1. A Convergent Solution to Tensor Subspace Learning

2. Concept Tensor Subspace Learning. Concept Tensor: multi-dimensional (or multi-way) arrays of components

3. Application Tensor Subspace Learning. Application real-world data are affected by multifarious factors for the person identification, we may have facial images of different ► views and poses ► lightening conditions ► expressions the observed data evolve differently along the variation of different factors ► image columns and rows

4. Application Tensor Subspace Learning. Application it is desirable to dig through the intrinsic connections among different affection factors of the data. Tensor provides a concise and effective representation. Illumination pose expression Image columns Image rows Images

5. Tensor Subspace Learning algorithms Traditional Tensor Discriminant algorithms Tensor Subspace Analysis He et.al Two-dimensional Linear Discriminant Analysis Discriminant Analysis with Tensor Representation Ye et.al Yan et.al project the tensor along different dimensions or ways projection matrices for different dimensions are derived iteratively solve an trace ratio optimization problem DO NOT CONVERGE !

6. Tensor Subspace Learning algorithms Graph Embedding – a general framework An undirected intrinsic graph G={X,W} is constructed to represent the pairwise similarities over sample data. A penalty graph or a scale normalization item is constructed to impose extra constraints on the transform. intrinsic graph penalty graph

7. Discriminant Analysis Objective Solve the projection matrices iteratively: leave one projection matrix as variable while keeping others as constant. No closed form solution Mode-k unfolding of the tensor

8. Objective Deduction Discriminant Analysis Objective Trace Ratio: General Formulation for the objectives of the Discriminant Analysis based Algorithms. DATER: TSA: Within Class Scatter of the unfolded data Between Class Scatter of the unfolded data Diagonal Matrix with weights Constructed from Image Manifold

9. Disagreement between the Objective and the Optimization Process Why do previous algorithms not converge? GEVD The conversion from Trace Ratio to Ratio Trace induces an inconsistency among the objectives of different dimensions!

10. from Trace Ratio to Trace Difference What will we do? from Trace Ratio to Trace Difference Objective: Define Then Trace Ratio Trace Difference Find So that

11. from Trace Ratio to Trace Difference What will we do? from Trace Ratio to Trace Difference Constraint Let We have Thus The Objective rises monotonously! Projection matrices of different dimensions share the same objective Whereare the leading eigen vectors of.

12. Main Algorithm Process Main Algorithm 1: Initialization. Initialize as arbitrary column orthogonal matrices. 2: Iterative optimization. For t=1, 2,..., Tmax, Do For k=1, 2,..., n, Do 1. Set. 2. Compute and. 3. Conduct Eigenvalue Decomposition: 4. Reshape the projection directions 5. 3: Output the projection matrices

13. Strict Monotony and Convergency If Then Thus orthogonal matrix Meanwhile, both and have been calculated from the leading eigenvectors of the projected samples. span the same space So The algorithm is strictly monotonous! Theorem[Meyer,1976]: Assume that the algorithm Ω is strictly monotonic with respect to J and it generates a sequence which lies in a compact set. If is normed, then. The algorithm is guaranteed to converge!

14. Hightlights of the Trace Ratio based algorithm Highlights of our algorithm The objective value is guaranteed to monotonously increase; and the multiple projection matrices are proved to converge. Only eigenvalue decomposition method is applied for iterative optimization, which makes the algorithm extremely efficient. Enhanced potential classification capability of the derived low- dimensional representation from the subspace learning algorithms. The algorithm does not suffer from the singularity problem that is often encountered by the traditional generalized eigenvalue decomposition method used to solve the ratio trace optimization problem.

15. Monotony of the Objective Experimental Results The traditional ratio trace based procedure does not converge, while our new solution procedure guarantees the monotonous increase of the objective function value and commonly our new procedure will converge after about 4-10 iterations. Moreover, the final converged value of the objective function from our new procedure is much larger than the value of the objective function for any iteration of the ratio trace based procedure.

16. Convergency of the Projection Matrices Experimental Results The projection matrices converge after 4-10 iterations for our new solution procedure; while for the traditional procedure, heavy oscillations exist and the solution does not converge.

17. Face Recognition Results Experimental Results 1. TMFA TR mostly outperforms all the other methods concerned in this work, with only one exception for the case G5P5 on the CMU PIE database. 2. For vector-based algorithms, the trace ratio based formulation is consistently superior to the ratio trace based one for subspace learning. 3. Tensor representation has the potential to improve the classification performance for both trace ratio and ratio trace formulations of subspace learning.

18. Summary A novel iterative procedure was proposed to directly optimize the objective function of general subspace learning based on tensor representation. The convergence of the projection matrices and the monotony property of the objective function value were proven. The first work to give a convergent solution for the general tensor-based subspace learning.

19. Thank You!