Lowner-John Ellipsoid

minimum volume surrounding ellipsoid

Consider a set of ellipsoid \(C\). Minimum volume ellipsoid \(\epsilon\) with \(C \subset \epsilon\). We can parameterize \(\epsilon\) as \(\epsilon = \qty {v \mid \norm{Av + b}_{2} \leq 1}\); where we assume \(A \in \mathcal{S}_{++}^{n}\).

The volume is proportional to \(\text{det} A^{-1}\). Thus to find minimal-volume ellipsoid, solve:

\begin{align} \min_{A,b}\quad & \log \text{det} A^{-1} \\ \textrm{s.t.} \quad & \text{sup}_{v \in C} \norm{A v + b}_{2} \leq 1 \end{align}

In convex optimization, relax and round / polishing is a procedure by which you perform a local search after coming up with a relaxation, and round into the actual feasible set (such as integers).

experiment design problem, but such that we ignore that \(m \lambda_{k}\) are integers.

Key Sequence

Notation

New Concepts

• minimum volume elipsolid
• centering
  • analytic center of inequalities
• classification
  • linear classification is convex
  • robust linear discrimination

Important Results / Claims

• relax and round

Questions

Interesting Factoids

• linear classification is convex

New Model

China would need to shift to new growth model… Since 1980, it relied on:

1. mobilization on two mechanisms
2. shift in control of resources to state to private

There may be thus diminishing returns.

a new growth model

• China’s capacity outstrips global demand
• incomes no longerr rising as rapidly
• foreign direct investment is slowing

Expansion of credit

China bank access accounts for a large percentage of global GDP; thus bad debt can accumulate and hurt. Banks is thus driving the infrastructure drive via credit. Credit growth comes up, GDP growdth comes down; bad state! Yet, almost 20% of enterprises are loosing money.