PhD Student at Carnegie Mellon University
The brain architecture is captured by structural neuroimaging techniques such as diffusion spectrum imaging (DSI), while neural activities are captured by functional neuroimaging techniques such as functional magnetic resonance imaging (fMRI). We are building a statistical model that relates the two different kinds of connectivity measurements based on recent developments in advanced processing techniques that extract local connectivity of the brain (Yeh et al. 2016). We hope this model to give an insight into the role of structure-function relationship in human cognition and action.
I wrote a brief review on known minimax rates and generalization error bounds for feedforward neural networks with nonlinear activation functions. The results suggest that (1) two-layer neural networks can avoid the curse of dimensionality and that (2) they are adaptive to an underlying sparse structure—if it exists. However, it is unclear whether these results generalize to deep neural networks.
We are studying a new type of high-dimensional regression model that fits an additive model where each component is either convex, concave, or identically zero. This has led to a challenging and fascinating problem we call “convexity pattern selection,” which is to infer the correct sparsity and convexity pattern of \(p\) variables, among the \(3^p\) possible patterns. Other shape constraints such as monotonicity can be used. These models extend the idea of sparse additive models (Ravikumar et al. 2009).
Sociologists design and conduct extensive surveys to study factors behind high crime rates or low income levels in certain neighborhoods. Aiming to build an effective alternative to these costly and time-consuming methods, we studied data-driven methods that model the latent factors using neighborhood-level Google Street View images.
We implemented a prediction model using the ImageNet-pretrained features of Caffe, an efficient convolutional neural network (CNN) implementation for image classification (Jia et al. 2014). We also collected survey data using the Amazon Mechanical Turk service where we asked people to compare the perceived safety and affluence given two images.
Combining the ideas of log-concave density estimation (Cule and Samworth 2009) and sum-of-squares (SOS) convexity (Ahmadi 2010, Lasserre 2009), we began to develop density estimation and graphical modeling methods using log-SOS-concave functions. We are studying a projected gradient descent approach using semidefinite programming (SDP) and time-varying sampling techniques (Narayanan and Rakhlin 2013).
As a full participant in the Mathematics REU program, I studied a kernel defined on bags of vectors using the Bhattacharyya distance, as proposed by Jebara and Kondor in 2003. I studied kernel methods such as kernel principal component analysis, the theory of reproducing kernel Hilbert spaces (RKHS), and regularized covariance estimation in an RKHS.
As an apprentice (first-year) participant in the Mathematics REU program, I studied one of Erdös' theorems that there exists a graph with arbitrarily large chromatic number and girth. In particular, I studied the proof of Kneser's conjecture, moment curves, Gale's theorem on the distribution of points on the sphere, Borsuk graphs, and Kneser graphs.