back to comfortable ground! - mit csailgroups.csail.mit.edu/.../13_optimal_transport.pdf · topics...
TRANSCRIPT
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Justin SolomonMIT, Spring 2017
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Back to comfortable ground!
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BIASED
…toward my own research!
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Understand geometry from a
“softened” probabilisticstandpoint.
Secondary goal:Application of machinery from previous lectures
(vector fields, geodesics, metric spaces, optimization…)
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x
½(x)
“Somewhere over here.”
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x
½(x)
“Exactly here.”
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x
½(x)
“One of these two places.”
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Which is closer, 1 or 2?
Query 1 2
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Which is closer, 1 or 2?
Query 1 2
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Which is closer, 1 or 2?
Query 1 2
p(x; y)
p1(x; y) p2(x; y)
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p1(x) p2(x)
Lp normKL divergence
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Which is closer, 1 or 2?
Query 1 2
p(x; y)
p1(x; y) p2(x; y)
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Neither!
Query 1 2
p(x; y)
p1(x; y) p2(x; y)
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Measured overlap, not displacement.
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Smaller bins worsen histogram distances
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Permuting histogram bins has
no effecton these distances.
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Image courtesy M. Cuturi
Geometric theory of probability
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Compare in this direction
Not in this direction
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Match mass from the distributions
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Match mass from the distributions
Cost to move mass 𝒎from 𝒙 to 𝒚:
𝒎 ⋅ 𝒅(𝒙, 𝒚)x
y
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Supply distribution 𝒑𝟎 Demand distribution 𝒑𝟏
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p q
𝒎 ⋅ 𝒅(𝒙, 𝒚)
Starts at 𝒑
Ends at 𝒒
Positive mass
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EMD is a metric when d(x,y) satisfies the triangle inequality.
“The Earth Mover's Distance as a Metric for Image Retrieval”Rubner, Tomasi, and Guibas; IJCV 40.2 (2000): 99—121.
Revised in:
“Ground Metric Learning”Cuturi and Avis; JMLR 15 (2014)
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Comparing histogram descriptorshttp://web.mit.edu/vondrick/ihog/
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Min-cost flow
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Step 1: Compute 𝑫𝒊𝒋
Step 2: Solve linear program
Simplex
Interior point
Hungarian algorithm
…
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Underlying map!
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Min-cost flow
Useful conclusions:
1. Practical
Can do better than generic solvers.
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Min-cost flow
Useful conclusions:
1. Practical
2. Theoretical
Can do better than generic solvers.
𝑻 ∈ 𝟎, 𝟏 𝒏×𝒏 usuallycontains 𝑶(𝒏) nonzeros.
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Can we optimize faster?
Is there a continuum interpretation?
What properties does this model exhibit?
We’ll answer them in parallel!
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-1 +2
-1Supply
Demand
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-1 +2
-1Supply
Demand
1
1
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-1 +2
-1
1
1
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Orient edges arbitrarily
-1 +2
-1 0
0 0
1
-1
00
2
0
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In computer science:
Network flow problem
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We used the structure of D.
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Probabilities advectalong the surface
Think of probabilities like a fluid
Solomon, Rustamov, Guibas, and Butscher.“Earth Mover’s Distances on Discrete Surfaces.”
SIGGRAPH 2014
“Eulerian”
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Total work
Advects from 𝝆𝟎 to 𝝆𝟏
“Beckmann problem”
Scales linearly
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http://users.cms.caltech.edu/~keenan/pdf/DGPDEC.pdf
Curl free
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Curl-free Div-free
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1.
2.
Sparse SPD linear solve for 𝒇
Unconstrained and convex optimization for 𝒈
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x y
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Proposition: Satisfies triangle inequality.
0 eigenfunctions 100 eigenfunctions
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No “displacement interpolation”McCann. “A Convexity Principle for Interacting Gases.” Advances in Mathematics 128 (1997).
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Monge-Kantorovich Problem
TRICKY
NOTATION
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Function from sets to probability
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x
y
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Continuous analog of EMDhttp://www.sciencedirect.com/science/article/pii/S152407031200029X#
Shortest path distance
Expectation
Geodesic distance d(x,y)
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Not always well-posed!Image courtesy M. Cuturi
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http://realgl.blogspot.com/2013/01/pdf-cdf-inv-cdf.html
PDF [CDF] CDF-1
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W1 ineffective for averaging tasks
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Mass moves along shortest paths
“Explains” shortest path.Image from “Optimal Transport with Proximal Splitting” (Papadakis, Peyré, and Oudet)
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Cuturi. “Sinkhorn distances: Lightspeed computation of optimal transport” (NIPS 2013)
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Prove on the board:
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Alternating projection
Sinkhorn & Knopp. "Concerning nonnegative matrices and doubly stochastic matrices". Pacific J. Math. 21, 343–348 (1967).
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1. Supply vector p2. Demand vector q
3. Multiplication by K
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Gaussian convolutionFish image from borisfx.com
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No need to store K
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No need to store K
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“Geodesics in heat”Crane, Weischedel, and Wardetzky; TOG 2013
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Replace K with heat kernel
Solomon et al. "Convolutional Wasserstein Distances: Efficient Optimal Transportation on
Geometric Domains." SIGGRAPH 2015.
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Similar problems, different algorithms
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Benamou & Brenier“A computational fluid mechanics solution of the
Monge-Kantorovich mass transfer problem”Numer. Math. 84 (2000), pp. 375-393
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Tangent space/inner product at 𝝁
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Consider set of distributions as a manifold
Tangent spaces from advection
Geodesics from displacement interpolation
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Giant field in modern math
Topics in Optimal TransportationVillani, 2003
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Example: Semi-discrete transport
Lévy. “A numerical algorithm for L2 semi-discrete optimal transport in 3D.” (2014)
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Slide courtesy M. Cuturi
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Slide courtesy M. Cuturi
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Learning
𝑣 ∈ 𝑉0
𝑣 ∉ 𝑉0“Wasserstein Propagation for Semi-Supervised Learning” (Solomon et al.)
“Fast Computation of Wasserstein Barycenters” (Cuturi and Doucet)
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Morphing and registration
“Displacement Interpolation Using Lagrangian Mass Transport” (Bonneel et al.)
“An Optimal Transport Approach to Robust Reconstruction and Simplification of 2D Shapes” (de Goes et al.)
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Graphics
“Earth Mover’s Distances on Discrete Surfaces” (Solomon et al.)
“Blue Noise Through Optimal Transport” (de Goes et al.)
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Vision and image processing
“Geodesic Shape Retrieval via Optimal Mass Transport” (Rabin, Peyré, and Cohen)
“Adaptive Color Transfer with Relaxed Optimal Transport” (Rabin, Ferradans, and Papadakis)
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Justin SolomonMIT, Spring 2017