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I am reading the book *Multidimensional Particle Swarm Optimization for Machine Learning and Pattern Recognition*.

They use $L_{2}$ *Minkowski norm* (Euclidean) as the distance metric in the feature space for *Long-Term ECG Classification*.

I am myself using just $L^{2}$ seminorm.

I did not find reason why they use Minkowski norm.

Little info here what is Minkowski space.

The book Riemann-Finsley Geometry 2005 says that

- Any two norms on finite dimensional space are equivalent
- show operator norm submultiplicative
- relation between norms of two vectors
- Why is this true: $\|x\|_1 \le \sqrt n \cdot \|x\|_2$?
- Gradient of the TV norm of an image
- Subordinate matrix norm

The book *The Geometry and Spacetime – An Introduction to Special and General Relavity 2000* says that

The Minkowski geometry of spacetime as the invariant theory of Lorentz

transformations, making constant comparisons with the familiar

Euclidean geometry of ordinary space as the invariant theory of

rotations.

The Minkowski space has been used in the inverse problem for nonlinear hyperbolic equations.

**What are the advantages of Minkowski norm to $L^{2}$ seminorm when considering ECG classification?**

- Optimising $x_1x_2+x_2x_3+\cdots+x_nx_1$ given certain constraints
- Why is this weighted least squares cost function a function of weights?
- Quasiconvex and lower semicontinuous function
- Minimizing $\cot^2 A +\cot^2 B + \cot^2 C$ for $A+B+C=\pi$
- Control on Conformal map
- Duality. Is this the correct Dual to this Primal L.P.?
- Optimization over union of convex sets
- Fundamental Optimization question consisting of two parts.
- Optimisation Problem on Cone
- Proof of Neumann Lemma

The authors of *Multidimensional Particle Swarm” use the ordinary Euclidean metric. They just give it a strange name “$L^2$ Minkowski norm”, in my opinion unnecessarily. But it may be common within the subject area.

The book *Riemann-Finsler Geometry* gives the definition of a related concept, but it’s not the definition the authors of the first book use. Notice that this is a book from a rather different area of mathematics.

The book *Geometry and Spacetime* is talking about yet different concept.

Minkowskian distance between two vectors $\mathbf{x}_i$ and $\mathbf{x}_j$ forms the general equation for $L^m$, which is defined as

\begin{equation}

d(\mathbf{x}_i,\mathbf{x}_j)=\biggl( \sum_{k=1}^p (x_{ik}-x_{jk})^m \biggr)^{1/m},

\end{equation}

where $p$ is the number of features.

The Euclidean and Manhattan distance are simply special cases of the $L^1$ and $L^2$ distance, respectively.

Quite basically, you can use almost any distance metric you want for the fitness function. For example, the dot product: $\mathbf{x}_i^T\mathbf{x}_j$, polynomial kernels such as $K(\mathbf{x}_i,\mathbf{x}_j)=(1+\mathbf{x}_i^T\mathbf{x}_j)^d$, or radial basis kernels:

$K(\mathbf{x}_i,\mathbf{x}_j)=\exp[-d(\mathbf{x}_i,\mathbf{x}_j)]$.

Minkowski is merely a general family of distances, and when working with metaheuristics such as PSO it is better to evaluate different distance metrics for their informativeness in class prediction (classification).

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