Connections to surface reconstruction:

• A problem that arises frequently in graphics is to estimate curves or surfaces given point clouds (e.g. in 3D scanning)
• Crust algorithm:
  • Let $S$ be the set of sample points
  • Compute the Voronoi diagram $\text{Vor}(S)$ and let $V$ be its set of Voronoi vertices.
  • Compute the Delaunay triangulation $\text{Del}(S \cup V)$.
  • The curve $P$ is composed of the edges of $\text{Del}(S \cup V)$ with endpoints in $S$.
• Smooth curve shortening: $$ \frac{\partial C}{\partial t} = \frac{\partial^2 C}{\partial s}$$ where $C(s,t)$ defines a geometric flow a curve at time $t$ parametrized by arclength $s$ (generalized from $C(s) = (x(s),y(s))$ to multiple time points)
• Midpoint transformation: a discrete curve defined by the points $(v_1,\dots,v_n)$ can be aggregated to remove noise by replacing vertices $v_i$ and $v_{i+1}$ with their average (i.e. the midpoint of the segment $v_iv_{i+1}$) $$ \frac{\partial v_i}{\partial t} = \frac{\partial^2 v_i}{\partial x^2}$$

Is anything here applicable to the string method?

• Analogy between motion planning and minimum-energy path finding

Connection to motion planning:

• Protein conformations are elements of a configuration space $\mathcal{C}$, the set of all possible configurations of its atoms (a.k.a. state space, moduli space)
• In motion planning, we want to find a path through configuration space from one point to another given some constraints (e.g. avoid obstacles, or we might associate a cost $L(c)$ with each intermediate configuration $c$ and want to minimize $\sum_c L(c)$ (e.g. free energy barriers)
• Cell decomposition - partition configuration space into a finite number of cells and find paths between cells
• Probabilistic roadmap algorithms for the protein folding problem, sampling-based motion planning - randomly generate samples, query if they are in $\mathcal{C}$, connect local points into a graph structure

Coarse-graining / enhanced sampling notes:

• See wiki page on rare event sampling: http://en.wikipedia.org/wiki/Rare_Event_Sampling
  • Stochastic Process Rare Event Sampling: http://scitation.aip.org/content/aip/journal/jcp/133/24/10.1063/1.3525099
• String method: given the locations of two metastable states, find the minimum energy path (MEP) between them
• Transition-path sampling
• Transition-interface sampling
• Umbrella sampling: add a bias potential to "flatten" large energy barriers. Set the bias to be a function of some order parameter
• Replica exchange (parallel tempering) http://en.wikipedia.org/wiki/Parallel_tempering (two temperatures in parallel) $$ p_{i \to j} = \min \left\{1, \frac{\exp \left( - \frac{E_j}{kT_i} \frac{E_i}{kT_j} \right) }{\exp \left( - \frac{E_i}{kT_i} \frac{E_j}{kT_j} \right) } \right\} $$
• Metadynamics
• CATCH UP ON THESE TOPICS: http://www.mi.fu-berlin.de/w/CompMolBio/MolecularModellingSimulation
• Spectral rate theory for two-state kinetics
• Weighted histogram
• Multistate Bennett Acceptance Ratio

For my sanity, I need to make an "MCMC Zoo" similar to Scott Aaronson's "Complexity Zoo"

Goose-chase notes:

• Any useful connections to work of Stephan Ermon? (using SAT-solvers as oracles in algorithms to estimate the partition-function) -- probably not, continuous vs. discrete space. Actually, there's a step in the projected Markov models paper that takes the product over a large number of distinct paths
• Any useful connections to work of Keenan Crane? (learning surfaces from point clouds)
• Useful connections to work of Byron Boots?
  • Certainly: Reduced Rank Hidden Markov Models (compare with "Projected and hidden Markov models for calculating kinetics and metastable states of complex molecules")
  • Possibly: Hilbert-Space Embeddings of HMMs http://machinelearning.wustl.edu/mlpapers/paper_files/icml2010_SongSGS10.pdf
    • Consider collaborations with Gunnar Rätsch re: kernel methods
• Lagrange multipliers: http://en.wikipedia.org/wiki/Lagrange_multiplier
• A geometrical approach to computing free-energy landscapes from short-ranged potentials http://www.pnas.org/content/110/1/E5.full.pdf
• Comparing a simple theoretical model for protein folding with all-atom molecular dynamics simulations http://www.pnas.org/content/110/44/17880.full.pdf+html
• Nonparametric modal regression?
• Probabilistic numerics blog: http://www.probabilistic-numerics.org/literature/index.html#ABC
• Glasgow Inference Group: http://www.dcs.gla.ac.uk/inference/people.cfm

MSM-learning notes

• Nonparametric estimation of a a Markov kernel using kernel density estimation:
  • http://en.wikipedia.org/wiki/Markov_kernel
  • http://www.sciencedirect.com/science/article/pii/S0165176509002560
    • Oops, not actually relevant
• Simultaneous learning of state-decompositions and transition-matrices?
• How would you sample over transition operators?
  • Simultaneously learn a low-dimensional representation and a transition operator that reproduces the observed dynamics?
• "Efficient Bayesian estimation of Markov model transition matrices with given stationary distribution"
  • Before reading: what I might do differently:
    • Incorporate geometric information about states
      • Direct pairwise RMSD -- likely noisy, but a good baseline
      • Cost of min-cost non-direct path between states -- estimating a "geodesic" in discretized conformation space. Smaller distances are more reliable
        • Efficient inference about Markov processes: http://projecteuclid.org/euclid.aoms/1177707039
        • Contains information on properties of stochastic matrices
        • READ CHAPTER 5 OF THE MSMs BOOK: contains 5 algorithms for Bayesian sampling of transition matrices (containing efficient Markov kernels for transition matrices that are reversible), plus some test problems
        • Variational Bayes?
        • http://www.gatsby.ucl.ac.uk/vbayes/
• Stochastic neighbor compression: http://jmlr.org/proceedings/papers/v32/kusner14.pdf
  • Huge improvement in speed and accuracy over KNN (useful for state decomposition?)
• Nonlinear metric learning: http://www1.cse.wustl.edu/~kilian/papers/chilmnn.pdf

  - Also useful for state decomposition? Learn a better predictor of kinetic distance than geometry. Also compare with Robert McGibbon's work on this

String Method

String method for the study of rare events (Weinan, Ren, and Vanden-Eijnden, 2002, Phys. Rev. B)

• plus two follow-up papers
• plus Science paper on melting

Project proposal: Reduced-rank projected Markov models

• Problem: discrete state decomposition

• Drawing on three papers:

Project proposal: Information-Geometric Sampling Methods for Transition Matrices

• When the number of states is large, posterior inference becomes intractable

State of the Art

• Taken from Chapter 5 of Intro to MSMs

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