Recent high-dimensional single-cell technologies such as mass cytometry are enabling time series experiments to monitor the temporal evolution of cell state distributions and to identify dynamically important cell states, such as fate decision states in differentiation. However, these technologies are destructive, and require analysis approaches that temporally map between cell state distributions across time points. Current approaches to approximate the single-cell time series as a dynamical system suffer from too restrictive assumptions about the type of kinetics, or link together pairs of sequential measurements in a discontinuous fashion. We propose Dynamic Distribution Decomposition (DDD), an operator approximation approach to infer a continuous distribution map between time points. On the basis of single-cell snapshot time series data, DDD approximates the continuous time Perron-Frobenius operator by means of a finite set of basis functions. This procedure can be interpreted as a continuous time Markov chain over a continuum of states. By only assuming a memoryless Markov (autonomous) process, the types of dynamics represented are more general than those represented by other common models, e.g., chemical reaction networks, stochastic differential equations. Furthermore, we can a posteriori check whether the autonomy assumptions are valid by calculation of prediction error-which we show gives a measure of autonomy within the studied system. The continuity and autonomy assumptions ensure that the same dynamical system maps between all time points, not arbitrarily changing at each time point. We demonstrate the ability of DDD to reconstruct dynamically important cell states and their transitions both on synthetic data, as well as on mass cytometry time series of iPSC reprogramming of a fibroblast system. We use DDD to find previously identified subpopulations of cells and to visualise differentiation trajectories. Dynamic Distribution Decomposition allows interpretation of high-dimensional snapshot time series data as a lowdimensional Markov process, thereby enabling an interpretable dynamics analysis for a variety of biological processes by means of identifying their dynamically important cell states.
Acceleration schemes can dramatically improve existing optimization procedures. In most of the work on these schemes, such as nonlinear generalized minimal residual (N-GMRES), acceleration is based on minimizing the 2 norm of some target on subspaces of ℝ n . There are many numerical examples that show how accelerating general-purpose and domain-specific optimizers with N-GMRES results in large improvements. We propose a natural modification to N-GMRES, which significantly improves the performance in a testing environment originally used to advocate N-GMRES. Our proposed approach, which we refer to as O-ACCEL (objective acceleration), is novel in that it minimizes an approximation to the objective function on subspaces of ℝ n . We prove that O-ACCEL reduces to the full orthogonalization method for linear systems when the objective is quadratic, which differentiates our proposed approach from existing acceleration methods. Comparisons with the limited-memory Broyden-Fletcher-Goldfarb-Shanno and nonlinear conjugate gradient methods indicate the competitiveness of O-ACCEL. As it can be combined with domain-specific optimizers, it may also be beneficial in areas where limited-memory Broyden-Fletcher-Goldfarb-Shanno and nonlinear conjugate gradient methods are not suitable.
This paper presents the results of the Dynamic Pricing Challenge, held on the occasion of the 17 th INFORMS Revenue Management and Pricing Section Conference on June 29-30, 2017 in Amsterdam, The Netherlands. For this challenge, participants submitted algorithms for pricing and demand learning of which the numerical performance was analyzed in simulated market environments. This allows consideration of market dynamics that are not analytically tractable or can not be empirically analyzed due to practical complications. Our findings implicate that the relative performance of algorithms varies substantially across different market dynamics, which confirms the intrinsic complexity of pricing and learning in the presence of competition.
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