A Convergent Solution to Tensor Subspace Learning Huan Wang1 Shuicheng Yan 2 Thomas Huang 2 Xiaoou Tang1,3 Chinese University University of Illinois of Hong Kong1 at Urbana Champaign 2 Microsoft Research Asia 3 Tensor Subspace Learning . Concept Tensor: multi-dimensional (or multi-way) arrays of components Concept 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 image columns and rows the observed data evolve differently along the variation of different factors
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. Images Image columns expression pose Image rows Illumination Application Traditional Tensor Discriminant algorithms Two-dimensional Linear Discriminant Analysis Ye et.al Discriminant Analysis with Tensor Representation Yan et.al Tensor Subspace Analysis He et.al
project the tensor along different dimensions or ways solve an trace ratio optimization problem projection matrices for different dimensions are derived iteratively DO NOT CONVERGE ! 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. penalty graph intrinsic graph Tensor Subspace Learning algorithms Discriminant Analysis Objective arg m ax U k |nk1 k n 2 p
( X X ) U | W ij i j k k 1 ij k n 2 ( X X ) U | i j i j k k 1 Wij No closed form solution Solve the projection matrices iteratively: leave one projection matrix as variable while keeping others as constant.
arg m ax U Yi ~ k k kT k kT kT k kT k 2 p U Y U Y
W i j i j ij k 2 U Y U Y i j i j Wij ~ Mode-k unfolding of the tensor Yi Yi X i 1 U 1... k 1 U k 1 k 1 U k 1... n U n Discriminant Analysis Objective kT k k kT
k Tr (U S U ) k k j k Tr (U S U ) arg m ax U p k k k k T j S i j Wij (Yi Y )(Yi Y ) p Sk
p k ij (Yi i j W k j k k T j Y )(Yi Y ) Trace Ratio: General Formulation for the objectives of the Discriminant Analysis based Algorithms. Within Class Scatter of the DATER: S unfolded data W Constructed from Image Manifold TSA: k S kp Sp Objective Deduction
Between Class Scatter of the unfolded data Diagonal Matrix with weights Why do previous algorithms not converge? k1T arg m ax U k1 k1 p Tr (U S k1U ) k1T k1 k1T Tr (U S U ) k1 1 k1T arg m ax Tr ((U S U ) U S kp1U k1 )
k1 k1 U k1 GEVD arg m ax U k2 k2T p k2T k2 k2 Tr (U S k2 U ) arg m ax Tr ((U k2 k2
k2 1 S U ) U U k2 Tr (U S U ) Tr ( A) Tr ( B ) k2T 1 Tr ( B A) k2T S kp2U k2 ) The conversion from Trace Ratio to Ratio Trace induces an inconsistency among the objectives of different dimensions! Disagreement between the Objective and the Optimization Process
What will we do? from Trace Ratio to Trace Difference kT Objective: arg mk ax U Tr (U S kpU k ) kT Tr (U S kU k ) g (U ) Tr (U T ( S kp S k )U ) Trace Ratio Define kT t kT t p k k k t k t
Tr (U S U ) Tr (U S U ) Find g (U ) g (U tk ) So that Then kT t Trace Difference Tr (U ( S kp S k )U tk ) 0 Tr (U ( S kp S k )U ) g (U tk ) 0 Tr (U T S kpU ) T k Tr (U S U ) g (U tk ) from Trace Ratio to Trace Difference What will we do? from Trace Ratio to Trace Difference
g (U ) Tr (U T ( S kp S k )U ) Constraint Thus kT p t 1 k kT k t 1 Tr (U S U tk1 ) T U U I k t 1 Tr (U S U ) U tk1 [u1 , u2 ,..., um' ] Let k
Where u1 , u2 ,..., umk' are the leading The Objective rises monotonously! p k ( S S ) . k eigen vectors of We have g (U tk1 ) g (U tk ) 0 Projection matrices of different dimensions share the same objective from Trace Ratio to Trace Difference 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. 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. Enhanced potential classification capability of the derived lowdimensional representation from the subspace learning algorithms. Hightlights of the Trace Ratio based algorithm 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. Monotony of the Objective 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. Convergency of the Projection Matrices 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. Face Recognition Results 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. Summary Thank You!
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