Computaional Complexity of Frobenius Norm

How can I calculate the computaional complexity of Frobenius norm of each column vector(M X 1) in a M X N matrix and finally sorting the norm values in descending order? To clarify I have N-column vectors in a matrix and I want to calculate the magnitude of each column and finally arranging in a descending order in an algorithm. So what will be the computational complexity of doing this task in a a Big-O notation?

Subset of $k$ vectors with shortest sum, with respect to $\ell_\infty$ norm

I have a collection of $ n$ vectors $ x_1, …, x_n \in \mathbb{R}_{\geq 0}^{d}$ . Given these vectors and an integer $ k$ , I want to find the subset of $ k$ vectors whose sum is shortest with respect to the uniform norm. That is, find the (possibly not unique) set $ W^* \subset \{x_1, …, x_n\}$ such that $ \left| W^* \right| = k$ and

$ $ W^* = \arg\min\limits_{W \subset \{x_1, …, x_n\} \land \left| W \right| = k} \left\lVert \sum\limits_{v \in W} v \right\rVert_{\infty}$ $

The brute-force solution to this problem takes $ O(dkn^k)$ operations – there are $ {n \choose k} = O(n^k)$ subsets to test, and each one takes $ O(dk)$ operations to compute the sum of the vectors and then find the uniform norm (in this case, just the maximum coordinate, since all vectors are non-negative).

My questions:

  1. Is there are a better algorithm than brute force? Approximation algorithms are okay.

One idea I had was to consider a convex relaxation where we assign each vector a fractional weight in $ [0, 1]$ and require that the weights sum to $ k$ . The resulting subset of $ \mathbb{R}^d$ spanned by all such weighted combinations is indeed convex. However, even if I we can find the optimum weight vector, I am not sure how to use this set of weights to choose a subset of $ k$ vectors. In other words, what integral rounding scheme to use?

I have also thought abut dynamic programming but I’m not sure if this would end up being faster in the worst-case.

  1. Consider a variation where we want to find the optimal subset for every $ k$ in $ [n]$ . Again, is there a better approach than solving the problem naively for each $ k$ ? I think there ought to be a way to use the information from runs on subsets of size $ k$ to those of size $ (k + 1)$ and so on.

  2. Consider the variation where instead of a subset size $ k$ , one is given some target norm $ r \in \mathbb{R}$ . The task is to find the largest subset of $ \{x_1, …, x_n\}$ whose sum has uniform norm $ \leq r$ . In principle one would have to search over $ O(2^n)$ subsets of the vectors. Do the algorithms change? Further, is the decision version (for example, we could ask if there exists a subset of size $ \geq k$ whose sum has uniform norm $ \leq r$ ) of the problem NP-hard?

Is it a norm in android apps to only have bottom navigation only on parent activities

I have checked multiple apps, and in most of them, bottom navigation is only visible on parent activities, when i go in any inner activity bottom navigation goes away, so my question is that is there any guideline regarding this available in material design, or everyone is just following this approach without any proper guideline,

and is it a right approach to have bottom navigation only on parent activities.

and in which cases we hide navigation on child activities, if not in all.

For example, i have an item in navigation Team, when user taps on it i open Team Screen, which contains 5 items, so when user further goes into these items i disable navigation, is it the correct approach?

Given a system in $\mathbb{F}_2$ in RREF, how do I find a solution of minimal norm?

I have a $ 12 \times 12$ (so not really large) system of linear equations in $ \mathbb{F}_2$ which I got to RREF through the usual row reduction. Suppose the system has multiple solutions, and call the unknowns $ x_i$ . What is the least expensive way to find a solution that minimizes the amount of $ x_i$ ‘s such that $ x_i = 1$ , or equivalently, a solution of minimal norm? Is this solution unique?

Moments of the Schatten norm of matrix

I am wondering what is the connection between order of the moment of the p-th shatten norm of the matrix and order if the shatten norm itself.

More precisely, why one would ever sick for the bound of the p-th moment of the p-shatyen norm of the matrix, why we would not consider the q-th moment of the p-th Shatten norm?

(see for example https://scholar.google.nl/scholar?hl=en&as_sdt=0%2C5&as_vis=1&q=moment+of+schatten+norm&btnG=#d=gs_qabs&u=%23p%3Dap9f83X_ylcJ)

Reference request: norm topology on M(X) vs. weak topology

Let $ (X,d)$ be a metric space and $ \mathcal{M}(X)$ be the space of regular (e.g. Radon) measures on $ X$ . There are two standard topologies on $ \mathcal{M}(X)$ : The (probabilist’s) weak topology and the strong norm topology, where the norm is the total variation norm.

Surprisingly, I have found very little discussion in the literature comparing these two topologies rigourously, besides the oft-cited claim that the norm topology is much stronger than the weak topology. I am looking for a reference that discusses and compares these topologies, esp. things like convergence, boundedness, open sets, projections, etc.

I am mostly concerned with probability measures $ \mathcal{P}(X)\subset\mathcal{M}(X)$ , but I am not sure how much of a difference this makes wrt topological concerns.

Luxemburg norm as argument of Young’s function: $\Phi\left(\lVert f \rVert_{L^{\Phi}}\right)$

Let $ \Phi$ be a Youngs’s function, i.e. $ $ \Phi(t) = \int_0^t \varphi(s) \,\mathrm d s$ $ for some $ \varphi$ satifying

  1. $ \varphi:[0,\infty)\to[0,\infty]$ is increasing
  2. $ \varphi$ is lower semi continuous
  3. $ \varphi(0) = 0$
  4. $ \varphi$ is neither identically zero nor identically infinite

and define the Luxemburg norm of $ f:\Omega\to\mathbb{R}$ as $ $ \lVert f \rVert_{L^{\Phi}} := \inf \left\{\gamma>0\,\middle|\, \int_{\Omega} \Phi\left(\frac {\lvert f(x)\rvert}{\gamma} \right)\,\mathrm{d}x\right\}.$ $


Question: What can we say about $ \Phi\left(\lVert f \rVert_{L^{\Phi}}\right)$ ? In particular, I’d like to know, if $ $ \Phi\left(\lVert f \rVert_{L^{\Phi}}\right) \leq C \int_{\Omega}\Phi(\lvert f(x)\rvert) \,\mathrm d x$ $ holds for some $ C$ independent of $ f$ .

Any idea or hint for a reference is welome!


Notes:

  • The above inequality trivially holds for $ \Phi(t) = t^p$ , where $ p>1$
  • Maybe it’s appropriate to consider this question in the more general framework of Musielak-Orlicz spaces. However, e.g. in Lebesgue and Sobolev Spaces with Variable Exponents I was unable to find an appropriate result.
  • I have asked this question on Math.Stackexchange without luck, so I’m trying here.

Schur norm of weighted Cauchy matrix

The Schur norm of a matrix $ A$ is defined to be $ \|A\|_S=\max\{\|A\circ X\|: \|X\|\leq 1\}$ , where $ \|\cdot \|$ is the operator norm of a matrix, i.e., the largest singular value.

Let $ a_1,\ldots, a_m, b_1,\ldots, b_n$ be positive reals.Let $ A$ be an $ m\times n$ matrix defined to be $ A_{i,j}=(a_i-b_j)/(a_i+b_j)$ .

My question is how to compute $ \|A\|_S$ . Is it upper bounded by an absolute constant independent of $ m, n$ ?

The norm squared of a moment map

I am studying the paper by E. Lerman: https://arxiv.org/pdf/math/0410568.pdf

Let $ (M,\sigma)$ be a connected symplectic manifold with an hamiltonien action of a compact Lie group $ G$ , so that there exist a moemnt map $ $ \mu : M\to\mathcal{G}^\ast$ $ $ \mathcal{G}^\ast$ being the dual of the Lie algebra of $ G$ . We assume that $ \mu$ is $ G$ -equivariant: $ $ \mu(g\cdot x)=\mathrm{Ad}_g^\ast\circ\mu(x)$ $ and that $ \mu$ is proper (the preimage of any compact is compact). Let $ f=\|\mu\|^2$ (for an Ad-invariant norm on $ \mathcal{G}^\ast$ ).

I know that the moment map is important by:

1- a convexity theorem of Atiyah and Guillemin-Sternberg.

2- symplectic reduction, where the quotient of the zero level of the moment map by the group makes it possible to construct new symplectic manifolds.

Hence my question:

What is the motivation to study the norm squared of a moment map? in particular, why is it important to know that the zero level set of the moment map is a retract by deformation of a piece of the manifold?

As I understand it, $ f$ behaves like a Morse-Bott function (Kirwan works) and that the stable manifold of a critical component of $ f$ is a submanifold. That the gradient flow of $ f$ is defined for all $ t\geq0$ . Here Lerman asserts that this is true because $ f$ is proper, but $ x^3$ is proper but its gradient $ -3x^2\partial_x$ is not defined for all $ t\geq0$ .

I think we have to show that $ \nabla_f$ is $ G$ -invariant and therefore complete.

That $ f$ is real analytic to show that the limit of a trajectory of any point $ \phi_t(x)$ is reduced to a point $ \phi_\infty(x)$ . That the applications $ t\to\phi_t(x)$ and $ x\to\phi_\infty(x)$ are continuous.

Linear regression: not noramalising by y’s norm

I was recently reading an article on Pearson correlation, and OLS coefficients. I came across the following section.

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I understand that using calculus we can arrive at an expression for finding a, the coefficient. The expression’s denominator turns out to not contain y’s norm. In the last paragraph of the excerpt, I could not understand the following line

Not normalizing for y is what you want for the linear regression

Why don’t we want to normalize for y? What is the physical/geometrical significance of this?