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We study how the posterior contraction rate under a Gaussian process (GP) prior depends on the intrinsic dimension of the predictors and smoothness of the regression function. An open question is whether a generic GP prior that does not incorporate knowledge of the intrinsic lower-dimensional structure of the predictors can attain an adaptive rate for a broad class of such structures. We show that this is indeed the case, establishing conditions under which the posterior contraction rates become adaptive to the intrinsic dimension $\varrho$ in terms of the covering number of the data domain $X$ (the Minkowski dimension), and prove the optimal posterior contraction rate $O(n^{-s/(2s +\varrho)})$, up to a logarithmic factor, assuming an approximation order $s$ of the reproducing kernel Hilbert space (RKHS) on ${X}$. When ${X}$ is a $\varrho$-dimensional compact smooth manifold, we study RKHS approximations to intrinsically defined $s$-order H\"older functions on the manifold for any positive $s$ by a novel analysis of kernel approximations on manifolds, leading to the optimal adaptive posterior contraction rate. We propose an empirical Bayes prior on the kernel bandwidth using kernel affinity and $k$-nearest neighbor statistics, eliminating the need for prior knowledge of the intrinsic dimension. The efficiency of the proposed Bayesian regression approach is demonstrated on various numerical experiments.

We construct and analyze a neural network two-sample test to determine whether two datasets came from the same distribution (null hypothesis) or not (alternative hypothesis). We perform time-analysis on a neural tangent kernel (NTK) two-sample test. In particular, we derive the theoretical minimum training time needed to ensure the NTK two-sample test detects a deviation-level between the datasets. Similarly, we derive the theoretical maximum training time before the NTK two-sample test detects a deviation-level. By approximating the neural network dynamics with the NTK dynamics, we extend this time-analysis to the realistic neural network two-sample test generated from time-varying training dynamics and finite training samples. A similar extension is done for the neural network two-sample test generated from time-varying training dynamics but trained on the population. To give statistical guarantees, we show that the statistical power associated with the neural network two-sample test goes to 1 as the neural network training samples and test evaluation samples go to infinity. Additionally, we prove that the training times needed to detect the same deviation-level in the null and alternative hypothesis scenarios are well-separated. Finally, we run some experiments showcasing a two-layer neural network two-sample test on a hard two-sample test problem and plot a heatmap of the statistical power of the two-sample test in relation to training time and network complexity.

We present a computationally efficient framework, called $\texttt{FlowDRO}$, for solving flow-based distributionally robust optimization (DRO) problems with Wasserstein uncertainty sets while aiming to find continuous worst-case distribution (also called the Least Favorable Distribution, LFD) and sample from it. The requirement for LFD to be continuous is so that the algorithm can be scalable to problems with larger sample sizes and achieve better generalization capability for the induced robust algorithms. To tackle the computationally challenging infinitely dimensional optimization problem, we leverage flow-based models and continuous-time invertible transport maps between the data distribution and the target distribution and develop a Wasserstein proximal gradient flow type algorithm. In theory, we establish the equivalence of the solution by optimal transport map to the original formulation, as well as the dual form of the problem through Wasserstein calculus and Brenier theorem. In practice, we parameterize the transport maps by a sequence of neural networks progressively trained in blocks by gradient descent. We demonstrate its usage in adversarial learning, distributionally robust hypothesis testing, and a new mechanism for data-driven distribution perturbation differential privacy, where the proposed method gives strong empirical performance on high-dimensional real data., Comment: IEEE Journal on Selected Areas in Information Theory (JSAIT). Accepted. 2024

Flow-based generative models enjoy certain advantages in computing the data generation and the likelihood, and have recently shown competitive empirical performance. Compared to the accumulating theoretical studies on related score-based diffusion models, analysis of flow-based models, which are deterministic in both forward (data-to-noise) and reverse (noise-to-data) directions, remain sparse. In this paper, we provide a theoretical guarantee of generating data distribution by a progressive flow model, the so-called JKO flow model, which implements the Jordan-Kinderleherer-Otto (JKO) scheme in a normalizing flow network. Leveraging the exponential convergence of the proximal gradient descent (GD) in Wasserstein space, we prove the Kullback-Leibler (KL) guarantee of data generation by a JKO flow model to be $O(\varepsilon^2)$ when using $N \lesssim \log (1/\varepsilon)$ many JKO steps ($N$ Residual Blocks in the flow) where $\varepsilon $ is the error in the per-step first-order condition. The assumption on data density is merely a finite second moment, and the theory extends to data distributions without density and when there are inversion errors in the reverse process where we obtain KL-$W_2$ mixed error guarantees. The non-asymptotic convergence rate of the JKO-type $W_2$-proximal GD is proved for a general class of convex objective functionals that includes the KL divergence as a special case, which can be of independent interest. The analysis framework can extend to other first-order Wasserstein optimization schemes applied to flow-based generative models.

Point process models are widely used for continuous asynchronous event data, where each data point includes time and additional information called "marks", which can be locations, nodes, or event types. This paper presents a novel point process model for discrete event data over graphs, where the event interaction occurs within a latent graph structure. Our model builds upon Hawkes's classic influence kernel-based formulation in the original self-exciting point processes work to capture the influence of historical events on future events' occurrence. The key idea is to represent the influence kernel by Graph Neural Networks (GNN) to capture the underlying graph structure while harvesting the strong representation power of GNNs. Compared with prior works focusing on directly modeling the conditional intensity function using neural networks, our kernel presentation herds the repeated event influence patterns more effectively by combining statistical and deep models, achieving better model estimation/learning efficiency and superior predictive performance. Our work significantly extends the existing deep spatio-temporal kernel for point process data, which is inapplicable to our setting due to the fundamental difference in the nature of the observation space being Euclidean rather than a graph. We present comprehensive experiments on synthetic and real-world data to show the superior performance of the proposed approach against the state-of-the-art in predicting future events and uncovering the relational structure among data.

The diffusion maps embedding of data lying on a manifold has shown success in tasks such as dimensionality reduction, clustering, and data visualization. In this work, we consider embedding data sets that were sampled from a manifold which is closed under the action of a continuous matrix group. An example of such a data set is images whose planar rotations are arbitrary. The G-invariant graph Laplacian, introduced in Part I of this work, admits eigenfunctions in the form of tensor products between the elements of the irreducible unitary representations of the group and eigenvectors of certain matrices. We employ these eigenfunctions to derive diffusion maps that intrinsically account for the group action on the data. In particular, we construct both equivariant and invariant embeddings, which can be used to cluster and align the data points. We demonstrate the utility of our construction in the problem of random computerized tomography.

The neural Ordinary Differential Equation (ODE) model has shown success in learning complex continuous-time processes from observations on discrete time stamps. In this work, we consider the modeling and forecasting of time series data that are non-stationary and may have sharp changes like spikes. We propose an RNN-based model, called RNN-ODE-Adap, that uses a neural ODE to represent the time development of the hidden states, and we adaptively select time steps based on the steepness of changes of the data over time so as to train the model more efficiently for the "spike-like" time series. Theoretically, RNN-ODE-Adap yields provably a consistent estimation of the intensity function for the Hawkes-type time series data. We also provide an approximation analysis of the RNN-ODE model showing the benefit of adaptive steps. The proposed model is demonstrated to achieve higher prediction accuracy with reduced computational cost on simulated dynamic system data and point process data and on a real electrocardiography dataset.

Flow-based models are widely used in generative tasks, including normalizing flow, where a neural network transports from a data distribution $P$ to a normal distribution. This work develops a flow-based model that transports from $P$ to an arbitrary $Q$ where both distributions are only accessible via finite samples. We propose to learn the dynamic optimal transport between $P$ and $Q$ by training a flow neural network. The model is trained to optimally find an invertible transport map between $P$ and $Q$ by minimizing the transport cost. The trained optimal transport flow subsequently allows for performing many downstream tasks, including infinitesimal density ratio estimation (DRE) and distribution interpolation in the latent space for generative models. The effectiveness of the proposed model on high-dimensional data is demonstrated by strong empirical performance on high-dimensional DRE, OT baselines, and image-to-image translation.

Graph Laplacian based algorithms for data lying on a manifold have been proven effective for tasks such as dimensionality reduction, clustering, and denoising. In this work, we consider data sets whose data points lie on a manifold that is closed under the action of a known unitary matrix Lie group G. We propose to construct the graph Laplacian by incorporating the distances between all the pairs of points generated by the action of G on the data set. We deem the latter construction the ``G-invariant Graph Laplacian'' (G-GL). We show that the G-GL converges to the Laplace-Beltrami operator on the data manifold, while enjoying a significantly improved convergence rate compared to the standard graph Laplacian which only utilizes the distances between the points in the given data set. Furthermore, we show that the G-GL admits a set of eigenfunctions that have the form of certain products between the group elements and eigenvectors of certain matrices, which can be estimated from the data efficiently using FFT-type algorithms. We demonstrate our construction and its advantages on the problem of filtering data on a noisy manifold closed under the action of the special unitary group SU(2).

Normalizing flow is a class of deep generative models for efficient sampling and likelihood estimation, which achieves attractive performance, particularly in high dimensions. The flow is often implemented using a sequence of invertible residual blocks. Existing works adopt special network architectures and regularization of flow trajectories. In this paper, we develop a neural ODE flow network called JKO-iFlow, inspired by the Jordan-Kinderleherer-Otto (JKO) scheme, which unfolds the discrete-time dynamic of the Wasserstein gradient flow. The proposed method stacks residual blocks one after another, allowing efficient block-wise training of the residual blocks, avoiding sampling SDE trajectories and score matching or variational learning, thus reducing the memory load and difficulty in end-to-end training. We also develop adaptive time reparameterization of the flow network with a progressive refinement of the induced trajectory in probability space to improve the model accuracy further. Experiments with synthetic and real data show that the proposed JKO-iFlow network achieves competitive performance compared with existing flow and diffusion models at a significantly reduced computational and memory cost., Comment: NeurIPS 2023 spotlight

Point process data are becoming ubiquitous in modern applications, such as social networks, health care, and finance. Despite the powerful expressiveness of the popular recurrent neural network (RNN) models for point process data, they may not successfully capture sophisticated non-stationary dependencies in the data due to their recurrent structures. Another popular type of deep model for point process data is based on representing the influence kernel (rather than the intensity function) by neural networks. We take the latter approach and develop a new deep non-stationary influence kernel that can model non-stationary spatio-temporal point processes. The main idea is to approximate the influence kernel with a novel and general low-rank decomposition, enabling efficient representation through deep neural networks and computational efficiency and better performance. We also take a new approach to maintain the non-negativity constraint of the conditional intensity by introducing a log-barrier penalty. We demonstrate our proposed method's good performance and computational efficiency compared with the state-of-the-art on simulated and real data.

Change-point detection, detecting an abrupt change in the data distribution from sequential data, is a fundamental problem in statistics and machine learning. CUSUM is a popular statistical method for online change-point detection due to its efficiency from recursive computation and constant memory requirement, and it enjoys statistical optimality. CUSUM requires knowing the precise pre- and post-change distribution. However, post-change distribution is usually unknown a priori since it represents anomaly and novelty. Classic CUSUM can perform poorly when there is a model mismatch with actual data. While likelihood ratio-based methods encounter challenges facing high dimensional data, neural networks have become an emerging tool for change-point detection with computational efficiency and scalability. In this paper, we introduce a neural network CUSUM (NN-CUSUM) for online change-point detection. We also present a general theoretical condition when the trained neural networks can perform change-point detection and what losses can achieve our goal. We further extend our analysis by combining it with the Neural Tangent Kernel theory to establish learning guarantees for the standard performance metrics, including the average run length (ARL) and expected detection delay (EDD). The strong performance of NN-CUSUM is demonstrated in detecting change-point in high-dimensional data using both synthetic and real-world data.

The Gaussian kernel and its traditional normalizations (e.g., row-stochastic) are popular approaches for assessing similarities between data points. Yet, they can be inaccurate under high-dimensional noise, especially if the noise magnitude varies considerably across the data, e.g., under heteroskedasticity or outliers. In this work, we investigate a more robust alternative -- the doubly stochastic normalization of the Gaussian kernel. We consider a setting where points are sampled from an unknown density on a low-dimensional manifold embedded in high-dimensional space and corrupted by possibly strong, non-identically distributed, sub-Gaussian noise. We establish that the doubly stochastic affinity matrix and its scaling factors concentrate around certain population forms, and provide corresponding finite-sample probabilistic error bounds. We then utilize these results to develop several tools for robust inference under general high-dimensional noise. First, we derive a robust density estimator that reliably infers the underlying sampling density and can substantially outperform the standard kernel density estimator under heteroskedasticity and outliers. Second, we obtain estimators for the pointwise noise magnitudes, the pointwise signal magnitudes, and the pairwise Euclidean distances between clean data points. Lastly, we derive robust graph Laplacian normalizations that accurately approximate various manifold Laplacians, including the Laplace Beltrami operator, improving over traditional normalizations in noisy settings. We exemplify our results in simulations and on real single-cell RNA-sequencing data. For the latter, we show that in contrast to traditional methods, our approach is robust to variability in technical noise levels across cell types.

Learning to differentiate model distributions from observed data is a fundamental problem in statistics and machine learning, and high-dimensional data remains a challenging setting for such problems. Metrics that quantify the disparity in probability distributions, such as the Stein discrepancy, play an important role in high-dimensional statistical testing. In this paper, we investigate the role of $L^2$ regularization in training a neural network Stein critic so as to distinguish between data sampled from an unknown probability distribution and a nominal model distribution. Making a connection to the Neural Tangent Kernel (NTK) theory, we develop a novel staging procedure for the weight of regularization over training time, which leverages the advantages of highly-regularized training at early times. Theoretically, we prove the approximation of the training dynamic by the kernel optimization, namely the ``lazy training'', when the $L^2$ regularization weight is large, and training on $n$ samples converge at a rate of ${O}(n^{-1/2})$ up to a log factor. The result guarantees learning the optimal critic assuming sufficient alignment with the leading eigen-modes of the zero-time NTK. The benefit of the staged $L^2$ regularization is demonstrated on simulated high dimensional data and an application to evaluating generative models of image data.

Bi-stochastic normalization provides an alternative normalization of graph Laplacians in graph-based data analysis and can be computed efficiently by Sinkhorn-Knopp (SK) iterations. This paper proves the convergence of bi-stochastically normalized graph Laplacian to manifold (weighted-)Laplacian with rates, when $n$ data points are i.i.d. sampled from a general $d$-dimensional manifold embedded in a possibly high-dimensional space. Under certain joint limit of $n \to \infty$ and kernel bandwidth $\epsilon \to 0$, the point-wise convergence rate of the graph Laplacian operator (under 2-norm) is proved to be $ O( n^{-1/(d/2+3)})$ at finite large $n$ up to log factors, achieved at the scaling of $\epsilon \sim n^{-1/(d/2+3)} $. When the manifold data are corrupted by outlier noise, we theoretically prove the graph Laplacian point-wise consistency which matches the rate for clean manifold data plus an additional term proportional to the boundedness of the inner-products of the noise vectors among themselves and with data vectors. Motivated by our analysis, which suggests that not exact bi-stochastic normalization but an approximate one will achieve the same consistency rate, we propose an approximate and constrained matrix scaling problem that can be solved by SK iterations with early termination. Numerical experiments support our theoretical results and show the robustness of bi-stochastically normalized graph Laplacian to high-dimensional outlier noise.

Spectral methods which represent data points by eigenvectors of kernel matrices or graph Laplacian matrices have been a primary tool in unsupervised data analysis. In many application scenarios, parametrizing the spectral embedding by a neural network that can be trained over batches of data samples gives a promising way to achieve automatic out-of-sample extension as well as computational scalability. Such an approach was taken in the original paper of SpectralNet (Shaham et al. 2018), which we call SpecNet1. The current paper introduces a new neural network approach, named SpecNet2, to compute spectral embedding which optimizes an equivalent objective of the eigen-problem and removes the orthogonalization layer in SpecNet1. SpecNet2 also allows separating the sampling of rows and columns of the graph affinity matrix by tracking the neighbors of each data point through the gradient formula. Theoretically, we show that any local minimizer of the new orthogonalization-free objective reveals the leading eigenvectors. Furthermore, global convergence for this new orthogonalization-free objective using a batch-based gradient descent method is proved. Numerical experiments demonstrate the improved performance and computational efficiency of SpecNet2 on simulated data and image datasets.

Graph prediction problems prevail in data analysis and machine learning. The inverse prediction problem, namely to infer input data from given output labels, is of emerging interest in various applications. In this work, we develop \textit{invertible graph neural network} (iGNN), a deep generative model to tackle the inverse prediction problem on graphs by casting it as a conditional generative task. The proposed model consists of an invertible sub-network that maps one-to-one from data to an intermediate encoded feature, which allows forward prediction by a linear classification sub-network as well as efficient generation from output labels via a parametric mixture model. The invertibility of the encoding sub-network is ensured by a Wasserstein-2 regularization which allows free-form layers in the residual blocks. The model is scalable to large graphs by a factorized parametric mixture model of the encoded feature and is computationally scalable by using GNN layers. The existence of invertible flow mapping is backed by theories of optimal transport and diffusion process, and we prove the expressiveness of graph convolution layers to approximate the theoretical flows of graph data. The proposed iGNN model is experimentally examined on synthetic data, including the example on large graphs, and the empirical advantage is also demonstrated on real-application datasets of solar ramping event data and traffic flow anomaly detection., Comment: Accepted at IEEE Journal on Selected Areas in Information Theory (JSAIT)---Special Issue Deep Learning for Inverse Problems

We propose an alternative approach to neural network training using the monotone vector field, an idea inspired by the seminal work of Juditsky and Nemirovski [Juditsky & Nemirovsky, 2019] developed originally to solve parameter estimation problems for generalized linear models (GLM) by reducing the original non-convex problem to a convex problem of solving a monotone variational inequality (VI). Our approach leads to computationally efficient procedures that converge fast and offer guarantee in some special cases, such as training a single-layer neural network or fine-tuning the last layer of the pre-trained model. Our approach can be used for more efficient fine-tuning of a pre-trained model while freezing the bottom layers, an essential step for deploying many machine learning models such as large language models (LLM). We demonstrate its applicability in training fully-connected (FC) neural networks, graph neural networks (GNN), and convolutional neural networks (CNN) and show the competitive or better performance of our approach compared to stochastic gradient descent methods on both synthetic and real network data prediction tasks regarding various performance metrics.

We present a method to capture groupings of similar calls and determine their relative spatial distribution from a collection of crime record narratives. We first obtain a topic distribution for each narrative, and then propose a nearest neighbors relative density estimation (kNN-RDE) approach to obtain spatial relative densities per topic. Experiments over a large corpus ($n=475,019$) of narrative documents from the Atlanta Police Department demonstrate the viability of our method in capturing geographic hot-spot trends which call dispatchers do not initially pick up on and which go unnoticed due to conflation with elevated event density in general., Comment: 9 pages, 12 figures

Self- and mutually-exciting point processes are popular models in machine learning and statistics for dependent discrete event data. To date, most existing models assume stationary kernels (including the classical Hawkes processes) and simple parametric models. Modern applications with complex event data require more general point process models that can incorporate contextual information of the events, called marks, besides the temporal and location information. Moreover, such applications often require non-stationary models to capture more complex spatio-temporal dependence. To tackle these challenges, a key question is to devise a versatile influence kernel in the point process model. In this paper, we introduce a novel and general neural network-based non-stationary influence kernel with high expressiveness for handling complex discrete events data while providing theoretical performance guarantees. We demonstrate the superior performance of our proposed method compared with the state-of-the-art on synthetic and real data.

We present a novel neural network Maximum Mean Discrepancy (MMD) statistic by identifying a new connection between neural tangent kernel (NTK) and MMD. This connection enables us to develop a computationally efficient and memory-efficient approach to compute the MMD statistic and perform NTK based two-sample tests towards addressing the long-standing challenge of memory and computational complexity of the MMD statistic, which is essential for online implementation to assimilating new samples. Theoretically, such a connection allows us to understand the NTK test statistic properties, such as the Type-I error and testing power for performing the two-sample test, by adapting existing theories for kernel MMD. Numerical experiments on synthetic and real-world datasets validate the theory and demonstrate the effectiveness of the proposed NTK-MMD statistic.

We present a study of a kernel-based two-sample test statistic related to the Maximum Mean Discrepancy (MMD) in the manifold data setting, assuming that high-dimensional observations are close to a low-dimensional manifold. We characterize the test level and power in relation to the kernel bandwidth, the number of samples, and the intrinsic dimensionality of the manifold. Specifically, when data densities $p$ and $q$ are supported on a $d$-dimensional sub-manifold ${M}$ embedded in an $m$-dimensional space and are H\"older with order $\beta$ (up to 2) on ${M}$, we prove a guarantee of the test power for finite sample size $n$ that exceeds a threshold depending on $d$, $\beta$, and $\Delta_2$ the squared $L^2$-divergence between $p$ and $q$ on the manifold, and with a properly chosen kernel bandwidth $\gamma$. For small density departures, we show that with large $n$ they can be detected by the kernel test when $\Delta_2$ is greater than $n^{- { 2 \beta/( d + 4 \beta ) }}$ up to a certain constant and $\gamma$ scales as $n^{-1/(d+4\beta)}$. The analysis extends to cases where the manifold has a boundary and the data samples contain high-dimensional additive noise. Our results indicate that the kernel two-sample test has no curse-of-dimensionality when the data lie on or near a low-dimensional manifold. We validate our theory and the properties of the kernel test for manifold data through a series of numerical experiments.

The Gaussian-smoothed optimal transport (GOT) framework, recently proposed by Goldfeld et al., scales to high dimensions in estimation and provides an alternative to entropy regularization. This paper provides convergence guarantees for estimating the GOT distance under more general settings. For the Gaussian-smoothed $p$-Wasserstein distance in $d$ dimensions, our results require only the existence of a moment greater than $d + 2p$. For the special case of sub-gamma distributions, we quantify the dependence on the dimension $d$ and establish a phase transition with respect to the scale parameter. We also prove convergence for dependent samples, only requiring a condition on the pairwise dependence of the samples measured by the covariance of the feature map of a kernel space. A key step in our analysis is to show that the GOT distance is dominated by a family of kernel maximum mean discrepancy (MMD) distances with a kernel that depends on the cost function as well as the amount of Gaussian smoothing. This insight provides further interpretability for the GOT framework and also introduces a class of kernel MMD distances with desirable properties. The theoretical results are supported by numerical experiments.

This work studies the spectral convergence of graph Laplacian to the Laplace-Beltrami operator when the graph affinity matrix is constructed from $N$ random samples on a $d$-dimensional manifold embedded in a possibly high dimensional space. By analyzing Dirichlet form convergence and constructing candidate approximate eigenfunctions via convolution with manifold heat kernel, we prove that, with Gaussian kernel, one can set the kernel bandwidth parameter $\epsilon \sim (\log N/ N)^{1/(d/2+2)}$ such that the eigenvalue convergence rate is $N^{-1/(d/2+2)}$ and the eigenvector convergence in 2-norm has rate $N^{-1/(d+4)}$; When $\epsilon \sim (\log N/N)^{1/(d/2+3)}$, both eigenvalue and eigenvector rates are $N^{-1/(d/2+3)}$. These rates are up to a $\log N$ factor and proved for finitely many low-lying eigenvalues. The result holds for un-normalized and random-walk graph Laplacians when data are uniformly sampled on the manifold, as well as the density-corrected graph Laplacian (where the affinity matrix is normalized by the degree matrix from both sides) with non-uniformly sampled data. As an intermediate result, we prove new point-wise and Dirichlet form convergence rates for the density-corrected graph Laplacian. Numerical results are provided to verify the theory.

Kernelized Gram matrix $W$ constructed from data points $\{x_i\}_{i=1}^N$ as $W_{ij}= k_0( \frac{ \| x_i - x_j \|^2} {\sigma^2} )$ is widely used in graph-based geometric data analysis and unsupervised learning. An important question is how to choose the kernel bandwidth $\sigma$, and a common practice called self-tuned kernel adaptively sets a $\sigma_i$ at each point $x_i$ by the $k$-nearest neighbor (kNN) distance. When $x_i$'s are sampled from a $d$-dimensional manifold embedded in a possibly high-dimensional space, unlike with fixed-bandwidth kernels, theoretical results of graph Laplacian convergence with self-tuned kernels have been incomplete. This paper proves the convergence of graph Laplacian operator $L_N$ to manifold (weighted-)Laplacian for a new family of kNN self-tuned kernels $W^{(\alpha)}_{ij} = k_0( \frac{ \| x_i - x_j \|^2}{ \epsilon \hat{\rho}(x_i) \hat{\rho}(x_j)})/\hat{\rho}(x_i)^\alpha \hat{\rho}(x_j)^\alpha$, where $\hat{\rho}$ is the estimated bandwidth function {by kNN}, and the limiting operator is also parametrized by $\alpha$. When $\alpha = 1$, the limiting operator is the weighted manifold Laplacian $\Delta_p$. Specifically, we prove the point-wise convergence of $L_N f $ and convergence of the graph Dirichlet form with rates. Our analysis is based on first establishing a $C^0$ consistency for $\hat{\rho}$ which bounds the relative estimation error $|\hat{\rho} - \bar{\rho}|/\bar{\rho}$ uniformly with high probability, where $\bar{\rho} = p^{-1/d}$, and $p$ is the data density function. Our theoretical results reveal the advantage of self-tuned kernel over fixed-bandwidth kernel via smaller variance error in low-density regions. In the algorithm, no prior knowledge of $d$ or data density is needed. The theoretical results are supported by numerical experiments on simulated data and hand-written digit image data.

Convolutional Neural Networks (CNNs) are known to be significantly over-parametrized, and difficult to interpret, train and adapt. In this paper, we introduce a structural regularization across convolutional kernels in a CNN. In our approach, each convolution kernel is first decomposed as 2D dictionary atoms linearly combined by coefficients. The widely observed correlation and redundancy in a CNN hint a common low-rank structure among the decomposed coefficients, which is here further supported by our empirical observations. We then explicitly regularize CNN kernels by enforcing decomposed coefficients to be shared across sub-structures, while leaving each sub-structure only its own dictionary atoms, a few hundreds of parameters typically, which leads to dramatic model reductions. We explore models with sharing across different sub-structures to cover a wide range of trade-offs between parameter reduction and expressiveness. Our proposed regularized network structures open the door to better interpreting, training and adapting deep models. We validate the flexibility and compatibility of our method by image classification experiments on multiple datasets and underlying network structures, and show that CNNs now maintain performance with dramatic reduction in parameters and computations, e.g., only 5\% parameters are used in a ResNet-18 to achieve comparable performance. Further experiments on few-shot classification show that faster and more robust task adaptation is obtained in comparison with models with standard convolutions.

Geometric variations like rotation, scaling, and viewpoint changes pose a significant challenge to visual understanding. One common solution is to directly model certain intrinsic structures, e.g., using landmarks. However, it then becomes non-trivial to build effective deep models, especially when the underlying non-Euclidean grid is irregular and coarse. Recent deep models using graph convolutions provide an appropriate framework to handle such non-Euclidean data, but many of them, particularly those based on global graph Laplacians, lack expressiveness to capture local features required for representation of signals lying on the non-Euclidean grid. The current paper introduces a new type of graph convolution with learnable low-rank local filters, which is provably more expressive than previous spectral graph convolution methods. The model also provides a unified framework for both spectral and spatial graph convolutions. To improve model robustness, regularization by local graph Laplacians is introduced. The representation stability against input graph data perturbation is theoretically proved, making use of the graph filter locality and the local graph regularization. Experiments on spherical mesh data, real-world facial expression recognition/skeleton-based action recognition data, and data with simulated graph noise show the empirical advantage of the proposed model.

Structured CNN designed using the prior information of problems potentially improves efficiency over conventional CNNs in various tasks in solving PDEs and inverse problems in signal processing. This paper introduces BNet2, a simplified Butterfly-Net and inline with the conventional CNN. Moreover, a Fourier transform initialization is proposed for both BNet2 and CNN with guaranteed approximation power to represent the Fourier transform operator. Experimentally, BNet2 and the Fourier transform initialization strategy are tested on various tasks, including approximating Fourier transform operator, end-to-end solvers of linear and nonlinear PDEs, and denoising and deblurring of 1D signals. On all tasks, under the same initialization, BNet2 achieves similar accuracy as CNN but has fewer parameters. And Fourier transform initialized BNet2 and CNN consistently improve the training and testing accuracy over the randomly initialized CNN.

The recent success of generative adversarial networks and variational learning suggests training a classifier network may work well in addressing the classical two-sample problem. Network-based tests have the computational advantage that the algorithm scales to large samples. This paper proposes a two-sample statistic which is the difference of the logit function, provided by a trained classification neural network, evaluated on the testing set split of the two datasets. Theoretically, we prove the testing power to differentiate two sub-exponential densities given that the network is sufficiently parametrized. When the two densities lie on or near to low-dimensional manifolds embedded in possibly high-dimensional space, the needed network complexity is reduced to only scale with the intrinsic dimensionality. Both the approximation and estimation error analysis are based on a new result of near-manifold integral approximation. In experiments, the proposed method demonstrates better performance than previous network-based tests using classification accuracy as the two-sample statistic, and compares favorably to certain kernel maximum mean discrepancy tests on synthetic datasets and hand-written digit datasets.

While generative adversarial networks (GANs) have revolutionized machine learning, a number of open questions remain to fully understand them and exploit their power. One of these questions is how to efficiently achieve proper diversity and sampling of the multi-mode data space. To address this, we introduce BasisGAN, a stochastic conditional multi-mode image generator. By exploiting the observation that a convolutional filter can be well approximated as a linear combination of a small set of basis elements, we learn a plug-and-played basis generator to stochastically generate basis elements, with just a few hundred of parameters, to fully embed stochasticity into convolutional filters. By sampling basis elements instead of filters, we dramatically reduce the cost of modeling the parameter space with no sacrifice on either image diversity or fidelity. To illustrate this proposed plug-and-play framework, we construct variants of BasisGAN based on state-of-the-art conditional image generation networks, and train the networks by simply plugging in a basis generator, without additional auxiliary components, hyperparameters, or training objectives. The experimental success is complemented with theoretical results indicating how the perturbations introduced by the proposed sampling of basis elements can propagate to the appearance of generated images., Comment: Published as a conference paper at ICLR 2020

In this paper, we consider domain-invariant deep learning by explicitly modeling domain shifts with only a small amount of domain-specific parameters in a Convolutional Neural Network (CNN). By exploiting the observation that a convolutional filter can be well approximated as a linear combination of a small set of dictionary atoms, we show for the first time, both empirically and theoretically, that domain shifts can be effectively handled by decomposing a convolutional layer into a domain-specific atom layer and a domain-shared coefficient layer, while both remain convolutional. An input channel will now first convolve spatially only with each respective domain-specific dictionary atom to "absorb" domain variations, and then output channels are linearly combined using common decomposition coefficients trained to promote shared semantics across domains. We use toy examples, rigorous analysis, and real-world examples with diverse datasets and architectures, to show the proposed plug-in framework's effectiveness in cross and joint domain performance and domain adaptation. With the proposed architecture, we need only a small set of dictionary atoms to model each additional domain, which brings a negligible amount of additional parameters, typically a few hundred.

Encoding the scale information explicitly into the representation learned by a convolutional neural network (CNN) is beneficial for many computer vision tasks especially when dealing with multiscale inputs. We study, in this paper, a scaling-translation-equivariant (ST-equivariant) CNN with joint convolutions across the space and the scaling group, which is shown to be both sufficient and necessary to achieve equivariance for the regular representation of the scaling-translation group ST . To reduce the model complexity and computational burden, we decompose the convolutional filters under two pre-fixed separable bases and truncate the expansion to low-frequency components. A further benefit of the truncated filter expansion is the improved deformation robustness of the equivariant representation, a property which is theoretically analyzed and empirically verified. Numerical experiments demonstrate that the proposed scaling-translation-equivariant network with decomposed convolutional filters (ScDCFNet) achieves significantly improved performance in multiscale image classification and better interpretability than regular CNNs at a reduced model size.

Variational autoencoders (VAEs) and generative adversarial networks (GANs) enjoy an intuitive connection to manifold learning: in training the decoder/generator is optimized to approximate a homeomorphism between the data distribution and the sampling space. This is a construction that strives to define the data manifold. A major obstacle to VAEs and GANs, however, is choosing a suitable prior that matches the data topology. Well-known consequences of poorly picked priors are posterior and mode collapse. To our knowledge, no existing method sidesteps this user choice. Conversely, $\textit{diffusion maps}$ automatically infer the data topology and enjoy a rigorous connection to manifold learning, but do not scale easily or provide the inverse homeomorphism (i.e. decoder/generator). We propose a method that combines these approaches into a generative model that inherits the asymptotic guarantees of $\textit{diffusion maps}$ while preserving the scalability of deep models. We prove approximation theoretic results for the dimension dependence of our proposed method. Finally, we demonstrate the effectiveness of our method with various real and synthetic datasets., Comment: 24 pages, 9 figures, 1 table; accepted to ECCV 2020

Given two symmetric and positive semidefinite square matrices $A, B$, is it true that any matrix given as the product of $m$ copies of $A$ and $n$ copies of $B$ in a particular sequence must be dominated in the spectral norm by the ordered matrix product $A^m B^n$? For example, is $$ \| AABAABABB \| \leq \| AAAAABBBB \|\ ? $$ Drury has characterized precisely which disordered words have the property that an inequality of this type holds for all matrices $A,B$. However, the $1$-parameter family of counterexamples Drury constructs for these characterizations is comprised of $3 \times 3$ matrices, and thus as stated the characterization applies only for $N \times N$ matrices with $N \geq 3$. In contrast, we prove that for $2 \times 2$ matrices, the general rearrangement inequality holds for all disordered words. We also show that for larger $N \times N$ matrices, the general rearrangement inequality holds for all disordered words, for most $A,B$ (in a sense of full measure) that are sufficiently small perturbations of the identity.

The extraction of clusters from a dataset which includes multiple clusters and a significant background component is a non-trivial task of practical importance. In image analysis this manifests for example in anomaly detection and target detection. The traditional spectral clustering algorithm, which relies on the leading $K$ eigenvectors to detect $K$ clusters, fails in such cases. In this paper we propose the {\it spectral embedding norm} which sums the squared values of the first $I$ normalized eigenvectors, where $I$ can be significantly larger than $K$. We prove that this quantity can be used to separate clusters from the background in unbalanced settings, including extreme cases such as outlier detection. The performance of the algorithm is not sensitive to the choice of $I$, and we demonstrate its application on synthetic and real-world remote sensing and neuroimaging datasets.

Deep networks, especially convolutional neural networks (CNNs), have been successfully applied in various areas of machine learning as well as to challenging problems in other scientific and engineering fields. This paper introduces Butterfly-Net, a low-complexity CNN with structured and sparse cross-channel connections, together with a Butterfly initialization strategy for a family of networks. Theoretical analysis of the approximation power of Butterfly-Net to the Fourier representation of input data shows that the error decays exponentially as the depth increases. Combining Butterfly-Net with a fully connected neural network, a large class of problems are proved to be well approximated with network complexity depending on the effective frequency bandwidth instead of the input dimension. Regular CNN is covered as a special case in our analysis. Numerical experiments validate the analytical results on the approximation of Fourier kernels and energy functionals of Poisson's equations. Moreover, all experiments support that training from Butterfly initialization outperforms training from random initialization. Also, adding the remaining cross-channel connections, although significantly increase the parameter number, does not much improve the post-training accuracy and is more sensitive to data distribution.

Explicit encoding of group actions in deep features makes it possible for convolutional neural networks (CNNs) to handle global deformations of images, which is critical to success in many vision tasks. This paper proposes to decompose the convolutional filters over joint steerable bases across the space and the group geometry simultaneously, namely a rotation-equivariant CNN with decomposed convolutional filters (RotDCF). This decomposition facilitates computing the joint convolution, which is proved to be necessary for the group equivariance. It significantly reduces the model size and computational complexity while preserving performance, and truncation of the bases expansion serves implicitly to regularize the filters. On datasets involving in-plane and out-of-plane object rotations, RotDCF deep features demonstrate greater robustness and interpretability than regular CNNs. The stability of the equivariant representation to input variations is also proved theoretically under generic assumptions on the filters in the decomposed form. The RotDCF framework can be extended to groups other than rotations, providing a general approach which achieves both group equivariance and representation stability at a reduced model size.

We study the effectiveness of various approaches that defend against adversarial attacks on deep networks via manipulations based on basis function representations of images. Specifically, we experiment with low-pass filtering, PCA, JPEG compression, low resolution wavelet approximation, and soft-thresholding. We evaluate these defense techniques using three types of popular attacks in black, gray and white-box settings. Our results show JPEG compression tends to outperform the other tested defenses in most of the settings considered, in addition to soft-thresholding, which performs well in specific cases, and yields a more mild decrease in accuracy on benign examples. In addition, we also mathematically derive a novel white-box attack in which the adversarial perturbation is composed only of terms corresponding a to pre-determined subset of the basis functions, of which a "low frequency attack" is a special case., Comment: added link to GitHub repository

Filters in a Convolutional Neural Network (CNN) contain model parameters learned from enormous amounts of data. In this paper, we suggest to decompose convolutional filters in CNN as a truncated expansion with pre-fixed bases, namely the Decomposed Convolutional Filters network (DCFNet), where the expansion coefficients remain learned from data. Such a structure not only reduces the number of trainable parameters and computation, but also imposes filter regularity by bases truncation. Through extensive experiments, we consistently observe that DCFNet maintains accuracy for image classification tasks with a significant reduction of model parameters, particularly with Fourier-Bessel (FB) bases, and even with random bases. Theoretically, we analyze the representation stability of DCFNet with respect to input variations, and prove representation stability under generic assumptions on the expansion coefficients. The analysis is consistent with the empirical observations., Comment: Published at ICML 2018

The paper introduces a new kernel-based Maximum Mean Discrepancy (MMD) statistic for measuring the distance between two distributions given finitely-many multivariate samples. When the distributions are locally low-dimensional, the proposed test can be made more powerful to distinguish certain alternatives by incorporating local covariance matrices and constructing an anisotropic kernel. The kernel matrix is asymmetric; it computes the affinity between $n$ data points and a set of $n_R$ reference points, where $n_R$ can be drastically smaller than $n$. While the proposed statistic can be viewed as a special class of Reproducing Kernel Hilbert Space MMD, the consistency of the test is proved, under mild assumptions of the kernel, as long as $\|p-q\| \sqrt{n} \to \infty $, and a finite-sample lower bound of the testing power is obtained. Applications to flow cytometry and diffusion MRI datasets are demonstrated, which motivate the proposed approach to compare distributions.

Let $(M,g)$ be a compact manifold and let $-\Delta \phi_k = \lambda_k \phi_k$ be the sequence of Laplacian eigenfunctions. We present a curious new phenomenon which, so far, we only managed to understand in a few highly specialized cases: the family of functions $f_N:M \rightarrow \mathbb{R}_{\geq 0}$ $$ f_N(x) = \sum_{k \leq N}{ \frac{1}{\sqrt{\lambda_k}} \frac{|\phi_k(x)|}{\|\phi_k\|_{L^{\infty}(M)}}}$$ seems strangely suited for the detection of anomalous points on the manifold. It may be heuristically interpreted as the sum over distances to the nearest nodal line and potentially hints at a new phenomenon in spectral geometry. We give rigorous statements on the unit square $[0,1]^2$ (where minima localize in $\mathbb{Q}^2$) and on Paley graphs (where $f_N$ recovers the geometry of quadratic residues of the underlying finite field $\mathbb{F}_p$). Numerical examples show that the phenomenon seems to arise on fairly generic manifolds., Comment: 11 pages, 7 figures

Community detection is a fundamental unsupervised learning problem for unlabeled networks which has a broad range of applications. Many community detection algorithms assume that the number of clusters $r$ is known apriori. In this paper, we propose an approach based on semi-definite relaxations, which does not require prior knowledge of model parameters like many existing convex relaxation methods and recovers the number of clusters and the clustering matrix exactly under a broad parameter regime, with probability tending to one. On a variety of simulated and real data experiments, we show that the proposed method often outperforms state-of-the-art techniques for estimating the number of clusters., Comment: 12 pages, 4 figure; AISTATS 2018