The L\'evy process has been widely applied to mathematical finance, quantum mechanics, peridynamics, and so on. However, calibrating the nonparametric multivariate distribution related to the L\'evy process from observations is a very challenging problem due to the lack of explicit distribution functions. In this paper, we propose a novel algorithm based on neural networks and automatic differentiation for solving this problem. We use neural networks to approximate the nonparametric part and discretize the characteristic exponents using accuracy numerical quadratures. Automatic differentiation is then applied to compute gradients and we minimize the mismatch between empirical and exact characteristic exponents using first-order optimization approaches. Another distinctive contribution of our work is that we made an effort to investigate the approximation ability of neural networks and the convergence behavior of algorithms. We derived the estimated number of neurons for a two-layer neural network. To achieve an accuracy of $\varepsilon$ with the input dimension $d$, it is sufficient to build $\mathcal{O}\left(\left(\frac{d}{\varepsilon} \right)^2\right)$ and $\mathcal{O}\left(\frac{d}{\varepsilon} \right)$ for the first and second layers. The numbers are polynomial in the input dimension compared to the exponential $\mathcal{O}\left(\varepsilon^{-d} \right)$ for one. We also give the convergence proof of the neural network concerning the training samples under mild assumptions and show that the RMSE decreases linearly in the number of training data in the consistency error dominancy region for the 2D problem. It is the first-ever convergence analysis for such an algorithm in literature to our best knowledge. Finally, we apply the algorithms to the stock markets and reveal some interesting patterns in the pairwise $\alpha$ index.

Kernel matrices are popular in machine learning and scientific computing, but they are limited by their quadratic complexity in both construction and storage. It is well-known that as one varies the kernel parameter, e.g., the width parameter in radial basis function kernels, the kernel matrix changes from a smooth low-rank kernel to a diagonally-dominant and then fully-diagonal kernel. Low-rank approximation methods have been widely-studied, mostly in the first case, to reduce the memory storage and the cost of computing matrix-vector products. Here, we use ideas from scientific computing to propose an extension of these methods to situations where the matrix is not well-approximated by a low-rank matrix. In particular, we construct an efficient block low-rank approximation method---which we call the Block Basis Factorization---and we show that it has $\mathcal{O}(n)$ complexity in both time and memory. Our method works for a wide range of kernel parameters, extending the domain of applicability of low-rank approximation methods, and our empirical results demonstrate the stability (small standard deviation in error) and superiority over current state-of-art kernel approximation algorithms.