The talk presents an extension for high dimensions of an idea from a recent result concerning near optimal adaptive finite element methods (AFEM). The usual adaptive strategy for finding conforming partitions in AFEM is ”mark → subdivide → complete”. In this strategy any element can be marked for subdivision but since the resulting partition often contains hanging nodes, additional elements have to be subdivided in the completion step to get a conforming partition. This process is very well understood for triangulations received via newest vertex bisection procedure. In particular, it is proven that the number of elements in the final partition is limited by constant times the number of marked cells. This motivated us [B., Fierro, Veeser, in preparation] to design a marking procedure that is limited only to cells of the partition whose subdivision will result in a conforming partition and therefore no completion step is necessary. We also proved that this procedure is near best in terms of both error of approximation and complexity. This result is formulated in terms of tree approximations and opens the possibility to design similar algorithms in high dimensions using sparse occupancy trees introduced in [B., Dahmen, Lamby, 2011]. The talk describes the framework of approximating high dimensional data using conforming sparse occupancy trees.
Information retrieval from graphs plays an increasingly important role in data science and machine learning. This talk focuses on two such examples. The first one concerns the graph cuts problem: how to find the optimal k-way graph cuts given an adjacency matrix. We present a convex relaxation of ratio cut and normalized cut, which gives rise to a rigorous theoretical analysis of graph cuts. We derive deterministic bounds of finding the optimal graph cuts via a spectral proximity condition which naturally depends on the intra-cluster and inter-cluster connectivity. Moreover, our theory provides theoretic guarantees for spectral clustering and community detection under stochastic block model. The second example is about the landscape of a nonconvex cost function arising from group synchronization and matrix completion. This function also appears as the energy function of coupled oscillators on networks. We study how the landscape of this function is related to the underlying network topologies. We prove that the optimization landscape has no spurious local minima if the underlying network is a deterministic dense graph or an Erdos-Renyi random graph. The results find applications in signal processing and dynamical systems on networks.
Inference (aka predictive modeling) is in the core of many data science problems. Traditional approaches could be either statistically or computationally efficient, however not necessarily both. The existing principles in deriving these models - such as the maximal likelihood estimation principle - may have been developed decades ago, and do not take into account the new aspects of the data, such as their large volume, variety, velocity and veracity. On the other hand, many existing empirical algorithms are doing extremely well in a wide spectrum of applications, such as the deep learning framework; however they do not have the theoretical guarantee like these classical methods. We aim to develop new algorithms that are both computationally efficient and statistically optimal. Such a work is fundamental in nature, however will have significant impacts in all data science problems that one may encounter in the society. Following the aforementioned spirit, I will describe a set of my past and current projects including L1-based relaxation, fast nonlinear correlation, optimality of detectability, and nonconvex regularization. All of them integrates statistical and computational considerations to develop data analysis tools.
We present a Hamiltonian formulation of the dynamics of the ``shape'' of N point vortices on the plane and the sphere: For example, if N=3, it is the dynamics of the shape of the triangle formed by three point vortices, regardless of the position and orientation of the triangle on the plane/sphere.For the planar case, reducing the basic equations of point vortex dynamics by the special Euclidean group SE(2) yields a Lie-Poisson equation for relative configurations of the vortices. Particularly, we show that the shape dynamics is periodic in certain cases. We extend the approach to the spherical case by first lifting the dynamics from the two-sphere to C^2 and then performing reductions by symmetries.
Neural networks with a large number of parameters admit a mean-field description, which has recently served as a theoretical explanation for the favorable training properties of "overparameterized" models. In this regime, gradient descent obeys a deterministic partial differential equation (PDE) that converges to a globally optimal solution for networks with a single hidden layer under appropriate assumptions. In this talk, we propose a non-local mass transport dynamics that leads to a modified PDE with the same minimizer. We implement this non-local dynamics as a stochastic neuronal birth-death process and we prove that it accelerates the rate of convergence in the mean-field limit. We subsequently realize this PDE with two classes of numerical schemes that converge to the mean-field equation, each of which can easily be implemented for neural networks with finite numbers of parameters. We illustrate our algorithms with two models to provide intuition for the mechanism through which convergence is accelerated. Joint work with G. Rotskoff (NYU), S. Jelassi (Princeton) and E. Vanden-Eijnden (NYU).
Abstract: The Euclidean distance geometry problem arises in a wide variety of applications, from determining molecular conformations in computational chemistry to localization in sensor networks. Instead of directly reconstruct the incomplete distance matrix, we consider a low-rank matrix completion method to reconstruct the associated Gram matrix with respect to a suitable basis. Computationally, simple and fast algorithms are designed to solve the proposed problem. Theoretically, the well known restricted isometry property (RIP) can not be satisfied in the scenario. Instead, a dual basis approach is considered to theoretically analyze the reconstruction problem. Furthermore, by introducing a new condition on the basis called the correlation condition, our theoretical analysis can be also extended to a more general setting to handle low-rank matrix completion problems under any given non-orthogonal basis. This new condition is polynomial time checkable and holds for many cases of deterministic basis where RIP might not hold or is NP-hard to verify. If time permits, I will also discuss a combination of low-rank matrix completion with geometric PDEs on point clouds to understanding manifold-structured data represented as incomplete inter-point distance data. This talk is based on:1. A. Tasissa, R. Lai, “Low-rank Matrix Completion in a General Non-orthogonal Basis”, arXiv:1812.05786 2018. 2. A. Tasissa, R. Lai, “Exact Reconstruction of Euclidean Distance Geometry Problem Using Low-rank Matrix Completion”, accepted, IEEE. Transaction on Information Theory, 2018. 3. R. Lai, J. Li, “Solving Partial Differential Equations on Manifolds From Incomplete Inter-Point Distance”, SIAM Journal on Scientific Computing, 39(5), pp. 2231-2256, 2017.
The Georgia Scientific Computing Symposium is a forum for professors, postdocs, graduate students and other researchers in Georgia to meet in an informal setting, to exchange ideas, and to highlight local scientific computing research. The symposium has been held every year since 2009 and is open to the entire research community.
This year, the symposium will be held on Saturday, February 16, 2019, at Georgia Institute of Technology. Please see
for more information
In this talk, we will discuss some advantages of using non-convex penalty functions in variational regularization problems and how to handle them using the so-called Convex-Nonconvex approach. In particular, TV-like non-convex penalty terms will be presented for the problems in segmentation and additive decomposition of scalar functions defined over a 2-manifold embedded in \R^3. The parametrized regularization terms are equipped by a free scalar parameter that allows to tune their degree of non-convexity. Appropriate numerical schemes based on the Alternating Directions Methods of Multipliers procedure are proposed to solve the optimization problems.
We will go to lunch together after the talk with the graduate students.
We introduce methods from convex optimization to solve the multi-marginal transport type problems arise in the context of density functional theory. Convex relaxations are used to provide outer approximation to the set of N-representable 2-marginals and 3-marginals, which in turn provide lower bounds to the energy. We further propose rounding schemes to obtain upper bound to the energy.