Linear Hamilton Jacobi Bellman Equations in High Dimensions

The Hamilton Jacobi Bellman Equation (HJB) provides the globally optimal solution to large classes of control problems. Unfortunately, this generality comes at a price, the calculation of such solutions is typically intractible for systems with more than moderate state space size due to the curse of dimensionality. This work combines recent results in the structure of the HJB, and its reduction to a linear Partial Differential Equation (PDE), with methods based on low rank tensor representations, known as a separated representations, to address the curse of dimensionality. The result is an algorithm to solve optimal control problems which scales linearly with the number of states in a system, and is applicable to systems that are nonlinear with stochastic forcing in finite-horizon, average cost, and first-exit settings. The method is demonstrated on inverted pendulum, VTOL aircraft, and quadcopter models, with system dimension two, six, and twelve respectively.Comment: 8 pages. Accepted to CDC 201

Parametric t-Distributed Stochastic Exemplar-centered Embedding

Parametric embedding methods such as parametric t-SNE (pt-SNE) have been widely adopted for data visualization and out-of-sample data embedding without further computationally expensive optimization or approximation. However, the performance of pt-SNE is highly sensitive to the hyper-parameter batch size due to conflicting optimization goals, and often produces dramatically different embeddings with different choices of user-defined perplexities. To effectively solve these issues, we present parametric t-distributed stochastic exemplar-centered embedding methods. Our strategy learns embedding parameters by comparing given data only with precomputed exemplars, resulting in a cost function with linear computational and memory complexity, which is further reduced by noise contrastive samples. Moreover, we propose a shallow embedding network with high-order feature interactions for data visualization, which is much easier to tune but produces comparable performance in contrast to a deep neural network employed by pt-SNE. We empirically demonstrate, using several benchmark datasets, that our proposed methods significantly outperform pt-SNE in terms of robustness, visual effects, and quantitative evaluations.Comment: fixed typo

Multidimensional approximation of nonlinear dynamical systems

A key task in the field of modeling and analyzing nonlinear dynamical systems is the recovery of unknown governing equations from measurement data only. There is a wide range of application areas for this important instance of system identification, ranging from industrial engineering and acoustic signal processing to stock market models. In order to find appropriate representations of underlying dynamical systems, various data-driven methods have been proposed by different communities. However, if the given data sets are high-dimensional, then these methods typically suffer from the curse of dimensionality. To significantly reduce the computational costs and storage consumption, we propose the method multidimensional approximation of nonlinear dynamical systems (MANDy) which combines data-driven methods with tensor network decompositions. The efficiency of the introduced approach will be illustrated with the aid of several high-dimensional nonlinear dynamical systems

On the mathematical Structure of Quantum Measurement Theory

We show that the key problems of quantum measurement theory, namely the reduction of the wave packet of a microsystem and the specification of its quantum state by a macroscopic measuring instrument, may be rigorously resolved within the traditional framework of the quantum mechanics of finite conservative systems. The argument is centred on the generic model of a microsystem, S, coupled to a finite macroscopic measuring instrument, I, which itself is an N-particle quantum system. The pointer positions of I correspond to the macrostates of this instrument, as represented by orthogonal subspaces of the Hilbert space of its pure states. These subspaces, or 'phase cells', are the simultaneous eigenspaces of a set of coarse grained intercommuting macroscopic observables, M, and, crucially, are of astronomically large dimensionalities, which incease exponentially with N. We formulate conditions on the conservative dynamics of the composite (S+I) under which it yields both a reduction of the wave packet describing the state of S and a one-to-one correspondence, following a measurement, between the pointer position of I and the resultant state of S; and we show that these conditions are fulfilled by the finite version of the Coleman-Hepp model.Comment: 20 pages, minor correstions installed, to appear in Rep. Math. Phy