arXiv subjects

Dongmian Zou

Publications and source records attributed to Dongmian Zou.

30 records · Page 2Linked to original sources

arXiv2023

Autoencoding Hyperbolic Representation for Adversarial Generation

With the recent advance of geometric deep learning, neural networks have been extensively used for data in non-Euclidean domains. In particular, hyperbolic neural networks have proved successful in processing hierarchical information of data. However, many hyperbolic neural networks are numerically unstable during training, which precludes using complex architectures. This crucial problem makes it difficult to build hyperbolic generative models for real and complex data. In this work, we propose a hyperbolic generative network in which we design novel architecture and layers to improve stability in training. Our proposed network contains three parts: first, a hyperbolic autoencoder (AE) that produces hyperbolic embedding for input data; second, a hyperbolic generative adversarial network (GAN) for generating the hyperbolic latent embedding of the AE from simple noise; third, a generator that inherits the decoder from the AE and the generator from the GAN. We call this network the hyperbolic AE-GAN, or HAEGAN for short. The architecture of HAEGAN fosters expressive representation in the hyperbolic space, and the specific design of layers ensures numerical stability. Experiments show that HAEGAN is able to generate complex data with state-of-the-art structure-related performance.

cs.LG↗

arXiv2022

Robust Vector Quantized-Variational Autoencoder

Image generative models can learn the distributions of the training data and consequently generate examples by sampling from these distributions. However, when the training dataset is corrupted with outliers, generative models will likely produce examples that are also similar to the outliers. In fact, a small portion of outliers may induce state-of-the-art generative models, such as Vector Quantized-Variational AutoEncoder (VQ-VAE), to learn a significant mode from the outliers. To mitigate this problem, we propose a robust generative model based on VQ-VAE, which we name Robust VQ-VAE (RVQ-VAE). In order to achieve robustness, RVQ-VAE uses two separate codebooks for the inliers and outliers. To ensure the codebooks embed the correct components, we iteratively update the sets of inliers and outliers during each training epoch. To ensure that the encoded data points are matched to the correct codebooks, we quantize using a weighted Euclidean distance, whose weights are determined by directional variances of the codebooks. Both codebooks, together with the encoder and decoder, are trained jointly according to the reconstruction loss and the quantization loss. We experimentally demonstrate that RVQ-VAE is able to generate examples from inliers even if a large portion of the training data points are corrupted.

cs.LG↗

arXiv2022

Graph Neural Network Based Node Deployment for Throughput Enhancement

The recent rapid growth in mobile data traffic entails a pressing demand for improving the throughput of the underlying wireless communication networks. Network node deployment has been considered as an effective approach for throughput enhancement which, however, often leads to highly non-trivial non-convex optimizations. Although convex approximation based solutions are considered in the literature, their approximation to the actual throughput may be loose and sometimes lead to unsatisfactory performance. With this consideration, in this paper, we propose a novel graph neural network (GNN) method for the network node deployment problem. Specifically, we fit a GNN to the network throughput and use the gradients of this GNN to iteratively update the locations of the network nodes. Besides, we show that an expressive GNN has the capacity to approximate both the function value and the gradients of a multivariate permutation-invariant function, as a theoretic support to the proposed method. To further improve the throughput, we also study a hybrid node deployment method based on this approach. To train the desired GNN, we adopt a policy gradient algorithm to create datasets containing good training samples. Numerical experiments show that the proposed methods produce competitive results compared to the baselines.

cs.NI↗

arXiv2021

Ensemble Riemannian Data Assimilation over the Wasserstein Space

In this paper, we present an ensemble data assimilation paradigm over a Riemannian manifold equipped with the Wasserstein metric. Unlike the Eulerian penalization of error in the Euclidean space, the Wasserstein metric can capture translation and difference between the shapes of square-integrable probability distributions of the background state and observations -- enabling to formally penalize geophysical biases in state-space with non-Gaussian distributions. The new approach is applied to dissipative and chaotic evolutionary dynamics and its potential advantages and limitations are highlighted compared to the classic variational and filtering data assimilation approaches under systematic and random errors.

stat.ME↗

arXiv2020

Regularized Variational Data Assimilation for Bias Treatment using the Wasserstein Metric

This paper presents a new variational data assimilation (VDA) approach for the formal treatment of bias in both model outputs and observations. This approach relies on the Wasserstein metric stemming from the theory of optimal mass transport to penalize the distance between the probability histograms of the analysis state and an a priori reference dataset, which is likely to be more uncertain but less biased than both model and observations. Unlike previous bias-aware VDA approaches, the new Wasserstein metric VDA (WM-VDA) dynamically treats systematic biases of unknown magnitude and sign in both model and observations through assimilation of the reference data in the probability domain and can fully recover the probability histogram of the analysis state. The performance of WM-VDA is compared with the classic three-dimensional VDA (3D-Var) scheme on first-order linear dynamics and the chaotic Lorenz attractor. Under positive systematic biases in both model and observations, we consistently demonstrate a significant reduction in the forecast bias and unbiased root mean squared error.

stat.ME↗

arXiv2019

Robust Subspace Recovery Layer for Unsupervised Anomaly Detection

We propose a neural network for unsupervised anomaly detection with a novel robust subspace recovery layer (RSR layer). This layer seeks to extract the underlying subspace from a latent representation of the given data and removes outliers that lie away from this subspace. It is used within an autoencoder. The encoder maps the data into a latent space, from which the RSR layer extracts the subspace. The decoder then smoothly maps back the underlying subspace to a "manifold" close to the original inliers. Inliers and outliers are distinguished according to the distances between the original and mapped positions (small for inliers and large for outliers). Extensive numerical experiments with both image and document datasets demonstrate state-of-the-art precision and recall.

cs.LG↗

arXiv2019

Encoding Robust Representation for Graph Generation

Generative networks have made it possible to generate meaningful signals such as images and texts from simple noise. Recently, generative methods based on GAN and VAE were developed for graphs and graph signals. However, the mathematical properties of these methods are unclear, and training good generative models is difficult. This work proposes a graph generation model that uses a recent adaptation of Mallat's scattering transform to graphs. The proposed model is naturally composed of an encoder and a decoder. The encoder is a Gaussianized graph scattering transform, which is robust to signal and graph manipulation. The decoder is a simple fully connected network that is adapted to specific tasks, such as link prediction, signal generation on graphs and full graph and signal generation. The training of our proposed system is efficient since it is only applied to the decoder and the hardware requirements are moderate. Numerical results demonstrate state-of-the-art performance of the proposed system for both link prediction and graph and signal generation.

cs.LG↗

arXiv2018

Graph Convolutional Neural Networks via Scattering

We generalize the scattering transform to graphs and consequently construct a convolutional neural network on graphs. We show that under certain conditions, any feature generated by such a network is approximately invariant to permutations and stable to graph manipulations. Numerical results demonstrate competitive performance on relevant datasets.

cs.IT↗

arXiv2018

On Lipschitz Bounds of General Convolutional Neural Networks

Many convolutional neural networks (CNNs) have a feed-forward structure. In this paper, a linear program that estimates the Lipschitz bound of such CNNs is proposed. Several CNNs, including the scattering networks, the AlexNet and the GoogleNet, are studied numerically and compared to the theoretical bounds. Next, concentration inequalities of the output distribution to a stationary random input signal expressed in terms of the Lipschitz bound are established. The Lipschitz bound is further used to establish a nonlinear discriminant analysis designed to measure the separation between features of different classes.

cs.IT↗

arXiv2017

Lipschitz Properties for Deep Convolutional Networks

In this paper we discuss the stability properties of convolutional neural networks. Convolutional neural networks are widely used in machine learning. In classification they are mainly used as feature extractors. Ideally, we expect similar features when the inputs are from the same class. That is, we hope to see a small change in the feature vector with respect to a deformation on the input signal. This can be established mathematically, and the key step is to derive the Lipschitz properties. Further, we establish that the stability results can be extended for more general networks. We give a formula for computing the Lipschitz bound, and compare it with other methods to show it is closer to the optimal value.

cs.LG↗

arXiv2015

On Lipschitz Analysis and Lipschitz Synthesis for the Phase Retrieval Problem

In this paper we prove two results regarding reconstruction from magnitudes of frame coefficients (the so called "phase retrieval problem"). First we show that phase retrievability as an algebraic property implies that nonlinear maps are bi-Lipschitz with respect to appropriate metrics on the quotient space. Second we prove that reconstruction can be performed using Lipschitz continuous maps. Specifically we show that when nonlinear analysis maps $α,β:\hat{H}\rightarrow R^m$ are injective, with $α(x)=(|\langle x,f_k\rangle |)_{k=1}^m$ and $β(x)=(|\langle x,f_k \rangle|^2)_{k=1}^m$, where $\{f_1,\ldots,f_m\}$ is a frame for a Hilbert space $H$ and $\hat{H}=H/T^1$, then $α$ is bi-Lipschitz with respect to the class of "natural metrics" $D_p(x,y)= min_φ || x-e^{iφ}y {||}_p$, whereas $β$ is bi-Lipschitz with respect to the class of matrix-norm induced metrics $d_p(x,y)=|| xx^*-yy^*{||}_p$. Furthermore, there exist left inverse maps $ω,ψ:R^m\rightarrow \hat{H}$ of $α$ and $β$ respectively, that are Lipschitz continuous with respect to the appropriate metric. Additionally we obtain the Lipschitz constants of these inverse maps in terms of the lower Lipschitz constants of $α$ and $β$. Surprisingly the increase in Lipschitz constant is a relatively small factor, independent of the space dimension or the frame redundancy.

math.FA↗

arXiv2014

Phase Retrieval using Lipschitz Continuous Maps

In this note we prove that reconstruction from magnitudes of frame coefficients (the so called "phase retrieval problem") can be performed using Lipschitz continuous maps. Specifically we show that when the nonlinear analysis map $α:{\mathcal H}\rightarrow\mathbb{R}^m$ is injective, with $(α(x))_k=| |^2$, where $\{f_1,\ldots,f_m\}$ is a frame for the Hilbert space ${\mathcal H}$, then there exists a left inverse map $ω:\mathbb{R}^m\rightarrow {\mathcal H}$ that is Lipschitz continuous. Additionally we obtain the Lipschitz constant of this inverse map in terms of the lower Lipschitz constant of $α$. Surprisingly the increase in Lipschitz constant is independent of the space dimension or frame redundancy.

math.FA↗