We’re rewarding the question askers & reputations are being recalculated! Read more.

2 added 1917 characters in body

Consider the system $$\tag 1 H\delta x=-g$$ where $$H$$ and $$g$$ are the Hessian and gradient of some cost function $$f$$ of the form $$f(x)=e(x)^Te(x)$$. The function $$e(x)=z-\hat{z}(x)$$ is an error function, $$z$$ is an observation (measurement) and $$\hat{z}$$ maps the estimated parameters to a measurement prediction.

This minimization is encountered in each iteration of many SLAM algorithms, e.g.one could think of $$H$$ as a bundle adjustment Hessian. Suppose $$x=(x_1,x_2)^T$$, and let $$x_2$$ be some variables that we seek to marginalize. Many authors claim that this marginalization is equivalent to solving a smaller liner system $$M\delta x_1=-b$$ where $$M$$ and $$g$$ are computed by applying Schur's complement to (1), i.e. if $$H= \begin{pmatrix} H_{11} & H_{12}\\ H_{21} & H_{22} \end{pmatrix}$$ then $$M=H_{11}-H_{12}H_{22}^{-1}H_{21}$$ and $$b=g_1-H_{12}H_{22}^{-1}g_2$$

I fail to understand why that is equivalent to marginalization... I understand the concept of marginalization for a Gaussian, and I know that schur's complement appears in the marginalization if we use the canonical representation (using an information matrix), but I don't see the link with the linear system.

Edit: I understand how Schur's complement appears in the process of marginalizing or conditioning $$p(a,b)$$ with $$a,b$$ Gaussian variables, as in the link supplied by Josh Vander Hook. I had come to the same conclusions, but using the canonical notation: If we express the Gaussian $$p(a,b)$$ in canonical form, then $$p(a)$$ is gaussian and its information matrix is the Schur complement of the information matrix of $$p(a,b)$$, etc. Now the problem is that I don't understand how Schur's complement appears in marginalization in bundle adjustment (for reference, in these recent papers: c-klam (page 3 if you want to look) and in this (part titled marginalization). In these papers, a single bundle adjustment (BA) iteration is performed in a manner similar to what I initially described in the question. I feel like there is a simple connection between marginalizing a Gaussian and the marginalization in BA that I am missing. For example, one could say that optimizing $$f$$ (one iteration) is equivalent to drawing a random variable following a denstiy $$e^{-\frac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu)}$$ where $$\Sigma$$ is the inverse of the Hessian $$H$$ of $$f$$, and $$\mu$$ is the true value for $$x$$ (or an approximation of that value), and that marginalizing this density is equivalent to using Schur's compelement in the bundle? I am really confused...

Consider the system $$\tag 1 H\delta x=-g$$ where $$H$$ and $$g$$ are the Hessian and gradient of some cost function $$f$$ of the form $$f(x)=e(x)^Te(x)$$. The function $$e(x)=z-\hat{z}(x)$$ is an error function, $$z$$ is an observation (measurement) and $$\hat{z}$$ maps the estimated parameters to a measurement prediction.

This minimization is encountered in each iteration of many SLAM algorithms, e.g.one could think of $$H$$ as a bundle adjustment Hessian. Suppose $$x=(x_1,x_2)^T$$, and let $$x_2$$ be some variables that we seek to marginalize. Many authors claim that this marginalization is equivalent to solving a smaller liner system $$M\delta x_1=-b$$ where $$M$$ and $$g$$ are computed by applying Schur's complement to (1), i.e. if $$H= \begin{pmatrix} H_{11} & H_{12}\\ H_{21} & H_{22} \end{pmatrix}$$ then $$M=H_{11}-H_{12}H_{22}^{-1}H_{21}$$ and $$b=g_1-H_{12}H_{22}^{-1}g_2$$

I fail to understand why that is equivalent to marginalization... I understand the concept of marginalization for a Gaussian, and I know that schur's complement appears in the marginalization if we use the canonical representation (using an information matrix), but I don't see the link with the linear system.

Consider the system $$\tag 1 H\delta x=-g$$ where $$H$$ and $$g$$ are the Hessian and gradient of some cost function $$f$$ of the form $$f(x)=e(x)^Te(x)$$. The function $$e(x)=z-\hat{z}(x)$$ is an error function, $$z$$ is an observation (measurement) and $$\hat{z}$$ maps the estimated parameters to a measurement prediction.

This minimization is encountered in each iteration of many SLAM algorithms, e.g.one could think of $$H$$ as a bundle adjustment Hessian. Suppose $$x=(x_1,x_2)^T$$, and let $$x_2$$ be some variables that we seek to marginalize. Many authors claim that this marginalization is equivalent to solving a smaller liner system $$M\delta x_1=-b$$ where $$M$$ and $$g$$ are computed by applying Schur's complement to (1), i.e. if $$H= \begin{pmatrix} H_{11} & H_{12}\\ H_{21} & H_{22} \end{pmatrix}$$ then $$M=H_{11}-H_{12}H_{22}^{-1}H_{21}$$ and $$b=g_1-H_{12}H_{22}^{-1}g_2$$

I fail to understand why that is equivalent to marginalization... I understand the concept of marginalization for a Gaussian, and I know that schur's complement appears in the marginalization if we use the canonical representation (using an information matrix), but I don't see the link with the linear system.

Edit: I understand how Schur's complement appears in the process of marginalizing or conditioning $$p(a,b)$$ with $$a,b$$ Gaussian variables, as in the link supplied by Josh Vander Hook. I had come to the same conclusions, but using the canonical notation: If we express the Gaussian $$p(a,b)$$ in canonical form, then $$p(a)$$ is gaussian and its information matrix is the Schur complement of the information matrix of $$p(a,b)$$, etc. Now the problem is that I don't understand how Schur's complement appears in marginalization in bundle adjustment (for reference, in these recent papers: c-klam (page 3 if you want to look) and in this (part titled marginalization). In these papers, a single bundle adjustment (BA) iteration is performed in a manner similar to what I initially described in the question. I feel like there is a simple connection between marginalizing a Gaussian and the marginalization in BA that I am missing. For example, one could say that optimizing $$f$$ (one iteration) is equivalent to drawing a random variable following a denstiy $$e^{-\frac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu)}$$ where $$\Sigma$$ is the inverse of the Hessian $$H$$ of $$f$$, and $$\mu$$ is the true value for $$x$$ (or an approximation of that value), and that marginalizing this density is equivalent to using Schur's compelement in the bundle? I am really confused...

1

# SLAM : Why is marginalization the same as schur's complement?

Consider the system $$\tag 1 H\delta x=-g$$ where $$H$$ and $$g$$ are the Hessian and gradient of some cost function $$f$$ of the form $$f(x)=e(x)^Te(x)$$. The function $$e(x)=z-\hat{z}(x)$$ is an error function, $$z$$ is an observation (measurement) and $$\hat{z}$$ maps the estimated parameters to a measurement prediction.

This minimization is encountered in each iteration of many SLAM algorithms, e.g.one could think of $$H$$ as a bundle adjustment Hessian. Suppose $$x=(x_1,x_2)^T$$, and let $$x_2$$ be some variables that we seek to marginalize. Many authors claim that this marginalization is equivalent to solving a smaller liner system $$M\delta x_1=-b$$ where $$M$$ and $$g$$ are computed by applying Schur's complement to (1), i.e. if $$H= \begin{pmatrix} H_{11} & H_{12}\\ H_{21} & H_{22} \end{pmatrix}$$ then $$M=H_{11}-H_{12}H_{22}^{-1}H_{21}$$ and $$b=g_1-H_{12}H_{22}^{-1}g_2$$

I fail to understand why that is equivalent to marginalization... I understand the concept of marginalization for a Gaussian, and I know that schur's complement appears in the marginalization if we use the canonical representation (using an information matrix), but I don't see the link with the linear system.