Heat Kernels and Diffusion Processes

Transcription

1 Heat Kernels and Diffusion Processes Definition: University of Alicante (Spain) Matrix Computing (subject 3168 Degree in Maths) 30 hours (theory)) + 15 hours (practical assignment)

2 Contents 1. Solving the Discrete Heat Equation 1. Heat Equation and the Laplacian 2. The Heat Kernel and its Interpretation 3. Behavior depending on time 4. The Path-length Distribution 2. PageRank and the Diffusion Process 1. PageRank 2. Page-Rank and Diffusion Rank 3. Flow Complexity 1. From Polytopal Complexity to Flowing Complexity 2. The Flow Conjecture and Complexity 3. Examples from Bioinformatics and 3D object recognition

3 Solving the Discrete Heat Equation The discrete Heat Equation: Being L the Laplacian of a graph G with n vertices, and K a nxn matrix parameterized by β (time or inverse temperature). The solution: Is the matrix exponentiation of the Laplacian: Using Taylor expansion: Spectral decompositiontaking the eigenvectors of L and the eigenvalues of L:

4 The Heat Kernel and its Interpretation Example #2: K(i,j) is the probability that a lazy random walk starting at i reaches j. Lazy random walks have a probability depending on β off staying at i. Simulation of a heat diffusion state starting by heat=1 at each vertex when β=0.

5 The Heat Kernel and its Interpretation More spectral definitions: K defines a doubly-stochastic matrix (sum of rows and cols = 1). As K is a kernel/gram matrix, K(i,j) represents a dot product (dissimilarity) in a given space. Behavior depending on time: Exercise #6 (proof) Path-length distribution: number of paths of length k:

6 PageRank Definition: [Page et al.,99][eiron et al,04] Idea: Quantify the average importance of a node (e.g. webpage) after a sequence of (probabilistic) transitions. Given a set V of vertices. let x a n= V indicator vector so that x i measures the importance of vertex i. Being A the adjacency matrix and α the probability of moving to another node (surf to another webpage) being 1 the vectors of all ones and g a randon vector, typically g=(1/n) 1 (no preferred starting node) and α=0.85.

7 PageRank Properties Over-democratic Internet: [Yang et al, 07] All nodes (pages) are born equally. This favors to manipulate the rank of a node by creating many links to it. Input-independent Given the transition matrix: independently of the (always non-zero) input, the iterative process will converge to the same stable distribution corresponding to the maximum eigenvalue 1 of P. This property makes impossible to set preferences (high initial values to trusted pages and low, even negative, for spam). Alternative? DiffussionRank!

8 DiffusionRank Definition: [Yang et al, 07] Assuming that the heat difference at a node between t and t + t, say f(t) and f(t + t) at a given node is the sum of the heat it receives from its neighboring nodes: we have that for t close to zero: For γ=0 no heat is diffused. Anti-manipulation ranking but network structure ignored For γ= DiffusionRank converges to PageRank. For γ=1 DiffusionRank works well in practice.

9 Polytopal Complexity: BvN Theorem Polytopal Complexity: [Escolano, Hancock and Lozano, 08] Following the Birkhoff-von Newmann theorem, any doublystochastic matrix (e.g. a diffusion kernel matrix) can be decomposed into a convex combination of permutation matrices:

10 Polytopal Complexity: Examples Star (40 nodes) Line (40 nodes)

11 Polytopal Complexity: Bvn Complexity Global Polytopal Complexity: It is given by the following multidimensional descriptor: When considered a function, it satisfies: Moreover and, are computed from:

12 Polytopal Complexity: Bvn Complexity Global Polytopal Complexity: The graph complexity trace is a signature of the interaction between the heat diffusion process and the structure/topology of the graph as the inverse temperature increases and thus the range of vertex interaction decreases. The signature can be also interpreted as a trajectory (or geodesic in the Polytope) between the vertex of the polytope encoding the identity matrix and the barycenter of the polytope. A typical singature is heavy tailed and monotonically increasing from 1 at until it reaches then a topological phase transition occurs and the signature descends towards zero at The interval encodes inter class variability whereas the other one encodes intra class variability.

13 Polytopal Complexity: PPIs

14 Polytopal Complexity: Diffusion process Line

15 The Flow Conjecture and Complexity Heat Flow (Definition): The flow of the heat kernel (DSM) at a given beta is defined by: The Flow Conjecture and Complexity The inverse temperature yields the maximum entropy of the pdf coming from the BvN decomposition and it is also a PTP, iff it also maximizes the heat flow and it is also a PTP. Flow Complexity

16 Polytopal Complexity: SHREC database

17 Polytopal to Flow: Definition Example of BvN decomposition: Polytopal Complexity: The Maximum Entropy BvN decomposition is unique but the problem is #P. The Constructive BvN decompostion is O(N 3 x γ).

18 Polytopal Complexity: 3D objects Extended Reeb Graphs (ERGs): [Biasotti, 04,05] Critical points -> Critical areas (maxima, minima, saddle) Track the evolution of level sets and form graphs. Different functions -> Different graphs (see conclusions).

19 Polytopal Complexity: 3D objects Integral geodesic distance: Computed for each v=vertex in the mesh The bi are an uniform sampling of all the vertices. The derived graph is invariant to translation at rotation (at least)

20 Polytopal Complexity: Similarity Matrix