Singular value decomposition is a factorization of a matrix, which is a generalization of the eigendecomposition of normal matricies (i.e. where \(A = V^{-1} D V\) when \(A\) is diagonalizable, i.e. by the spectral theorem possible when matricies are normal).

Definitions

Singular value decomposition Every \(m \times n\) matrix has a factorization of the form:

\begin{equation} M = U D^{\frac{1}{2}} V^{*} \end{equation}

where, \(U\) is an unitary matrix, \(D^{\frac{1}{2}}\) a diagonalish (i.e. rectangular diagonal) matrix with non-negative numbers on its diagonal called singular values, which are the positive square roots of eigenvalues of \(M^{* }M\) — meaning the diagonal of \(D^{\frac{1}{2}}\) is non-negative (\(\geq 0\)). Finally, \(V\) is formed columns of orthonormal bases of eigenvectors of \(M^{*}M\).

The SIR Model is a model to show how diseases spread.

• Susceptible – # of susceptible people
• Infectious — # of infectious people
• Removed — # of removed people

Compartmental SIR model

S => I => R [ => S]

So then, the question is: what is the transfer rate between populations between these compartments?

Parameters:

• \(R_0\) “reproductive rate”: the number of people that one infectious person will infect over the duration of their entire infectious period, if the rest of the population is entirely susceptible (only appropriate for a short duration)
• \(D\) “duration”: duration of the infectious period
• \(N\) “number”: population size (fixed)

Transition I to R:

skinning is a notion in graphics which solves Cloth Fitting Task which adds a prior \(S\qty(\theta)\) which is not learned to make the learning of \(D\qty(\theta)\) easier

\begin{equation} \phi = f_{}\qty(\theta) = S \qty(\theta) + D \qty(\theta) \end{equation}

so then, when we make a model, keeping around \(S\qty(\theta)\) which is learned by physical dynamics, and constraining a displacement map \(D\qty(\theta)\) within each square \(S\qty(\theta)\).

We can then rasterize the diffs into RGB space into a grid, then we can run a CNN.

https://en.wikipedia.org/wiki/Sliding_mode_control

it has a property of “chattering”, which was fixed by applying tanh.

Problem: we have a drone in an elevator; the elevator moves and hence the drone has a non-internal reference frame.

For a dynamical system which has uncertainty term which is not observable to the controller, one could control for position and velocity well with a Sliding Mode Controller

Sliding mode observer controls for the disturbance in the non-inertial reference frame.