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4

Let's solve a more general problem (Least Squares with Linear Equality Constraints): $$ \begin{alignat*}{3} \arg \min_{x} & \quad & \frac{1}{2} \left\| A x - b \right\|_{2}^{2} \\ \text{subject to} & \quad & C x = d \end{alignat*} $$ The Lagrangian is given by: $$ L \left( x, \nu \right) = \frac{1}{2} \left\| A x - b \right\|_{2}^{2} + {\...


4

One way to interpret the Tikhonov Regularization is using the Maximum A Posteriori (MAP) framework. Lets' say we have a model of the form: $$ \boldsymbol{y} = H \boldsymbol{x} + \boldsymbol{n} $$ Where $ \boldsymbol{n} \sim N \left( 0, {\sigma}_{n}^{2} \right) $, namely Additive White Gaussian Noise, and the prior knowledge about $ \boldsymbol{x} $ is $ \...


3

I suggest this reference regarding the comparison between least-squares and Kalman filters : Fundamentals of Kalman Filtering: A Practical Approach by P. Zarchan & H. Mussof Especially Chapter 3 (Recursive Least-Squares Filtering) and Chapter 4 (Polynomial Kalman Filters). In Chapter 4, the authors show that the discrete (time) n-th order polynomial ...


2

This is an example of the Fidelity Term and Prior Term model. In many Inverse Problems we assume some model on the additive noise. This part is modeled by the Fidelity Term ($ \mathcal{D} \left(A \boldsymbol{f}, \tilde{\boldsymbol{g}} \right) $ in your example). For Gaussian Noise it is given by Least Squares Term: $$ \frac{1}{2} {\left\| A \boldsymbol{f} - \...


2

Note that for a small sampling interval $T$, $\big(d[k+1]-d[k]\big)/T$ is a good approximation for the velocity. So if you fit $au[k]+b$ to a given set of measurements $v[k]$, it is valid to conclude $$d[k+1]=d[k]+T\big(au[k]+b\big)\tag{1}$$ In the text you refer to they might have normalized $T$, so it changes the units without changing the values of $a$ ...


1

Ideas: 1) Find the minimum eigen vector of $R_1$ and assign this to $w$. This will minimize $ J_1$ , doesn't maximize $J_2$. But when $R_1$ and $R_2$ are positive definite or full rank matrices and computation is an issue, this is a decent solution. 2) Form a new objective $J_1 - J_2$ and minimize this with the given constraint 3) Try and formulate as ...


1

I will give you a hint: you can first relax this problem to be a convex optimization problem by editing the second constraint as $$Xw <= \vec{1}$$ where the inequality is elementwise, then form the dual problem or the lagrangian as it is known popularly $$ w^TRw + \lambda^T(Xw -1) \tag{1}$$ where $$\lambda <=\vec{0}$$ Differencate (1) with respect ...


1

$$ J(h) = \int_{R^+UR-}|e^Th-F_d|^2d\omega\tag{1}\\ = \int(e^Th-F_d)^H(e^Th-F_d)d\omega\\ = \int((e^Th)^H(e^Th) + F_d^HF_d -(e^Th)^HF_d - F_d^He^Th)d\omega\\ = h^H(\int(e^T)^He^Td\omega) h + \int |F_d|^2d\omega -\int (2 Re\{(e^Th)^HF_d\})d\omega $$ The second term in above integral is the integral of $L_2$ norm of $F_d$ in the region $R^+ U R^-$. Since $F_d(...


1

The problem that recursive least squares (RLS) can solve can be formulated as recursively solving for $\hat{\theta}$, such that it is the least squares solution to $$ \hat{\theta}_n = \arg\min_x \sum_{k=0}^n w[k]\,\|z[k] - \phi[k]^\top x\|_2^2, $$ where $w[k]$ are weights, $z[k]$ and $\phi[k]$ are known and $z[k]$ is assumed to be generated by using $\phi[...


1

The reason x[n] must be white is because the solution will effectively spectrally weight the channel response based on the amount of energy present in each spectral frequency location. A white noise source provides equal weight to all frequencies. If energy is not present in any particular frequency bin, a proper solution cannot be found for that frequency. ...


1

$$(A^TA)^{-1}A^T(y + noise) = \hat x$$ where $noise$ is vector of the same size as $y$ with all elements equal to unknown constant. That is all. You can get your estimated solution as a function of noise mean position in explicit form.


1

I can add that LMS algorithm has a sample-based update.


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