### 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

### 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. $$\dot{x}=Ax\tag{4.22}$$ 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