Graph Embedding and Extensions: A General Framework for Dimensionality Reduction Keywords: Dimensionality reduction, manifold learning, subspace learning, graph embedding framework. 1.Introduction Techniques for dimensionality reduction Linear: PCA/LDA/LPP... Nonlinear: ISOMAP/Laplacian Eigenmap/LLE... Linear Nonlinear: kernel trick
Graph embedding framework A unified view for understanding and explaining many po pular algorithms such as the ones mentioned above. A platform for developing new dimension reduction algori thms. 2.Graph embedding 2.1Graph embedding Let X [ x1 ,...x N ], xi R m
need to find m is often very large so we F : x y, y R m ' , m' m N N G
X , W , W R Intrinsic graph: --similarity matrix
Penalty graph: G p X , W p , W p R N N --the similarity to be suppressed in the dimension-reduced feature space Y Our graph-preserving criterion is: 2 * y arg min yi y j Wij arg min y T Ly y T By d
y T By d i j L D W , Dii Wij i j L is called Laplacian matrix B typically is diagonal for scale normalization or Lmatrix of the penalty graph Linearization:
y X T w 2 T w* arg min yi y j Wij arg min w XLX w w ' XBXwd i i or w ' w d w ' XBXwd or w ' w d
Kernelization: : x T T 2 T * arg min K i K j Wij arg min KLK ' KBK d i i
or ' K d ' KBK d or ' K d k ( xi , x j ) ( xi ) ( xi ) Both can be obtained by solving: Lv Bv L L, XLX T , KLK ; B I , B, K , XBX T Tensorization:
( w1...wn )* arg min X w ...w f ( w1 ...wn ) d i j i 1 n
2 X j w1... wn Wij 2.2General Framework for Dimensionality Reductio n The adjacency graphs for PCA and LDA. (a) Constraint and intrinsic graph in PCA. (b) Penalty and intrinsic graphs in LDA. 2.3 Related Works and Discussions 2.3.1 Kernel Interpretation and Out-of-Sample Extensi
on Ham et al. [13] proposed a kernel interpretation of KPC A,ISOMAP, LLE, and Laplacian Eigenmap Bengio et al. [4] presented a method for computing the l ow dimensional representation of out-of-sample data. Comparison: Kernel Interpretation Graph embeding normalized similarity matrix unsupervised learning laplacian matrix
both supervised&unsupervised 2.3.2 Brands Work [5] Brands Work can be viewed as a special case of the graph embedding framework y * arg max y T Wy y T Dy 1 y * arg min y T ( D W ) y
y T Dy 1 2.3.3 Laplacian Eigenmap [3] and LPP [10] Single graph B=D Nonnegative similarity matrix Although [10] attempts to use LPP to explain PC A and LDA, this explanation is incomplete. The constraint matrix B is fixed to D in LPP, while the constraint matrix of LDA is comes from a penalty graph t hat connects all samples with equal weights;hence, LPP cannot explain LPP. Also,a minimization algorithm, does not explain why PCA maximizes the objective function.
3 MARGINAL FISHER ANALYSIS 3.1 Marginal Fisher Analysis Limitation of LDA:data distribution assumption limited available projection directions MFA overcomed the limitation by characterizing intraclass c ompactness and interclass separability. intrinsic graph: each sample is connected to its k1 nearest neighbors of the same class (intraclass compactness) penalty graph: each sample is connected to its k2 nearest neighbors of other classes
(interclass separability) Procedure of MFA PCA projection Constructing the intraclass compactness and int erclass separability graphs. Marginal Fisher Criterion Output the final linear projection direction Advantages of MFA The available projection directions are
much greater than that of LDA There is no assumption on the data distribution of each class Without prior information on data distributions KMFA Projection direction: The distance between sample xi and xj is For a new data point x, its projection to the derived
optimal direction is obtained as TMFA: 4.Experiments 4.1face recognition 4.1.1 MFA>Fisherface(LDA+PCA)>PCA PCA+MFA>PCA+LDA>PCA 4.1.2 Kernel trick KDA>LDA,KMFA>MFA
KMFA>PCA,Fisherface,LPP Trainingset Adequate: LPP > Fisherface ,PCA Inadequate: Fisherface > LPP>PCA anyway, MFA>=LPP Performance can be substantially improved by e xploring a certain range of PCA dimensions first. PCA+MFA>MFA,Bayesian face >PCA,Fisherface,LPP Tensor representation brings encouraging impro vements compared with vector-based algorithms
it is critical to collect sufficient samples for all su bjects! 4.2 A Non-Gaussian Case 5.CONCLUSION AND FUTURE WORK All possible extensions of the algorithms m entioned in this paper Combination of the kernel trick and tensori zation The selection of parameters k1 and k2 How to utilize higher order statistics of the
data set in the graph embedding framewor k?
