Questions tagged [least-squares]

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2answers
48 views

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 ...
1
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3answers
69 views

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)=\...
2
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1answer
284 views

Minimise $\sum_i \begin{Vmatrix}\boldsymbol{x}_i \\ \boldsymbol{y}_i\end{Vmatrix}$

Consider column vectors $\boldsymbol{z}_i$, $\quad i=1,\dots,n$. Each $\boldsymbol{z}_i$ has $j$ elements and can be expressed as $\boldsymbol{z}_i = \begin{bmatrix} \boldsymbol{x}_i \\ \boldsymbol{y}...
5
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2answers
190 views

Symmetric linear least-squares solution

Given tall matrices $A$ and $Y$ and the following overdetermined linear system in square matrix $X$ $$AX=Y$$ is there an explicit formula for the least-squares solution if $X$ is constrained to be ...
0
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1answer
180 views

How to determine the damping factor in Levenberg-Marquardt?

From the algorithm, we can see that it tries different damping factor until it gets a good one by the error. Is the damping factor related to the eigenvalues of the Hessian matrix?
2
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0answers
17 views

Difference Between Total Least Squares Plane and Plane Orthogonal to Principal Axis of Inertia Tensor

Given a finite set $P$ of points in $\mathbb{E}^3$ , one can calculate an approximating plane either as the solution of a Total Least Squares problem or by interpreting the problem physically, ...
1
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1answer
79 views

Minimization Proof of Conditioning on Gaussian is Gaussian

It is well known that $E[X|X+Y]$ is Gaussian if both $X$ and $Y$ are, and the result can be derived using standard density arguments. However, how can one prove it by only resulting to optimization ...
1
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0answers
92 views

Choice of residual function for least squares error minimization

Good morning, I have the a set of data $(\sigma,D,\alpha_0)_i$, $i=1...n$ data. I want to determine two parameters $K_{IC}$, $C_f$ in the basic equation given as $K_{IC} = \sigma \sqrt{D} k_0(\...
2
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2answers
140 views

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 ...
8
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3answers
226 views

Regularized linear vs. RKHS-regression

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)...
5
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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{...
4
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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) \...
3
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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 ...
2
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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,\...
4
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1answer
10k views

Gauss-Newton vs Gradient Descent vs Levenberg-Marquadt for least squared method

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 ...
0
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0answers
64 views

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 ...
1
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1answer
98 views

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 ...
1
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1answer
155 views

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 $\|a+Qb\|_2$?
2
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1answer
201 views

How to force least squares solution matrix to be diagonal? [closed]

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 $X$ ...
7
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1answer
157 views

Least-squares solution of systems of Sylvester equations

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 $A_{i}X+XB_{i}=C_{i}$ ...