y) = E[X ∈Ci|Y = y]Ci = xi : src(xi) = xmin,sink(xi) = xmaxri(y) = E[X ∈Ci|Y = y]
1
X ∼ Uniform([0,1]d)
P(X ∈ Sd) =πd/2
Γ(d/2+1)2d
limd→∞ P(X ∈ Sd) = 0minx = ∑i, j[δ (yi,y j)−d(xi,x j)]2
var(PY )X = PYyi = ∑N
k=0 P(Ck|xi)(ak +bkxi)ri(y) = E[X ∈Ci|Y = y]Ci = xi : src(xi) = xmin,sink(xi) = xmaxri(y) = E[X ∈Ci|Y = y]
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Properties• Only relies on accurate local distances
• Shortcuts in graph - very bad approximation
• Quality measure based on graph embedding
• Hard to detect
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DimensionD
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Properties
• Classical multidimensional scaling is not minimizing
• Optimization based approaches
X ∼ Uniform([0,1]d)
P(X ∈ Sd) =πd/2
Γ(d/2+1)2d
limd→∞ P(X ∈ Sd) = 0minx = ∑i, j[δ (yi,y j)−d(xi,x j)]2
var(PY )X = PYyi = ∑N
k=0 P(Ck|xi)(ak +bkxi)ri(y) = E[X ∈Ci|Y = y]Ci = xi : src(xi) = xmin,sink(xi) = xmaxri(y) = E[X ∈Ci|Y = y]
1
A. Agarwal, J. Phillips and S. Venkatasubramanian, Universal Multi-Dimensional Scaling, Conference on Knowledge Discovery and Data Mining 2010
J. Kruskal, Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis, Psychometrika 1964
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Laplacian Eigenmaps• Given a manifold find functions
such that is minimized
• The low dimensional embedding is
• Small gradient implies that close by points will be mapped close together
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fM∇ f (y)2dy M ∆ f
min f E[g( f (Y ))−Y2]minz1,...,zn ∑ig( f (yi))− yi2
φ(r, t) = r + t
0 v(φ(r,τ),τ) dτv(r,τ)Q
d(yi,y j)2 = minv 1
0 v(r,τ)Q dτsuch that
Ωyi(φ(r,1))− y j(r))2
2 dr = 0(1)
φ(r,1)≈ v(r,0) = u(r), and d(yi,y j)2 ≈minu
Ωu(r)Q dr,subject to
Ωyi(r +u(r))− y j(r))2
2 dr ≤ ε (2)
da(yi,y j)2 = minu
Ω ||u(r)||2Q drsuch that
Ωyi(r +u(r))− y j(r))2
2 dr ≤ ε (3)
limd→∞ P(X ∈ Sd) = 0d(yi,y j) = 1
2(da(yi,y j)+da(y j,yi)) .(4)
1
fM∇ f (y)2dy M ∆ f
min f E[g( f (Y ))−Y2]minz1,...,zn ∑ig( f (yi))− yi2
φ(r, t) = r + t
0 v(φ(r,τ),τ) dτv(r,τ)Q
d(yi,y j)2 = minv 1
0 v(r,τ)Q dτsuch that
Ωyi(φ(r,1))− y j(r))2
2 dr = 0(1)
φ(r,1)≈ v(r,0) = u(r), and d(yi,y j)2 ≈minu
Ωu(r)Q dr,subject to
Ωyi(r +u(r))− y j(r))2
2 dr ≤ ε (2)
da(yi,y j)2 = minu
Ω ||u(r)||2Q drsuch that
Ωyi(r +u(r))− y j(r))2
2 dr ≤ ε (3)
limd→∞ P(X ∈ Sd) = 0d(yi,y j) = 1
2(da(yi,y j)+da(y j,yi)) .(4)
1
f : M → RM∇ f (y)2dy M ∆ f
min f E[g( f (Y ))−Y2]minz1,...,zn ∑ig( f (yi))− yi2
φ(r, t) = r + t
0 v(φ(r,τ),τ) dτv(r,τ)Q
d(yi,y j)2 = minv 1
0 v(r,τ)Q dτsuch that
Ωyi(φ(r,1))− y j(r))2
2 dr = 0(1)
φ(r,1)≈ v(r,0) = u(r), and d(yi,y j)2 ≈minu
Ωu(r)Q dr,subject to
Ωyi(r +u(r))− y j(r))2
2 dr ≤ ε (2)
da(yi,y j)2 = minu
Ω ||u(r)||2Q drsuch that
Ωyi(r +u(r))− y j(r))2
2 dr ≤ ε (3)
limd→∞ P(X ∈ Sd) = 0d(yi,y j) = 1
2(da(yi,y j)+da(y j,yi)) .(4)
1
f : M → RM∇ f (y)2dy
M∆ fx = [ f1(y), . . . , fn(y)] ∈ Rn
min f E[g( f (Y ))−Y2]minz1,...,zn ∑ig( f (yi))− yi2
φ(r, t) = r + t
0 v(φ(r,τ),τ) dτv(r,τ)Q
d(yi,y j)2 = minv 1
0 v(r,τ)Q dτsuch that
Ωyi(φ(r,1))− y j(r))2
2 dr = 0(1)
φ(r,1)≈ v(r,0) = u(r), and d(yi,y j)2 ≈minu
Ωu(r)Q dr,subject to
Ωyi(r +u(r))− y j(r))2
2 dr ≤ ε (2)
1
f1f2x = [ f1(y), f2(y)]f : M → RM∇ f (y)2dy
M∆ fx = [ f1(y), . . . , fn(y)] ∈ Rn
min f E[g( f (Y ))−Y2]minz1,...,zn ∑ig( f (yi))− yi2
φ(r, t) = r + t
0 v(φ(r,τ),τ) dτv(r,τ)Q
d(yi,y j)2 = minv 1
0 v(r,τ)Q dτsuch that
Ωyi(φ(r,1))− y j(r))2
2 dr = 0(1)
1
f1f2x = [ f1(y), f2(y)]f : M → RM∇ f (y)2dy
M∆ fx = [ f1(y), . . . , fn(y)] ∈ Rn
min f E[g( f (Y ))−Y2]minz1,...,zn ∑ig( f (yi))− yi2
φ(r, t) = r + t
0 v(φ(r,τ),τ) dτv(r,τ)Q
d(yi,y j)2 = minv 1
0 v(r,τ)Q dτsuch that
Ωyi(φ(r,1))− y j(r))2
2 dr = 0(1)
1
f1f2x = [ f1(y), f2(y)]f : M → RM∇ f (y)2dy
M∆ fx = [ f1(y), . . . , fn(y)] ∈ Rn
min f E[g( f (Y ))−Y2]minz1,...,zn ∑ig( f (yi))− yi2
φ(r, t) = r + t
0 v(φ(r,τ),τ) dτv(r,τ)Q
d(yi,y j)2 = minv 1
0 v(r,τ)Q dτsuch that
Ωyi(φ(r,1))− y j(r))2
2 dr = 0(1)
1
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Properties
• Again only local distances important
• No quality measure of the embedding
f1f2x = [ f1(y), f2(y)]f : M → RM∇ f (y)2dy
M∆ fx = [ f1(y), . . . , fn(y)] ∈ Rn
min f E[g( f (Y ))−Y2]minz1,...,zn ∑ig( f (yi))− yi2
φ(r, t) = r + t
0 v(φ(r,τ),τ) dτv(r,τ)Q
d(yi,y j)2 = minv 1
0 v(r,τ)Q dτsuch that
Ωyi(φ(r,1))− y j(r))2
2 dr = 0(1)
1
f1f2x = [ f1(y), f2(y)]f : M → RM∇ f (y)2dy
M∆ fx = [ f1(y), . . . , fn(y)] ∈ Rn
min f E[g( f (Y ))−Y2]minz1,...,zn ∑ig( f (yi))− yi2
φ(r, t) = r + t
0 v(φ(r,τ),τ) dτv(r,τ)Q
d(yi,y j)2 = minv 1
0 v(r,τ)Q dτsuch that
Ωyi(φ(r,1))− y j(r))2
2 dr = 0(1)
1
f1f2x = [ f1(y), f2(y)]f : M → RM∇ f (y)2dy
M∆ fx = [ f1(y), . . . , fn(y)] ∈ Rn
min f E[g( f (Y ))−Y2]minz1,...,zn ∑ig( f (yi))− yi2
φ(r, t) = r + t
0 v(φ(r,τ),τ) dτv(r,τ)Q
d(yi,y j)2 = minv 1
0 v(r,τ)Q dτsuch that
Ωyi(φ(r,1))− y j(r))2
2 dr = 0(1)
1
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Eigenfunction Issue
• Minimzing
• Orthogonality constraint on f in function space (not geometrically on manifold)
• Eigenvectors with higher frequency along same extension on the manifold can have smaller cost
fM∇ f (y)2dy M ∆ f
min f E[g( f (Y ))−Y2]minz1,...,zn ∑ig( f (yi))− yi2
φ(r, t) = r + t
0 v(φ(r,τ),τ) dτv(r,τ)Q
d(yi,y j)2 = minv 1
0 v(r,τ)Q dτsuch that
Ωyi(φ(r,1))− y j(r))2
2 dr = 0(1)
φ(r,1)≈ v(r,0) = u(r), and d(yi,y j)2 ≈minu
Ωu(r)Q dr,subject to
Ωyi(r +u(r))− y j(r))2
2 dr ≤ ε (2)
da(yi,y j)2 = minu
Ω ||u(r)||2Q drsuch that
Ωyi(r +u(r))− y j(r))2
2 dr ≤ ε (3)
limd→∞ P(X ∈ Sd) = 0d(yi,y j) = 1
2(da(yi,y j)+da(y j,yi)) .(4)
1
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Eigenfunction Issue
• B is orthogonal to A (in function space)
• Cost of B less than C (the desired eigenvector)
Samuel Gerber, Tolga Tasdizen, Ross Whitaker, Robust Non-linear Dimensionality Reduction using Successive 1-Dimensional Laplacian Eigenmaps, ICML 2007
x
yx
y
x
y
y
fx
fx
f
B
C
A
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Conditional Expectation Manifolds
Manifold learning as unsupervised non-parametric model fitting
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Principal Curves/SurfacesCurve through the middle of a density
T. Hastie, W. Stuetzle, Principal curvesJournal of the American Statistical Association 1989
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Principal Surface Definition• Minimal orthogonal projection onto surface
• Principal surface iff conditional expectation of the projection equal to surface
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Principal Surface Estimation
• Principal surfaces are extremal points of (objective function)
• Pick a parametrized surface model
• Optimize over parameters of
• Unfortunately principal surfaces are all saddle points of
• Projection is a non-linear optimization problem
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Conditional Expectation Manifolds (CEM)
• Define a coordinate mapping
• Model surface as conditional expectation of coordinate mapping.
• Optimize coordinate mapping
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CEM Estimation
• Coordinate mapping as kernel regression
s
Samuel Gerber, Tolga Tasdizen, Ross Whitaker "Dimensionality Reduction and Principal Surfaces via Kernel Map Manifolds", (ICCV 2009)
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CEM Estimation
• Conditional expectation estimated with kernel regression
s
Samuel Gerber, Tolga Tasdizen, Ross Whitaker "Dimensionality Reduction and Principal Surfaces via Kernel Map Manifolds", (ICCV 2009)
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Some results
• Effect of optimization
Input Initial MSE 8.6 Optimized MSE 2.6
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Some results• 1965 images of different facial expression (20x28)
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Work in Progress• Saddle point property of extrema is
problematic for model selection
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Figure 2: Minimization of d(λ ,Y )2 with automatic bandwidth selection starting fromσg = 1 and σλ = 0.1. (a) fitted curve with optimization path and (b) train and test errorwith points indicating minimal train and test error, respectively.
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Figure 3: Minimization of q(λ ,Y )2 with automatic bandwidth selection starting fromσg = 1 and σλ = 0.1. (a) fitted curve with optimization path and (b) train and test errorwith points indicating minimal train and test error, respectively.
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Work in Progress• Conditional expectation manifolds pave
way for other objective functions
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Figure 4: Minimization of d(λ ,Y )2 with automatic bandwidth selection starting fromσg = 0.1 and σλ = 0.1. (a) fitted curve with optimization path and (b) train and testerror with points indicating minimal train and test error, respectively.
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q(!,
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!!
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(a) (b)
Figure 5: Minimization of q(λ ,Y )2 with automatic bandwidth selection starting fromσg = 0.1 and σλ = 0.1. (a) fitted curve with optimization path and (b) train and testerror with points indicating minimal train and test error, respectively.
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Brain Population Analysis
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Motivation
• Proof of concept
• Conditional expectation manifold for brain images
• Non-linearity in shape space
• Natural extension at the time from single atlas to multiple atlases to continuum
• Simplify statistics on shape spaces
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Measuring Shape Differences
• Euclidean space does not capture changes in shape
• Distance based on measuring length of transformation
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• Diffeomorphic transform
• Riemannian metric ( )
• Geodesics on diffeomorphic transformations
• Induces metric on images
φ(r, t) = r + t
0 v(φ(r,τ),τ) dτv(r,τ)Q
d(yi,y j)2 = minv 1
0 v(r,τ)Q dτsuch that
Ωyi(φ(r,1))− y j(r))2
2 dr = 0(1)
φ(r,1)≈ v(r,0) = u(r), and d(yi,y j)2 ≈minu
Ωu(r)Q dr,subject to
Ωyi(r +u(r))− y j(r))2
2 dr ≤ ε (2)
da(yi,y j)2 = minu
Ω ||u(r)||2Q drsuch that
Ωyi(r +u(r))− y j(r))2
2 dr ≤ ε (3)
limd→∞ P(X ∈ Sd) = 0d(yi,y j) = 1
2(da(yi,y j)+da(y j,yi)) .(4)Q = α∇+(1−α)Iu(r)2
Q = α||∇u(r)||2 +(1−α)||u(r)||2M ∆ f (y) f (y)dy
1
φ(r, t) = r + t
0 v(φ(r,τ),τ) dτv(r,τ)Q
d(yi,y j)2 = minv 1
0 v(r,τ)Q dτsuch that
Ωyi(φ(r,1))− y j(r))2
2 dr = 0(1)
φ(r,1)≈ v(r,0) = u(r), and d(yi,y j)2 ≈minu
Ωu(r)Q dr,subject to
Ωyi(r +u(r))− y j(r))2
2 dr ≤ ε (2)
da(yi,y j)2 = minu
Ω ||u(r)||2Q drsuch that
Ωyi(r +u(r))− y j(r))2
2 dr ≤ ε (3)
limd→∞ P(X ∈ Sd) = 0d(yi,y j) = 1
2(da(yi,y j)+da(y j,yi)) .(4)Q = α∇+(1−α)Iu(r)2
Q = α||∇u(r)||2 +(1−α)||u(r)||2M ∆ f (y) f (y)dy
1
φ(r, t) = r + t
0 v(φ(r,τ),τ) dτv(r,τ)Q
d(yi,y j)2 = minv 1
0 v(r,τ)Q dτsuch that
Ωyi(φ(r,1))− y j(r))2
2 dr = 0(1)
φ(r,1)≈ v(r,0) = u(r), and d(yi,y j)2 ≈minu
Ωu(r)Q dr,subject to
Ωyi(r +u(r))− y j(r))2
2 dr ≤ ε (2)
da(yi,y j)2 = minu
Ω ||u(r)||2Q drsuch that
Ωyi(r +u(r))− y j(r))2
2 dr ≤ ε (3)
limd→∞ P(X ∈ Sd) = 0d(yi,y j) = 1
2(da(yi,y j)+da(y j,yi)) .(4)Q = α∇+(1−α)Iu(r)2
Q = α||∇u(r)||2 +(1−α)||u(r)||2M ∆ f (y) f (y)dy
1
Large Deformation Diffeomorphic Metric
φ(r, t) = r + t
0 v(φ(r,τ),τ) dτv(r,τ)Q
d(yi,y j)2 = minv 1
0 v(r,τ)Q dτsuch that
Ωyi(φ(r,1))− y j(r))2
2 dr = 0(1)
φ(r,1)≈ v(r,0) = u(r), and d(yi,y j)2 ≈minu
Ωu(r)Q dr,subject to
Ωyi(r +u(r))− y j(r))2
2 dr ≤ ε (2)
da(yi,y j)2 = minu
Ω ||u(r)||2Q drsuch that
Ωyi(r +u(r))− y j(r))2
2 dr ≤ ε (3)
limd→∞ P(X ∈ Sd) = 0d(yi,y j) = 1
2(da(yi,y j)+da(y j,yi)) .(4)Q = α∇+(1−α)Iu(r)2
Q = α||∇u(r)||2 +(1−α)||u(r)||2M ∆ f (y) f (y)dy
1
φ(r, t) = r + t
0 v(φ(r,τ),τ) dτv(r,τ)Q
d(yi,y j)2 = minv 1
0 v(r,τ)Q dτsuch that
Ωyi(φ(r,1))− y j(r))2
2 dr = 0(1)
φ(r,1)≈ v(r,0) = u(r), and d(yi,y j)2 ≈minu
Ωu(r)Q dr,subject to
Ωyi(r +u(r))− y j(r))2
2 dr ≤ ε (2)
da(yi,y j)2 = minu
Ω ||u(r)||2Q drsuch that
Ωyi(r +u(r))− y j(r))2
2 dr ≤ ε (3)
limd→∞ P(X ∈ Sd) = 0d(yi,y j) = 1
2(da(yi,y j)+da(y j,yi)) .(4)Q = α∇+(1−α)Iu(r)2
Q = α||∇u(r)||2 +(1−α)||u(r)||2M ∆ f (y) f (y)dy
1
φ(r, t) = r + t
0 v(φ(r,τ),τ) dτv(r,τ)Q
d(yi,y j)2 = minv 1
0 v(r,τ)Q dτsuch that
Ωyi(φ(r,1))− y j(r))2
2 dr = 0(1)
φ(r,1)≈ v(r,0) = u(r), and d(yi,y j)2 ≈minu
Ωu(r)Q dr,subject to
Ωyi(r +u(r))− y j(r))2
2 dr ≤ ε (2)
da(yi,y j)2 = minu
Ω ||u(r)||2Q drsuch that
Ωyi(r +u(r))− y j(r))2
2 dr ≤ ε (3)
limd→∞ P(X ∈ Sd) = 0d(yi,y j) = 1
2(da(yi,y j)+da(y j,yi)) .(4)Q = α∇+(1−α)Iu(r)2
Q = α||∇u(r)||2 +(1−α)||u(r)||2M ∆ f (y) f (y)dy
1
φ(r, t) = r + t
0 v(φ(r,τ),τ) dτv(r,τ)Q
d(yi,y j)2 = minv 1
0 v(r,τ)Q dτsuch that
Ωyi(φ(r,1))− y j(r))2
2 dr = 0(1)
φ(r,1)≈ v(r,0) = u(r), and d(yi,y j)2 ≈minu
Ωu(r)Q dr,subject to
Ωyi(r +u(r))− y j(r))2
2 dr ≤ ε (2)
da(yi,y j)2 = minu
Ω ||u(r)||2Q drsuch that
Ωyi(r +u(r))− y j(r))2
2 dr ≤ ε (3)
limd→∞ P(X ∈ Sd) = 0d(yi,y j) = 1
2(da(yi,y j)+da(y j,yi)) .(4)Q = α∇+(1−α)Iu(r)2
Q = α||∇u(r)||2 +(1−α)||u(r)||2M ∆ f (y) f (y)dy
1
φ(r, t) = r + t
0 v(φ(r,τ),τ) dτv(r,τ)Q
d(yi,y j)2 = minv 1
0 v(r,τ)Q dτsuch that
Ωyi(φ(r,1))− y j(r))2
2 dr = 0(1)
φ(r,1)≈ v(r,0) = u(r), and d(yi,y j)2 ≈minu
Ωu(r)Q dr,subject to
Ωyi(r +u(r))− y j(r))2
2 dr ≤ ε (2)
da(yi,y j)2 = minu
Ω ||u(r)||2Q drsuch that
Ωyi(r +u(r))− y j(r))2
2 dr ≤ ε (3)
limd→∞ P(X ∈ Sd) = 0d(yi,y j) = 1
2(da(yi,y j)+da(y j,yi)) .(4)Q = α∇+(1−α)Iu(r)2
Q = α||∇u(r)||2 +(1−α)||u(r)||2M ∆ f (y) f (y)dy
1
d(e,φ)2 = minv t
0
Ωv(r,τ)Qdr dτd(yi,y j)2 = minv
10 v(r,τ)Q dτ
such that
Ωyi(φ(r,1))− y j(r))22 dr = 0
(4)
φ(r,1)≈ v(r,0) = u(r), and d(yi,y j)2 ≈minu
Ωu(r)Q dr,subject to
Ωyi(r +u(r))− y j(r))2
2 dr ≤ ε (5)
da(yi,y j)2 = minu
Ω ||u(r)||2Q drsuch that
Ωyi(r +u(r))− y j(r))2
2 dr ≤ ε (6)
limd→∞ P(X ∈ Sd) = 0d(yi,y j) = 1
2(da(yi,y j)+da(y j,yi)) .(7)Q = α∇+(1−α)Iu(r)2
Q = α||∇u(r)||2 +(1−α)||u(r)||2M ∆ f (y) f (y)dy
var(PY )X = PY
2
φ(r, t) = r + t
0 v(φ(r,τ),τ) dτv(r,τ)Q
d(yi,y j)2 = minv 1
0 v(r,τ)Q dτsuch that
Ωyi(φ(r,1))− y j(r))2
2 dr = 0(1)
φ(r,1)≈ v(r,0) = u(r), and d(yi,y j)2 ≈minu
Ωu(r)Q dr,subject to
Ωyi(r +u(r))− y j(r))2
2 dr ≤ ε (2)
da(yi,y j)2 = minu
Ω ||u(r)||2Q drsuch that
Ωyi(r +u(r))− y j(r))2
2 dr ≤ ε (3)
limd→∞ P(X ∈ Sd) = 0d(yi,y j) = 1
2(da(yi,y j)+da(y j,yi)) .(4)Q = α∇+(1−α)Iu(r)2
Q = α||∇u(r)||2 +(1−α)||u(r)||2M ∆ f (y) f (y)dy
1
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Manifold in Brain SpaceSpace of Smooth Images
Manifold induced by
diffeomorphic image
metric
Learned data
manifold
Samples/images
Frechet mean on
metric manifold
Frechet mean on
data manifold
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Data set:spiral segments Manifold mean Diffeomorphic mean
mean on
metric manifold
Manifold in Brain Space
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Approximating the Diffeomorphic Metric
• For small deformations work in tangent space
• Distance defined by
• For symmetry
φ(r, t) = r + t
0 v(φ(r,τ),τ) dτv(r,τ)Q
d(yi,y j)2 = minv 1
0 v(r,τ)Q dτsuch that
Ωyi(φ(r,1))− y j(r))2
2 dr = 0(1)
φ(r,1)≈ v(r,0) = u(r), and d(yi,y j)2 ≈minu
Ωu(r)Q dr,subject to
Ωyi(r +u(r))− y j(r))2
2 dr ≤ ε (2)
da(yi,y j)2 = minu
Ω ||u(r)||2Q drsuch that
Ωyi(r +u(r))− y j(r))2
2 dr ≤ ε (3)
limd→∞ P(X ∈ Sd) = 0d(yi,y j) = 1
2(da(yi,y j)+da(y j,yi)) .(4)Q = α∇+(1−α)Iu(r)2
Q = α||∇u(r)||2 +(1−α)||u(r)||2M ∆ f (y) f (y)dy
1
φ(r, t) = r + t
0 v(φ(r,τ),τ) dτv(r,τ)Q
d(yi,y j)2 = minv 1
0 v(r,τ)Q dτsuch that
Ωyi(φ(r,1))− y j(r))2
2 dr = 0(1)
φ(r,1)≈ v(r,0) = u(r), and d(yi,y j)2 ≈minu
Ωu(r)Q dr,subject to
Ωyi(r +u(r))− y j(r))2
2 dr ≤ ε (2)
da(yi,y j)2 = minu
Ω ||u(r)||2Q drsuch that
Ωyi(r +u(r))− y j(r))2
2 dr ≤ ε (3)
limd→∞ P(X ∈ Sd) = 0d(yi,y j) = 1
2(da(yi,y j)+da(y j,yi)) .(4)Q = α∇+(1−α)Iu(r)2
Q = α||∇u(r)||2 +(1−α)||u(r)||2M ∆ f (y) f (y)dy
1
φ(r, t) = r + t
0 v(φ(r,τ),τ) dτv(r,τ)Q
d(yi,y j)2 = minv 1
0 v(r,τ)Q dτsuch that
Ωyi(φ(r,1))− y j(r))2
2 dr = 0(1)
φ(r,1)≈ v(r,0) = u(r), and d(yi,y j)2 ≈minu
Ωu(r)Q dr,subject to
Ωyi(r +u(r))− y j(r))2
2 dr ≤ ε (2)
da(yi,y j)2 = minu
Ω ||u(r)||2Q drsuch that
Ωyi(r +u(r))− y j(r))2
2 dr ≤ ε (3)
limd→∞ P(X ∈ Sd) = 0d(yi,y j) = 1
2(da(yi,y j)+da(y j,yi)) .(4)Q = α∇+(1−α)Iu(r)2
Q = α||∇u(r)||2 +(1−α)||u(r)||2M ∆ f (y) f (y)dy
1
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Manifold Representation
• Represent manifold as conditional expectation of some function
• Non euclidean space use Frechet mean
f(y) = ∑ni=1
Ky(d(y,yi))zi∑n
j=1 Ky(d(y,y j)).(1)
g(x) = argminy ∑ni=1
Kx(x− f (yi))2)∑n
j=1 Kx(x− f (y j))2)d(y,yi)2 .(2)
ym = argminy∈M ∑ni=1 wid(y,yi)2 , (3)
g(x) = E[Y | f (y) = x]M∇ f (y)2dy
M∆ fx = [ f1(y), . . . , fn(y)] ∈ Rn
min f E[g( f (Y ))−Y2]minz1,...,zn ∑ig( f (yi))− yi2
φ(r, t) = r + t
0 v(φ(r,τ),τ) dτv(r,τ)Q
1
f(y) = ∑ni=1
Ky(d(y,yi))zi∑n
j=1 Ky(d(y,y j)).(1)
g(x) = argminy ∑ni=1
Kx(x− f (yi))2)∑n
j=1 Kx(x− f (y j))2)d(y,yi)2 .(2)
ym = argminy∈M ∑ni=1 wid(y,yi)2 , (3)
g(x) = E[Y | f (y) = x]M∇ f (y)2dy
M∆ fx = [ f1(y), . . . , fn(y)] ∈ Rn
min f E[g( f (Y ))−Y2]minz1,...,zn ∑ig( f (yi))− yi2
φ(r, t) = r + t
0 v(φ(r,τ),τ) dτv(r,τ)Q
1
f(y) = ∑ni=1
Ky(d(y,yi))zi∑n
j=1 Ky(d(y,y j)).(1)
g(x) = argminy ∑ni=1
Kx(x− f (yi))2)∑n
j=1 Kx(x− f (y j))2)d(y,yi)2 .(2)
ym = argminy∈M ∑ni=1 wid(y,yi)2 , (3)
g(x) = E[Y | f (y) = x]M∇ f (y)2dy
M∆ fx = [ f1(y), . . . , fn(y)] ∈ Rn
min f E[g( f (Y ))−Y2]minz1,...,zn ∑ig( f (yi))− yi2
φ(r, t) = r + t
0 v(φ(r,τ),τ) dτv(r,τ)Q
1
B. Davis, P. Fletcher, E. Bullitt, S. Joshi, Population shape regression from random design data, ICCV 2007
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Manifold Representation• Compute embedding based on pairwise
distance matrix (isomap)
• Define coordinate mapping based kernel map manifold approach
f(y) = ∑ni=1
Ky(d(y,yi))zi∑n
j=1 Ky(d(y,y j)).(1)
g(x) = argminy ∑ni=1
Kx(x− f (yi))2)∑n
j=1 Kx(x− f (y j))2)d(y,yi)2 .(2)
ym = argminy∈M ∑ni=1 wid(y,yi)2 , (3)
g(x) = E[Y | f (y) = x]M∇ f (y)2dy
M∆ fx = [ f1(y), . . . , fn(y)] ∈ Rn
min f E[g( f (Y ))−Y2]minz1,...,zn ∑ig( f (yi))− yi2
φ(r, t) = r + t
0 v(φ(r,τ),τ) dτv(r,τ)Q
1
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• In all steps:
• Large distances have negligible effect
Manifold Representation
f(y) = ∑ni=1
Ky(d(y,yi))zi∑n
j=1 Ky(d(y,y j)).(1)
g(x) = argminy ∑ni=1
Kx(x− f (yi))2)∑n
j=1 Kx(x− f (y j))2)d(y,yi)2 .(2)
ym = argminy∈M ∑ni=1 wid(y,yi)2 , (3)
g(x) = E[Y | f (y) = x]M∇ f (y)2dy
M∆ fx = [ f1(y), . . . , fn(y)] ∈ Rn
min f E[g( f (Y ))−Y2]minz1,...,zn ∑ig( f (yi))− yi2
φ(r, t) = r + t
0 v(φ(r,τ),τ) dτv(r,τ)Q
1
f(y) = ∑ni=1
Ky(d(y,yi))zi∑n
j=1 Ky(d(y,y j)).(1)
g(x) = argminy ∑ni=1
Kx(x− f (yi))2)∑n
j=1 Kx(x− f (y j))2)d(y,yi)2 .(2)
ym = argminy∈M ∑ni=1 wid(y,yi)2 , (3)
g(x) = E[Y | f (y) = x]M∇ f (y)2dy
M∆ fx = [ f1(y), . . . , fn(y)] ∈ Rn
min f E[g( f (Y ))−Y2]minz1,...,zn ∑ig( f (yi))− yi2
φ(r, t) = r + t
0 v(φ(r,τ),τ) dτv(r,τ)Q
1
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Results• OASIS data set
• 416 subjects, age 16 to 80
• 100 subjects diagnosed with mild to moderate dementia
• ADNI data set
• 156 Subjects, age 57 to 88
• 38 normal, 84 MCI, 34 early AD
20 22 24 26 28 300
5
10
15
20
25
30 MMSE Histogram
10 15 20 25 300
20
40
60
80
100
120
140 MMSE Histogram
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OASIS 2D Embedding
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Manifold Fit - OASIS• Measure reconstruction error
• Comparison to PCA
• Comparison of different metrics
• Scale by average nearest neighbor distance
f(y) = ∑ni=1
Ky(d(y,yi))zi∑n
j=1 Ky(d(y,y j)).(1)
g(x) = argminy ∑ni=1
Kx(x− f (yi))2)∑n
j=1 Kx(x− f (y j))2)d(y,yi)2 .(2)
ym = argminy∈M ∑ni=1 wid(y,yi)2 , (3)
g(x) = E[Y | f (y) = x]M∇ f (y)2dy
error = ∑i d(g( f (yi)),yi)∑i d(nn(yi),yi)
∆ fx = [ f1(y), . . . , fn(y)] ∈ Rn
min f E[g( f (Y ))−Y2]minz1,...,zn ∑ig( f (yi))− yi2
φ(r, t) = r + t
0 v(φ(r,τ),τ) dτ
1
Manifold Model PCA
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Manifold Fit - ADNI
1.07 0.81 1.23Projection distance
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Statistical Analysis - OASIS• Linear regression on age, MMSE, CDR
• Comparison to PCA and age as predictor
• Controlled for age - BIC to select best model
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Statistical Analysis - OASIS
• Restricted to subjects age above 60
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Statistical Analysis - ADNI
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Reconstructions -ADNI
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ADNI - Statistics
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Extensions• Different Metrics?
• Transformation based metric is expensive
• No optimization of conditional expectation manifold
• Embedding/Statistics including metric tensor.
• Adding supervision
• Fit manifold with respect to a clinical predictor
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Thank you
This work is supported by
NIH/NCBC grant U54-EB005149NSF grant CCF-073222
NIBIB grant 5RO1EB007688-02
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Thoughts on Manifold Learning• For which applications / tasks is manifold learning
effective?
• Purely unsupervised tasks are rare
• Exploratory analysis
• In supervised settings:
• Manifold learning as regularization
• Feature extraction
• Stratified, non flat-able manifolds and detection of non-manifold structure