# Questions tagged [least-squares]

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### Non-parametric regression and curvature

Given a finite set of points $(x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)$ in the plane, Linear Regression tells us how to find the straight line "$y=a+bx$" best approximating the given points, in the ...
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### RKHS/non-parametric regression with missing response values

I am interested in doing RKHS regression with missing response variables. Given input-output pairs $(x_i,y_i)$, I want to estimate a function $f(\cdot)$ as follows \begin{equation}f(x)\approx u(x)=\...
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### In practice, what's the fastest method to find a least square solution rather than using SVD decompostion?

I'm working on a real-time implementation of Lucas-Kanade for optical flow. However, the SVD decomposition to do achieve the least square method to reduce the error seems to take too much time. A ...
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I'm studying the difference between regularization in RKHS regression and linear regression, but I have a hard time grasping the crucial difference between the two. Given input-output pairs $(x_i,y_i)... 1answer 169 views ### Rank-constrained least-squares solution of the Sylvester matrix equation For the Sylvester matrix equation$AX+XB=C$, I want to find the least-squares solution$X$via $$\begin{array}{ll} \text{minimize} & \| AX + XB - C \|_{\text{F}}^2\\ \text{subject to} & \mbox{... 1answer 231 views ### Least square solution to AXB+CXD=E I am trying to find the least-squares solution X of the following matrix equation$$AXB+CXD=E$$Of course, I know that this equation can be written in the form$$(B^T \otimes A+D^T \otimes C) \... 2answers 300 views ### Standard solution to semidefinite program [closed] I have an optimization problem of the following form $$\text{minimize} \,\|Qa-b\|_2 \quad \text{ subject to } Q \succeq 0$$ where$a,b \in \mathbb{R}^n$are given and the$n \times n$square matrix ... 1answer 168 views ### Reconstruct matrix given all differences of neighbors We have an unknown$m\times n$matrix$X=(x_{ij})_{i=1,j=1}^{m,n}$. Assume we are given measurements of the differences $$x_{i,j+1}-x_{i,j}$$ and $$x_{i+1,j}-x_{i,j}$$ for all$(i,j)\in \{1,\...
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I need to clarify some idea I have in my mind about linear and non-linear regressions. Whatever I now about this topic comes from the book of Taylor "Introduction to error analysis": a set of ...
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### Why is ideal wavelet selection a least-squares estimate?

In their classic paper "Ideal spatial adaptation by wavelet shrinkage" (http://biomet.oxfordjournals.org/content/81/3/425.short?rss=1&ssource=mfr), Donoho and Johnstone make the following ...
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### MSE of measurable function is still conditional expectation

Motivation Then the usual stochastic filtering problem says that: $$\operatorname{argmin}_{Z \in L^2(\mathscr{G}_t)}\,\mathbb{E}[(Y_t-Z_t)^2],$$ where $\mathscr{G}_t$ is the $\sigma$-algebra ...
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### Least squares problem with constrained solution [closed]

If $a_{m\times 1}$ and $Q_{m\times n}$ ($m<n$) are known, and we know every element of $b$ is between $[-1\ \ 1]$, how to determine $b$ to minimize $\|a+Qb\|_2$?
I have the following matrix equation $$AX=B$$ given $8 \times 3$ matrices $A$ and $B$. $X$ is a $3 \times 3$ diagonal matrix whose main diagonal contains the $3$ unknowns. Whenever I solve for $X$ ...
The Sylvester equation $AX+XB=C$ has been studied quite a lot and there are known algorithms for solving it. But has the situation where (an over-determined) system of equations $A_{i}X+XB_{i}=C_{i}$ ...