Quantifying Meibomian Gland Morphology Using Artificial Intelligence.

SIGNIFICANCE: Quantifying meibomian gland morphology from meibography images is used for the diagnosis, treatment, and management of meibomian gland dysfunction in clinics. A novel and automated method is described for quantifying meibomian gland morphology from meibography images. PURPOSE: Meibomian gland morphological abnormality is a common clinical sign of meibomian gland dysfunction, yet there exist no automated methods that provide standard quantifications of morphological features for individual glands. This study introduces an automated artificial intelligence approach to segmenting individual meibomian gland regions in infrared meibography images and analyzing their morphological features. METHODS: A total of 1443 meibography images were collected and annotated. The dataset was then divided into development and evaluation sets. The development set was used to train and tune deep learning models for segmenting glands and identifying ghost glands from images, whereas the evaluation set was used to evaluate the performance of the model. The gland segmentations were further used to analyze individual gland features, including gland local contrast, length, width, and tortuosity. RESULTS: A total of 1039 meibography images (including 486 upper and 553 lower eyelids) were used for training and tuning the deep learning model, whereas the remaining 404 images (including 203 upper and 201 lower eyelids) were used for evaluations. The algorithm on average achieved 63% mean intersection over union in segmenting glands, and 84.4% sensitivity and 71.7% specificity in identifying ghost glands. Morphological features of each gland were also fed to a support vector machine for analyzing their associations with ghost glands. Analysis of model coefficients indicated that low gland local contrast was the primary indicator for ghost glands. CONCLUSIONS: The proposed approach can automatically segment individual meibomian glands in infrared meibography images, identify ghost glands, and quantitatively analyze gland morphological features.

Forward Operator Estimation in Generative Models with Kernel Transfer Operators.

Generative models (e.g., variational autoencoders, flow-based generative models, GANs) usually involve finding a mapping from a known distribution, e.g. Gaussian, to an estimate of the unknown data-generating distribution. This process is often carried out by searching over a class of non-linear functions (e.g., representable by a deep neural network). While effective in practice, the associated runtime/memory costs can increase rapidly, and will depend on the performance desired in an application. We propose a much cheaper (and simpler) strategy to estimate this mapping based on adapting known results in kernel transfer operators. We show that if some compromise in functionality (and scalability) is acceptable, our proposed formulation enables highly efficient distribution approximation and sampling, and offers surprisingly good empirical performance which compares favorably with powerful baselines.

On the Versatile Uses of Partial Distance Correlation in Deep Learning.

Comparing the functional behavior of neural network models, whether it is a single network over time or two (or more networks) during or post-training, is an essential step in understanding what they are learning (and what they are not), and for identifying strategies for regularization or efficiency improvements. Despite recent progress, e.g., comparing vision transformers to CNNs, systematic comparison of function, especially across different networks, remains difficult and is often carried out layer by layer. Approaches such as canonical correlation analysis (CCA) are applicable in principle, but have been sparingly used so far. In this paper, we revisit a (less widely known) from statistics, called distance correlation (and its partial variant), designed to evaluate correlation between feature spaces of different dimensions. We describe the steps necessary to carry out its deployment for large scale models - this opens the door to a surprising array of applications ranging from conditioning one deep model w.r.t. another, learning disentangled representations as well as optimizing diverse models that would directly be more robust to adversarial attacks. Our experiments suggest a versatile regularizer (or constraint) with many advantages, which avoids some of the common difficulties one faces in such analyses .

VolterraNet: A Higher Order Convolutional Network With Group Equivariance for Homogeneous Manifolds.

Convolutional neural networks have been highly successful in image-based learning tasks due to their translation equivariance property. Recent work has generalized the traditional convolutional layer of a convolutional neural network to non-euclidean spaces and shown group equivariance of the generalized convolution operation. In this paper, we present a novel higher order Volterra convolutional neural network (VolterraNet) for data defined as samples of functions on Riemannian homogeneous spaces. Analagous to the result for traditional convolutions, we prove that the Volterra functional convolutions are equivariant to the action of the isometry group admitted by the Riemannian homogeneous spaces, and under some restrictions, any non-linear equivariant function can be expressed as our homogeneous space Volterra convolution, generalizing the non-linear shift equivariant characterization of Volterra expansions in euclidean space. We also prove that second order functional convolution operations can be represented as cascaded convolutions which leads to an efficient implementation. Beyond this, we also propose a dilated VolterraNet model. These advances lead to large parameter reductions relative to baseline non-euclidean CNNs. To demonstrate the efficacy of the VolterraNet performance, we present several real data experiments involving classification tasks on spherical-MNIST, atomic energy, Shrec17 data sets, and group testing on diffusion MRI data. Performance comparisons to the state-of-the-art are also presented.

SurReal: Complex-Valued Learning as Principled Transformations on a Scaling and Rotation Manifold.

Complex-valued data are ubiquitous in signal and image processing applications, and complex-valued representations in deep learning have appealing theoretical properties. While these aspects have long been recognized, complex-valued deep learning continues to lag far behind its real-valued counterpart. We propose a principled geometric approach to complex-valued deep learning. Complex-valued data could often be subject to arbitrary complex-valued scaling; as a result, real and imaginary components could covary. Instead of treating complex values as two independent channels of real values, we recognize their underlying geometry: we model the space of complex numbers as a product manifold of nonzero scaling and planar rotations. Arbitrary complex-valued scaling naturally becomes a group of transitive actions on this manifold. We propose to extend the property instead of the form of real-valued functions to the complex domain. We define convolution as the weighted Fréchet mean on the manifold that is equivariant to the group of scaling/rotation actions and define distance transform on the manifold that is invariant to the action group. The manifold perspective also allows us to define nonlinear activation functions, such as tangent ReLU and G -transport, as well as residual connections on the manifold-valued data. We dub our model SurReal, as our experiments on MSTAR and RadioML deliver high performance with only a fractional size of real- and complex-valued baseline models.

Transformer for 3D Point Clouds.

Deep neural networks are widely used for understanding 3D point clouds. At each point convolution layer, features are computed from local neighbourhoods of 3D points and combined for subsequent processing in order to extract semantic information. Existing methods adopt the same individual point neighborhoods throughout the network layers, defined by the same metric on the fixed input point coordinates. This common practice is easy to implement but not necessarily optimal. Ideally, local neighborhoods should be different at different layers, as more latent information is extracted at deeper layers. We propose a novel end-to-end approach to learn different non-rigid transformations of the input point cloud so that optimal local neighborhoods can be adopted at each layer. We propose both linear (affine) and non-linear (projective and deformable) spatial transformers for 3D point clouds. With spatial transformers on the ShapeNet part segmentation dataset, the network achieves higher accuracy for all categories, with 8 percent gain on earphones and rockets in particular. Our method also outperforms the state-of-the-art on other point cloud tasks such as classification, detection, and semantic segmentation. Visualizations show that spatial transformers can learn features more efficiently by dynamically altering local neighborhoods according to the geometry and semantics of 3D shapes in spite of their within-category variations.

Equivariance Allows Handling Multiple Nuisance Variables When Analyzing Pooled Neuroimaging Datasets.

Pooling multiple neuroimaging datasets across institutions often enables improvements in statistical power when evaluating associations (e.g., between risk factors and disease outcomes) that may otherwise be too weak to detect. When there is only a single source of variability (e.g., different scanners), domain adaptation and matching the distributions of representations may suffice in many scenarios. But in the presence of more than one nuisance variable which concurrently influence the measurements, pooling datasets poses unique challenges, e.g., variations in the data can come from both the acquisition method as well as the demographics of participants (gender, age). Invariant representation learning, by itself, is ill-suited to fully model the data generation process. In this paper, we show how bringing recent results on equivariant representation learning (for studying symmetries in neural networks) instantiated on structured spaces together with simple use of classical results on causal inference provides an effective practical solution. In particular, we demonstrate how our model allows dealing with more than one nuisance variable under some assumptions and can enable analysis of pooled scientific datasets in scenarios that would otherwise entail removing a large portion of the samples. Our code is available on https://github.com/vsingh-group/DatasetPooling.

Understanding Uncertainty Maps in Vision with Statistical Testing.

Quantitative descriptions of confidence intervals and uncertainties of the predictions of a model are needed in many applications in vision and machine learning. Mechanisms that enable this for deep neural network (DNN) models are slowly becoming available, and occasionally, being integrated within production systems. But the literature is sparse in terms of how to perform statistical tests with the uncertainties produced by these overparameterized models. For two models with a similar accuracy profile, is the former model's uncertainty behavior better in a statistically significant sense compared to the second model? For high resolution images, performing hypothesis tests to generate meaningful actionable information (say, at a user specified significance level α=0.05) is difficult but needed in both mission critical settings and elsewhere. In this paper, specifically for uncertainties defined on images, we show how revisiting results from Random Field theory (RFT) when paired with DNN tools (to get around computational hurdles) leads to efficient frameworks that can provide a hypothesis test capabilities, not otherwise available, for uncertainty maps from models used in many vision tasks. We show via many different experiments the viability of this framework.

ManifoldNet: A Deep Neural Network for Manifold-Valued Data With Applications.

Geometric deep learning is a relatively nascent field that has attracted significant attention in the past few years. This is partly due to the availability of data acquired from non-euclidean domains or features extracted from euclidean-space data that reside on smooth manifolds. For instance, pose data commonly encountered in computer vision reside in Lie groups, while covariance matrices that are ubiquitous in many fields and diffusion tensors encountered in medical imaging domain reside on the manifold of symmetric positive definite matrices. Much of this data is naturally represented as a grid of manifold-valued data. In this paper we present a novel theoretical framework for developing deep neural networks to cope with these grids of manifold-valued data inputs. We also present a novel architecture to realize this theory and call it the ManifoldNet. Analogous to vector spaces where convolutions are equivalent to computing weighted sums, manifold-valued data 'convolutions' can be defined using the weighted Fréchet Mean ([Formula: see text]). (This requires endowing the manifold with a Riemannian structure if it did not already come with one.) The hidden layers of ManifoldNet compute [Formula: see text]s of their inputs, where the weights are to be learnt. This means the data remain manifold-valued as they propagate through the hidden layers. To reduce computational complexity, we present a provably convergent recursive algorithm for computing the [Formula: see text]. Further, we prove that on non-constant sectional curvature manifolds, each [Formula: see text] layer is a contraction mapping and provide constructive evidence for its non-collapsibility when stacked in layers. This captures the two fundamental properties of deep network layers. Analogous to the equivariance of convolution in euclidean space to translations, we prove that the [Formula: see text] is equivariant to the action of the group of isometries admitted by the Riemannian manifold on which the data reside. To showcase the performance of ManifoldNet, we present several experiments using both computer vision and medical imaging data sets.

An Online Riemannian PCA for Stochastic Canonical Correlation Analysis.

We present an efficient stochastic algorithm (RSG+) for canonical correlation analysis (CCA) using a reparametrization of the projection matrices. We show how this reparametrization (into structured matrices), simple in hindsight, directly presents an opportunity to repurpose/adjust mature techniques for numerical optimization on Riemannian manifolds. Our developments nicely complement existing methods for this problem which either require O(d 3) time complexity per iteration with O ( 1 t ) convergence rate (where d is the dimensionality) or only extract the top 1 component with O ( 1 t ) convergence rate. In contrast, our algorithm offers an improvement: it achieves O(d 2 k) runtime complexity per iteration for extracting the top k canonical components with O ( 1 t ) convergence rate. While our paper focuses more on the formulation and the algorithm, our experiments show that the empirical behavior on common datasets is quite promising. We also explore a potential application in training fair models with missing sensitive attributes.

Flow-based Generative Models for Learning Manifold to Manifold Mappings.

Many measurements or observations in computer vision and machine learning manifest as non-Euclidean data. While recent proposals (like spherical CNN) have extended a number of deep neural network architectures to manifold-valued data, and this has often provided strong improvements in performance, the literature on generative models for manifold data is quite sparse. Partly due to this gap, there are also no modality transfer/translation models for manifold-valued data whereas numerous such methods based on generative models are available for natural images. This paper addresses this gap, motivated by a need in brain imaging - in doing so, we expand the operating range of certain generative models (as well as generative models for modality transfer) from natural images to images with manifold-valued measurements. Our main result is the design of a two-stream version of GLOW (flow-based invertible generative models) that can synthesize information of a field of one type of manifold-valued measurements given another. On the theoretical side, we introduce three kinds of invertible layers for manifold-valued data, which are not only analogous to their functionality in flow-based generative models (e.g., GLOW) but also preserve the key benefits (determinants of the Jacobian are easy to calculate). For experiments, on a large dataset from the Human Connectome Project (HCP), we show promising results where we can reliably and accurately reconstruct brain images of a field of orientation distribution functions (ODF) from diffusion tensor images (DTI), where the latter has a 5 × faster acquisition time but at the expense of worse angular resolution.

Simpler Certified Radius Maximization by Propagating Covariances.

One strategy for adversarially training a robust model is to maximize its certified radius - the neighborhood around a given training sample for which the model's prediction remains unchanged. The scheme typically involves analyzing a "smoothed" classifier where one estimates the prediction corresponding to Gaussian samples in the neighborhood of each sample in the mini-batch, accomplished in practice by Monte Carlo sampling. In this paper, we investigate the hypothesis that this sampling bottleneck can potentially be mitigated by identifying ways to directly propagate the covariance matrix of the smoothed distribution through the network. To this end, we find that other than certain adjustments to the network, propagating the covariances must also be accompanied by additional accounting that keeps track of how the distributional moments transform and interact at each stage in the network. We show how satisfying these criteria yields an algorithm for maximizing the certified radius on datasets including Cifar-10, ImageNet, and Places365 while offering runtime savings on networks with moderate depth, with a small compromise in overall accuracy. We describe the details of the key modifications that enable practical use. Via various experiments, we evaluate when our simplifications are sensible, and what the key benefits and limitations are.

A Variational Approximation for Analyzing the Dynamics of Panel Data.

Panel data involving longitudinal measurements of the same set of participants taken over multiple time points is common in studies to understand childhood development and disease modeling. Deep hybrid models that marry the predictive power of neural networks with physical simulators such as differential equations, are starting to drive advances in such applications. The task of modeling not just the observations but the hidden dynamics that are captured by the measurements poses interesting statistical/computational questions. We propose a probabilistic model called ME-NODE to incorporate (fixed + random) mixed effects for analyzing such panel data. We show that our model can be derived using smooth approximations of SDEs provided by the Wong-Zakai theorem. We then derive Evidence Based Lower Bounds for ME-NODE, and develop (efficient) training algorithms using MC based sampling methods and numerical ODE solvers. We demonstrate ME-NODE's utility on tasks spanning the spectrum from simulations and toy data to real longitudinal 3D imaging data from an Alzheimer's disease (AD) study, and study its performance in terms of accuracy of reconstruction for interpolation, uncertainty estimates and personalized prediction.

Intrinsic Grassmann Averages for Online Linear, Robust and Nonlinear Subspace Learning.

Principal component analysis (PCA) and Kernel principal component analysis (KPCA) are fundamental methods in machine learning for dimensionality reduction. The former is a technique for finding this approximation in finite dimensions and the latter is often in an infinite dimensional reproducing Kernel Hilbert-space (RKHS). In this paper, we present a geometric framework for computing the principal linear subspaces in both (finite and infinite) situations as well as for the robust PCA case, that amounts to computing the intrinsic average on the space of all subspaces: the Grassmann manifold. Points on this manifold are defined as the subspaces spanned by K-tuples of observations. The intrinsic Grassmann average of these subspaces are shown to coincide with the principal components of the observations when they are drawn from a Gaussian distribution. We show similar results in the RKHS case and provide an efficient algorithm for computing the projection onto the this average subspace. The result is a method akin to KPCA which is substantially faster. Further, we present a novel online version of the KPCA using our geometric framework. Competitive performance of all our algorithms are demonstrated on a variety of real and synthetic data sets.

Nyströmformer: A Nystöm-based Algorithm for Approximating Self-Attention.

Transformers have emerged as a powerful tool for a broad range of natural language processing tasks. A key component that drives the impressive performance of Transformers is the self-attention mechanism that encodes the influence or dependence of other tokens on each specific token. While beneficial, the quadratic complexity of self-attention on the input sequence length has limited its application to longer sequences - a topic being actively studied in the community. To address this limitation, we propose Nyströmformer - a model that exhibits favorable scalability as a function of sequence length. Our idea is based on adapting the Nyström method to approximate standard self-attention with O(n) complexity. The scalability of Nyströmformer enables application to longer sequences with thousands of tokens. We perform evaluations on multiple downstream tasks on the GLUE benchmark and IMDB reviews with standard sequence length, and find that our Nyströmformer performs comparably, or in a few cases, even slightly better, than standard self-attention. On longer sequence tasks in the Long Range Arena (LRA) benchmark, Nyströmformer performs favorably relative to other efficient self-attention methods. Our code is available at https://github.com/mlpen/Nystromformer.

Scaling Recurrent Models via Orthogonal Approximations in Tensor Trains.

Modern deep networks have proven to be very effective for analyzing real world images. However, their application in medical imaging is still in its early stages, primarily due to the large size of three-dimensional images, requiring enormous convolutional or fully connected layers - if we treat an image (and not image patches) as a sample. These issues only compound when the focus moves towards longitudinal analysis of 3D image volumes through recurrent structures, and when a point estimate of model parameters is insufficient in scientific applications where a reliability measure is necessary. Using insights from differential geometry, we adapt the tensor train decomposition to construct networks with significantly fewer parameters, allowing us to train powerful recurrent networks on whole brain image volume sequences. We describe the "orthogonal" tensor train, and demonstrate its ability to express a standard network layer both theoretically and empirically. We show its ability to effectively reconstruct whole brain volumes with faster convergence and stronger confidence intervals compared to the standard tensor train decomposition. We provide code and show experiments on the ADNI dataset using image sequences to regress on a cognition related outcome.

A Deep Learning Approach for Meibomian Gland Atrophy Evaluation in Meibography Images.

PURPOSE: To develop a deep learning approach to digitally segmenting meibomian gland atrophy area and computing percent atrophy in meibography images. METHODS: A total of 706 meibography images with corresponding meiboscores were collected and annotated for each one with eyelid and atrophy regions. The dataset was then divided into the development and evaluation sets. The development set was used to train and tune the deep learning model, while the evaluation set was used to evaluate the performance of the model. RESULTS: Four hundred ninety-seven meibography images were used for training and tuning the deep learning model while the remaining 209 images were used for evaluations. The algorithm achieves 95.6% meiboscore grading accuracy on average, largely outperforming the lead clinical investigator (LCI) by 16.0% and the clinical team by 40.6%. Our algorithm also achieves 97.6% and 95.4% accuracy for eyelid and atrophy segmentations, respectively, as well as 95.5% and 66.7% mean intersection over union accuracies (mean IU), respectively. The average root-mean-square deviation (RMSD) of the percent atrophy prediction is 6.7%. CONCLUSIONS: The proposed deep learning approach can automatically segment the total eyelid and meibomian gland atrophy regions, as well as compute percent atrophy with high accuracy and consistency. This provides quantitative information of the gland atrophy severity based on meibography images. TRANSLATIONAL RELEVANCE: Based on deep neural networks, the study presents an accurate and consistent gland atrophy evaluation method for meibography images, and may contribute to improved understanding of meibomian gland dysfunction.

Dilated Convolutional Neural Networks for Sequential Manifold-valued Data.

Efforts are underway to study ways via which the power of deep neural networks can be extended to non-standard data types such as structured data (e.g., graphs) or manifold-valued data (e.g., unit vectors or special matrices). Often, sizable empirical improvements are possible when the geometry of such data spaces are incorporated into the design of the model, architecture, and the algorithms. Motivated by neuroimaging applications, we study formulations where the data are sequential manifold-valued measurements. This case is common in brain imaging, where the samples correspond to symmetric positive definite matrices or orientation distribution functions. Instead of a recurrent model which poses computational/technical issues, and inspired by recent results showing the viability of dilated convolutional models for sequence prediction, we develop a dilated convolutional neural network architecture for this task. On the technical side, we show how the modules needed in our network can be derived while explicitly taking the Riemannian manifold structure into account. We show how the operations needed can leverage known results for calculating the weighted Fréchet Mean (wFM). Finally, we present scientific results for group difference analysis in Alzheimer's disease (AD) where the groups are derived using AD pathology load: here the model finds several brain fiber bundles that are related to AD even when the subjects are all still cognitively healthy.

A geometric framework for statistical analysis of trajectories with distinct temporal spans.

Analyzing data representing multifarious trajectories is central to the many fields in Science and Engineering; for example, trajectories representing a tennis serve, a gymnast's parallel bar routine, progression/remission of disease and so on. We present a novel geometric algorithm for performing statistical analysis of trajectories with distinct number of samples representing longitudinal (or temporal) data. A key feature of our proposal is that unlike existing schemes, our model is deployable in regimes where each participant provides a different number of acquisitions (trajectories have different number of sample points or temporal span). To achieve this, we develop a novel method involving the parallel transport of the tangent vectors along each given trajectory to the starting point of the respective trajectories and then use the span of the matrix whose columns consist of these vectors, to construct a linear subspace in R m . We then map these linear subspaces (possibly of distinct dimensions) of R m on to a single high dimensional hypersphere. This enables computing group statistics over trajectories by instead performing statistics on the hypersphere (equipped with a simpler geometry). Given a point on the hypersphere representing a trajectory, we also provide a "reverse mapping" algorithm to uniquely (under certain assumptions) reconstruct the subspace that corresponds to this point. Finally, by using existing algorithms for recursive Fréchet mean and exact principal geodesic analysis on the hypersphere, we present several experiments on synthetic and real (vision and medical) data sets showing how group testing on such diversely sampled longitudinal data is possible by analyzing the reconstructed data in the subspace spanned by the first few principal components.

Feature selection using a neural framework with controlled redundancy.

We first present a feature selection method based on a multilayer perceptron (MLP) neural network, called feature selection MLP (FSMLP). We explain how FSMLP can select essential features and discard derogatory and indifferent features. Such a method may pick up some useful but dependent (say correlated) features, all of which may not be needed. We then propose a general scheme for dealing with feature selection with "controlled redundancy" (CoR). The proposed scheme, named as FSMLP-CoR, can select features with a controlled redundancy both for classification and function approximation/prediction type problems. We have also proposed a new more effective training scheme named mFSMLP-CoR. The idea is general in nature and can be used with other learning schemes also. We demonstrate the effectiveness of the algorithms using several data sets including a synthetic data set. We also show that the selected features are adequate to solve the problem at hand. Here, we have considered a measure of linear dependency to control the redundancy. The use of nonlinear measures of dependency, such as mutual information, is straightforward. Here, there are some advantages of the proposed schemes. They do not require explicit evaluation of the feature subsets. Here, feature selection is integrated into designing of the decision-making system. Hence, it can look at all features together and pick up whatever is necessary. Our methods can account for possible nonlinear subtle interactions between features, as well as that between features, tools, and the problem being solved. They can also control the level of redundancy in the selected features. Of the two learning schemes, mFSMLP-CoR, not only improves the performance of the system, but also significantly reduces the dependency of the network's behavior on the initialization of connection weights.