Alexander Cloninger

Stefan C. Schonsheck, Scott Mahan, Timo Klock et al.

Autoencoding is a popular method in representation learning. Conventional autoencoders employ symmetric encoding-decoding procedures and a simple Euclidean latent space to detect hidden low-dimensional structures in an unsupervised way. Some modern approaches to novel data generation such as generative adversarial networks askew this symmetry, but still employ a pair of massive networks--one to generate the image and another to judge the images quality based on priors learned from a training set. This work introduces a chart autoencoder with an asymmetric encoding-decoding process that can incorporate additional semi-supervised information such as class labels. Besides enhancing the capability for handling data with complicated topological and geometric structures, the proposed model can successfully differentiate nearby but disjoint manifolds and intersecting manifolds with only a small amount of supervision. Moreover, this model only requires a low-complexity encoding operation, such as a locally defined linear projection. We discuss the approximation power of such networks and derive a bound that essentially depends on the intrinsic dimension of the data manifold rather than the dimension of ambient space. Next we incorporate bounds for the sampling rate of training data need to faithfully represent a given data manifold. We present numerical experiments that verify that the proposed model can effectively manage data with multi-class nearby but disjoint manifolds of different classes, overlapping manifolds, and manifolds with non-trivial topology. Finally, we conclude with some experiments on computer vision and molecular dynamics problems which showcase the efficacy of our methods on real-world data.

Alexander Cloninger, Keaton Hamm, Varun Khurana et al.

We introduce LOT Wassmap, a computationally feasible algorithm to uncover low-dimensional structures in the Wasserstein space. The algorithm is motivated by the observation that many datasets are naturally interpreted as probability measures rather than points in $\mathbb{R}^n$, and that finding low-dimensional descriptions of such datasets requires manifold learning algorithms in the Wasserstein space. Most available algorithms are based on computing the pairwise Wasserstein distance matrix, which can be computationally challenging for large datasets in high dimensions. Our algorithm leverages approximation schemes such as Sinkhorn distances and linearized optimal transport to speed-up computations, and in particular, avoids computing a pairwise distance matrix. We provide guarantees on the embedding quality under such approximations, including when explicit descriptions of the probability measures are not available and one must deal with finite samples instead. Experiments demonstrate that LOT Wassmap attains correct embeddings and that the quality improves with increased sample size. We also show how LOT Wassmap significantly reduces the computational cost when compared to algorithms that depend on pairwise distance computations.

Robi Bhattacharjee, Alexander Cloninger, Yoav Freund et al.

Effective resistance (ER) is an attractive way to interrogate the structure of graphs. It is an alternative to computing the eigenvectors of the graph Laplacian. One attractive application of ER is to point clouds, i.e. graphs whose vertices correspond to IID samples from a distribution over a metric space. Unfortunately, it was shown that the ER between any two points converges to a trivial quantity that holds no information about the graph's structure as the size of the sample increases to infinity. In this study, we show that this trivial solution can be circumvented by considering a region-based ER between pairs of small regions rather than pairs of points and by scaling the edge weights appropriately with respect to the underlying density in each region. By keeping the regions fixed, we show analytically that the region-based ER converges to a non-trivial limit as the number of points increases to infinity. Namely the ER on a metric space. We support our theoretical findings with numerical experiments.

Chester Holtz, Pengwen Chen, Alexander Cloninger et al.

Motivated by the need to address the degeneracy of canonical Laplace learning algorithms in low label rates, we propose to reformulate graph-based semi-supervised learning as a nonconvex generalization of a \emph{Trust-Region Subproblem} (TRS). This reformulation is motivated by the well-posedness of Laplacian eigenvectors in the limit of infinite unlabeled data. To solve this problem, we first show that a first-order condition implies the solution of a manifold alignment problem and that solutions to the classical \emph{Orthogonal Procrustes} problem can be used to efficiently find good classifiers that are amenable to further refinement. To tackle refinement, we develop the framework of Sequential Subspace Optimization for graph-based SSL. Next, we address the criticality of selecting supervised samples at low-label rates. We characterize informative samples with a novel measure of centrality derived from the principal eigenvectors of a certain submatrix of the graph Laplacian. We demonstrate that our framework achieves lower classification error compared to recent state-of-the-art and classical semi-supervised learning methods at extremely low, medium, and high label rates.

Dhruv Kohli, Gal Mishne, Alexander Cloninger

Given a set of overlapping local views (patches) of a dataset, we consider the problem of finding a rigid alignment of the views that minimizes a $2$-norm based alignment error. In general, the views are noisy and a perfect alignment may not exist. In this work, we characterize the non-degeneracy of an alignment in the noisy setting based on the kernel and positivity of a certain matrix. This leads to a polynomial time algorithm for testing the non-degeneracy of a given alignment. Subsequently, we focus on Riemannian gradient descent for minimizing the alignment error, providing a sufficient condition on an alignment for the algorithm to converge (locally) linearly to it. \revadd{Additionally, we provide an exact recovery and noise stability analysis of the algorithm}. In the case of noiseless views, a perfect alignment exists, resulting in a realization of the points that respects the geometry of the views. Under a mild condition on the views, we show that a non-degenerate perfect alignment \revadd{characterizes the infinitesimally rigidity of a realization, and thus the local rigidity of a generic realization}. By specializing the non-degeneracy conditions to the noiseless case, we derive necessary and sufficient conditions on the overlapping structure of the views for \revadd{a perfect alignment to be non-degenerate and equivalently, for the resulting realization to be infinitesimally rigid}. Similar results are also derived regarding the uniqueness of a perfect alignment and global rigidity.

Chester Holtz, Gal Mishne, Alexander Cloninger

Probabilistic generative models provide a flexible and systematic framework for learning the underlying geometry of data. However, model selection in this setting is challenging, particularly when selecting for ill-defined qualities such as disentanglement or interpretability. In this work, we address this gap by introducing a method for ranking generative models based on the training dynamics exhibited during learning. Inspired by recent theoretical characterizations of disentanglement, our method does not require supervision of the underlying latent factors. We evaluate our approach by demonstrating the need for disentanglement metrics which do not require labels\textemdash the underlying generative factors. We additionally demonstrate that our approach correlates with baseline supervised methods for evaluating disentanglement. Finally, we show that our method can be used as an unsupervised indicator for downstream performance on reinforcement learning and fairness-classification problems.

Varun Khurana, Xiuyuan Cheng, Alexander Cloninger

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.

Nicholas Karris, Evangelos A. Nikitopoulos, Ioannis G. Kevrekidis et al.

We develop an Euler-type method to predict the evolution of a time-dependent probability measure without explicitly learning an operator that governs its evolution. We use linearized optimal transport theory to prove that the measure-valued analog of Euler's method is first-order accurate when the measure evolves ``smoothly.'' In applications of interest, however, the measure is an empirical distribution of a system of stochastic particles whose behavior is only accessible through an agent-based micro-scale simulation. In such cases, this empirical measure does not evolve smoothly because the individual particles move chaotically on short time scales. However, we can still perform our Euler-type method, and when the particles' collective distribution approximates a measure that \emph{does} evolve smoothly, we observe that the algorithm still accurately predicts this collective behavior over relatively large Euler steps, thus reducing the number of micro-scale steps required to step forward in time. In this way, our algorithm provides a ``macro-scale timestepper'' that requires less micro-scale data to still maintain accuracy, which we demonstrate with three illustrative examples: a biological agent-based model, a model of a PDE, and a model of Langevin dynamics.

Alireza Pirhadi, Mohammad Hossein Moslemi, Alexander Cloninger et al.

Ensuring Conditional Independence (CI) constraints is pivotal for the development of fair and trustworthy machine learning models. In this paper, we introduce \sys, a framework that harnesses optimal transport theory for data repair under CI constraints. Optimal transport theory provides a rigorous framework for measuring the discrepancy between probability distributions, thereby ensuring control over data utility. We formulate the data repair problem concerning CIs as a Quadratically Constrained Linear Program (QCLP) and propose an alternating method for its solution. However, this approach faces scalability issues due to the computational cost associated with computing optimal transport distances, such as the Wasserstein distance. To overcome these scalability challenges, we reframe our problem as a regularized optimization problem, enabling us to develop an iterative algorithm inspired by Sinkhorn's matrix scaling algorithm, which efficiently addresses high-dimensional and large-scale data. Through extensive experiments, we demonstrate the efficacy and efficiency of our proposed methods, showcasing their practical utility in real-world data cleaning and preprocessing tasks. Furthermore, we provide comparisons with traditional approaches, highlighting the superiority of our techniques in terms of preserving data utility while ensuring adherence to the desired CI constraints.

Sawyer Robertson, Zhengchao Wan, Alexander Cloninger

The fields of effective resistance and optimal transport on graphs are filled with rich connections to combinatorics, geometry, machine learning, and beyond. In this article we put forth a bold claim: that the two fields should be understood as one and the same, up to a choice of $p$. We make this claim precise by introducing the parameterized family of $p$-Beckmann distances for probability measures on graphs and relate them sharply to certain Wasserstein distances. Then, we break open a suite of results including explicit connections to optimal stopping times and random walks on graphs, graph Sobolev spaces, and a Benamou-Brenier type formula for $2$-Beckmann distance. We further explore empirical implications in the world of unsupervised learning for graph data and propose further study of the usage of these metrics where Wasserstein distance may produce computational bottlenecks.

Zhaiming Shen, Alex Havrilla, Rongjie Lai et al.

Transformers serve as the foundational architecture for large language and video generation models, such as GPT, BERT, SORA and their successors. Empirical studies have demonstrated that real-world data and learning tasks exhibit low-dimensional structures, along with some noise or measurement error. The performance of transformers tends to depend on the intrinsic dimension of the data/tasks, though theoretical understandings remain largely unexplored for transformers. This work establishes a theoretical foundation by analyzing the performance of transformers for regression tasks involving noisy input data on a manifold. Specifically, the input data are in a tubular neighborhood of a manifold, while the ground truth function depends on the projection of the noisy data onto the manifold. We prove approximation and generalization errors which crucially depend on the intrinsic dimension of the manifold. Our results demonstrate that transformers can leverage low-complexity structures in learning task even when the input data are perturbed by high-dimensional noise. Our novel proof technique constructs representations of basic arithmetic operations by transformers, which may hold independent interest.

Jun Linwu, Varun Khurana, Nicholas Karris et al.

The pyLOT library offers a Python implementation of linearized optimal transport (LOT) techniques and methods to use in downstream tasks. The pipeline embeds probability distributions into a Hilbert space via the Optimal Transport maps from a fixed reference distribution, and this linearization allows downstream tasks to be completed using off the shelf (linear) machine learning algorithms. We provide a case study of performing ML on 3D scans of lemur teeth, where the original questions of classification, clustering, dimension reduction, and data generation reduce to simple linear operations performed on the LOT embedded representations.

Scott Mahan, Caroline Moosmüller, Alexander Cloninger

We introduce Deep Set Linearized Optimal Transport, an algorithm designed for the efficient simultaneous embedding of point clouds into an $L^2-$space. This embedding preserves specific low-dimensional structures within the Wasserstein space while constructing a classifier to distinguish between various classes of point clouds. Our approach is motivated by the observation that $L^2-$distances between optimal transport maps for distinct point clouds, originating from a shared fixed reference distribution, provide an approximation of the Wasserstein-2 distance between these point clouds, under certain assumptions. To learn approximations of these transport maps, we employ input convex neural networks (ICNNs) and establish that, under specific conditions, Euclidean distances between samples from these ICNNs closely mirror Wasserstein-2 distances between the true distributions. Additionally, we train a discriminator network that attaches weights these samples and creates a permutation invariant classifier to differentiate between different classes of point clouds. We showcase the advantages of our algorithm over the standard deep set approach through experiments on a flow cytometry dataset with a limited number of labeled point clouds.

Tongtong Liang, Esha Singh, Rahul Parhi et al.

We study how architectural inductive bias reshapes the implicit regularization induced by the edge-of-stability phenomenon in gradient descent. Prior work has established that for fully connected networks, the strength of this regularization is governed solely by the global input geometry; consequently, it is insufficient to prevent overfitting on difficult distributions such as the high-dimensional sphere. In this paper, we show that locality and weight sharing fundamentally change this picture. Specifically, we prove that provided the receptive field size $m$ remains small relative to the ambient dimension $d$, these networks generalize on spherical data with a rate of $n^{-\frac{1}{6} +O(m/d)}$, a regime where fully connected networks provably fail. This theoretical result confirms that weight sharing couples the learned filters to the low-dimensional patch manifold, thereby bypassing the high dimensionality of the ambient space. We further corroborate our theory by analyzing the patch geometry of natural images, showing that standard convolutional designs induce patch distributions that are highly amenable to this stability mechanism, thus providing a systematic explanation for the superior generalization of convolutional networks over fully connected baselines.

Tianxiang Wang, Yingtong Ke, Dhananjay Bhaskar et al.

Single-cell technologies generate high-dimensional point clouds of cells, enabling detailed characterization of complex patient states and treatment responses. Yet each patient is represented by an irregular point cloud rather than a simple vector, making it difficult to directly quantify and compare biological differences between individuals. Nonlinear methods such as kernels and neural networks achieve predictive accuracy but act as black boxes, offering little biological interpretability. To address these limitations, we adapt the Linear Optimal Transport (LOT) framework to this setting, embedding irregular point clouds into a fixed-dimensional Euclidean space while preserving distributional structure. This embedding provides a principled linear representation that preserves optimal transport geometry while enabling downstream analysis. It also forms a registration between any two patients, enabling direct comparison of their cellular distributions. Within this space, LOT enables: (i) \textbf{accurate and interpretable classification} of COVID-19 patient states, where classifier weights map back to specific markers and spatial regions driving predictions; and (ii) \textbf{synthetic data generation} for patient-derived organoids, exploiting the linearity of the LOT embedding. LOT barycenters yield averaged cellular profiles representing combined conditions or samples, supporting drug interaction testing. Together, these results establish LOT as a unified framework that bridges predictive performance, interpretability, and generative modeling. By transforming heterogeneous point clouds into structured embeddings directly traceable to the original data, LOT opens new opportunities for understanding immune variation and treatment effects in high-dimensional biological systems.

Tongtong Liang, Alexander Cloninger, Rahul Parhi et al.

Understanding generalization in overparameterized neural networks hinges on the interplay between the data geometry, neural architecture, and training dynamics. In this paper, we theoretically explore how data geometry controls this implicit bias. This paper presents theoretical results for overparameterized two-layer ReLU networks trained below the edge of stability. First, for data distributions supported on a mixture of low-dimensional balls, we derive generalization bounds that provably adapt to the intrinsic dimension. Second, for a family of isotropic distributions that vary in how strongly probability mass concentrates toward the unit sphere, we derive a spectrum of bounds showing that rates deteriorate as the mass concentrates toward the sphere. These results instantiate a unifying principle: When the data is harder to "shatter" with respect to the activation thresholds of the ReLU neurons, gradient descent tends to learn representations that capture shared patterns and thus finds solutions that generalize well. On the other hand, for data that is easily shattered (e.g., data supported on the sphere) gradient descent favors memorization. Our theoretical results consolidate disparate empirical findings that have appeared in the literature.

Dhruv Kohli, Sawyer J. Robertson, Gal Mishne et al.

Estimating the tangent spaces of a data manifold is a fundamental problem in data analysis. The standard approach, Local Principal Component Analysis (LPCA), struggles in high-noise settings due to a critical trade-off in choosing the neighborhood size. Selecting an optimal size requires prior knowledge of the geometric and noise characteristics of the data that are often unavailable. In this paper, we propose a spectral method, Laplacian Eigenvector Gradient Orthogonalization (LEGO), that utilizes the global structure of the data to guide local tangent space estimation. Instead of relying solely on local neighborhoods, LEGO estimates the tangent space at each data point by orthogonalizing the gradients of low-frequency eigenvectors of the graph Laplacian. We provide two theoretical justifications of our method. First, a differential geometric analysis on a tubular neighborhood of a manifold shows that gradients of the low-frequency Laplacian eigenfunctions of the tube align closely with the manifold's tangent bundle, while an eigenfunction with high gradient in directions orthogonal to the manifold lie deeper in the spectrum. Second, a random matrix theoretic analysis also demonstrates that low-frequency eigenvectors are robust to sub-Gaussian noise. Through comprehensive experiments, we demonstrate that LEGO yields tangent space estimates that are significantly more robust to noise than those from LPCA, resulting in marked improvements in downstream tasks such as manifold learning, boundary detection, and local intrinsic dimension estimation.

Sawyer Jack Robertson, Chester Holtz, Zhengchao Wan et al.

We study the problem of semi-supervised learning on graphs in the regime where data labels are scarce or possibly corrupted. We propose an approach called $p$-conductance learning that generalizes the $p$-Laplace and Poisson learning methods by introducing an objective reminiscent of $p$-Laplacian regularization and an affine relaxation of the label constraints. This leads to a family of probability measure mincut programs that balance sparse edge removal with accurate distribution separation. Our theoretical analysis connects these programs to well-known variational and probabilistic problems on graphs (including randomized cuts, effective resistance, and Wasserstein distance) and provides motivation for robustness when labels are diffused via the heat kernel. Computationally, we develop a semismooth Newton-conjugate gradient algorithm and extend it to incorporate class-size estimates when converting the continuous solutions into label assignments. Empirical results on computer vision and citation datasets demonstrate that our approach achieves state-of-the-art accuracy in low label-rate, corrupted-label, and partial-label regimes.

Andrew Dennehy, Xiaoyu Zou, Shabnam J. Semnani et al.

DBSCAN and OPTICS are powerful algorithms for identifying clusters of points in domains where few assumptions can be made about the structure of the data. In this paper, we leverage these strengths and introduce a new algorithm, LINSCAN, designed to seek lineated clusters that are difficult to find and isolate with existing methods. In particular, by embedding points as normal distributions approximating their local neighborhoods and leveraging a distance function derived from the Kullback Leibler Divergence, LINSCAN can detect and distinguish lineated clusters that are spatially close but have orthogonal covariances. We demonstrate how LINSCAN can be applied to seismic data to identify active faults, including intersecting faults, and determine their orientation. Finally, we discuss the properties a generalization of DBSCAN and OPTICS must have in order to retain the stability benefits of these algorithms.

Varun Khurana, Harish Kannan, Alexander Cloninger et al.

In this paper we study supervised learning tasks on the space of probability measures. We approach this problem by embedding the space of probability measures into $L^2$ spaces using the optimal transport framework. In the embedding spaces, regular machine learning techniques are used to achieve linear separability. This idea has proved successful in applications and when the classes to be separated are generated by shifts and scalings of a fixed measure. This paper extends the class of elementary transformations suitable for the framework to families of shearings, describing conditions under which two classes of sheared distributions can be linearly separated. We furthermore give necessary bounds on the transformations to achieve a pre-specified separation level, and show how multiple embeddings can be used to allow for larger families of transformations. We demonstrate our results on image classification tasks.

Srinjoy Das, Hrushikesh Mhaskar, Alexander Cloninger

This paper introduces kdiff, a novel kernel-based measure for estimating distances between instances of time series, random fields and other forms of structured data. This measure is based on the idea of matching distributions that only overlap over a portion of their region of support. Our proposed measure is inspired by MPdist which has been previously proposed for such datasets and is constructed using Euclidean metrics, whereas kdiff is constructed using non-linear kernel distances. Also, kdiff accounts for both self and cross similarities across the instances and is defined using a lower quantile of the distance distribution. Comparing the cross similarity to self similarity allows for measures of similarity that are more robust to noise and partial occlusions of the relevant signals. Our proposed measure kdiff is a more general form of the well known kernel-based Maximum Mean Discrepancy (MMD) distance estimated over the embeddings. Some theoretical results are provided for separability conditions using kdiff as a distance measure for clustering and classification problems where the embedding distributions can be modeled as two component mixtures. Applications are demonstrated for clustering of synthetic and real-life time series and image data, and the performance of kdiff is compared to competing distance measures for clustering.

Andreas Oslandsbotn, Zeljko Kereta, Valeriya Naumova et al.

Kernel ridge regression (KRR) is a popular scheme for non-linear non-parametric learning. However, existing implementations of KRR require that all the data is stored in the main memory, which severely limits the use of KRR in contexts where data size far exceeds the memory size. Such applications are increasingly common in data mining, bioinformatics, and control. A powerful paradigm for computing on data sets that are too large for memory is the streaming model of computation, where we process one data sample at a time, discarding each sample before moving on to the next one. In this paper, we propose StreaMRAK - a streaming version of KRR. StreaMRAK improves on existing KRR schemes by dividing the problem into several levels of resolution, which allows continual refinement to the predictions. The algorithm reduces the memory requirement by continuously and efficiently integrating new samples into the training model. With a novel sub-sampling scheme, StreaMRAK reduces memory and computational complexities by creating a sketch of the original data, where the sub-sampling density is adapted to the bandwidth of the kernel and the local dimensionality of the data. We present a showcase study on two synthetic problems and the prediction of the trajectory of a double pendulum. The results show that the proposed algorithm is fast and accurate.

Jinjie Zhang, Harish Kannan, Alexander Cloninger et al.

We propose the use of low bit-depth Sigma-Delta and distributed noise-shaping methods for quantizing the Random Fourier features (RFFs) associated with shift-invariant kernels. We prove that our quantized RFFs -- even in the case of $1$-bit quantization -- allow a high accuracy approximation of the underlying kernels, and the approximation error decays at least polynomially fast as the dimension of the RFFs increases. We also show that the quantized RFFs can be further compressed, yielding an excellent trade-off between memory use and accuracy. Namely, the approximation error now decays exponentially as a function of the bits used. Moreover, we empirically show by testing the performance of our methods on several machine learning tasks that our method compares favorably to other state of the art quantization methods in this context.

Dhruv Kohli, Alexander Cloninger, Gal Mishne

We present Low Distortion Local Eigenmaps (LDLE), a manifold learning technique which constructs a set of low distortion local views of a dataset in lower dimension and registers them to obtain a global embedding. The local views are constructed using the global eigenvectors of the graph Laplacian and are registered using Procrustes analysis. The choice of these eigenvectors may vary across the regions. In contrast to existing techniques, LDLE can embed closed and non-orientable manifolds into their intrinsic dimension by tearing them apart. It also provides gluing instruction on the boundary of the torn embedding to help identify the topology of the original manifold. Our experimental results will show that LDLE largely preserved distances up to a constant scale while other techniques produced higher distortion. We also demonstrate that LDLE produces high quality embeddings even when the data is noisy or sparse.

Caroline Moosmüller, Alexander Cloninger

Discriminating between distributions is an important problem in a number of scientific fields. This motivated the introduction of Linear Optimal Transportation (LOT), which embeds the space of distributions into an $L^2$-space. The transform is defined by computing the optimal transport of each distribution to a fixed reference distribution, and has a number of benefits when it comes to speed of computation and to determining classification boundaries. In this paper, we characterize a number of settings in which LOT embeds families of distributions into a space in which they are linearly separable. This is true in arbitrary dimension, and for families of distributions generated through perturbations of shifts and scalings of a fixed distribution.We also prove conditions under which the $L^2$ distance of the LOT embedding between two distributions in arbitrary dimension is nearly isometric to Wasserstein-2 distance between those distributions. This is of significant computational benefit, as one must only compute $N$ optimal transport maps to define the $N^2$ pairwise distances between $N$ distributions. We demonstrate the benefits of LOT on a number of distribution classification problems.

Alexander Cloninger, Timo Klock

We study the approximation of two-layer compositions $f(x) = g(φ(x))$ via deep networks with ReLU activation, where $φ$ is a geometrically intuitive, dimensionality reducing feature map. We focus on two intuitive and practically relevant choices for $φ$: the projection onto a low-dimensional embedded submanifold and a distance to a collection of low-dimensional sets. We achieve near optimal approximation rates, which depend only on the complexity of the dimensionality reducing map $φ$ rather than the ambient dimension. Since $φ$ encapsulates all nonlinear features that are material to the function $f$, this suggests that deep nets are faithful to an intrinsic dimension governed by $f$ rather than the complexity of the domain of $f$. In particular, the prevalent assumption of approximating functions on low-dimensional manifolds can be significantly relaxed using functions of type $f(x) = g(φ(x))$ with $φ$ representing an orthogonal projection onto the same manifold.

Alexander Cloninger, Hrushikesh Mhaskar

We consider the problem of classification of points sampled from an unknown probability measure on a Euclidean space. We study the question of querying the class label at a very small number of judiciously chosen points so as to be able to attach the appropriate class label to every point in the set. Our approach is to consider the unknown probability measure as a convex combination of the conditional probabilities for each class. Our technique involves the use of a highly localized kernel constructed from Hermite polynomials, in order to create a hierarchical estimate of the supports of the constituent probability measures. We do not need to make any assumptions on the nature of any of the probability measures nor know in advance the number of classes involved. We give theoretical guarantees measured by the $F$-score for our classification scheme. Examples include classification in hyper-spectral images and MNIST classification.

Alexander Potapov, Ian Colbert, Ken Kreutz-Delgado et al.

Stochastic-sampling-based Generative Neural Networks, such as Restricted Boltzmann Machines and Generative Adversarial Networks, are now used for applications such as denoising, image occlusion removal, pattern completion, and motion synthesis. In scenarios which involve performing such inference tasks with these models, it is critical to determine metrics that allow for model selection and/or maintenance of requisite generative performance under pre-specified implementation constraints. In this paper, we propose a new metric for evaluating generative model performance based on $p$-values derived from the combined use of Maximum Mean Discrepancy (MMD) and permutation-based (PT-based) resampling, which we refer to as PT-MMD. We demonstrate the effectiveness of this metric for two cases: (1) Selection of bitwidth and activation function complexity to achieve minimum power-at-performance for Restricted Boltzmann Machines; (2) Quantitative comparison of images generated by two types of Generative Adversarial Networks (PGAN and WGAN) to facilitate model selection in order to maximize the fidelity of generated images. For these applications, our results are shown using Euclidean and Haar-based kernels for the PT-MMD two sample hypothesis test. This demonstrates the critical role of distance functions in comparing generated images against their corresponding ground truth counterparts as what would be perceived by human users.

Xiuyuan Cheng, Alexander Cloninger

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.

Saeed Vahidian, Baharan Mirzasoleiman, Alexander Cloninger

In a number of situations, collecting a function value for every data point may be prohibitively expensive, and random sampling ignores any structure in the underlying data. We introduce a scalable optimization algorithm with no correction steps (in contrast to Frank-Wolfe and its variants), a variant of gradient ascent for coreset selection in graphs, that greedily selects a weighted subset of vertices that are deemed most important to sample. Our algorithm estimates the mean of the function by taking a weighted sum only at these vertices, and we provably bound the estimation error in terms of the location and weights of the selected vertices in the graph. In addition, we consider the case where nodes have different selection costs and provide bounds on the quality of the low-cost selected coresets. We demonstrate the benefits of our algorithm on the semi-supervised node classification of graph convolutional neural network, point clouds and structured graphs, as well as sensor placement where the cost of placing sensors depends on the location of the placement. We also elucidate that the empirical convergence of our proposed method is faster than random selection and various clustering methods while still respecting sensor placement cost. The paper concludes with validation of the developed algorithm on both synthetic and real datasets, demonstrating that it outperforms the current state of the art.

Henry Li, Ofir Lindenbaum, Xiuyuan Cheng et al.

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.

Alexander Cloninger

In this paper, we bound the error induced by using a weighted skeletonization of two data sets for computing a two sample test with kernel maximum mean discrepancy. The error is quantified in terms of the speed in which heat diffuses from those points to the rest of the data, as well as how at the weights on the reference points are, and gives a non-asymptotic, non-probabilistic bound. The result ties into the problem of the eigenvector triple product, which appears in a number of important problems. The error bound also suggests an optimization scheme for choosing the best set of reference points and weights. The method is tested on a several two sample test examples.

Alexander Cloninger, Stefan Steinerberger

We discuss the geometry of Laplacian eigenfunctions $-Δφ= λφ$ on compact manifolds $(M,g)$ and combinatorial graphs $G=(V,E)$. The 'dual' geometry of Laplacian eigenfunctions is well understood on $\mathbb{T}^d$ (identified with $\mathbb{Z}^d$) and $\mathbb{R}^n$ (which is self-dual). The dual geometry is of tremendous role in various fields of pure and applied mathematics. The purpose of our paper is to point out a notion of similarity between eigenfunctions that allows to reconstruct that geometry. Our measure of 'similarity' $ α(φ_λ, φ_μ)$ between eigenfunctions $φ_λ$ and $φ_μ$ is given by a global average of local correlations $$ α(φ_λ, φ_μ)^2 = \| φ_λ φ_μ \|_{L^2}^{-2}\int_{M}{ \left( \int_{M}{ p(t,x,y)( φ_λ(y) - φ_λ(x))( φ_μ(y) - φ_μ(x)) dy} \right)^2 dx},$$ where $p(t,x,y)$ is the classical heat kernel and $e^{-t λ} + e^{-t μ} = 1$. This notion recovers all classical notions of duality but is equally applicable to other (rough) geometries and graphs; many numerical examples in different continuous and discrete settings illustrate the result.

Xiuyuan Cheng, Alexander Cloninger, Ronald R. Coifman

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.

Alexander Cloninger, Brita Roy, Carley Riley et al.

We address the problem of defining a network graph on a large collection of classes. Each class is comprised of a collection of data points, sampled in a non i.i.d. way, from some unknown underlying distribution. The application we consider in this paper is a large scale high dimensional survey of people living in the US, and the question of how similar or different are the various counties in which these people live. We use a co-clustering diffusion metric to learn the underlying distribution of people, and build an approximate earth mover's distance algorithm using this data adaptive transportation cost.

Alexander Cloninger

We consider the problem of constructing diffusion operators high dimensional data $X$ to address counterfactual functions $F$, such as individualized treatment effectiveness. We propose and construct a new diffusion metric $K_F$ that captures both the local geometry of $X$ and the directions of variance of $F$. The resulting diffusion metric is then used to define a localized filtration of $F$ and answer counterfactual questions pointwise, particularly in situations such as drug trials where an individual patient's outcomes cannot be studied long term both taking and not taking a medication. We validate the model on synthetic and real world clinical trials, and create individualized notions of benefit from treatment.

Alexander Cloninger, Stefan Steinerberger

Spectral embedding uses eigenfunctions of the discrete Laplacian on a weighted graph to obtain coordinates for an embedding of an abstract data set into Euclidean space. We propose a new pre-processing step of first using the eigenfunctions to simulate a low-frequency wave moving over the data and using both position as well as change in time of the wave to obtain a refined metric to which classical methods of dimensionality reduction can then applied. This is motivated by the behavior of waves, symmetries of the wave equation and the hunting technique of bats. It is shown to be effective in practice and also works for other partial differential equations -- the method yields improved results even for the classical heat equation.

Jared Katzman, Uri Shaham, Jonathan Bates et al.

Medical practitioners use survival models to explore and understand the relationships between patients' covariates (e.g. clinical and genetic features) and the effectiveness of various treatment options. Standard survival models like the linear Cox proportional hazards model require extensive feature engineering or prior medical knowledge to model treatment interaction at an individual level. While nonlinear survival methods, such as neural networks and survival forests, can inherently model these high-level interaction terms, they have yet to be shown as effective treatment recommender systems. We introduce DeepSurv, a Cox proportional hazards deep neural network and state-of-the-art survival method for modeling interactions between a patient's covariates and treatment effectiveness in order to provide personalized treatment recommendations. We perform a number of experiments training DeepSurv on simulated and real survival data. We demonstrate that DeepSurv performs as well as or better than other state-of-the-art survival models and validate that DeepSurv successfully models increasingly complex relationships between a patient's covariates and their risk of failure. We then show how DeepSurv models the relationship between a patient's features and effectiveness of different treatment options to show how DeepSurv can be used to provide individual treatment recommendations. Finally, we train DeepSurv on real clinical studies to demonstrate how it's personalized treatment recommendations would increase the survival time of a set of patients. The predictive and modeling capabilities of DeepSurv will enable medical researchers to use deep neural networks as a tool in their exploration, understanding, and prediction of the effects of a patient's characteristics on their risk of failure.

Uri Shaham, Alexander Cloninger, Ronald R. Coifman

We discuss approximation of functions using deep neural nets. Given a function $f$ on a $d$-dimensional manifold $Γ\subset \mathbb{R}^m$, we construct a sparsely-connected depth-4 neural network and bound its error in approximating $f$. The size of the network depends on dimension and curvature of the manifold $Γ$, the complexity of $f$, in terms of its wavelet description, and only weakly on the ambient dimension $m$. Essentially, our network computes wavelet functions, which are computed from Rectified Linear Units (ReLU)

Alexander Cloninger, Ronald R. Coifman, Nicholas Downing et al.

In this paper, we build an organization of high-dimensional datasets that cannot be cleanly embedded into a low-dimensional representation due to missing entries and a subset of the features being irrelevant to modeling functions of interest. Our algorithm begins by defining coarse neighborhoods of the points and defining an expected empirical function value on these neighborhoods. We then generate new non-linear features with deep net representations tuned to model the approximate function, and re-organize the geometry of the points with respect to the new representation. Finally, the points are locally z-scored to create an intrinsic geometric organization which is independent of the parameters of the deep net, a geometry designed to assure smoothness with respect to the empirical function. We examine this approach on data from the Center for Medicare and Medicaid Services Hospital Quality Initiative, and generate an intrinsic low-dimensional organization of the hospitals that is smooth with respect to an expert driven function of quality.

Gal Mishne, Uri Shaham, Alexander Cloninger et al.

Non-linear manifold learning enables high-dimensional data analysis, but requires out-of-sample-extension methods to process new data points. In this paper, we propose a manifold learning algorithm based on deep learning to create an encoder, which maps a high-dimensional dataset and its low-dimensional embedding, and a decoder, which takes the embedded data back to the high-dimensional space. Stacking the encoder and decoder together constructs an autoencoder, which we term a diffusion net, that performs out-of-sample-extension as well as outlier detection. We introduce new neural net constraints for the encoder, which preserves the local geometry of the points, and we prove rates of convergence for the encoder. Also, our approach is efficient in both computational complexity and memory requirements, as opposed to previous methods that require storage of all training points in both the high-dimensional and the low-dimensional spaces to calculate the out-of-sample-extension and the pre-image.