A generalization of the Sherman–Morrison–Woodbury formula

We present a unified online statistical framework for quantifying a collection of binary images. Since medical image segmentation is often done semi-automatically, the resulting binary images may be available in a sequential manner. Further, modern medical imaging datasets are too large to fit into a computer’s memory. Thus, there is a need to develop an iterative analysis framework where the final statistical maps are updated sequentially each time a new image is added to the analysis. We propose a new algorithm for online statistical inference and apply to characterize mandible growth during the first two decades of life.

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“…Subsequently, we have

W_{m - 1}^{'} λ_{m - 1} + z_{m - 1}^{'} y_{m} = false(W_{m - 1} + z_{m}^{'} z_{m} false) λ_{m} .

Using Woodbury formula [6],

{false(W_{m - 1} + z_{m}^{'} z_{m} false)}^{- 1} = W_{m - 1}^{- 1} - c_{m} W_{m - 1}^{- 1} z_{m}^{'},

where

c_{m} = 1 / false(1 + z_{m} W_{m - 1} z_{m}^{'} false)

is scalar. Then we have the explicit online algorithm for updating the parameter vector:

λ_{m} = false(I - W_{m - 1}^{- 1} z_{m}^{'} y_{m} - c_{m} W_{m - 1}^{- 1} z_{m}^{'} W_{m - 1}^{'} false) λ_{m - 1} - c_{m} W_{m - 1}^{- 1} z_{m}^{'} z_{m}^{'} y_{m},

where I is the identity matrix of size k × k .…”

Section: Online Algorithm For Linear Regressionmentioning

confidence: 99%

Online Statistical Inference for Large-Scale Binary Images

Chung

Chuang

Vorperian

2017

Lecture Notes in Computer Science

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“…The diagonal r×r matrix B records whether a bond α(i) is switched from weak to strong (Bαα = ks − kw = +0.99) or vice versa (Bαα = −0.99), and M is a n d ×r matrix whose r columns are the bond vectors mα for the switched bonds α(i) . This allows one to calculate changes in the Green function more efficiently using the Woodbury formula (61,62),…”

Section: Evolving the Green Function Using Dyson's And Woodbury's Formentioning

confidence: 99%

Green function of correlated genes and the mechanical evolution of protein

Dutta

Eckmann

Libchaber

et al. 2018

Preprint

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There has been growing evidence that cooperative interactions and configurational rearrangements underpin protein functions. But in spite of vast genetic and structural data, the information-dense, heterogeneous nature of protein has held back the progress in understanding the underlying principles. Here we outline a general theory of protein that quantitatively links sequence, dynamics and function: The protein is a strongly-coupled amino acid network whose interactions and large-scale motions are captured by the mechanical propagator, also known as the Green function. The propagator relates the gene to the connectivity of the amino acid network and the transmission of forces through the protein. How well the force pattern conforms to the collective modes of the functional protein is measured by the fitness. Mutations introduce localized perturbations to the propagator which scatter the force field. The emergence of function is manifested by a topological transition when a band of such perturbations divides the protein into subdomains. Epistasis quantifies how much the combined effect of multiple mutations departs from additivity. We find that epistasis is the nonlinearity of the Green function, which corresponds to a sum over multiple scattering paths passing through the localized perturbations. We apply this mechanical framework to the simulations of protein evolution, and observe long-range epistasis which facilitates collective functional modes. Our model lays the foundation for understanding the protein as an evolved state of matter and may be a prototype for other strongly-correlated living systems.Protein evolution | Mechanical epistasis | Genotype-to-phenotype map | Green function | Dimensional reduction

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“…We can use Sherman-Morrison formula [9] to make the above updating depend on the inverse of B k , that iŝ…”

Section: First Type Of Bc (Bc1)mentioning

confidence: 99%

Two improved classes of Broyden's methods for solving nonlinear systems of equations

Al-Towaiq¹,

Hour²

2017

J. Math. Computer Sci.

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In this paper, we propose two efficient algorithms based on Broyden's methods by using the central finite difference and modification of Newton's method for solving systems of nonlinear equations. The most significant features of these algorithms are their simplicity and excellent accuracy. Some numerical examples are given to test the validity of the proposed algorithms and for comparison reasons. Superior results show the efficiency and accuracy of the proposed algorithms and a tremendous improvements in Broyden's methods.

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A generalization of the Sherman–Morrison–Woodbury formula

Abstract: a b s t r a c tIn this paper, we develop conditions under which the Sherman-Morrison-Woodbury formula can be represented in the Moore-Penrose inverse and the generalized Drazin inverse forms. These results generalize the original Sherman-Morrison-Woodbury formula.

Cited by 93 publications

References 7 publications

Online Statistical Inference for Large-Scale Binary Images

Online Statistical Inference for Large-Scale Binary Images

Green function of correlated genes and the mechanical evolution of protein

Two improved classes of Broyden's methods for solving nonlinear systems of equations

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