![Page 1: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/1.jpg)
Dimensionality Reduction:From Data Representation to General Framework
Dong XU
School of Computer EngineeringNanyang Technological University, Singapore
![Page 2: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/2.jpg)
What is Dimensionality Reduction?
PCA LDA
Examples: 2D space to 1D space
![Page 3: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/3.jpg)
What is Dimensionality Reduction?
Example: 3D space to 2D space
ISOMAP: Geodesic Distance PreservingJ. Tenenbaum et al., 2000
![Page 4: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/4.jpg)
Why Conduct Dimensionality Reduction?
Pose Variation
Expression Variation
LPP, 2003He et al.
Uncover intrinsic structure
Visualization Feature Extraction Computation Efficiency Broad Applications
Face RecognitionHuman Gait Recognition
CBIR
![Page 5: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/5.jpg)
OutlineDimensionality Reduction for Tensor-
based ObjectsGraph Embedding: A General Framework
for Dimensionality Reduction
![Page 6: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/6.jpg)
OutlineDimensionality Reduction for Tensor-
based ObjectsGraph Embedding: A General Framework
for Dimensionality Reduction
![Page 7: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/7.jpg)
What is Tensor?
Tensors are arrays of numbers which transform in certain ways under coordinate transformations.
1m
2m3m
1m
Vector Matrix 3rd-order Tensor
2m
1m
1 2 3m m m X R1 2m mX R1mxR
![Page 8: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/8.jpg)
2
1
m
ij ik kjk
Y X U
100
100
100
100
100
100
100
10
100
10
100
10
100
10
100
10
100
10
=
=
=
Definition of Mode-k Product
(100)
2 )'(10m2 (100)m
2m(100)
1m(100)
1m
2 )'(10m
1m
2 (100)m
1(100)m
3 (40)m
2 )'(10m
2 (100)m
k U Y XNotation:
Product for two Matrices
Original Matrix
New Matrix
=1(100)m
3 (40)m
2 )'(10m
Y XUProjection
Matrix
Original Tensor
New Tensor
Projection Matrix
Projection:high-dimensional space
-> low-dimensional space
Reconstruction:low-dimensional space
-> high-dimensional space
![Page 9: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/9.jpg)
Data Representation in Dimensionality Reduction Vector Matrix 3rd-order Tensor
High Dimension
Low Dimension
Examples
PCA, LDA Rank-1 Decomposition, 2001
A. Shashuaand A. Levin
Low rank approximation of matrix
J. Ye
Tensorface, 2002M. Vasilescu andD. Terzopoulos
Our WorkXu et al., 2005Yan et al., 2005
Filtered Image Video SequenceGray-level Image
![Page 10: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/10.jpg)
What is Gabor Features?Gabor features can improve recognition performance in comparison to grayscale features. C Liu and H Wechsler, T-IP, 2002
Gabor Wavelet KernelsEight Orientations
Five Scales
Input: Grayscale
ImageOutput:
40 Gabor-filteredImages
…
![Page 11: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/11.jpg)
Why Represent Objects as Tensorsinstead of Vectors?
Natural RepresentationGray-level Images (2D structure)Videos (3D structure)Gabor-filtered Images (3D structure)
Enhance Learnability in Real ApplicationCurse of Dimensionality (Gabor-filtered image: 100*100*40 -> Vector: 400,000)Small sample size problem
Reduce Computation Cost
![Page 12: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/12.jpg)
Concurrent Subspace Analysis as an Example(Criterion: Optimal Reconstruction)
1m
10040
100
1U
2U
The reconstructed sample
Input sample
Projection Matrices?
Sample in Low-dimensional space
3U
1m
10
10
10
Dimensionality Reduction
1m
10040
100
Reconstruction
D. Xu, S. Yan, H. Zhang and et al., CVPR, 20053
1
* 31
21 1 1 3 3 3
|
( | )
arg min || ... ||k k
k k
i iiU
U
U U U U
X X
Objective Function:
![Page 13: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/13.jpg)
Connection to Previous Work –Tensorface(M. Vasilescu and D. Terzopoulos, 2002)
. . .From an algorithmic view or mathematics view, CSA and Tensorface are both variants of Rank-(R1,R2,…,Rn) decomposition.
Tensorface CSA
Motivation Characterize external factors Characterize internal factorsInput: Gray-level Image Vector Matrix
Input: Gabor-filtered Image (Video Sequence )
Not address 3rd-order tensor
When equal to PCA The number of images per person are only one or are a
prime numberNever
Number of Images per Person for Training
Lots of images per person One image per person
![Page 14: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/14.jpg)
Experiments: Database Description
Number of Persons (Images per person)
Image Size (Pixels)
Example Images
Simulated Video Sequence
60 (1) 646413
ORL database 40 (10) 5646
CMU PIE-1 sub-database
60 (10) 6464
CMU PIE-2 sub-databases
60 (10) 6464
![Page 15: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/15.jpg)
Experiments: Object Reconstruction (1)Input: Gabor-filtered imagesORL databaseCMU PIE-1 database
Objective Evaluation Criterion:Root Mean Squared Error (RMSE) and Compression Ratio (CR)
ORL database CMU PIE-1 database
![Page 16: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/16.jpg)
Experiments: Object Reconstruction (2)
Original Images
Reconstructed Images from PCA
Reconstructed Images from CSA
Input: Simulated video sequence
![Page 17: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/17.jpg)
Experiments: Face Recognition
Input: Gray-level images and Gabor-filtered imagesORL databaseCMU PIE database
Algorithm CMU PIE-1 CMU PIE-2 ORL
PCA (Gray-level feature) 70.1% 28.3% 76.9%
PCA (Gabor feature) 80.1% 42.0% 86.6%CSA (Ours) 90.5% 59.4 % 94.4%
![Page 18: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/18.jpg)
Summary
• This is the first work to address dimensionality reduction with a tensor representation of arbitrary order.
• Opens a new research direction.
![Page 19: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/19.jpg)
Bilinear and Tensor Subspace Learning (New Research Direction)
• Concurrent Subspace Analysis (CSA), CVPR 2005 and T-CSVT 2008• Discriminant Analysis with Tensor Representation (DATER): CVPR 2005
and T-IP 2007• Rank-one Projections with Adaptive Margins (RPAM): CVPR 2006 and T-
SMC-B 2007• Enhancing Tensor Subspace Learning by Element Rearrangement: CVPR
2007 and T-PAMI 2009• Discriminant Locally Linear Embedding with High Order Tensor Data
(DLLE/T): T-SMC-B 2008• Convergent 2D Subspace Learning with Null Space Analysis (NS2DLDA): T-
CSVT 2008• Semi-supervised Bilinear Subspace Learning: T-IP 2009• Applications in Human Gait Recognition
– CSA+DATER: T-CSVT 2006– Tensor Marginal Fisher Analysis (TMFA): T-IP 2007
Other researchers also published several papers along this direction!!!
![Page 20: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/20.jpg)
Human Gait Recognition: Basic Modules
(a) (d): The extracted silhouettes from one probe and gallery video;
(b) (c): The gray-level Gait Energy Images (GEI).
![Page 21: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/21.jpg)
Human Gait Recognition with Matrix Representation
D. Xu, S. Yan, H. Zhang and et al., T-CSVT, 2006
![Page 22: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/22.jpg)
USF HumanIDExperiment (Probe) # of Probe
SetsDifference between Gallery and
Probe SetA (G, A, L, NB, M/N) 122 ViewB (G, B, R, NB, M/N) 54 ShoeC (G, B, L, NB, M/N) 54 View and ShoeD (C, A, R, NB, M/N) 121 SurfaceE (C, B, R, NB, M/N) 60 Surface and ShoeF (C, A, L, NB, M/N) 121 Surface and ViewG (C, B, L, NB, M/N) 60 Surface, Shoe, and ViewH (G, A, R, BF, M/N) 120 BriefcaseI (G, B, R, BF, M/N) 60 Briefcase and ShoeJ (G, A, L, BF, M/N) 120 Briefcase and ViewK (G, A/B, R, NB, N) 33 Time, Shoe, and ClothingL (C, A/B, R, NB, N) 33 Time, Shoe, Clothing, and Surface
1. Shoe types: A or B; 2. Carrying: with or without a briefcase; 3. Time: May or November; 4. Surface: grass or concrete; 5. Viewpoint: left or right
![Page 23: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/23.jpg)
Human Gait Recognition: Our Contributions
*The DNGR method additionally uses the manually annotated silhouettes, which are not publicly available.
Methods Average Rank‐1 Results (%)
Our Recent Work (Ours, TIP 2012) 70.07
DNGR (Sarkar’s group, TPAMI 2006) 62.81Image‐to‐Class distance (Ours, TCSVT 2010) 61.19
GTDA (Maybank’s group, TPAMI 2007) 60.58
Bilinear Subspace Learning method 2: MMFA (Ours, TIP 2007)
59.9%
Bilinear Subspace Learning method 1: CSA + DATER (Ours, TCSVT 2006)
58.5%
PCA+LDA (Bhanu’s group, TPAMI 2006) 57.70%
Top ranked results on the benchmark USF HumanID dataset
![Page 24: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/24.jpg)
How to Utilize More Correlations?
PixelRearrangement
Sets of highlycorrelated pixels
Columns of highlycorrelated pixels
Pixel Rearrangement
Potential Assumption in Previous Tensor-based Subspace Learning:
Intra-tensor correlations: Correlations among the features within certain tensor dimensions, such as rows, columns and Gabor features…
D. Xu, S. Yan et al., T-PAMI 2009
![Page 25: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/25.jpg)
Problem Definition• The task of enhancing correlation/redundancy among
2nd–order tensor is to search for a pixel rearrangement operator R, such that
* 2
, 1arg min{ min || || }
NR T R Ti iR U V i
R X UU X VV
1. is the rearranged matrix from sample2. The column numbers of U and V are predefined
iXRiX
After pixel rearrangement, we can use the rearranged tensors as input for concurrent subspace analysis
![Page 26: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/26.jpg)
Rearrangement Results
26
![Page 27: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/27.jpg)
Reconstruction Visualization
27
![Page 28: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/28.jpg)
Reconstruction Visualization
28
![Page 29: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/29.jpg)
OutlineDimensionality Reduction for Tensor-
based ObjectsGraph Embedding: A General Framework
for Dimensionality Reduction
![Page 30: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/30.jpg)
Representative Previous Work
ISOMAP: Geodesic Distance Preserving
J. Tenenbaum et al., 2000
LLE: Local Neighborhood Relationship Preserving
S. Roweis & L. Saul, 2000
LE/LPP: Local Similarity Preserving, M. Belkin, P. Niyogi et al., 2001, 2003
PCA LDA
![Page 31: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/31.jpg)
Dimensionality Reduction Algorithms
• Any common perspective to understand and explain these dimensionality reduction algorithms? Or any unified formulationthat is shared by them?
• Any general tool to guide developing new algorithms for dimensionality reduction?
Statistics-based Geometry-based
PCA/KPCA LDA/KDA … ISOMAP LLE LE/LPP …
Matrix Tensor
Hundreds
![Page 32: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/32.jpg)
Our Answers
Direct Graph Embedding
1minT
T
y B yy Ly
Original PCA & LDA,ISOMAP, LLE,
Laplacian Eigenmap
Linearization
PCA, LDA, LPP
wXy T
Kernelization
KPCA, KDA
)( iii xw
Tensorization
CSA, DATER
nnii wwwy 2
21
1X
Type
Formulation
Example
S. Yan, D. Xu et al., CVPR 2005, T-PAMI 2007
![Page 33: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/33.jpg)
Direct Graph Embedding
1 2[ , ,..., ]NX x x x 1 2[ , ,..., ]TNy y y y
Data in high-dimensional space and low-dimensional space (assumed as 1D space here):
L, B: Laplacian matrix from S, SP;
[ , ]i ijG SxIntrinsic Graph:
Penalty Graph
S, SP: Similarity matrix (graph edge)
[ , ]P PijiG Sx
, ii ijj iL D S D S i
Similarity in high dimensional space
![Page 34: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/34.jpg)
Direct Graph Embedding -- Continued
1 2[ , ,..., ]NX x x x 1 2[ , ,..., ]TNy y y y
* 2
1 1 1 1
arg min || || arg mini j ijy y or y y ori jy By y B y
y y y S y L y
* 2
1 1
arg min || ||i j ijy y or i jy By
y y y S
Data in high-dimensional space and low-dimensional space (assumed as 1D space here):
L, B: Laplacian matrix from S, SP;[ , ]i ijG Sx
Criterion to Preserve Graph Similarity:
Intrinsic Graph:
Penalty Graph
S, SP: Similarity matrix (graph edge)
Special case B is Identity matrix (Scale normalization)
[ , ]P PijiG Sx
Problem: It cannot handle new test data.
, ii ijj iL D S D S i
Similarity in high
dimensional space
![Page 35: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/35.jpg)
Linearization
y X w
*
1 1
arg minw w or
w XBX w
w w XL X w
Linear mapping function
Objective function in Linearization
Intrinsic Graph
Penalty Graph
Problem: linear mapping function is not enough to preserve the real nonlinear structure?
![Page 36: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/36.jpg)
Kernelization
: ix Ff
the original input space to anotherhigher dimensional Hilbert space.
Nonlinear mapping:
( , ) ( ) ( )k x y x y ( , )ij i jK k x x
( )i iiw x
*
1 1
arg minK orKBK
a KLK
Kernel matrix:
Constraint:
Objective function in Kernelization
Intrinsic Graph
Penalty Graph
)( ix
![Page 37: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/37.jpg)
Tensorization
Low dimensional representation is obtained as:
Objective function in Tensorization
1 21 2 ... n
i i ny w w w X
1 1 2 1 2
1
* 21 2 1 2
( ,..., ) 1( ,..., ) arg min || ... ... ||n n n
ni n j n ij
f w w i jw w w w w w w w S
X X
1 1 2
1 1 2 1 2
21 21
21 2 1 2
( ,..., ) || ... ||
( ,..., ) || ... ... ||
n n
n n n
Ni n iii
Pi n j n ij
i j
f w w w w w B or
f w w w w w w w w S
X
X Xwhere
Intrinsic Graph
Penalty Graph
![Page 38: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/38.jpg)
Common Formulation
Tensorization1 1 2 1 2
1
* 21 2 1 2
( ,..., ) 1( ,..., ) arg min || ... ... ||n n n
ni n j n ij
f w w i j
w w w w w w w w S
X X
1 1 2
1 1 2 1 2
21 21
21 2 1 2
( ,..., ) || ... ||
( ,..., ) || ... ... ||
n n
n n n
Ni n iii
Pi n j n ij
i j
f w w w w w B or
f w w w w w w w w S
X
X Xwhere
Linearization
Kernelization
Direct Graph Embedding
L, B: Laplacian matrix from S, SP;
S, SP: Similarity matrixIntrinsic graph
Penalty graph
*
1 1
arg minw w or
w XBX w
w w XL X w
*
1 1
arg minK orKBK
a KLK
*
1 1
arg miny y ory By
y y L y
![Page 39: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/39.jpg)
A General Framework for Dimensionality Reduction
Algorithm S & B Definition Embedding Type
PCA/KPCA/CSA L/K/T
LDA/KDA/DATER L/K/T
ISOMAP D
LLE D
LE/LPPif ; B=D
D/L
1 , ;NijS i j B I
1, ,
i j iij l l l NS n B I ee
( ) , ;ij G ijS D i j B I
;S M M M M B I
2exp{ || || / }ij i jS x x t
|| ||i jx x
D: Direct Graph Embedding L: LinearizationK: KernelizationT: Tensorization
![Page 40: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/40.jpg)
New Dimensionality Reduction Algorithm: Marginal Fisher Analysis
ijS
Important Information for face recognition:
1) Label information2) Local manifold structure (neighborhood or margin)
1: if xi is among the k1-nearest neighbors of xj in the same class;0 : otherwise
1: if the pair (i,j) is among the k2 shortest pairs among the data set;0: otherwise
PijS
![Page 41: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/41.jpg)
Marginal Fisher Analysis: Advantage
No Gaussian distribution assumption
![Page 42: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/42.jpg)
Experiments: Face Recognition
PIE-1 G3/P7 G4/P6PCA+LDA (Linearization) 65.8% 80.2%
PCA+MFA (Ours) 71.0% 84.9%KDA (Kernelization) 70.0% 81.0%
KMFA (Ours) 72.3% 85.2%DATER-2 (Tensorization) 80.0% 82.3%
TMFA-2 (Ours) 82.1% 85.2%
ORL G3/P7 G4/P6PCA+LDA (Linearization) 87.9% 88.3%
PCA+MFA (Ours) 89.3% 91.3%KDA (Kernelization) 87.5% 91.7%
KMFA (Ours) 88.6% 93.8%DATER-2 (Tensorization) 89.3% 92.0%
TMFA-2 (Ours) 95.0% 96.3%
![Page 43: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/43.jpg)
Summary
• Optimization framework that unifies previous dimensionality reduction algorithms as special cases.
• A new dimensionality reduction algorithm: Marginal Fisher Analysis.
![Page 44: Dong XU School of Computer Engineering Nanyang ...icapr15/DongXu_Keynote.pdf · School of Computer Engineering Nanyang Technological University, Singapore. ... (Maybank’s group,](https://reader033.vdocument.in/reader033/viewer/2022051803/5b0b98277f8b9a99488e2748/html5/thumbnails/44.jpg)