A theorem on the trace of certain matrix products and some applications

“…Note that the mode of the distribution is characterized by the parameter Ψ and does not depend on the parameter ν. The proof of the theorem depends crucially on a strong result, a type of rearrangement inequality proved in Kristof (1969).…”

Section: Theorem 2 Let D ∈ R Pmentioning

confidence: 99%

“…The proof of part (a) Theorem 6 is similar to that of the proof of Theorem 4. The proof for part (b) of Theorem 6 is more involved and depends on several key results, including the rearrangement inequality by Kristof (1969), the log convexity of 0…”

Section: (B) For Any Open Setmentioning

confidence: 99%

Conjugate Priors and Posterior Inference for the Matrix Langevin Distribution on the Stiefel Manifold

Pal¹,

Sengupta²,

Mitra³

et al. 2020

Bayesian Anal.

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Directional data emerges in a wide array of applications, ranging from atmospheric sciences to medical imaging. Modeling such data, however, poses unique challenges by virtue of their being constrained to non-Euclidean spaces like manifolds. Here, we present a unified Bayesian framework for inference on the Stiefel manifold using the Matrix Langevin distribution. Specifically, we propose a novel family of conjugate priors and establish a number of theoretical properties relevant to statistical inference. Conjugacy enables translation of these properties to their corresponding posteriors, which we exploit to develop the posterior inference scheme. For the implementation of the posterior computation, including the posterior sampling, we adopt a novel computational procedure for evaluating the hypergeometric function of matrix arguments that appears as normalization constants in the relevant densities.

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“…Note that by cyclic perturbation which retains the trace unchanged and due to ED = 0 3p 2 ×(3p 2 −r) , we have Tr(Y D A) = Tr(YA D ) = Tr((I 3p 2 ×3p 2 − EE )YA D ) = Tr(UΣV VU ) = Tr(Σ). For every D satisfying that D D = I (3p 2 −r)×(3p 2 −r) , E D = 0 3p 2 ×(3p 2 −r) , we have Tr(Y DA) = Tr((I 3p 2 ×3p 2 − EE )YA D ) = Tr(UΣV D ) = Tr(ΣV D U).By using a generalization version[59] of the Kristof's Theorem[60], we have Tr(Y DA) = Tr(ΣV D U) ≤ Tr(Σ). The equality is obtained at V D U = I (3p 2 −r)×(3p 2 −r) , i.e., D = UV =D.…”

mentioning

confidence: 99%

External Prior Guided Internal Prior Learning for Real-World Noisy Image Denoising

Zhang

2018

IEEE Trans. on Image Process.

143

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Most of existing image denoising methods learn image priors from either external data or the noisy image itself to remove noise. However, priors learned from external data may not be adaptive to the image to be denoised, while priors learned from the given noisy image may not be accurate due to the interference of corrupted noise. Meanwhile, the noise in real-world noisy images is very complex, which is hard to be described by simple distributions such as Gaussian distribution, making real-world noisy image denoising a very challenging problem. We propose to exploit the information in both external data and the given noisy image, and develop an external prior guided internal prior learning method for real-world noisy image denoising. We first learn external priors from an independent set of clean natural images. With the aid of learned external priors, we then learn internal priors from the given noisy image to refine the prior model. The external and internal priors are formulated as a set of orthogonal dictionaries to efficiently reconstruct the desired image. Extensive experiments are performed on several real-world noisy image datasets. The proposed method demonstrates highly competitive denoising performance, outperforming state-of-the-art denoising methods including those designed for real-world noisy images.

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A theorem on the trace of certain matrix products and some applications

Cited by 30 publications

References 13 publications

Some Relationships between Factors and Components

Some Relationships between Factors and Components

Conjugate Priors and Posterior Inference for the Matrix Langevin Distribution on the Stiefel Manifold

External Prior Guided Internal Prior Learning for Real-World Noisy Image Denoising

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