**(1) New bounds on the capacity of hexagonal graphs**

In this work, we propose a new method to derive rigorous bounds on \eta, the growth rate of the logarithm of the number of independent sets on a hexagonal lattice. Specifically, we prove that 1.546440708536001 <= \eta <= 1.5513, which improves upon the best known 1.5463 <= \eta <= 1.5527. Our lower bound matches the numerical estimate of Baxter up to 9 digits after the decimal point.

Zhun Deng, Jie Ding, Kathryn Heal, Vahid Tarokh, The Number of Independent Sets In Hexagonal Graphs. pdf

**(2) Evolutionary Spectra for non-stationary processes**

In this work, we propose a new inference procedure for understanding non-stationary processes, under the framework of evolutionary spectra proposed by Priestley. We propose a new estimator of the evolutionary spectral density that improves upon the state-of-the-art, and analyze its bias/variance/resolution tradeoff. Based on the new estimator, we further propose a non-parametric stationarity test. pdf

**(3) Nonparametric methods**

In this joint work with Zhun Deng and Enmao Diao, we design an nonparametric distance between two density functions, where the distance is in the form of L2 distance weighted by an appropriately chosen kernel function. We construct a nonparametric estimator and rigorously characterize its large sample asymptotic performance. Our design aims to amplify the difference between distributions by focusing on a specific data domain. Thus, it can be applied to identify whether two sets of data are generated from the same underlying distribution, and specifically, to test independence of two random variables.

Zhun Deng, Jie Ding, Enmao Diao, Vahid Tarokh, Detecting Local Closeness with Weighted L2 Divergence. pdf

**(4) Quadruple sequence conjecture and generalization of Euler’s four-square identity**

In the study of g-design (proposed in the work of *block design*), I made the conjecture that “there exists four binary sequences that are complementary (elaborated in the work of *complementary lattice array*) and of length k, for any positive integer k. This turns out to be intimately related to the unsolved *Hadamard conjecture*. In an ongoing attempt to prove the conjecture, I discovered a generalization of *Euler’s four-square identity*, which seems quite interesting in its own right. The paper is in preparation.

**(5) Online learning for online object recognition**

Using a tactile glove with seven sensors, we collect data from a 7-dimensional time-series, where each dimension represents the pressure sequence applied to one sensor. In this project, we built a reference dictionary from historic data and online recognize the unknown object being grasped in an adaptive and computationally efficient manner.

Shahin Shahrampour, Mohammad Noshad, Jie Ding, Vahid Tarokh, Online Learning for Multimodal Data Fusion with Application to Object Recognition. pdf