RAID-G: Robust Estimation of Approximate
Infinite Dimensional Gaussian with
Application to Material Recognition
Presented by Qilong Wang and Peihua Li
Dalian University of Technology
http://ice.dlut.edu.cn/PeihuaLi/
Kernel SVM
Infinite-dimensional Covariance for classification
Hand-crafted features
Image
[23]
[20]
[17]
Feature extraction
Feature mapping in
RKHS
Image model
M. T. Harandi, M. Salzmann, and F. M. Porikli. Bregman divergences for infinite dimensional covariance matrices.
In CVPR, 2014
Ha Quang Minh, Marco San Biagio and Vittorio Murino. Log-Hilbert-Schmidt metric between positive definite
operators on Hilbert spaces. In NIPS, 2014.
M. Faraki, M. Harandi, and F. Porikli. Approximate infinitedimensional region covariance descriptors for image
classification. In ICASSP, 2015
Comparison of different infinite-dimensional image representation
Methods
Zhou et al.
[PAMI06]
Representation
Estimator
RBF kernel
(no explicit mapping)
Ledoit-Wolf
estimator
Covariance
Faraki et al.
[17]
RIAD-G
(Ours)
Linear
SVM?
Gaussian
Harandi et al.
[23]
Log-HS
[20]
Kernels & Mapping
Random Fourier transform
Nystrom method
for RBF kernel
Gaussian
Hellinger’s kernel
𝒳2 kernel
Computationally very expensive and unscalable
Feature mapping by RBF kernel & kernel SVM [20,23]
● Inefficient approximate mappings [17]
●
Unrobust covariance estimation [17,20,23]
No
Σ + λI
Yes
vN-MLE
Yes
Hard to use very
high-dimensional
features
Proposed RAID-G ̶ Our doings
CNN features
Hellinger’s kernel
Chi-Square kernel
Image
Feature extraction
Analytical, finitedimensional mapping
φHel (xk ) = xk
Gaussian
representation
φˆ Chi (xk ) = xk  L, 2Lsech(Lπ)cos(Llog(xk ))
2Lsech(Lπ)sin(Llog(xk ))
T
[RAID-G] Qilong Wang, Peihua Li, Wangmeng Zuo, and Lei Zhang. RAID-G: Robust Estimation of Approximate
Infinite Dimensional Gaussian with Application to Material Recognition, In CVPR, 2016
Proposed RAID-G
~500 features
CNN
High-dimension &
small sample problem
I am
robust
Hellinger’s kernel
~ Chi-Square kernel
Image
Feature extraction
Analytical, finitedimensional mapping
φHel (xk ) = xk
Robust Gaussian
φˆ Chi (xk ) = xk  L, 2Lsech(Lπ)cos(Llog(xk ))
2Lsech(Lπ)sin(Llog(xk ))
T
[RAID-G] Qilong Wang, Peihua Li, Wangmeng Zuo, and Lei Zhang. RAID-G: Robust Estimation of Approximate
Infinite Dimensional Gaussian with Application to Material Recognition, In CVPR, 2016
Proposed RAID-G ̶ robust estimator
Classical MLE(not robust in our case)
where
Robust vN-MLE
where
is the von Neumann divergence between matrices.
[RAID-G] Qilong Wang, Peihua Li, Wangmeng Zuo, and Lei Zhang. RAID-G: Robust Estimation of Approximate
Infinite Dimensional Gaussian with Application to Material Recognition, In CVPR, 2016
Proposed RAID-G ̶ Flowchart
Approximate RKHS

Mapping
.
.
.
...
Robust
Estimation
...

.
.
.
.
.
.
Mapping
Deep CNN features
RIAD-G
 Modeling images with RIAD-G in multiscale manner
 VGG-VD16 without fine-tuning
Gaussian Embedding
& vectorization
Linear
SVM
Results on Material Recognition




Gaussian representation always performs better than covariance.
The vN-MLE estimator significantly improves the classical MLE.
Feature mapping by kernel embeddings brings benefits.
RAID-G outperforms FV-CNN, achieving state-of-the-art results.
[RAID-G] Qilong Wang, Peihua Li, Wangmeng Zuo, and Lei Zhang. RAID-G: Robust Estimation of Approximate
Infinite Dimensional Gaussian with Application to Material Recognition, In CVPR, 2016
Comparison with other robust estimators
[LW]
O. Ledoit and M. Wolf. A well-conditioned estimator for large-dimensional covariance matrices. J. Multivariate
Analysis, 2004.
[Stein ] M. J. Daniels and R. E. Kass. Shrinkage estimators for covariance matrices. Biometrics, 2001.
[MMSE] Y. Chen, A. Wiesel, et al. Shrinkage algorithms for MMSE covariance estimation. IEEE TSP, 2010.
[EL-SP] E. Yang, A. Lozano, et al. Elementary estimators for sparse covariance matrices and other structured moments. In
ICML, 2014.
[vN-MLE]Qilong Wang, Peihua Li, Wangmeng Zuo, and Lei Zhang. RAID-G: Robust Estimation of Approximate Infinite
Dimensional Gaussian with Application to Material Recognition, In CVPR, 2016 .
Comparison with other explicit kernel mappings
[17]
M. Faraki, M. Harandi, and F. Porikli. Approximate infinitedimensional region covariance descriptors for image
classification. In ICASSP, 2015
[RAID-G] Qilong Wang, Peihua Li, Wangmeng Zuo, and Lei Zhang. RAID-G: Robust Estimation of Approximate Infinite
Dimensional Gaussian with Application to Material Recognition, In CVPR, 2016
Comparison with infinite dimensional
covariance on KTH-TIPS 2b
 [23, 20] are slightly better than RAID-G with hand-crafted features.
 RAID-G outperforms [23,20] by 7% with CNN features, while [23,20]
involves unaffordable cost with CNN features.
[23]
M. T. Harandi, M. Salzmann, and F. M. Porikli. Bregman divergences for infinite dimensional covariance matrices.
In CVPR, 2014
[20]
Ha Quang Minh, Marco San Biagio and Vittorio Murino. Log-Hilbert-Schmidt metric between positive definite
operators on Hilbert spaces. In NIPS, 2014.
[RAID-G] Qilong Wang, Peihua Li, Wangmeng Zuo, and Lei Zhang. RAID-G: Robust Estimation of Approximate Infinite
Dimensional Gaussian with Application to Material Recognition, In CVPR, 2016 .
Summary
RAID-G is a very competitive image representation
√ Efficient (analytic feature mapping & covariance computation)
√ Scalable (linear SVM)
√ Superior performance
Robust estimation is critical for higher dimension problem (>512)
Analytical, explicit feature mappings improve performance