Everything you wanted to know about VAMP but were afraid to ask - PowerPoint PPT Presentation

Everything you wanted to know about VAMP but were afraid to ask Brooke Husic Stanford/FU Berlin PyEMMA Workshop February 21, 2019

First of all V ariational A pproach for M arkov P rocesses Key papers: Wu & Noé 2017, arXiv:1707.04659, “Variational approach…” Paul et al, arXiv:1811.12551, “Identification of kinetic…”

First of all V ariational Real answer A pproach for Value M arkov Guesses P rocesses Key papers: Wu & Noé 2017, arXiv:1707.04659, “Variational approach…” Paul et al, arXiv:1811.12551, “Identification of kinetic…”

First of all V ariational Our data: Z 1 , Z 2 , …, Z t − 2 , Z t − 1 , Z t , Z t+1 A pproach for M arkov P rocesses A B C D E Key papers: Wu & Noé 2017, arXiv:1707.04659, “Variational approach…” Paul et al, arXiv:1811.12551, “Identification of kinetic…”

First of all V ariational Our data: Z 1 , Z 2 , …, Z t − 2 , Z t − 1 , Z t , Z t+1 A pproach for M arkov [Z t , Z t+1 ] Z 1 Z 2 P rocesses Z 2 Z 3 [Z t , Z t+ ! ] Z 3 Z 4 X = Y = Z 4 Z 5 Key papers: Wu & Noé 2017, arXiv:1707.04659, “Variational approach…” Z t − 1 Z t Paul et al, arXiv:1811.12551, “Identification of kinetic…”

Some history Atomic positions Atomic positions Atomic positions Hand-selected Pairwise Large sets of features RMSD features Dimensionality reduction State decomposition MSM MSM MSM Zwanzig dPCA, tICA MSMBuilder 1983 2005, 2011, 2013 2009 Figure from: Husic & Pande 2018, JACS, “Markov State Models: From an Art to a Science”

The problem Try 5 different S e a r c h 1 0 d i featurizations? n f f u e m r e b n e t r s o f c l u s t e r s ? Featurization Clustering MSM Raw ▷ internal coordinate ▷ algorithm ⊠ # Do chi angles help for a Trajectories ▷ number of system timescales ▷ transformations clusters Compare 3 dihedral featurization? ⊠ lag time different TICA lag times? Dimensionality Reduction ▷ PCA, TICA ▷ TICA lag time, # components Need a method to objectively evaluate modeling choices! Figure from: Husic & Pande 2017, J Chem Phys, “MSM lag time cannot be used for variational model selection”

Back to history Training Atomic positions Atomic positions Atomic positions Atomic positions Atomic positions set ① ② Hand-selected Pairwise Large sets of Large sets of features RMSD features features Variational evaluation validation Neural Cross network Dimensionality Dimensionality reduction reduction State decomposition Validation MSM MSM MSM MSM MSM set Zwanzig dPCA, TICA VAMPnets MSMBuilder VAC GMRQ, VAMP ③ 1983 2005, 2011, 2013 2017 2009 2013 2015, 2017 Figure from: Husic & Pande 2018, JACS, “Markov State Models: From an Art to a Science”

Let’s make sure we’re clear on MSMs This is it! This *is* the MSM. ★ Thermodynamics (populations!) ★ Kinetics (transition probabilities!) ★ Dynamical processes (eigenvectors!) ★ Pathways (TPT!) Transition matrix Figure from: Husic & Pande 2018, JACS, “Markov State Models: From an Art to a Science”

The VAC T( ! ) Transition matrix Key papers: Noé & Nüske 2013, Multiscale Model Simul, “A Variational Approach…” Nüske et al 2014, J Chem Theory Comput, “Variational Approach…”

The VAC T( ! ) ψ i = λ i ψ i t i = – ! / ln | λ i | The eigenvalues have special properties according to the Eigenvalues: related to timescales Perron-Frobenius theorem: - They are real - There is a unique Eigenvectors: dynamical processes maximum eigenvalue of 1 - All other eigenvalues have absolute values below 1 Key papers: Noé & Nüske 2013, Multiscale Model Simul, “A Variational Approach…” Nüske et al 2014, J Chem Theory Comput, “Variational Approach…”

The VAC T( ! ) ψ i = λ i ψ i t i = – ! / ln | λ i | Unknown true eigenvalues m m The variational ⋀ Σ λ i ≤ Σ λ i principle is for the eigenvalues i=1 i=1 Eigenvalue predictions from MSM Key papers: Noé & Nüske 2013, Multiscale Model Simul, “A Variational Approach…” Nüske et al 2014, J Chem Theory Comput, “Variational Approach…”

The VAC T( ! ) ψ i = λ i ψ i t i = – ! / ln | λ i | IMPORTANT: This ① Unknown true score is only for the eigenvalues m m transition matrix ⋀ Σ λ i ≤ Σ λ i SCORE = defined at the given lag time ! i=1 i=1 Eigenvalue predictions from MSM Key papers: Noé & Nüske 2013, Multiscale Model Simul, “A Variational Approach…” Nüske et al 2014, J Chem Theory Comput, “Variational Approach…”

Reminder Try 5 different S e a ✅‍ r c h 1 0 d ✅‍ i featurizations? n f f u e m r e b n e t r s o f c l u s t e r s ? Featurization Clustering MSM Raw ▷ internal coordinate ▷ algorithm ⊠ # Do chi angles help for a Trajectories ▷ number of system timescales ▷ transformations clusters Compare 3 dihedral featurization? ⊠ lag time Check 5 different MSM different TICA lag times? Dimensionality Reduction lag times? ✅‍ ▷ PCA, TICA ✅‍ ▷ TICA lag time, # components Eligible regime for scoring MSMs Figure from: Husic & Pande 2017, J Chem Phys, “MSM lag time cannot be used for variational model selection”

Cross validation Unknown true eigenvalues m m ⋀ Σ λ i ≤ Σ λ i SCORE = This method will i=1 i=1 have a problem with overfitting ② Eigenvalue predictions from MSM validation set Data: Training set Validation set Apply MSM and score eigenvalues Make MSM (is there enough sampling?) some number of Key paper: iterations with ⨉ different sets McGibbon & Pande 2015, J Chem Phys, “Variational cross-validation…”

An example From Husic et al 2016, J Chem Phys, “Optimized parameter selection…”

Finally: the VAMP! T( ! ) The transition matrix has certain properties due to the reversibility assumption. This includes having an eigendecomposition. Transition matrix Key papers: Wu & Noé 2017, arXiv:1707.04659, “Variational approach…” Paul et al, arXiv:1811.12551, “Identification of kinetic…”

Finally: the VAMP! m m ⋀ Σ σ i ≤ Σ σ i { φ i , σ i , φ i } K( ! ) SCORE = i=1 i=1 ③ Consider now a different matrix that is not necessarily However, it will always reversible. ? have a singular value It may not have an decomposition . eigendecomposition anymore, or its eigendecomposition may not be useful. The VAMP uses more Transition matrix general math to score models that may not be Key papers: reversible Wu & Noé 2017, arXiv:1707.04659, “Variational approach…” Paul et al, arXiv:1811.12551, “Identification of kinetic…”

What we’ve learned… - We have many choices when we make Markov state models - Luckily, we have the VAC to evaluate different choices objectively - But not the MSM lag time, of course . - We just have to do it under cross-validation to avoid overfitting - We can use the VAMP in the more general, nonreversible case - Which is the same as the VAC when we have an MSM! - With an objective metric, can’t we just make models automatically..? - Stay tuned!

Paper highlights VAC theory Noé & Nüske 2013, Multiscale Model Simul, “A Variational Approach…” Nüske et al 2014, J Chem Theory Comput, “Variational Approach…” Cross-validation McGibbon & Pande 2015, J Chem Phys, “Variational cross-validation…” VAMP theory Wu & Noé 2017, arXiv:1707.04659, “Variational approach…” Paul et al, arXiv:1811.12551, “Identification of kinetic…” General overview/history of MSMs Husic & Pande 2018, JACS, “Markov State Models: From an Art to a Science” General overview of ML methods Noé 2018, arXiv:1812.07669, “Machine learning…”