[Work Log] Branching curve clique-tree
Thinking about attachments.
- Organize into connected components.
Reviewing ML code, with considerations to generalizing for attachments.
Can we exploit the structure in the branching-prior covariance matrix to speed-up inversion in the marginal likelihood? No obvious way.
Why did we abandon the clique-tree method? Because it wasn't clear that the perturb-models would be compatible with it. Now, it seems like it might be, as long as we set the markov-order high enough. More importantly, it might be necessary, because after adding attachment, the covariance matrix is growing too large to allow direct evaluation.
Construct clique tree per-track, conditioned on parent point. In case of base-curves, the parent point is the prior.
New fields
- branch parent
- curve index
- point index
- branch distance
- i.e. starting index of self
- gaussian structure
- in canonical form
- no position variance
- no branching variance
- yes branch distance
Inferring branch distance
The reconstructed lateral branch curves are usually have gaps of several millimeters from their parent branch. Assigning an index of zero to the first point will almost certainly result in a worse marginal likelihood, because these gaps are not well-modelled.
Instead, we should attempt to infer the branch distance(or better yet, try to marginalize over it).
Spent some time trying to derive a "fast update" for covariance matrices for inferring branch distance. In this scheme, the matrices are computed assuming branch distance is zero, and then a easily computed delta matrix is added to account for non-zero branch distance.
After some work, it looks like the smoothing variance kernel is too complicated to permit such a method. It's just easier to recompute the entire covariance matrix.
Constructing branching clique-tree
- Construct individual cliques trees, assuming zero position variance (but including branch distance). Call this the "raw prior."
- Given a branch-point-index, compute the markov-blanket of the data points on the parent.
- Given raw prior and markov-blanket, construct conditional clique node from raw clique node.
- Given clique tree, multiply and marginalize from tips to root.
All cliques will now be stored in track structure (currently named Corrs).
How to construct conditional prior?
- \(\mathcal{MB}\) - indices of the parent curve markov blanket, centered at branch point.
- \(y_\mathcal{MB}\) - observations in the markov blanket
- \(x_b\) - branch point
- \(\Sigma_b\) - branch point predictive covariance, conditioned on markov blanket
- \(x_c\) - child curve points, relative to branch point.
- \(x_C\) - child curve points, relative to world origin
- \(N\) - number of points in child curve.
- \(\Sigma_c\) - curve predictive covariance, conditioned on branch point
- \(\Sigma_C\) - curve predictive covariance, conditioned on markov blanket
The conditional child curve distribution is
The marginal over the child curve arises due to the linear combination of random variables \(x_c\) and \(x_b\): \[ x_C = x_c + \left ( \begin{array}{c}I\\I\\ \ldots \\I \end{array}\right) x_b \] The corresponding distribution is: \[ p(x_C | y_\mathcal{MB}) = \mathcal{N}(\mu_C, \Sigma_C) \] where \[ \begin{align} \mu_C &= \left( \begin{array}{c}I\\I\\ \ldots \\I \end{array}\right) \mu_b \\ \Sigma_C &= \Sigma_c + \left( \begin{array}{c}I\\I\\ \ldots \\I \end{array}\right) \Sigma_b \left ( I \; I \ldots I \right ) \end{align} \]