Approximate Inference: Decomposition Methods with Applications to Computer Vision Kyomin Jung ( KAIST ) Joint work with Pushmeet Kohli (Microsoft Research) Devavrat Shah (MIT) 서울대학교 July 30 th 2009

Graphical Model A probabilistic model for which a graph denotes the conditional independence structure between random variables.  Bayesian network (directed graph)  Markov Random Field (undirected graph) Recently successful in machine learning

Graphical Model A probabilistic model for which a graph denotes the conditional independence structure between random variables.  Bayesian network (directed graph)

Graphical Model A probabilistic model for which a graph denotes the conditional independence structure between random variables.  Markov Random Field (undirected graph)

Markov Random Field (MRF) Developed from Ising model in statistical physics. Applications  computer vision, error correct coding, speech recognition, gene finding etc. Many heuristics for inference problems in MRF are devised.  Theoretical guarantee for the correctness of those algorithms are not known much. Our goal : designing simple algorithms for inference with provable error bound by utilizing structures of the MRF.

Outline Problem Statement and an Example Relevant Work Our Algorithms for Approximate Inference  Efficient algorithms based on local updates  When the MRF is defined on a graph with polynomial growth, our algorithm achieves approximation within arbitrary accuracy Applications  Image denoising  Image segmentation Conclusion

Markov Random Field (MRF) A collection of random variables, defined on a graph G. The probability distribution of at vertex is dependent only on its neighbors : Graph G

Markov Random Field (MRF) A collection of random variables, defined on a graph G. The probability distribution of at vertex is dependent only on its neighbors : Graph G

Markov Random Field (MRF) A collection of random variables, defined on a graph G. The probability distribution of at vertex is dependent only on its neighbors : Graph G

pair-wise MRF if for some and Pair-wise MRF Z is called the partition function.

Computing Maximum A Posteriori MAP(Maximum A Posteriori) assignment  Most likely assignment (mode of the distribution)  NP-hard even for simple graphs like grid. Our goal  For a given, compute approximation of MAP : such that

Example : Image denoising We want to restore a binary (-1/+1) image Y of size with noise added. Consider Y as an element of Use an MRF model to restore the original image. The underlying graph is a grid graph of size

Example : Image denoising Utilizes two properties of the original image Similar to Y It is smooth, i.e. number of edges with different color is small Define the following MRF, where MAP assignment : original image

Computing partition function Z Equivalent to computing marginal probability approximation of log Z is useful for many applications including statistical physics, computer vision. Our goal: compute such that

Relevant Work Belief Propagation (BP)  BP and its variants like Tree-Reweighted algorithm have been very successful when G does not have many small cycles. Ex) good when G is locally tree-like, and the MRF has correlation decay [Jordan, Tatikonda ‘99].  When G has lots of small cycles, their correctness are not known.  Pearl [‘88], Weiss [‘00], Yididia and Freeman [‘02], Wainwright, Jaakkola and Willsky [‘03]

Relevant Work Markov Chain Monte Carlo  Computing approximation of log Z  key is to prove rapid mixing property which is non-trivial.  Jerrum and Sinclair [‘89], Dyer, Frieze and Kannan [‘91] Recent development  Weitz [‘06] using self-avoiding walk tree approach  Deterministic computation for Z for graphs with degree <6  Cannot be applied to graphs with higher degree.

Our approach Computing approximation of MAP and log-partition function for general graphs are NP-hard. Many real applications of MRF model are defined on polynomially growing graphs. We utilize structural properties of the polynomially growing graphs to obtain approximation algorithms.

Polynomially growing graph ball of radius r around v w.r.t. the shortest path distance of G. G

Polynomially growing graph G

G (A sequence of) graph is polynomially growing if there is constants s.t. for all

Outline of our algorithm : MAP Begin with a random assignment. Choose an arbitrary order of vertices With the given vertex as a center, choose a ball of radius r, where r is chosen from a geometric distribution. Compute a MAP inside the ball while fixing the assignment outside the ball. Update by the computed MAP inside the ball. Output We show is an approximation of MAP.

Our MAP Algorithm

for

Our MAP Algorithm

for

Property of the geometric distribution For any Hence, for any edge e, Pr[ e is on the boundary of B(v,r)] Pr[e is inside the ball B(v,r)]

Proof for MAP Algorithm Consider an imaginary boundary of the algorithm as follows

Proof for MAP Algorithm Consider an imaginary boundary of the algorithm as follows

Proof for MAP Algorithm For any edge e of the graph G Pr[ e belongs to the boundary of the algorithm] Polynomial growth  Size of each ball is small  computation is efficient

Proof of approximation If we restrict to a region R, it is a MAP assignment in R with some fixed assignment outside R. Also, restricted to the region R is a MAP assignment in R with another fixed assignment outside R. region

Proof of approximation We show the following Lemma : if the total differences of the potential functions for two MRFs and on R is small, the difference between the probabilities induced by the MAP assignments for and on R is small. region

Proof of approximation By this lemma and the fact that for any edge e of G, Pr[ e belongs to the boundary of the algorithm] we obtain that the sum of the differences of the probabilities for all regions induced by and is small. region

Theorem [Jung, Shah] For the computation of MAP, our algorithm achieves approximate solution in expectation and it runs in time

Outline of Our Algorithm : log-partition function Obtain a random graph decomposition by removing some edges. Compute the log-partition function inside each connected component, while replacing the potential functions of the removed boundary edges of the component by a constant. Summand the computed values and output it. we show that the output is an approximation of the log-partition function.

Graph decomposition

for Graph decomposition

for Graph decomposition

for Graph decomposition

for Graph decomposition

Proof of approximation bounds Ex, for the upper bound, where R is regions and B is the set of boundary edges.

Theorem [Jung, Shah] For the computation of log Z, our algorithm outputs approximate upper bound and lower bound of log Z in expectation, and it runs in time

Application to Image Processing In computer vision, the underlying graph is a grid Relevant Problems  Image Denoising  Image segmentation/ reconstruction Detect a specific object in an image Ex) face recognition, medical image process

We require the ratio of specific part of an object is close to a fixed ratio Ex) Face segmentation Fix ratios of eye, nose, mouth, etc. For the computation of MAP with fixed ratio, we provide an algorithm that outputs approximate solution in time, where k is the number of objects*. MRF with fixed ratio * Joint work with Kohli

Future work Adaptation of existing algorithms to computations in each component Learning underlying Markov Random Field Understanding limitations of inference algorithms Thank you