Stochastic gradient descent based algorithms are typically used as the general optimization tools for most deep learning models. A Restricted Boltzmann Machine (RBM) is a probabilistic generative model that can be stacked to construct deep architectures. For RBM with Bernoulli inputs, non-Euclidean algorithm such as stochastic spectral descent (SSD) has been specifically designed to speed up th...

Suppose that m,n ∈ N and that A : R → R is a linear operator. It is shown here that if k, r ∈ N satisfy k < r 6 rank(A) then there exists a subset σ ⊆ {1, . . . ,m} with |σ| = k such that the restriction of A to R ⊆ R is invertible, and moreover the operator norm of the inverse A−1 : A(R) → R is at most a constant multiple of the quantity √ mr/((r − k) ∑m i=r si(A) 2), where s1(A) > . . . > sm(...

In this paper we present novel strategies for completion of 5D pre-stack seismic data, viewed as a 5D tensor or as a set of 4D tensors across temporal frequencies. In contrast to existing complexity penalized algorithms for seismic data completion, which employ matrix analogues of tensor decompositions such as HOSVD or use overlapped Schatten norms from different unfoldings (or matricization) o...

Rapid advances in sensor, wireless communication, cloud computing and data science have brought unprecedented amount of to assist transportation engineers researchers making better decisions. However, traffic reality often has corrupted or incomplete values due detector communication malfunctions. Data imputation is thus required ensure the effectiveness downstream data-driven applications. To ...

We consider convex sets whose modulus of convexity is uniformly quadratic. First, we observe several interesting relations between different positions of such “2-convex” bodies; in particular, the isotropic position is a finite volume-ratio position for these bodies. Second, we prove that high dimensional 2-convex bodies posses one-dimensional marginals that are approximately Gaussian. Third, w...

Let $||X||_p=\text{Tr}[(X^\ast X)^{p/2}]^{1/p}$ denote the $p$-Schatten norm of a matrix $X\in M_{n\times n}(\mathbb{C})$, and $\sigma(X)$ singular values with $\uparrow$ $\downarrow$ indicating its increasing or decreasing rearrangements. We wish to examine inequalities between $||A+B||_p^p+||A-B||_p^p$, $||\sigma_\downarrow(A)+\sigma_\downarrow(B)||_p^p+||\sigma_\downarrow(A)-\sigma_\downarro...

We provide a new proof of Volberg’s Theorem characterizing thin interpolating sequences as those for which the Gram matrix associated to the normalized reproducing kernels is a compact perturbation of the identity. In the same paper, Volberg characterized sequences for which the Gram matrix is a compact perturbation of a unitary as well as those for which the Gram matrix is a Schatten-2 class p...

Learning linear combinations of multiple kernels is an appealing strategy when the right choice of features is unknown. Previous approaches to multiple kernel learning (MKL) promote sparse kernel combinations to support interpretability and scalability. Unfortunately, this `1-norm MKL is rarely observed to outperform trivial baselines in practical applications. To allow for robust kernel mixtur...