matrix equations - Deep Learning Book - Early Stopping and L2 Regularization

In Goodfellow et al.'s Deep Learning, on page 245, when proving the equivalence of early stopping and $L_2$ regularization,The equation (7.41) is given as$$\mathbf{Q}^T \tilde{\mathbf{w}} = (\Lambda + \alpha\mathbf{I})^{-1}\Lambda\mathbf{Q}^T\mathbf{w}^*$$And it is rearranged (Equation (7.42)) to $$\mathbf{Q}^T \tilde{\mathbf{w}} = [\mathbf{I} - (\Lambda + \alpha\mathbf{I})^{-1}\alpha] \mathbf{Q}^T\mathbf{w}^*$$Can anyone please show me how this rearrangement is done?Thank you....Read more

matrix equations - How to derive the solution of Tikhonov Regularization via SVD

The solution to Tikhonov Regularization is$$x=(A^HA+\sigma^2_{min}I)^{-1}A^Hb$$where $\sigma^2_{min}$ is the minimum of the singular values of $A$.Then we apply $SVD$ to $A$ such that,$$A=U\Sigma V^H$$then the solution is,$$x=(V\Sigma^2 V^H+\sigma^2_{min} I)^{-1}V\Sigma U^Hb$$But on the textbook, it says that the solution could be simplified such as$$x=V(\Sigma^2+\sigma^2_{min} I)^{-1}\Sigma U^Hb$$I cannot find the solution above myself. I guess Woodbury Matrix Indentity should be helpful...Thanks!...Read more

linear algebra - Deriving the fundamental equation (with regards to computer vision)

I'm having a hard time understanding how a few equations are being derived. So the fundamental equation is an equation that relates corresponding points in stereo images. Anyway, that's the basic background. The question is more on the matrix manipulation. So I'm trying to get (1) and (2) into (3) and I have(1) $\tilde m_1 = A_1[I\hspace{5 pt}0]\tilde M $(2) $\tilde m_2 = A_2[R\hspace{5 pt}t]\tilde M$ Eliminating $M$ from the above equations you can obtain(3) $\tilde m_2^T \hspace{2 pt} A_2^{-T} \hspace{2 pt} [t]_x\hspace{2 pt} R \hspace{2 pt}A...Read more

How to solve the following quadratic matrix equation?

Solve the following matrix equation in $D$ $$ A=D^{T}(DVD^{T}+\alpha\lambda_{\max}(D^{T}D)I)^{-1}D$$ where $I$ is the identity matrix, $A$ and $V$ are known matrices, $\alpha$ is a known real positive constant, and $\lambda_{\max}$ denotes the maximum eigenvalue.I tried to use SVD decomposition of $A$ in order to find the solution $D$, but it seems like the matrix $A$ has to respect a condition in order to have a solution $D$. Can anyone help me find this condition please? Does anyone have any idea how to solve it in order to find $D$, pl...Read more

Solve for matrix that is hidden inside a scalar

Let $X$ an invertible $n\times n$ matrix, parameter vectors $P, K$ $n\times 1$, vector $\Omega$, $1\times k$ and matrix $\Lambda$ $n\times k$. I would like to solve with respect to $X$ the equation:$$P^TX^{-1}KP^{T}X^{-1}\Lambda=\Omega$$The gist of my question is how does one take out the unknown matrix from a scalar product....Read more

solving for scalar involving matrix equations

consider following matrix equation$\boldsymbol{1}^T \Sigma^{-1} ( \bar{\textbf{x}} - n \boldsymbol{1}. c) = 0 $ here $\boldsymbol{1} , \bar{\textbf{x}}$ are vectors of dimension $n \times 1$ and $\Sigma$ is invertible matrix of dimension $n \times n$ and n,c are constantsis there any way to solve it for scalar c ?$\boldsymbol{1}^T \Sigma^{-1}\bar{\textbf{x}} = (n.c)\boldsymbol{1}^T \Sigma^{-1}\boldsymbol{1}$and I am struck at here...Read more

Isolate vector in matrix equation

I have this equation:$$H=\frac{t\cdot n^T}{n^T\cdot x}$$with $t$, $n$ and $x$ being $3\times 1$ column vectors and $H$ a $3\times 3$ matrix, and where $\cdot$ is matrix multiplication. Notice that on the RHS the numerator results in a $3\times 3$ matrix and the denominator in a scalar.How can I isolate the vector $n$? UPDATE:This is the part of a larger equation that if I can solve I can solve the original. I've been messing around but I'm stuck and I don't know how to solve for vector n. I have tried to use sympy to isolate n but there I have ...Read more

nonlinear optimization - Solve equation with matrix variable

I want to solve a matrix $\Omega$ from a equation $\sum_k (\Omega + \Theta_k)^{-1} = Q$. The $Q$ and $\Theta, \forall k=1...K$ are known, and are positive definite matrices. $\Omega$ also has to be positive definite. all matrices are large (a few thousands of columns and rows). My questions are:(1) Is there a closed-form solution? How do I simplify the sum of the inverse of two matrix sum?(2) I'm OK to go for a numerical solution. But how do I define this problem? An optimization problem to minimize something like $f(\Omega) = ||\sum_k (\Omega ...Read more

matrix equations - Stability issue with least squares solver

I am using a least squares solver to auto-calibrate delta printers in firmware. The system measures height errors at 16 points, computes the height derivative with respect to the calibration parameters, and uses linear least squares to adjust the calibration parameters so as to minimise the sum of the squares of the height errors.I have just added another two calibration parameters to the original 6, and the problem I am getting is that these extra 2 are jumping around when I repeat the calibration operation. Here is a typical calculation, with...Read more

Least squares equations and Matrix Algebra

I have a question regarding the manipulation of linear least squares equations. Suppose $x$ satisfies$$A x = b$$in a least-squares sense, where $A$ is $m \times n$ with $m > n$, $x$ is $n \times 1$ and $b$ is $m \times 1$. If I use a $QR$ factorization of $A$, I can write\begin{align*}A x &= b \\QRx &= b \\Rx &= Q^T b \\x &= R^{-1} Q^T b\end{align*}and so it seems I can apply $Q^T$ and $R^{-1}$ to the rhs side of the equation without issue, since this is the well-known $QR$ least squares solution.However, now consider the fol...Read more

stability theory - Lyapunov Matrix Equation Theorem REDUNDANT?

I read the book A Linear Systems Primer by Antsaklis, Panos J., and Anthony N. Michel (Vol. 1. Boston: Birkhäuser, 2007) and I think a part of a theorem about the Lyapunov Matrix Equation seems wordy. \begin{equation} \dot{x}=Ax\tag{4.22} \end{equation} Theorem 4.29. Assume that the matrix A [for system (4.22)] has no eigenvalues with real part equal to zero. If all the eigenvalues of A have negative real parts, or if at least one of the eigenvalues of A has a positive real part,then there exists a quadratic Lyapunov function \begin{equation...Read more

least squares - Minimum norm solutions: Integral Equation

I wish to find the matrix that minimises the following Loss function$$\mathbf{A}^* = \text{argmin}_\mathbf{A} \int_{\mathbf{x}\in\mathbb{X}} \|\mathbf{A}\mathbf{x}-\mathbf{y}(\mathbf{x})\|^2d\mathbf{x}$$Is there a closed form solution or expression?I tried to find a solution based on the somewhat similar and simpler Linear Least Squares problem$$\mathbf{A}^* = \text{argmin}_\mathbf{A} \|\mathbf{A}\mathbf{X}-\mathbf{Y}\|^2$$where $\mathbf{X}$ is a matrix where each column is sampled from the $\mathbb{X}$ space, and $\mathbf{Y}$ is the matrix wit...Read more

matrix equations - Analytical solution to nonlinear least-squares problem

I have a data set which can be fit well to a single gaussian model, with dependent variables $y_i$ and independent variables $x_i$, with $i=1...N$. I want to avoid using a nonlinear fitting library, and to this end I worked through the math and ended up with an pseudo-analytical solution to the least-squares problem. If we set up$y=A\exp\left(\frac{-(x-\mu)}{2\sigma^2}\right)$Then taking $\ln$ of both sides and some straightforward matrix algebra gives the following matrix equation:$ \left( \begin{array}{c}-\frac{1}{2\sigma^2} \\\frac{\mu}{\sig...Read more