Tensor Sparse Coding for Positive Definite Matrices

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1. What contributions have the authors mentioned in the paper "Tensor sparse coding for positive definite matrices" ?

2. What future works have the authors mentioned in the paper "Tensor sparse coding for positive definite matrices" ?

3. What are the datasets used for sparse coding?

4. What is the tangent space of SPD matrices?

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This paper proposes a novel sparse coding technique for positive definite matrices, which respects the structure of the Riemannian manifold and preserves the positivity of their eigenvalues, without resorting to vectorization.. This work serves to bridge the gap between the sparse modeling paradigm and the space of positive definite matrices.

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Future work involves applying the above techniques to areas such as Diffusion Tensor Imaging.

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The datasets used are comprised of region covariance descriptors from various applications such as human appearance modeling, texture classification and face recognition.

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The tangent space of SPD matrices at any point on the manifold is Sn, the space of n × n symmetric matrices, and the tangent operator is the matrix logarithm.

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At each iteration of the experiment, 4 out of 7 images from each subject are taken for training, and the remaining 3 are used as test images, yielding a total of ( 7 3 ) = 35 different train-test splits.

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Each texture class corresponds to one training image of 256×256 pixels, which is broken down into non-overlapping blocks of 32 × 32 pixels.

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The Gabor filters [26] corresponding to 8 orientations (u = 0, . . . , 7) and 5 scales (v = 0, . . . , 4) are applied to each image, resulting in 40 different filter responses guv .

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There are 64 covariance descriptors from each texture class, of which 8 descriptors from each class are chosen for training, and the remaining are used for testing.

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In [10], the authors track non-rigid objects with an update mechanism based on Lie algebra defined at the tangent space of the identity matrix. [11] present a boosting framework over region covariances.

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Given an image I, let φ define a function that extracts an ndimensional feature vector zi from each pixel i ∈ I, such thatφ(I, xi, yi) = zi,where zi ∈ Rn, and (xi, yi) is the location of the ith pixel.

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The performance of the tensor sparse coding in terms of accuracy of reconstruction, sparsity of the decomposition, as well as variations for different input parameters is analyzed.

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The Logdet divergence, as the authors will elaborate next, is a well-suited distortion measure, not only due to its significant relation with Wishart and Gaussian distributions, but also because it results in a well-known and tractable convex optimization problem.

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throughout the rest of this work, the authors adhere to normalization by trace: tr(Ai) = 1.The authors evaluate the tensor sparse coding algorithm in a classification framework, where the training data is used as a dictionary A, and the test point S is approximated by a sparse non-negative linear combination of the dictionary atoms.

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The `1-constrained sparse coding is used for both the tensor and vector cases, and the constraint T is varied as a fraction of the true required ‘budget’ ‖x∗‖1.

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The above formulations of tensor sparse coding fall under a general class of optimization problems known as determinant maximization (MAXDET) problems [15], of which semi-definite programming (SDP) and linearprogramming (LP) are special cases.

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A given image region R is represented by the n × n covariance matrix CR of the feature vectors {zi}|R|i=1 of the pixels in region R. Thus the region covariance descriptor is given byCR = 1|R| − 1|R|∑i=1(zi − µR) (zi − µR)T ,where, µR is the mean vector,µR = 1|R||R|∑i=1zi.

Accepted Answer

Wang and Wu [30] perform object tracking using region covariances by incrementally learning a low-dimensional model for the covariances in an adaptive manner.

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More recently in the image processing and computer vision community, a new feature known as the region covariance descriptor has been introduced [9], [10], which represents an image region by the covariance of n-dimensional feature vectors at each pixel in that region.

Tensor Sparse Coding for Positive Definite Matrices

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Most frequently asked questions

1. What contributions have the authors mentioned in the paper "Tensor sparse coding for positive definite matrices" ?

2. What future works have the authors mentioned in the paper "Tensor sparse coding for positive definite matrices" ?

3. What are the datasets used for sparse coding?

4. What is the tangent space of SPD matrices?

5. How many different train-test splits are used for each face?

6. How many blocks of pixels is the training image of each texture class?

7. How many different filter responses are applied to each face image?

8. How many descriptors are used for training?

9. What is the boosting framework for non-rigid objects?

10. What is the function that extracts the feature vector zi from each pixel?

11. What is the performance of the tensor sparse coding?

12. Why is the logdet divergence a well-suited distortion measure?

13. What is the way to normalize the tensor sparse coding algorithm?

14. What is the constraint used for the tensor and vector sparse coding?

15. What is the definition of a tensor sparse coding problem?

16. What is the CR of the feature vectors zi|R|i=?

17. How do they learn a low-dimensional model for the covariances?

18. What is the definition of a region covariance descriptor?

Figures

Citations

Covariance, Subspace, and Intrinsic CramrRao Bounds

Kernel Sparse Subspace Clustering on Symmetric Positive Definite Manifolds

Multi-sensor fusion approach with fault detection and exclusion based on the KullbackLeibler Divergence

Kernel Sparse Subspace Clustering on Symmetric Positive Definite Manifolds

Modified Tensor Locality Preserving Projection for Dimensionality Reduction of Hyperspectral Images

References

LIBSVM: A library for support vector machines

Regression Shrinkage and Selection via the Lasso

An Introduction To Compressive Sampling

Robust Face Recognition via Sparse Representation

Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit

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