Amit Singer

The narrow escape problem for diffusion in cellular microdomains.

The study of the diffusive motion of ions or molecules in confined biological microdomains requires the derivation of the explicit dependence of quantities, such as the decay rate of the population or the forward chemical reaction rate constant on the geometry of the domain. Here, we obtain this explicit dependence for a model of a Brownian particle (ion, molecule, or protein) confined to a bounded domain (a compartment or a cell) by a reflecting boundary, except for a small window through which it can escape. We call the calculation of the mean escape time the narrow escape problem. This time diverges as the window shrinks, thus rendering the calculation a singular perturbation problem. Here, we present asymptotic formulas for the mean escape time in several cases, including regular domains in two and three dimensions and in some singular domains in two dimensions. The mean escape time comes up in many applications, because it represents the mean time it takes for a molecule to hit a target binding site. We present several applications in cellular biology: calcium decay in dendritic spines, a Markov model of multicomponent chemical reactions in microdomains, dynamics of receptor diffusion on the surface of neurons, and vesicle trafficking inside a cell.

Graph Laplacian Tomography From Unknown Random Projections

We introduce a graph Laplacian-based algorithm for the tomographic reconstruction of a planar object from its projections taken at random unknown directions. A Laplace-type operator is constructed on the data set of projections, and the eigenvectors of this operator reveal the projection orientations. The algorithm is shown to successfully reconstruct the Shepp-Logan phantom from its noisy projections. Such a reconstruction algorithm is desirable for the structuring of certain biological proteins using cryo-electron microscopy.

Vector Diffusion Maps and the Connection Laplacian

We introduce vector diffusion maps (VDM), a new mathematical framework for organizing and analyzing massive high-dimensional data sets, images, and shapes. VDM is a mathematical and algorithmic generalization of diffusion maps and other nonlinear dimensionality reduction methods, such as LLE, ISOMAP, and Laplacian eigenmaps. While existing methods are either directly or indirectly related to the heat kernel for functions over the data, VDM is based on the heat kernel for vector fields. VDM provides tools for organizing complex data sets, embedding them in a low-dimensional space, and interpolating and regressing vector fields over the data. In particular, it equips the data with a metric, which we refer to as the vector diffusion distance. In the manifold learning setup, where the data set is distributed on a low-dimensional manifold ℳ d embedded in ℝ p , we prove the relation between VDM and the connection Laplacian operator for vector fields over the manifold.

A remark on global positioning from local distances

Finding the global positioning of points in Euclidean space from a local or partial set of pairwise distances is a problem in geometry that emerges naturally in sensor networks and NMR spectroscopy of proteins. We observe that the eigenvectors of a certain sparse matrix exactly match the sought coordinates. This translates to a simple and efficient algorithm that is robust to noisy distance data.

Global Motion Estimation from Point Matches

Multiview structure recovery from a collection of images requires the recovery of the positions and orientations of the cameras relative to a global coordinate system. Our approach recovers camera motion as a sequence of two global optimizations. First, pair wise Essential Matrices are used to recover the global rotations by applying robust optimization using either spectral or semi definite programming relaxations. Then, we directly employ feature correspondences across images to recover the global translation vectors using a linear algorithm based on a novel decomposition of the Essential Matrix. Our method is efficient and, as demonstrated in our experiments, achieves highly accurate results on collections of real images for which ground truth measurements are available.

A Cheeger Inequality for the Graph Connection Laplacian

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Diffusion Interpretation of Nonlocal Neighborhood Filters for Signal Denoising

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Detecting intrinsic slow variables in stochastic dynamical systems by anisotropic diffusion maps

synchronization problem consists of estimating a set of

Uniqueness of Low-Rank Matrix Completion by Rigidity Theory

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Sensor network localization by eigenvector synchronization over the euclidean group

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