Olivier Duchenne ， Armand Joulin ， Jean Ponce Willow Lab ， ICCV2011

 Many applications: 1. Object recognition 2. Text categorization 3. time-series prediction 4. Gene expression profile analysis......

 Given a set of data (x 1, y 1 ), (x 2, y 2 ),..., (x n, y n ), the Kernel Method maps them into a potentially much higher dimensional feature space F.

 For a given learning problem one now considers the same algorithm in instead of R N, one works with the sample The kernel method seeks a pattern among the data in the feature sapce.

Idea: The nonlinear problem in a lower space can be solved by a linear method in a higher space. Example:

 【 Kernel function 】 A kernel function is a function k that for all x, z ∈ X satisfies where is a mapping from X to an (inner product) feature space F 

 The computation of a scalar product between two feature space vectors, can be readily reformulated in terms of a kernel function k

Is necessary? Not necessary What kind of k can be used? symmetric positive semi-definite ( kernel matrix ) Given a feature mapping, caan we compute the inner product in feature space? Yes Given a kernel function k, whether a feature mapping is existence? Yes [Mercer’s theorem]

 Linear Kernel  Polynomial Kernel  RBF (Gaussian) Kernel  Inverse multiquadric Kernel

 Kernel matrix  Consider the problem of finding a real-valued linear function that best intopolates a given training set S = {(x 1, y 1 ), (x 2, y 2 ),..., (x l, y l )} (least square)

 Dual form where K is the kernel matrix.

CAT DINOSAUR PAND A

 Feature correspondences can be used to construct an image comparison kernel that is appropriate for SVM-based classification, and often outperforms BOFs.  Image representations that enforce some degree of spatial consistency usually perform better in image classification tasks than pure bags of features that discard all spatial information.

 We need to design a good image similarity measure: ≈ ?

Graph-matching Method in this paper Sparse Features NN Classifier Slow Use pair-wise Information Lower performance As Dense SVM Classifier Fast enough Use pair-wise Information State-of-the-art performance

 An image I = a graph G = Nodes + Edges A node n=d n (x n,y n ) represent a region of I,  Each region is represented by a image Feature vector F n,e.g. SIFT....

Matching two iamges is realized by maximizing the energy function: