Mauro Maggioni

Determination of reaction coordinates via locally scaled diffusion map.

We present a multiscale method for the determination of collective reaction coordinates for macromolecular dynamics based on two recently developed mathematical techniques: diffusion map and the determination of local intrinsic dimensionality of large datasets. Our method accounts for the local variation of molecular configuration space, and the resulting global coordinates are correlated with the time scales of the molecular motion. To illustrate the approach, we present results for two model systems: all-atom alanine dipeptide and coarse-grained src homology 3 protein domain. We provide clear physical interpretation for the emerging coordinates and use them to calculate transition rates. The technique is general enough to be applied to any system for which a Boltzmann-sampled set of molecular configurations is available.

Diffusion Maps, Reduction Coordinates, and Low Dimensional Representation of Stochastic Systems

The concise representation of complex high dimensional stochastic systems via a few reduced coordinates is an important problem in computational physics, chemistry, and biology. In this paper we use the first few eigenfunctions of the backward Fokker–Planck diffusion operator as a coarse-grained low dimensional representation for the long-term evolution of a stochastic system and show that they are optimal under a certain mean squared error criterion. We denote the mapping from physical space to these eigenfunctions as the diffusion map. While in high dimensional systems these eigenfunctions are difficult to compute numerically by conventional methods such as finite differences or finite elements, we describe a simple computational data-driven method to approximate them from a large set of simulated data. Our method is based on defining an appropriately weighted graph on the set of simulated data and computing the first few eigenvectors and eigenvalues of the corresponding random walk matrix on this graph...

Manifold parametrizations by eigenfunctions of the Laplacian and heat kernels

We use heat kernels or eigenfunctions of the Laplacian to construct local coordinates on large classes of Euclidean domains and Riemannian manifolds (not necessarily smooth, e.g., with 𝒞α metric). These coordinates are bi-Lipschitz on large neighborhoods of the domain or manifold, with constants controlling the distortion and the size of the neighborhoods that depend only on natural geometric properties of the domain or manifold. The proof of these results relies on novel estimates, from above and below, for the heat kernel and its gradient, as well as for the eigenfunctions of the Laplacian and their gradient, that hold in the non-smooth category, and are stable with respect to perturbations within this category. Finally, these coordinate systems are intrinsic and efficiently computable, and are of value in applications.

Multiscale approximation with hierarchical radial basis functions networks

An approximating neural model, called hierarchical radial basis function (HRBF) network, is presented here. This is a self-organizing (by growing) multiscale version of a radial basis function (RBF) network. It is constituted of hierarchical layers, each containing a Gaussian grid at a decreasing scale. The grids are not completely filled, but units are inserted only where the local error is over threshold. This guarantees a uniform residual error and the allocation of more units with smaller scales where the data contain higher frequencies. Only local operations, which do not require any iteration on the data, are required; this allows to construct the network in quasi-real time. Through harmonic analysis, it is demonstrated that, although a HRBF cannot be reduced to a traditional wavelet-based multiresolution analysis (MRA), it does employ Riesz bases and enjoys asymptotic approximation properties for a very large class of functions. HRBF networks have been extensively applied to the reconstruction of three-dimensional (3-13) models from noisy range data. The results illustrate their power in denoising the original data, obtaining an effective multiscale reconstruction of better quality than that obtained by MRA.

Tensor-CUR Decompositions for Tensor-Based Data

Motivated by numerous applications in which the data may be modeled by a variable subscripted by three or more indices, we develop a tensor-based extension of the matrix CUR decomposition. The tensor-CUR decomposition is most relevant as a data analysis tool when the data consist of one mode that is qualitatively different from the others. In this case, the tensor-CUR decomposition approximately expresses the original data tensor in terms of a basis consisting of underlying subtensors that are actual data elements and thus that have a natural interpretation in terms of the processes generating the data. Assume the data may be modeled as a

Tensor-CUR decompositions for tensor-based data

(2+1)

Research on online social networks: time to face the real challenges

-tensor, i.e., an

Estimation of intrinsic dimensionality of samples from noisy low-dimensional manifolds in high dimensions with multiscale SVD

m \times n \times p

Genomic Characterization of Large Heterochromatic Gaps in the Human Genome Assembly

tensor

Polymer reversal rate calculated via locally scaled diffusion map

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