Riemannian Sparse Coding for Positive Definite Matrices Anoop Cherian1 Suvrit Sra2

1LEAR Project-team, Inria Grenoble - Rhône-Alpes 2MPI for Intelligent Systems, Tϋbingen, Germany

Introduction

Covariance Descriptors

Geometry of Positive Definite Matrices

Our Contributions

1. Riemannian Sparse Coding Formulation

2. Efficient Optimization

Experiments and Results

Region covariance descriptors are covariance matrices used as visual data descriptors in several computer vision applications. Examples include people tracking, diffusion MRI, object recognition, etc.

vec

Say d different features from patches

xi ԑ d

For p pixels

in a patch

T

iixx

pX ))((

1

1

where μ is the mean of xi’s

Covariance Descriptor

Advantages

• Multi-feature fusion

• Compact

• Real-time computable

• Robust to static noise

• Robust to illumination

• Robust to affine transforms

X is in the space of d x d symmetric positive definite (SPD) matrices (Sd

++) and is called the Covariance Descriptor

• SPD matrices form a manifold (Riemannian manifold) in the Euclidean space due to their positive definiteness property • Distances along the manifold are not straight lines, but curved geodesics!

Due to the great success of sparse coding for vectorial data, we propose a novel scheme for sparse coding SPD matrices using a dictionary whose atoms are SPD matrices. 1. Our novel sparse coding formulation uses a loss function defined

on the affine invariant Riemannian distance (which is the natural distance on this Riemannian manifold)

2. We propose a computationally efficient scheme for optimization

3. We show conditions at which our formulation is convex

4. Experiments on several computer vision applications demonstrate

significant promise of our approach over the state of the art.

Let be a dictionary with n atoms B1, B2, ..., Bn, each Bi ԑ Sd++ . Let X be the input

matrix to be sparse coded. Then, our sparse coding objective is:

Affine invariant Riemannian Distance Sparsity

Prior Work Loss function based on

Sivalingam et al., ECCV, 2010 LogDet divergence

Sra and Cherian, ECML, 2011 Frobenius distance

Ho et al., ICML, 2013 Log-Euclidean distance

Harandi et al., ECCV 2012, Jayasumana et al., CVPR, 2013

Li et al., CVPR, 2013

Kernel methods

In contrast to prior methods, our formulation is based on the intrinsic geometry of the SPD manifold (while other methods use proxy distances).

We use projected gradient descent for minimizing (1), which has the following sequence of iterations over the sparse coefficient vector α: There are three major computational challenges for each iteration in (2)

(1)

(2)

1. Computing the step size ηk -- we use Spectral Projected Gradient [Birgin et al. ‘01]

2. Computing the gradient (αk) efficiently (See paper for the algorithm) 3. Projection step -- we truncate the negative values for efficiency.

Computational Complexity: (nd2 + d3) for computing gradient in each iteration

3. Convexity Properties

The objective in (1) is not convex in α. But surprisingly, it becomes convex under some constraints!

where dR is the affine invariant Riemannian distance as shown in (1) above.

See paper for the proof.

Simulated Experiments

Matrix size fixed at 10 x 10 Number of atoms fixed at 200

Real Data Experiments

Dataset Features Dims #classes Dataset Size Method of Evaluation

Brodatz Texture SIG 5 x 5 110 10,000 SVM classifier

ETH80 object SIGC + Laws 19 x 19 8 3,280 SVM classifier

ETHZ people SIGC + Hue 18 x 18 146 8,580 NN Classifier

RGB-D object SIG + 3G 18 x 18 51 15,000 NN Classifier

S=Spatial features, I=Intensity, C=color, G=Gradient, Laws=Texture filters, 3G=3D gradients We used 80-20% for training and testing SVM resp., we used 20-80% for NN classifier.

Brodatz Textures ETH80 Objects

ETHZ Person Re-identification RGB-D Objects

LE-SC : Guo et al. AVSS’10, K-Stein: Harandi et al. ECCV’12, K-LE-SC: Li et al, ICCV’13, TSC: Sivalingam et al. ECCV’10, GDL: Sra and Cherian, ECML’11