Sensor Network Localization via Riemannian Conjugate Gradient and Rank Reduction: An Extended Version

This paper addresses the Sensor Network Localization (SNL) problem using received signal strength. The SNL is formulated as an Euclidean Distance Matrix Completion (EDMC) problem under the unit ball sample model. Using the Burer-Monteiro factorization type cost function, the EDMC is solved by Riemannian conjugate gradient with Hager-Zhang line search method on a quotient manifold. A "rank reduction" preprocess is proposed for proper initialization and to achieve global convergence with high probability. Simulations on a synthetic scene show that our approach attains better localization accuracy and is computationally efficient compared to several baseline methods. Characterization of a small local basin of attraction around the global optima of the s-stress function under Bernoulli sampling rule and incoherence matrix completion framework is conducted for the first time. Theoretical result conjectures that the Euclidean distance problem with a structure-less sample mask can be effectively handled using spectral initialization followed by vanilla first-order methods. This preliminary analysis, along with the aforementioned numerical accomplishments, provides insights into revealing the landscape of the s-stress function and may stimulate the design of simpler algorithms to tackle the non-convex formulation of general EDMC problems.

Generalization Error Curves for Analytic Spectral Algorithms under Power-law Decay

The generalization error curve of certain kernel regression method aims at determining the exact order of generalization error with various source condition, noise level and choice of the regularization parameter rather than the minimax rate. In this work, under mild assumptions, we rigorously provide a full characterization of the generalization error curves of the kernel gradient descent method (and a large class of analytic spectral algorithms) in kernel regression. Consequently, we could sharpen the near inconsistency of kernel interpolation and clarify the saturation effects of kernel regression algorithms with higher qualification, etc. Thanks to the neural tangent kernel theory, these results greatly improve our understanding of the generalization behavior of training the wide neural networks. A novel technical contribution, the analytic functional argument, might be of independent interest.

Optimal Rates of Kernel Ridge Regression under Source Condition in Large Dimensions

Motivated by the studies of neural networks (e.g.,the neural tangent kernel theory), we perform a study on the large-dimensional behavior of kernel ridge regression (KRR) where the sample size $n \asymp d^{\gamma}$ for some $\gamma > 0$. Given an RKHS $\mathcal{H}$ associated with an inner product kernel defined on the sphere $\mathbb{S}^{d}$, we suppose that the true function $f_{\rho}^{*} \in [\mathcal{H}]^{s}$, the interpolation space of $\mathcal{H}$ with source condition $s>0$. We first determined the exact order (both upper and lower bound) of the generalization error of kernel ridge regression for the optimally chosen regularization parameter $\lambda$. We then further showed that when $0<s\le1$, KRR is minimax optimal; and when $s>1$, KRR is not minimax optimal (a.k.a. he saturation effect). Our results illustrate that the curves of rate varying along $\gamma$ exhibit the periodic plateau behavior and the multiple descent behavior and show how the curves evolve with $s>0$. Interestingly, our work provides a unified viewpoint of several recent works on kernel regression in the large-dimensional setting, which correspond to $s=0$ and $s=1$ respectively.

On the Asymptotic Learning Curves of Kernel Ridge Regression under Power-law Decay

The widely observed 'benign overfitting phenomenon' in the neural network literature raises the challenge to the 'bias-variance trade-off' doctrine in the statistical learning theory. Since the generalization ability of the 'lazy trained' over-parametrized neural network can be well approximated by that of the neural tangent kernel regression, the curve of the excess risk (namely, the learning curve) of kernel ridge regression attracts increasing attention recently. However, most recent arguments on the learning curve are heuristic and are based on the 'Gaussian design' assumption. In this paper, under mild and more realistic assumptions, we rigorously provide a full characterization of the learning curve: elaborating the effect and the interplay of the choice of the regularization parameter, the source condition and the noise. In particular, our results suggest that the 'benign overfitting phenomenon' exists in very wide neural networks only when the noise level is small.

Optimal Rate of Kernel Regression in Large Dimensions

We perform a study on kernel regression for large-dimensional data (where the sample size $n$ is polynomially depending on the dimension $d$ of the samples, i.e., $n\asymp d^{\gamma}$ for some $\gamma >0$ ). We first build a general tool to characterize the upper bound and the minimax lower bound of kernel regression for large dimensional data through the Mendelson complexity $\varepsilon_{n}^{2}$ and the metric entropy $\bar{\varepsilon}_{n}^{2}$ respectively. When the target function falls into the RKHS associated with a (general) inner product model defined on $\mathbb{S}^{d}$, we utilize the new tool to show that the minimax rate of the excess risk of kernel regression is $n^{-1/2}$ when $n\asymp d^{\gamma}$ for $\gamma =2, 4, 6, 8, \cdots$. We then further determine the optimal rate of the excess risk of kernel regression for all the $\gamma>0$ and find that the curve of optimal rate varying along $\gamma$ exhibits several new phenomena including the {\it multiple descent behavior} and the {\it periodic plateau behavior}. As an application, For the neural tangent kernel (NTK), we also provide a similar explicit description of the curve of optimal rate. As a direct corollary, we know these claims hold for wide neural networks as well.

On the Optimality of Misspecified Kernel Ridge Regression

In the misspecified kernel ridge regression problem, researchers usually assume the underground true function $f_{\rho}^{*} \in [\mathcal{H}]^{s}$, a less-smooth interpolation space of a reproducing kernel Hilbert space (RKHS) $\mathcal{H}$ for some $s\in (0,1)$. The existing minimax optimal results require $\|f_{\rho}^{*}\|_{L^{\infty}}<\infty$ which implicitly requires $s > \alpha_{0}$ where $\alpha_{0}\in (0,1)$ is the embedding index, a constant depending on $\mathcal{H}$. Whether the KRR is optimal for all $s\in (0,1)$ is an outstanding problem lasting for years. In this paper, we show that KRR is minimax optimal for any $s\in (0,1)$ when the $\mathcal{H}$ is a Sobolev RKHS.

Statistical Optimality of Deep Wide Neural Networks

In this paper, we consider the generalization ability of deep wide feedforward ReLU neural networks defined on a bounded domain $\mathcal X \subset \mathbb R^{d}$. We first demonstrate that the generalization ability of the neural network can be fully characterized by that of the corresponding deep neural tangent kernel (NTK) regression. We then investigate on the spectral properties of the deep NTK and show that the deep NTK is positive definite on $\mathcal{X}$ and its eigenvalue decay rate is $(d+1)/d$. Thanks to the well established theories in kernel regression, we then conclude that multilayer wide neural networks trained by gradient descent with proper early stopping achieve the minimax rate, provided that the regression function lies in the reproducing kernel Hilbert space (RKHS) associated with the corresponding NTK. Finally, we illustrate that the overfitted multilayer wide neural networks can not generalize well on $\mathbb S^{d}$.

Kernel interpolation generalizes poorly

One of the most interesting problems in the recent renaissance of the studies in kernel regression might be whether the kernel interpolation can generalize well, since it may help us understand the `benign overfitting henomenon' reported in the literature on deep networks. In this paper, under mild conditions, we show that for any $\varepsilon>0$, the generalization error of kernel interpolation is lower bounded by $\Omega(n^{-\varepsilon})$. In other words, the kernel interpolation generalizes poorly for a large class of kernels. As a direct corollary, we can show that overfitted wide neural networks defined on sphere generalize poorly.

An Integrated Enhancement Solution for 24-hour Colorful Imaging

The current industry practice for 24-hour outdoor imaging is to use a silicon camera supplemented with near-infrared (NIR) illumination. This will result in color images with poor contrast at daytime and absence of chrominance at nighttime. For this dilemma, all existing solutions try to capture RGB and NIR images separately. However, they need additional hardware support and suffer from various drawbacks, including short service life, high price, specific usage scenario, etc. In this paper, we propose a novel and integrated enhancement solution that produces clear color images, whether at abundant sunlight daytime or extremely low-light nighttime. Our key idea is to separate the VIS and NIR information from mixed signals, and enhance the VIS signal adaptively with the NIR signal as assistance. To this end, we build an optical system to collect a new VIS-NIR-MIX dataset and present a physically meaningful image processing algorithm based on CNN. Extensive experiments show outstanding results, which demonstrate the effectiveness of our solution.

Instance-level Human Parsing via Part Grouping Network

Instance-level human parsing towards real-world human analysis scenarios is still under-explored due to the absence of sufficient data resources and technical difficulty in parsing multiple instances in a single pass. Several related works all follow the "parsing-by-detection" pipeline that heavily relies on separately trained detection models to localize instances and then performs human parsing for each instance sequentially. Nonetheless, two discrepant optimization targets of detection and parsing lead to suboptimal representation learning and error accumulation for final results. In this work, we make the first attempt to explore a detection-free Part Grouping Network (PGN) for efficiently parsing multiple people in an image in a single pass. Our PGN reformulates instance-level human parsing as two twinned sub-tasks that can be jointly learned and mutually refined via a unified network: 1) semantic part segmentation for assigning each pixel as a human part (e.g., face, arms); 2) instance-aware edge detection to group semantic parts into distinct person instances. Thus the shared intermediate representation would be endowed with capabilities in both characterizing fine-grained parts and inferring instance belongings of each part. Finally, a simple instance partition process is employed to get final results during inference. We conducted experiments on PASCAL-Person-Part dataset and our PGN outperforms all state-of-the-art methods. Furthermore, we show its superiority on a newly collected multi-person parsing dataset (CIHP) including 38,280 diverse images, which is the largest dataset so far and can facilitate more advanced human analysis. The CIHP benchmark and our source code are available at http://sysu-hcp.net/lip/.