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Spacey random walks CAM Colloquium

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Slides from my talk on spacey random walks at the Cornell CAM Colloquium on October 27, 2017.

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Spacey random walks CAM Colloquium

  1. 1. CAM Colloquium October 27, 2017 Slides. bit.ly/arb-cam-colloq Joint work with David Gleich (Purdue) & Lek-Heng Lim (U. Chicago) Spacey random walks and tensor eigenvectors Austin R. Benson Cornell University
  2. 2. 2 [Kemeny-Snell 76] “In the land of Oz they never have two nice days in a row. If they have a nice day, they are just as likely to have snow as rain the next day. If they have snow or rain, they have an even chain of having the same the next day. If there is a change from snow or rain, only half of the time is this change to a nice day.” Column-stochastic in my talk (since I’m a linear algebra person). Background. Markov chains, matrices, and eigenvectors have a long-standing relationship. Equations for stationary distribution x. The vector x is an eigenvector. Px = x.
  3. 3. 1. Start with a Markov chain 2. Inquire about the stationary distribution 3. Discover an eigenvector problem on the transition matrix 3 In general, {Zt} will be a stochastic process in this talk. This is the limiting fraction of time spent in each state. Background. Markov chains, matrices, and eigenvectors have a long-standing relationship.
  4. 4. 4 Higher-order means keeping more history on the same state space. Better model for several applications…  traffic flow in airport networks [Rosvall+ 14]  web browsing behavior [Pirolli-Pitkow 99; Chierichetti+ 12]  DNA sequences [Borodovsky-McIninch 93; Ching+ 04] Rosvall et al., Nature Comm., 2014. second- order MC Background. Higher-order Markov chains are useful for many data problems.
  5. 5. 5 1 3 2 P  Transition probability tensor [Li-Cui-Ng 13; Culp-Pearson-Zhang 17],  stochastic tensor [Yang-Yang 11]  stochastic hypermatrix [Benson-Gleich-Lim 17] For our purposes, “tensors” are just multi-way arrays of numbers (tensor ⟷ hypermatrix). A is a third-order n x n x n tensor → Ai,j,k is a (real) number, 1 < i, j, k < n. 1 3 2 A 2 1 A (a matrix is just a second-order tensor) Background. The transition probabilities of higher-order Markov chains can be represented by a tensor.
  6. 6. 6 A tensor eigenpair for a tensor A is a solution (x, 𝜆) to the following system of polynomial equations [Lim 05, Qi 05]. technically called an l2 or z tensor eigenpair—there are a few types of tensor eigenvectors (see new Qi-Luo 2017 book!) Analogous to matrix case, eigenpairs are stationary points of the Lagrangian for a generalized Rayleigh quotient [Lim 05]. Background. Tensors also have eigenvectors. tensor eigenvector matrix eigenvector
  7. 7. 7 However, there are few results connecting tensors and higher-order Markov chains.
  8. 8. 8 Do tensor eigenvectors tell us anything about higher-order Markov chains? 1. Start with a Markov chain 2. Inquire about stationary dist. 3. Discover a matrix eigenvector problem on the transition matrix 1. Start with a higher-order MC 2. Inquire about stationary dist. 3. Discover a tensor eigenvector problem on the transition tensor ?
  9. 9. 9 Second-order Markov chains have stationary distribution on pairs of states. Li-Ng approx. gives tensor eigenvectors. 1 3 2 P The stationary distribution on pairs of states is still a matrix eigenvector. [Li-Ng 14] Rank-1 approximation Xi,j = xixj gives a “distribution” on the original states as a tensor eigenvector.
  10. 10. 10 1 3 2 P Higher-order Markov chains and tensor eigenvectors. The Li and Ng “stationary distribution” This tensor eigenvector x has been studied algebraically…  Is nonnegative and sums to 1 ⟶ is stochastic [Li-Ng 14]  Almost always exists [Li-Ng 14]  …but might not be unique  Can sometimes be computed [Chu-Wu 14; Gleich-Lim-Yu 15] Nagging question. What is the stochastic process underlying this tensor eigenvector?
  11. 11. 11 Do tensor eigenvectors tell us anything about higher-order Markov chains? 1. Start with a Markov chain 2. Inquire about stationary dist. 3. Discover a matrix eigenvector problem on the transition matrix 1. Start with a higher-order MC 2. Inquire about stationary dist. 3. Discover a tensor eigenvector problem on the transition tensor of a related stochastic proc.
  12. 12. 12 What is the stochastic process whose stationary distribution is the tensor eigenvector Px2 = x?
  13. 13. 1. Start with the transition probabilities of a higher-order Markov chain 2. Upon arriving at state Zt = j, we space out and forget about coming from Zt-1 = k. 3. We still think that we are higher-order so we draw a random state r from our history and “pretend” that Zt-1 = r. 13 The spacey random walk.
  14. 14. 10 12 4 9 7 11 4 Zt-1 Zt Yt Key inisght [Benson-Gleich-Lim 17] Limiting distributions of this process are tensor eigenvectors of P. Higher-order Markov chain transition probabilities. 14 The spacey random walk. 1 3 2 P
  15. 15. 15 Fraction of time spent at state k up to time t The spacey random walk is a type of vertex-reinforced random walk. Vertex-reinforced random walks [Diaconis 88; Pemantle 92, 07; Benaïm 97] Ft is the 𝜎-algebra generated by the history up to time t {Z1, …, Zt} M(wt) is a column stochastic transition matrix that depends on wt Spacey random walks come from a particular map M that depends on P. 1 3 2 P 2 1 M(wt )
  16. 16. 16 Theorem [Benaïm97] heavily paraphrased In a discrete VRRW, the long-term behavior of the occupancy distribution wt follows the long-term behavior of the following dynamical system Key idea. we study convergence of the dynamical system for our particular map M Stationary distributions of vertex-reinforced random walks follow the trajectories of ODEs. 𝛑 maps a column stochastic matrix to its stationary distribution.
  17. 17. 17 Dynamical system for VRRWs Map for spacey random walks Stationary point Tensor eigenvector! (but not all are attractors) From continuous time dynamical systems to tensor eigenvectors.
  18. 18. 18 1. If the higher-order Markov chain is really just a first-order chain, then the SRW is identical to the first-order chain. 2. SRWs are asymptotically first-order Markovian. wt converges to w ⟶ dynamics converge to M(w) 3. Stationary distributions only need O(n) memory unlike higher- order Markov chains. 4. Nearly all 2 x 2 x 2 x … x 2 SRWs converge. 5. SRWs generalize Pólya urns processes. 6. Some convergence guarantees with Forward Euler integration and a new algorithm for computing the eigenvector. Theory of spacey random walks.
  19. 19. 19 Key idea. reduced dynamics to 1-dimensional ODE 1 3 2 P Dynamics of two-state spacey random walks. Unfolding of P. Then we can just write out our dynamics…
  20. 20. 20 stable stable unstable Theorem [Benson-Gleich- Lim 17] The dynamics of almost every 2 x 2 x … x 2 spacey random walk (of any order) converges to a stable equilibrium point. Dynamics of two-state spacey random walks.
  21. 21. 21 1. If the higher-order Markov chain is really just a first-order chain, then the SRW is identical to the first-order chain. 2. SRWs are asymptotically first-order Markovian. wt converges to w ⟶ dynamics converge to M(w) 3. Stationary distributions only need O(n) memory unlike higher- order Markov chains. 4. Nearly all 2 x 2 x 2 x … x 2 SRWs converge. 5. SRWs generalize Pólya urns processes. 6. Some convergence guarantees with Forward Euler integration and a new algorithm for computing the eigenvector. Theory of spacey random walks.
  22. 22. 22 Draw ball at random Put ball back with another of the same color Two-state second-order spacey random walk …converges! Spacey random walks and Pólya urn processes.
  23. 23. 23 …converges! Spacey random walks and more exotic Pólya urn processes. Draw m balls randomly with replacement. Put ball back with color C(b1, b2, …, bm) = purple. Two-state (m+1)th-order spacey random walk b1 b2 bm …
  24. 24. 24 1. If the higher-order Markov chain is really just a first-order chain, then the SRW is identical to the first-order chain. 2. SRWs are asymptotically first-order Markovian. wt converges to w ⟶ dynamics converge to M(w) 3. Stationary distributions only need O(n) memory unlike higher-order Markov chains. 4. Nearly all 2 x 2 x 2 x … x 2 SRWs converge. 5. SRWs generalize Pólya urns processes. 6. New methods to compute the eigenvector with some convergence guarantees. Theory of spacey random walks.
  25. 25. 25 Our stochastic viewpoint gives a new approach. We simply numerically integrate the dynamical system (works for our stochastic tensors). Current tensor eigenvector computation algorithms are algebraic, look like generalizations of matrix power method, shifted iteration, Newton iteration. [Lathauwer-Moore-Vandewalle 00, Regalia-Kofidis 00, Li-Ng 13; Chu-Wu 14; Kolda-Mayo 11, 14] Computing tensor eigenvectors. Higher-order power method Dynamical system Many known convergence issues! Empirical observation integrating the dynamical system with ODE45() in MATLAB/Julia always converges–tested for a wide variety of synthetic and real-world data (even when state-of-the-art general algorithms diverge!)
  26. 26. Theorem [Benson-Gleich-Lim 17] If a < ½, then the dynamical system converges to a unique fixed point, and numerical integration using forward Euler with step size h < (1 – a) / (1 – 2a) converges to this 26 Similar to the PageRank modification to a Markov chain. 1. With probability a, follow the spacey random walk 2. With probability 1 – a, teleport to random node. The spacey random surfer offers additional structure. 1 3 2 P 1 3 2 E = + all ones tensortransition tensorSRS tensor Pa 1 3 2 Pa
  27. 27. 27 1. Does the dynamical system always converge? Some types of VRRWs land in periodic orbits, but we have a special class. 2. Does convergence of the power method imply convergence of the dynamical system or convergence of numerical integration? Interlude. Open computational questions. 3. Is there a stochastic tensor P for which ODE45() integration fails? 4. What is the computational complexity of computing a fixed point of the dynamical system? PPAD complete? FP exists by Brouwer's theorem; similar results in algorithmic game theory [Etessami-Yannakakis 10] 1 3 2 P
  28. 28. 28 1. Modeling transportation systems [Benson-Gleich-Lim 17] The SRW describes taxi cab trajectories. 2. Clustering multi-dimensional nonnegative data [Wu-Benson-Gleich 16] The SRW provides a new spectral clustering methodology. 3. Ranking multi-relational data [Gleich-Lim-Yu 15] The spacey random surfer is the stochastic process underlying the “multilinear PageRank vector”. 4. Population genetics. The spacey random walk traces the lineage of alleles in a random mating model. The stationary distribution is the Hardy–Weinberg equilibrium. Applications of spacey random walks.
  29. 29. 29 1,2,2,1,5,4,4,… 1,2,3,2,2,5,5,… 2,2,3,3,3,3,2,… 5,4,5,5,3,3,1,… Model people by locations.  A passenger with location k is drawn at random.  The taxi picks up the passenger at location j.  The taxi drives the passenger to location i with probability Pi,j,k Approximate location dist. by history ⟶ spacey random walk. Urban Computing Microsoft Asia nyc.gov Spacey random walk model for taxi trajectories.
  30. 30. 30 x(1), x(2), x(3), x(4),… Maximum likelihood estimation problem convex objective linear constraints Spacey random walk model for taxi trajectories.
  31. 31.  One year of 1000 taxi trajectories in NYC.  States are neighborhoods in Manhattan.  Learn tensor P under spacey random walk model from training data of 800 taxis.  Evaluation RMSE on test data of 200 taxis. 31 NYC taxi data supports the SRW hypothesis
  32. 32. 32 1. Modeling transportation systems [Benson-Gleich-Lim 17] The SRW describes taxi cab trajectories. 2. Clustering multi-dimensional nonnegative data [Wu-Benson-Gleich 16] The SRW provides a new spectral clustering methodology. 3. Ranking multi-relational data [Gleich-Lim-Yu 15] The spacey random surfer is the stochastic process underlying the “multilinear PageRank vector”. 4. Population genetics. The spacey random walk traces the lineage of alleles in a random mating model. The stationary distribution is the Hardy–Weinberg equilibrium. Applications of spacey random walks.
  33. 33. 33 Connecting spacey random walks to clustering. Joint work with Tao Wu, Purdue Spacey random walks with stationary distributions are asymptotically Markov chains  occupancy vector wt converges to w ⟶ dynamics converge to M(w) This connects to spectral clustering on graphs.  Eigenvectors of the normalized Laplacian of a graph are eigenvectors of the random walk matrix. 1 3 2 P 2 1 M(wt ) General tensor spectral co-clustering for higher-order data, Wu-Benson-Gleich, NIPS, 2016.
  34. 34. 34 We use the random walk connection to spectral clustering to cluster nonnegative tensor data. [i1, i2, …, in]3 [i1, i2, …, in1 ] x [j1, j2, …, jn2 ] x [k1, k2, …, kn3 ] If the data is a brick, we symmetrize before normalization. [Ragnarsson-Van Loan 2011] Generalization of If the data is a symmetric cube, we can normalize it to get a transition tensor P.
  35. 35. 35 Input. Nonnegative brick of data. 1. Symmetrize the brick (if necessary) 2. Normalize to a stochastic tensor 3. Estimate the stationary distribution of the spacey random walk (or a generalization for sparse data—super-spacey random walk) 4. Form the asymptotic Markov model 5. Bisect indices using eigenvector of the asymptotic Markov model 6. Recurse Output. Partition of indices. The clustering methodology. 1 3 2 T
  36. 36. 36 Ti,j,k = #(flights between airport i and airport j on airline k) Clustering airline-airport-airport networks. UNCLUSTERED no apparent structure CLUSTERED diagonal structure evident
  37. 37. 37 “best” clusters  pronouns & articles (the, we, he, …)  prepositions & link verbs (in, of, as, to, …) fun 3-gram clusters  {cheese, cream, sour, low-fat, frosting, nonfat, fat-free}  {bag, plastic, garbage, grocery, trash, freezer} fun 4-gram cluster  {german, chancellor, angela, merkel, gerhard, schroeder, helmut, kohl} Ti,j,k = #(consecutive co-occurrences of words i, j, k in corpus) Ti,j,k,l = #(consecutive co-occurrences of words i, j, k, l in corpus) Data from Corpus of Contemporary American English (COCA) www.ngrams.info Clustering n-grams in natural language.
  38. 38. Spacey random walks Stochastic processes for understanding eigenvectors of transition probability tensors and analyzing higher-order data  The spacey random walk: a stochastic process for higher-order data. Austin Benson, David Gleich, and Lek-Heng Lim. SIAM Review (Research Spotlights), 2017. https://github.com/arbenson/spacey-random-walks  General tensor spectral co-clustering for higher-order data. Tao Wu, Austin Benson, and David Gleich. In Proceedings of NIPS, 2016. https://github.com/wutao27/GtensorSC http://cs.cornell.edu/~arb @austinbenson arb@cs.cornell.edu Thanks! Austin Benson

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