We want to solve huge POMDP in the real world, but the belief states are huge. Notably, reachable beliefs are very small given an initial belief.

## Why is vanilla PCA bad

PCA as a denoising procedure: the underlying data is some data which is normally noised. This is not strictly true, the points don’t have normal noise.

## Better PCA: E-PCA

Instead of Euclidean distance, we use

\begin{equation} L(U,V) = \mid X-UV\mid^{2} \end{equation}

as a metric, where:

\(U\) the feature

specifically:

\begin{equation} F(z) - yz + F^{*}(y) \end{equation}

where \(F\) is any convex objective that is problem specific that you choose,

**Bregman Divergence** forces the underlying matricies’ bases to be non-negative

## Overall Methods

- collect sample beliefs
- apply the belifs into E-PCA
- Discretize the E-PCA’d belifs into a new state space \(S\)
- Recalculate R (\(R(b) = b \cdot R(s)\)) and T (we simply sample \(b,o\) and calculate \(update(b,a,o)\)) for that state space S; congratulations, you are now solving an MDP
- value iteration