lsc directions

2021-04-17
1 min read

Disentangling Images with Lie Group Transformations and Sparse Coding

Directions for future work…

Background

We represent some image $I \in \mathbb{R^D}$ as

$$ I = WR(s)W^T\phi\alpha + \epsilon $$

Where $\phi \in \mathbb{R^{DxK}}$ is our dictionary of templates and $\alpha \in \mathbb{R^K}$ is our code.

$$ WR(s)W^T\phi\alpha + \epsilon $$

Few notes:

  • Each template, or column of $\mathbb{R^{DxK}}$, has unit L2 norm. This ensures that each discrete pattern is qualitatively unique irrespective of scaling.

Transform distribution proposal

We recognize that an overcomplete template basis necessitates learning smaller and more specific parts.

Perhaps we were able to learn a single transform on our composition of parts because each part was really a digit. And our code amounted to just choosing the correct digit.

With a true sparse-code, our parts will be more fickle, and we will want far more control over the manipulation of their poses for accurate reconstruction.

The following formulation is

Our objective is assign a transform $ T(s)=W R(s) W^T $ to each template in $\phi$.

As a summation:

$$ I = \sum_{k=1}^{K=20} \alpha_k W_k R(s_k) W_k^T \phi_k : W \in \mathbb{R^{Dx2}}$$