Motivation Problem Formulation ‘ 1 -Minimization Classification Experiments Future Directions Random Feature Selection for Robust Face Recognition Allen Y. Yang <[email protected]> Department of EECS, UC Berkeley with Shankar Sastry, Yi Ma, & John Wright HSN MIT Review, Sep 24, 2007 Allen Y. Yang < [email protected]> Random Feature Selection for Robust Face Recognition
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Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
Random Feature Selection for Robust Face Recognition
Allen Y. Yang <[email protected]>
Department of EECS, UC Berkeleywith Shankar Sastry, Yi Ma, & John Wright
HSN MIT Review, Sep 24, 2007
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
Next Generation Sensor Networks
1 Transition from dedicated sensor networks to general-purpose sensor networks.
2 Similar revolutions in IT industry:Computer:
Services:
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
Next Generation Sensor Networks
1 Transition from dedicated sensor networks to general-purpose sensor networks.
2 Similar revolutions in IT industry:Computer:
Services:
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
Advances in Sensor Networks
1 Proliferation
(a) Ubiquitous (b) Mobile (c) Personal
2 More powerful processing units
Intel XScale 600MHz CPU
Memory: 16 MB ROM, 64 MB RAM
Resolution: 1280× 1024 up to 15 fps
Integrated microphone, infrared, motionsensors.
IEEE 802.15.4 protocal, 250 kbps.
Figure: Next generation Berkeleywireless camera mote.
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
Advances in Sensor Networks
1 Proliferation
(a) Ubiquitous (b) Mobile (c) Personal
2 More powerful processing units
Intel XScale 600MHz CPU
Memory: 16 MB ROM, 64 MB RAM
Resolution: 1280× 1024 up to 15 fps
Integrated microphone, infrared, motionsensors.
IEEE 802.15.4 protocal, 250 kbps.
Figure: Next generation Berkeleywireless camera mote.
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
On-Demand Surveillance
1 Distributed recognition system:Multi-tasking.Adaptive to environments.
2 Adaptive feature selection is critical for on-demand surveillance.Thermometer, infrared: simple thresholding.
Face IDs:Action: spatial-temporal features in video sequences.
3 Sensor-server network
On-demand surveillance and band-limited channels present a conundrum.
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
On-Demand Surveillance
1 Distributed recognition system:Multi-tasking.Adaptive to environments.
2 Adaptive feature selection is critical for on-demand surveillance.Thermometer, infrared: simple thresholding.
Face IDs:Action: spatial-temporal features in video sequences.
3 Sensor-server network
On-demand surveillance and band-limited channels present a conundrum.
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
On-Demand Surveillance
1 Distributed recognition system:Multi-tasking.Adaptive to environments.
2 Adaptive feature selection is critical for on-demand surveillance.Thermometer, infrared: simple thresholding.
Face IDs:Action: spatial-temporal features in video sequences.
3 Sensor-server network
On-demand surveillance and band-limited channels present a conundrum.
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
Contributions
Qualification for feature selection
Data independent.
Application independent.
Fast to generate and compute.
Accurate in preserving true data structures.
Contributions
1 New framework for object/face recognition via compressed sensing.
2 Classification is encoded in a (global) sparse representation.
3 Random projection as universal dimensionality redunction.
4 Efficient solution via `1-minimization outperforms classical algorithms.
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
Contributions
Qualification for feature selection
Data independent.
Application independent.
Fast to generate and compute.
Accurate in preserving true data structures.
Contributions
1 New framework for object/face recognition via compressed sensing.
2 Classification is encoded in a (global) sparse representation.
3 Random projection as universal dimensionality redunction.
4 Efficient solution via `1-minimization outperforms classical algorithms.
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
Representation using Linear Models
1 Representation of samples in vector form y ∈ RD .
Figure: Stacking of 2-D image.
2 Recognition (supervised learning)Training: For K classes, collect samples {v1,1, · · · , v1,n1
}, · · · , {vK,1, · · · , vK,nK}.
Test: Present a new y, solve for label(y) ∈ [1, 2, · · · ,K ].
3 Subspace model for face recognition: [Belhumeur et al. 1997, Basri & Jocobs 2003]
y = αi,1vi,1 + αi,2vi,2 + · · ·+ αi,n1vi,ni
,= Aiαi ,
where Ai = [vi,1, vi,2, · · · , vi,ni].
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
Representation using Linear Models
1 Representation of samples in vector form y ∈ RD .
Figure: Stacking of 2-D image.
2 Recognition (supervised learning)Training: For K classes, collect samples {v1,1, · · · , v1,n1
}, · · · , {vK,1, · · · , vK,nK}.
Test: Present a new y, solve for label(y) ∈ [1, 2, · · · ,K ].
3 Subspace model for face recognition: [Belhumeur et al. 1997, Basri & Jocobs 2003]
y = αi,1vi,1 + αi,2vi,2 + · · ·+ αi,n1vi,ni
,= Aiαi ,
where Ai = [vi,1, vi,2, · · · , vi,ni].
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
Representation using Linear Models
1 Representation of samples in vector form y ∈ RD .
Figure: Stacking of 2-D image.
2 Recognition (supervised learning)Training: For K classes, collect samples {v1,1, · · · , v1,n1
}, · · · , {vK,1, · · · , vK,nK}.
Test: Present a new y, solve for label(y) ∈ [1, 2, · · · ,K ].
3 Subspace model for face recognition: [Belhumeur et al. 1997, Basri & Jocobs 2003]
y = αi,1vi,1 + αi,2vi,2 + · · ·+ αi,n1vi,ni
,= Aiαi ,
where Ai = [vi,1, vi,2, · · · , vi,ni].
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
Recognition via Sparse Representation
1 The label of y is unknown:
y = [A1,A2, · · · ,AK ]
α1α2
...αK
,= Ax0.
Over-determined system: A ∈ RD×n, where D � n = n1 + · · ·+ nK .
2 x0 encodes membership of y: If y belongs to Subject 1,
x0 =
α100
...0
∈ Rn.
That is, y should be only represented using the same subject!
If we recover sparse x0, recognition is solved! Not so fast!!
Directly solving A is expensive: D > 7× 104 for a 320× 240 grayscale image.
x0 is sparse: 1K
terms non-zero.
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
Recognition via Sparse Representation
1 The label of y is unknown:
y = [A1,A2, · · · ,AK ]
α1α2
...αK
,= Ax0.
Over-determined system: A ∈ RD×n, where D � n = n1 + · · ·+ nK .
2 x0 encodes membership of y: If y belongs to Subject 1,
x0 =
α100
...0
∈ Rn.
That is, y should be only represented using the same subject!
If we recover sparse x0, recognition is solved! Not so fast!!
Directly solving A is expensive: D > 7× 104 for a 320× 240 grayscale image.
x0 is sparse: 1K
terms non-zero.
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
Recognition via Sparse Representation
1 The label of y is unknown:
y = [A1,A2, · · · ,AK ]
α1α2
...αK
,= Ax0.
Over-determined system: A ∈ RD×n, where D � n = n1 + · · ·+ nK .
2 x0 encodes membership of y: If y belongs to Subject 1,
x0 =
α100
...0
∈ Rn.
That is, y should be only represented using the same subject!
If we recover sparse x0, recognition is solved! Not so fast!!
Directly solving A is expensive: D > 7× 104 for a 320× 240 grayscale image.
x0 is sparse: 1K
terms non-zero.
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
Dimensionality Redunction
1 Dimensionality redunctionConstruct linear projection R ∈ Rd×D , d is the feature dimension.
y.
= Ry = RAx0 = Ax0.
A ∈ Rd×n, but x0 is unchanged.
2 Holistic features
Eigenfaces [Turk & Pentland 1991]
Fisherfaces [Belhumeur et al.1997]
Laplacianfaces [He et al.2005]
3 Partial features
4 Unconventional features
Downsampled faces
Randomfaces
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
Dimensionality Redunction
1 Dimensionality redunctionConstruct linear projection R ∈ Rd×D , d is the feature dimension.
y.
= Ry = RAx0 = Ax0.
A ∈ Rd×n, but x0 is unchanged.
2 Holistic features
Eigenfaces [Turk & Pentland 1991]
Fisherfaces [Belhumeur et al.1997]
Laplacianfaces [He et al.2005]
3 Partial features
4 Unconventional features
Downsampled faces
Randomfaces
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
Dimensionality Redunction
1 Dimensionality redunctionConstruct linear projection R ∈ Rd×D , d is the feature dimension.
y.
= Ry = RAx0 = Ax0.
A ∈ Rd×n, but x0 is unchanged.
2 Holistic features
Eigenfaces [Turk & Pentland 1991]
Fisherfaces [Belhumeur et al.1997]
Laplacianfaces [He et al.2005]
3 Partial features
4 Unconventional features
Downsampled faces
Randomfaces
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
Dimensionality Redunction
1 Dimensionality redunctionConstruct linear projection R ∈ Rd×D , d is the feature dimension.
y.
= Ry = RAx0 = Ax0.
A ∈ Rd×n, but x0 is unchanged.
2 Holistic features
Eigenfaces [Turk & Pentland 1991]
Fisherfaces [Belhumeur et al.1997]
Laplacianfaces [He et al.2005]
3 Partial features
4 Unconventional features
Downsampled faces
Randomfaces
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
Randomfaces
Definition
Consider a projection matrix R ∈ Rd×D whose entries are independent Gaussian samples, andeach row is normalized to unit length. These row vectors are called d Randomfaces in RD .
Properties of Randomfaces (Universal Projection?)
1 Domain independent!
2 Data independent!
3 Fast to generate and compute!
High accuracy?
Not necessary: May underperform.
Condition 1: Signal is sparse.
Condition 2: `1-Minimization.
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
Randomfaces
Definition
Consider a projection matrix R ∈ Rd×D whose entries are independent Gaussian samples, andeach row is normalized to unit length. These row vectors are called d Randomfaces in RD .
Properties of Randomfaces (Universal Projection?)
1 Domain independent!
2 Data independent!
3 Fast to generate and compute!
High accuracy?
Not necessary: May underperform.
Condition 1: Signal is sparse.
Condition 2: `1-Minimization.
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
Randomfaces
Definition
Consider a projection matrix R ∈ Rd×D whose entries are independent Gaussian samples, andeach row is normalized to unit length. These row vectors are called d Randomfaces in RD .
Properties of Randomfaces (Universal Projection?)
1 Domain independent!
2 Data independent!
3 Fast to generate and compute!
High accuracy?
Not necessary: May underperform.
Condition 1: Signal is sparse.
Condition 2: `1-Minimization.
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
`0-Minimization
1 Underdetermined system y = Ax0 ∈ Rd : A ∈ Rd×n.Ask for the sparsest solution.
2 `0-Minimizationx0 = arg min
x‖x‖0 s.t. y = Ax.
‖ · ‖0 simply counts the number of nonzero terms.
3 `0-Ball
Optimization over `0-ball iscombinatorial.
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
`0-Minimization
1 Underdetermined system y = Ax0 ∈ Rd : A ∈ Rd×n.Ask for the sparsest solution.
2 `0-Minimizationx0 = arg min
x‖x‖0 s.t. y = Ax.
‖ · ‖0 simply counts the number of nonzero terms.
3 `0-Ball
Optimization over `0-ball iscombinatorial.
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
`0-Minimization
1 Underdetermined system y = Ax0 ∈ Rd : A ∈ Rd×n.Ask for the sparsest solution.
2 `0-Minimizationx0 = arg min
x‖x‖0 s.t. y = Ax.
‖ · ‖0 simply counts the number of nonzero terms.
3 `0-Ball
Optimization over `0-ball iscombinatorial.
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
`1/`0 Equivalence
1 If x0 is sparse enough, program (P0) is equivalent to
(P1) min ‖x‖1 s.t. y = Ax.
‖x‖1 = |x1|+ |x2|+ · · ·+ |xn|.
2 `1-Ball
`1-Minimization is linear (matchingpursuit, basis pursuit)
Solution equal to `0-minimization.
3 Equivalence conditionSpark condition (sufficient): [Donoho 2002]Equivalence breakdown point (asymptotic): [Donoho 2004].k-neighborlyness (iff): [Donoho preprint].
Given y = Ax0, there exists ρ(A), if ‖x0‖0 < ρ,
(1) `1-solution is unique, (2) x1 = x0.
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
`1/`0 Equivalence
1 If x0 is sparse enough, program (P0) is equivalent to
(P1) min ‖x‖1 s.t. y = Ax.
‖x‖1 = |x1|+ |x2|+ · · ·+ |xn|.2 `1-Ball
`1-Minimization is linear (matchingpursuit, basis pursuit)
Solution equal to `0-minimization.
3 Equivalence conditionSpark condition (sufficient): [Donoho 2002]Equivalence breakdown point (asymptotic): [Donoho 2004].k-neighborlyness (iff): [Donoho preprint].
Given y = Ax0, there exists ρ(A), if ‖x0‖0 < ρ,
(1) `1-solution is unique, (2) x1 = x0.
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
`1/`0 Equivalence
1 If x0 is sparse enough, program (P0) is equivalent to
(P1) min ‖x‖1 s.t. y = Ax.
‖x‖1 = |x1|+ |x2|+ · · ·+ |xn|.2 `1-Ball
`1-Minimization is linear (matchingpursuit, basis pursuit)
Solution equal to `0-minimization.
3 Equivalence conditionSpark condition (sufficient): [Donoho 2002]Equivalence breakdown point (asymptotic): [Donoho 2004].k-neighborlyness (iff): [Donoho preprint].
Given y = Ax0, there exists ρ(A), if ‖x0‖0 < ρ,
(1) `1-solution is unique, (2) x1 = x0.
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
`1-Minimization for Noisy Case
1 Introduce noise in the data
y = Ax0 + z ∈ Rd , where ‖z‖2 < ε.
2 `1-Norm near solution(P′1) min ‖x‖1 s.t. ‖y − Ax‖2 ≤ ε.
3 `1-Ball
`1-Minimization is quadratic
Stability: bounded data noise givesbounded estimation error ‖x1 − x0‖2
With large probability, there exist ρ(A) and ζ > 0, if ‖x0‖0 < ρ,
‖x1 − x0‖2 ≤ ζε.
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
`1-Minimization for Noisy Case
1 Introduce noise in the data
y = Ax0 + z ∈ Rd , where ‖z‖2 < ε.
2 `1-Norm near solution(P′1) min ‖x‖1 s.t. ‖y − Ax‖2 ≤ ε.
3 `1-Ball
`1-Minimization is quadratic
Stability: bounded data noise givesbounded estimation error ‖x1 − x0‖2
With large probability, there exist ρ(A) and ζ > 0, if ‖x0‖0 < ρ,
‖x1 − x0‖2 ≤ ζε.
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
`1-Minimization for Noisy Case
1 Introduce noise in the data
y = Ax0 + z ∈ Rd , where ‖z‖2 < ε.
2 `1-Norm near solution(P′1) min ‖x‖1 s.t. ‖y − Ax‖2 ≤ ε.
3 `1-Ball
`1-Minimization is quadratic
Stability: bounded data noise givesbounded estimation error ‖x1 − x0‖2
With large probability, there exist ρ(A) and ζ > 0, if ‖x0‖0 < ρ,
‖x1 − x0‖2 ≤ ζε.
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
Classification via Sparse Representation
1 Given an estimated ε, solve (P′1)⇒ x1.
2 Project x1 onto face subspaces:
δ1(x1) =
α10
...0
, δ2(x1) =
0α2
...0
, · · · , δK (x1) =
00
...αK
. (1)
3 Define residual ri = ‖y − Aδi (x1)‖2 for Subject i :
id(y) = arg mini=1,··· ,K{ri}
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
Classification via Sparse Representation
1 Given an estimated ε, solve (P′1)⇒ x1.
2 Project x1 onto face subspaces:
δ1(x1) =
α10
...0
, δ2(x1) =
0α2
...0
, · · · , δK (x1) =
00
...αK
. (1)
3 Define residual ri = ‖y − Aδi (x1)‖2 for Subject i :
id(y) = arg mini=1,··· ,K{ri}
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
Classification via Sparse Representation
1 Given an estimated ε, solve (P′1)⇒ x1.
2 Project x1 onto face subspaces:
δ1(x1) =
α10
...0
, δ2(x1) =
0α2
...0
, · · · , δK (x1) =
00
...αK
. (1)
3 Define residual ri = ‖y − Aδi (x1)‖2 for Subject i :
id(y) = arg mini=1,··· ,K{ri}
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
Outlier Rejection
`1-Coefficients for invalid images
Outlier Rejection
When `1-solution is not sparse or concentrated to one subspace, the test sample is invalid.
Sparsity Concentration Index: SCI(x).
=K ·maxi ‖δi (x)‖1/‖x‖1 − 1
K − 1∈ [0, 1].
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
Outlier Rejection
`1-Coefficients for invalid images
Outlier Rejection
When `1-solution is not sparse or concentrated to one subspace, the test sample is invalid.
Sparsity Concentration Index: SCI(x).
=K ·maxi ‖δi (x)‖1/‖x‖1 − 1
K − 1∈ [0, 1].
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
Randomface Ensembles
1 Blessing of dimensionality
When D is large for RD , with overwhelmingprobability, the row vectors in R ∈ Rd×D areindependent.
Subspace structures and sample membership arepreserved.
2 Randomface ensemble (For a particular random matrix R, the choice could be bad):Generate multiple projections: R1,R2, · · · ,Rl .For each Rj , solve
min ‖x‖1 s.t. ‖Rj y − RjAx‖2 ≤ εCompute an averaged residual function
ri = mean{r1i , r
2i , · · · , r
li }.
id(y) = arg mini=1,··· ,K{ri}
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
Randomface Ensembles
1 Blessing of dimensionality
When D is large for RD , with overwhelmingprobability, the row vectors in R ∈ Rd×D areindependent.
Subspace structures and sample membership arepreserved.
2 Randomface ensemble (For a particular random matrix R, the choice could be bad):Generate multiple projections: R1,R2, · · · ,Rl .For each Rj , solve
min ‖x‖1 s.t. ‖Rj y − RjAx‖2 ≤ εCompute an averaged residual function
ri = mean{r1i , r
2i , · · · , r
li }.
id(y) = arg mini=1,··· ,K{ri}
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
Extended Yale B 38 Subjects (Illumination Variance)
Table: Nearest Neighbor (Left) and Nearest Subspace (Right).
Dimension 30 56 120 504
Eigen [%] 74.3 81.4 85.5 88.4Laplacian [%] 77.1 83.5 87.2 90.7Random [%] 70.3 75.6 78.8 79Down [%] 51.7 62.6 71.6 78Fisher [%] 87.6 N/A N/A N/A
Dimension 30 56 120 504
Eigen [%] 89.9 91.1 92.5 93.2Laplacian [%] 89 90.4 91.9 93.4Random [%] 87.3 91.5 93.9 94.1Down [%] 80.8 88.2 91.1 93.4Fisher [%] 81.9 N/A N/A N/A
Table: `1-Minimization
Dimension 30 56 120 504
Eigen [%] 86.5 91.6 94 96.8Laplacian [%] 87.5 91.7 94 96.5Random [%] 82.6 91.5 95.5 98.1Down [%] 74.6 86.2 92.1 97.1Fisher [%] 86.9 N/A N/A N/A
Ensemble [%] 90.7 94.1 96.4 98.3
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
AR Database 100 Subjects (Illumination and Expression Variance)
Table: Nearest Neighbor (Left) and Nearest Subspace (Right).
Dimension 30 54 130 540
Eigen [%] 68.1 74.8 79.3 80.5Laplacian [%] 73.1 77.1 83.8 89.7Random [%] 56.7 63.7 71.4 75Down [%] 51.7 60.9 69.2 73.7Fisher [%] 83.4 86.8 N/A N/A
Dimension 30 54 130 540
Eigen [%] 64.1 77.1 82 85.1Laplacian [%] 66 77.5 84.3 90.3Random [%] 59.2 68.2 80 83.3Down [%] 56.2 67.7 77 82.1Fisher [%] 80.3 85.8 N/A N/A
Table: `1-Minimization.
Dimension 30 54 130 540
Eigen [%] 71.1 80 85.7 92Laplacian [%] 73.7 84.7 91 94.3Random [%] 57.8 75.5 87.6 94.7Down [%] 46.8 67 84.6 93.9Fisher [%] 87 92.3 N/A N/A
Ensemble [%] 78.5 85.8 91.2 95
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
Partial Features
Features Nose Right Eye Mouth & ChinDimension 4,270 5,040 12,936
`1-Minimization [%] 87.3 93.7 98.3NS [%] 83.7 78.6 94.4NN [%] 49.2 68.8 72.7
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
ROC Curves on AR database
Figure: ROC curve on Eigenfaces and AR database.
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
Motivation Problem Formulation `1-Minimization Classification Experiments Future Directions
Future Directions
1 Facial occlusion.
2 Pose variations.
3 Action segmentation and recognition.
References
1 Face recognition
http://www.eecs.berkeley.edu/~yang/software/face_recognition/
Feature selection in face recognition: A sparse representation perspective. UC Berkeley Tech ReportUCB/EECS-2007-99.
2 Compressed sensing
Candes, Compressive sampling, 2006.Donoho, For most large underdetermined systems of equations, the minimal `1-norm near-solution approximates thesparsest near-solution, 2004.Donoho, Neighborly polytopes and sparse solution of underdetermined linear equations, 2004.
3 Random projections of smooth manifolds
Baraniuk & Wakin, Random projection of smooth manifolds, 2006.Baraniuk et al., The Johnson-Lindenstrauss lemma meets compressed sensing, 2007.
Allen Y. Yang <[email protected]> Random Feature Selection for Robust Face Recognition
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