Yuxiang Wang

Kaiqi Zhang, Yu-Xiang Wang

We study the theory of neural network (NN) from the lens of classical nonparametric regression problems with a focus on NN's ability to adaptively estimate functions with heterogeneous smoothness -- a property of functions in Besov or Bounded Variation (BV) classes. Existing work on this problem requires tuning the NN architecture based on the function spaces and sample size. We consider a "Parallel NN" variant of deep ReLU networks and show that the standard $\ell_2$ regularization is equivalent to promoting the $\ell_p$-sparsity ($0<p<1$) in the coefficient vector of an end-to-end learned function bases, i.e., a dictionary. Using this equivalence, we further establish that by tuning only the regularization factor, such parallel NN achieves an estimation error arbitrarily close to the minimax rates for both the Besov and BV classes. Notably, it gets exponentially closer to minimax optimal as the NN gets deeper. Our research sheds new lights on why depth matters and how NNs are more powerful than kernel methods.

Veeranjaneyulu Sadhanala, Yu-Xiang Wang, Addison J. Hu et al.

We study a multivariate version of trend filtering, called Kronecker trend filtering or KTF, for the case in which the design points form a lattice in $d$ dimensions. KTF is a natural extension of univariate trend filtering (Steidl et al., 2006; Kim et al., 2009; Tibshirani, 2014), and is defined by minimizing a penalized least squares problem whose penalty term sums the absolute (higher-order) differences of the parameter to be estimated along each of the coordinate directions. The corresponding penalty operator can be written in terms of Kronecker products of univariate trend filtering penalty operators, hence the name Kronecker trend filtering. Equivalently, one can view KTF in terms of an $\ell_1$-penalized basis regression problem where the basis functions are tensor products of falling factorial functions, a piecewise polynomial (discrete spline) basis that underlies univariate trend filtering. This paper is a unification and extension of the results in Sadhanala et al. (2016, 2017). We develop a complete set of theoretical results that describe the behavior of $k^{\mathrm{th}}$ order Kronecker trend filtering in $d$ dimensions, for every $k \geq 0$ and $d \geq 1$. This reveals a number of interesting phenomena, including the dominance of KTF over linear smoothers in estimating heterogeneously smooth functions, and a phase transition at $d=2(k+1)$, a boundary past which (on the high dimension-to-smoothness side) linear smoothers fail to be consistent entirely. We also leverage recent results on discrete splines from Tibshirani (2020), in particular, discrete spline interpolation results that enable us to extend the KTF estimate to any off-lattice location in constant-time (independent of the size of the lattice $n$).

Dheeraj Baby, Yu-Xiang Wang

We consider the problem of online forecasting of sequences of length $n$ with total-variation at most $C_n$ using observations contaminated by independent $σ$-subgaussian noise. We design an $O(n\log n)$-time algorithm that achieves a cumulative square error of $\tilde{O}(n^{1/3}C_n^{2/3}σ^{4/3} + C_n^2)$ with high probability.We also prove a lower bound that matches the upper bound in all parameters (up to a $\log(n)$ factor). To the best of our knowledge, this is the first \emph{polynomial-time} algorithm that achieves the optimal $O(n^{1/3})$ rate in forecasting total variation bounded sequences and the first algorithm that \emph{adapts to unknown} $C_n$. Our proof techniques leverage the special localized structure of Haar wavelet basis and the adaptivity to unknown smoothness parameters in the classical wavelet smoothing [Donoho et al., 1998]. We also compare our model to the rich literature of dynamic regret minimization and nonstationary stochastic optimization, where our problem can be treated as a special case. We show that the workhorse in those settings --- online gradient descent and its variants with a fixed restarting schedule --- are instances of a class of \emph{linear forecasters} that require a suboptimal regret of $\tildeΩ(\sqrt{n})$. This implies that the use of more adaptive algorithms is necessary to obtain the optimal rate.

Xuemei Xie, Yuxiang Wang, Guangming Shi et al.

Conventional compressive sensing (CS) reconstruction is very slow for its characteristic of solving an optimization problem. Convolu- tional neural network can realize fast processing while achieving compa- rable results. While CS image recovery with high quality not only de- pends on good reconstruction algorithms, but also good measurements. In this paper, we propose an adaptive measurement network in which measurement is obtained by learning. The new network consists of a fully-connected layer and ReconNet. The fully-connected layer which has low-dimension output acts as measurement. We train the fully-connected layer and ReconNet simultaneously and obtain adaptive measurement. Because the adaptive measurement fits dataset better, in contrast with random Gaussian measurement matrix, under the same measuremen- t rate, it can extract the information of scene more efficiently and get better reconstruction results. Experiments show that the new network outperforms the original one.