Probability/Statistics Seminar
Brown University, Division of Applied Mathematics
Abstract: Modern neurophysiological techniques permit the simultaneous recording of the electrical activity of hundreds of individual neurons (and this number is quickly growing). The high dimensionality of the resulting data sets, combined with the peculiarities of neural activity patterns, create challenging statistical issues.
One of the more vexing statistical problems involves the assumption of repeatable experimental trials (that is, identically distributed data). This assumption is crucial for many statistical methods. It is unlikely to be valid in practice. As is often pointed out in the literature, non-repeatable trials can create strong statistical artifacts, leading to erroneous physiological conclusions. The main purpose of this talk is to describe two types of statistical techniques that allow for non-repeatable trials.
The first technique tests the hypothesis that the number of action potentials (electrical spikes) in a given time interval follows a Poisson distribution (a common model in neuroscience). This is a straightforward test if the data are independent and identically distributed. It is less straightforward, however, if each trial is Poisson with a different mean (non-repeatable trials). The second technique tests for the presence of unusual spiking patterns in the data. Making this precise takes care, especially for non-repeatable trials.
Brown University, Mathematics Department, Special Lecture
Probability/Statistics Seminar
Brown University, Division of Applied Mathematics
Abstract: Tikhonov regularization is a formalization of inductive learning which includes support vector machines, regularized least squares, and other classification and regression problems. For a family of kernel functions that includes the Gaussian, parameterized by a "bandwith" parameter \sigma, we characterize the limiting solution for large \sigma . We find that the regression functions obtained in the large \sigma limit are polynomials, the degree being dependent upon the regularization parameter.
The proof of this result rests on two key ideas. One of them is epi-convergence, a notion of functional convergence under which limits of minimizers converge to minimizers of limits. The other is a value-based formulation of regression, which allows us to track the regression function while the kernel is changing, making it evident why this phenomenon occurs for every Tikhonov regularization problem.
keywords: Tikhonov regularization, Gaussian kernel, kernel machines
Brown Analysis Seminar
Probability/Statistics Seminar
Brown University, Division of Applied Mathematics
Abstract: In certain problems in a variety of applied probability settings (from probabilistic analysis of algorithms to statistical physics), a central theme is to solve some fixed-point-equation on an appropriate space of probabilities, we call such an equation a Recursive Distributional Equation (RDE). Exploiting the natural recursive structure one can associate with a solution of a RDE a tree index random process which we call a Recursive Tree Process (RTP). In some sense if a RDE has a solution then the corresponding RTP is an almost sure representation of it.
In this talk we will discuss several examples where such processes arise naturally. We will outline some basic general theory with the main objective of determining possible influence of the boundary at infinity on the root for a recursive tree process. We will explore two aspects, the question on endogeny: the RTP being measurable with respect to the associated innovation process (the data available from the RDE), and the question of having a non-trivial tail for the RTP. We will give necessary and sufficient conditions for endogeny and tail-triviality, and will indicate some non-trivial applications of this theory.
(Some part of this talk is based on a joint work with Professor David J. Aldous).
Probability/Statistics Seminar
Brown University, Division of Applied Mathematics
Abstract: The study of diffusion operators of manifolds, graphs and ``data sets'' is useful for the analysis of the structure of the underlying space and of functions on the space. This in turn has many and important applications to disparate fields including partial differential equations, machine learning, dynamical and control systems, data analysis.
We discuss old and new ideas and algorithms for multiscale analysis associated to such diffusion operators. Given a local operator $T$ on a manifold or a graph, with large powers of low rank, we present a general multiresolution construction for efficiently computing, representing and compressing $T^t$. This allows the computation, to high precision, of functions of the operator, notably the associated Green's function, in compressed form, and their fast application. The dyadic powers of $T$ can be used to induce a multiresolution analysis, as in classical Littlewood-Paley and wavelet theory: we construct, with efficient and stable algorithms, scaling function and wavelet bases associated to this multiresolution analysis, together with the corresponding downsampling operators. This allows to extend multiscale signal processing to general spaces (such as manifolds and graphs) in a very natural way, with corresponding efficient algorithms.
We will sketch motivating applications, which include function approximation, denoising, and learning on data sets, model reduction for complex stochastic dynamical systems, multiscale analysis of Markov chains and Markov decision processes, multiscale analysis of complex networks (e.g. corpora of documents).
PDE Seminar
Brown University
Biomedical Engineering Seminar
Abstract: Hydrogels are widely used biomaterials since they are structurally similar to the extracellular matrix of many tissues, can immobilize cells and biologically active macromolecules, and may be delivered into the body in a minimally invasive manner. We are designing hydrogel biomaterials from the perspective of directing cell fate, and have demonstrated that the presentation of cell adhesion peptides and growth factors, in concert with gel physical properties (stiffness and degradation rate of gels) allows one to regulate gene expression in vitro and tissue regeneration in vivo. FRET techniques have been developed to probe receptor-ligand interactions in this system, and provide quantitative relations between cell-gel interactions and resultant control over cell fate. Altogether, these studies both improve our understanding of how adhesion regulates cell function, and provide new design criteria for cell-interactive biomaterials.
Cognitive & Linguistic Sciences
Job Talk
Position in Computational Modeling
Director -- Artificial Intelligence & Robotics Laboratory | |
Abstract: The term "model" is attributed to a large set of theoretical constructs in the cognetive science literature. One may encounter "algorithmic models," "connectionist models," "dynamical systems models," "evolutionary models," "stochastic models," and many others. Given this wealth, the question arises what all these models have in common, i.e., what makes them "models," and, furthermore, what they can do for cognitive science?
In the first part of my presentation, I will start wiath a brief summary of the intuitions, behind the notion of "model," which I take to be shared by the different kinds, and outline the general steps involved in the process of fitting a model to human data, friefly arguing for the strengths of embodied distributed parallel processing models, In the second part, I will present three different kinds of models I have developed: (1) a connectionist model of bilingual language processing, (2) an agent-based model of conflict resolution, and (3) a robotic platform for affective natural language interactions with people. I will use the first model to illustrate in some detail the process of model development (including fitting model parameters to human data and making predictions based on model outcomes). The second model is intended to demonstrate the utility of agent-based simulations for the exploration of large parameters spaces that can then lead to the formulation of general principles. Finally, I will argue that complex robots provide an excellent platform for the development of embodied models of affect, language, and social interaction. As such, they can also be used to perform real-time experiments with people, which, in turn, may lead to new insights into the critical factors involved in successful interactions of people with artifacts. In the third and final part, I will summarize my past and current activities and provide an outlook of future directions.
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