0
votes
0answers
20 views

Minimum in complex inner product vector space

I'm stuck at this problem, can someone give me a hint? Let $x_i$ and $y_i$ ($i=\overline{1,n}$) be vectors in an infinite dimensional vector space $V$ with inner product $(,)$ satisfy: ...
3
votes
0answers
39 views

Find closest vector to a given vector from a particular set of vector

Let $x=\left(x_t\right)_{t=1}^n$ be a vector such that $$ x_t = \prod_{i=1}^t u_i, \tag{1} $$ where each parameters $u_i$ can take any of two value $$ u_i \in \left\{a,b \right\} = \left\{ 1.3, 0.8 ...
0
votes
2answers
25 views

how to impose binarity constraint in a vector

This is part of a homework problem. In an optimization problem, I need to have a K dimensional vector S, such that each entry of the vector is either 0 or 1, and $l_1$ norm of S is <= K. I can't ...
0
votes
0answers
13 views

find medoid of a set of objects described by their non-metric distances.

suppose i have a set of objects, their coordinates are unknown to me, I am given their symmetric pairwise distances, you can imagine a matrix of size n-by-n where (i, j)th element is the distance ...
0
votes
1answer
30 views

Expressing a vector as the best linear combination of “random” vectors

Suppose I have something like: $\vec{v} = \langle 1, 2, 3, 4, 5 \rangle$ and I have a set of vectors (these are all just made up numbers): $\vec{w_1} = \langle 3, 7, -2, -4, 8 \rangle$ $\vec{w_2} ...
3
votes
1answer
266 views

What are non-orthogonal eigenvectors?

Given a symmetric matrix $A$, the maximum of the trace, $Tr(Z^TAZ)$ under the assumption that $Z^TZ=I$ occurs when $Z$ has the eigenvectors of $A$, as $Tr(U^TAU)= \lambda_1 +\lambda_2+...\lambda_ d$ ...
0
votes
1answer
40 views

Directive on Dimensionality Reduction

I have a data set (24 data records) which is in $\mathbb{R}^{13}$ and I need to project it to a lower dimension (at least to $\mathbb{R}^{3}$). My objective of the dimensionality reduction is to ...
0
votes
1answer
69 views

Extrema of a vector norm under two inner-product constraints.

If $\langle\vec{A},\vec{V}\rangle=1\; ,\; \langle\vec{B},\vec{V}\rangle=c$, then: \begin{align} max\left \| \vec{V} \right \|_{1}=?\;\;\;min\left \| \vec{V} \right \|_{1}=? \end{align} Consider the ...
0
votes
1answer
56 views

Convex Function of Two Vectors

Let $f:\mathbb{R}^M\times\mathbb{R}^N\rightarrow\mathbb{R}$ be a mapping such that for $\mathbf{Y}\in\mathbb{R}^N$ constant, $f(\mathbf{X}, \mathbf{Y})$ is a convex function of $\mathbf{X}$ and for ...
1
vote
1answer
113 views

On norm selection for the solution of an overdetermined linear system

I am considering the following linear system: $Ax = b$ Where: $A$ is $9000 \times 139$ $x$ is $139 \times 1$ and sparse $b$ is $9000 \times 1$ Most of the resources I have found online point to ...
0
votes
1answer
76 views

Find a number that minimizes distance to a vector of sets of numbers

Assumptions $V$ is a vector of sets $V_1,V_2,...,V_n$ of numbers: $V=[V_1, V_2,..., V_n]^T, \forall_{i=1..n}V_i\subset\mathbb{R}$ $c\in\mathbb{R}$ is constant $d(V,c)$ is an error metric: ...
1
vote
0answers
193 views

Gradient Descent for Primal Kernel SVM with Soft-Margin(Hinge) Loss

Given the primal objective $$F({\bf a})=L\sum_{i,j}a_{i}a_{j}k(x_i,x_j) + \sum_{i}max(0, 1-y_i \sum_{j}a_jk(x_i,x_j)$$ for the soft margin SVM, where ${\bf a}=(a_1,...,a_N)$, N being the number of ...
1
vote
0answers
119 views

constrained optimization of dot product

Given a real matrix $A$ find a positive vector $x$ of unit length ($x^T x = 1$) for which $x^T A^T A x$ is minimal (closest to $0$). A has size about $1000 \times 20$ and can be written as $[ A_P | ...
2
votes
1answer
307 views

Find equation of line such that area formed by line & positive coordinate axis is minimal

Find equation of line passing through $(20,12)$ such that the area of the triangle formed by the line and the positive axis is smallest possible. Also: $\frac{x}{a}+\frac{x}{b}=1$ where $a, b$ are ...
0
votes
1answer
155 views

parametrize hypersphere

I want to find an $n$-vector $\hat{\theta}$ that maximizes a function $f(\theta)$ subject to the $p$-norm constraint $||\theta||_p = c$. Is there a general parametrization of $p$-norm hyperspheres ...
1
vote
2answers
211 views

Finding $\min_{\mathbf x} (\mathbf y - \mathbf G\mathbf x)^T(\mathbf y - \mathbf G\mathbf x)$

Let $\mathbf G$ be a given $m \times n$ matrix, $\mathbf y$ a given $m \times 1$ column vector and $\mathbf x$ an unknown $n \times 1$ column vector such that $\mathbf x \ge 0$. 1) How do you find ...