# Browsing by Subject "cs.LG"

Now showing 1 - 20 of 28

###### Results Per Page

###### Sort Options

Item Open Access A Priori Generalization Analysis of the Deep Ritz Method for Solving High Dimensional Elliptic EquationsLu, Jianfeng; Lu, Yulong; Wang, MinThis paper concerns the a priori generalization analysis of the Deep Ritz Method (DRM) [W. E and B. Yu, 2017], a popular neural-network-based method for solving high dimensional partial differential equations. We derive the generalization error bounds of two-layer neural networks in the framework of the DRM for solving two prototype elliptic PDEs: Poisson equation and static Schr\"odinger equation on the $d$-dimensional unit hypercube. Specifically, we prove that the convergence rates of generalization errors are independent of the dimension $d$, under the a priori assumption that the exact solutions of the PDEs lie in a suitable low-complexity space called spectral Barron space. Moreover, we give sufficient conditions on the forcing term and the potential function which guarantee that the solutions are spectral Barron functions. We achieve this by developing a new solution theory for the PDEs on the spectral Barron space, which can be viewed as an analog of the classical Sobolev regularity theory for PDEs.Item Open Access A stochastic version of Stein Variational Gradient Descent for efficient samplingLi, L; Li, Y; Liu, JG; Liu, Z; Lu, JWe propose in this work RBM-SVGD, a stochastic version of Stein Variational Gradient Descent (SVGD) method for efficiently sampling from a given probability measure and thus useful for Bayesian inference. The method is to apply the Random Batch Method (RBM) for interacting particle systems proposed by Jin et al to the interacting particle systems in SVGD. While keeping the behaviors of SVGD, it reduces the computational cost, especially when the interacting kernel has long range. Numerical examples verify the efficiency of this new version of SVGD.Item Open Access Adaptive Hyper-box Matching for Interpretable Individualized Treatment Effect Estimation.(CoRR, 2020) Morucci, Marco; Orlandi, Vittorio; Rudin, Cynthia; Roy, Sudeepa; Volfovsky, AlexanderWe propose a matching method for observational data that matches units with others in unit-specific, hyper-box-shaped regions of the covariate space. These regions are large enough that many matches are created for each unit and small enough that the treatment effect is roughly constant throughout. The regions are found as either the solution to a mixed integer program, or using a (fast) approximation algorithm. The result is an interpretable and tailored estimate of a causal effect for each unit.Item Open Access Approximation of Functions over Manifolds: A Moving Least-Squares ApproachSober, B; Aizenbud, Y; Levin, DWe present an algorithm for approximating a function defined over a $d$-dimensional manifold utilizing only noisy function values at locations sampled from the manifold with noise. To produce the approximation we do not require any knowledge regarding the manifold other than its dimension $d$. We use the Manifold Moving Least-Squares approach of (Sober and Levin 2016) to reconstruct the atlas of charts and the approximation is built on-top of those charts. The resulting approximant is shown to be a function defined over a neighborhood of a manifold, approximating the originally sampled manifold. In other words, given a new point, located near the manifold, the approximation can be evaluated directly on that point. We prove that our construction yields a smooth function, and in case of noiseless samples the approximation order is $\mathcal{O}(h^{m+1})$, where $h$ is a local density of sample parameter (i.e., the fill distance) and $m$ is the degree of a local polynomial approximation, used in our algorithm. In addition, the proposed algorithm has linear time complexity with respect to the ambient-space's dimension. Thus, we are able to avoid the computational complexity, commonly encountered in high dimensional approximations, without having to perform non-linear dimension reduction, which inevitably introduces distortions to the geometry of the data. Additionaly, we show numerical experiments that the proposed approach compares favorably to statistical approaches for regression over manifolds and show its potential.Item Open Access Butterfly-Net: Optimal Function Representation Based on Convolutional Neural NetworksLi, Y; Cheng, X; Lu, JDeep 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 hard-coded weights and sparse across-channel connections, which aims at an optimal hierarchical function representation of the input signal. 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. Due to the ability of Butterfly-Net to approximate Fourier and local Fourier transforms, the result can be used for approximation upper bound for CNNs in a large class of problems. The analysis results are validated in numerical experiments on the approximation of a 1D Fourier kernel and of solving a 2D Poisson's equation.Item Open Access Complexity of zigzag sampling algorithm for strongly log-concave distributionsLu, Jianfeng; Wang, LihanWe study the computational complexity of zigzag sampling algorithm for strongly log-concave distributions. The zigzag process has the advantage of not requiring time discretization for implementation, and that each proposed bouncing event requires only one evaluation of partial derivative of the potential, while its convergence rate is dimension independent. Using these properties, we prove that the zigzag sampling algorithm achieves $\varepsilon$ error in chi-square divergence with a computational cost equivalent to $O\bigl(\kappa^2 d^\frac{1}{2}(\log\frac{1}{\varepsilon})^{\frac{3}{2}}\bigr)$ gradient evaluations in the regime $\kappa \ll \frac{d}{\log d}$ under a warm start assumption, where $\kappa$ is the condition number and $d$ is the dimension.Item Open Access Cross-Domain Multitask Learning with Latent Probit ModelsHan, S; Liao, X; Carin, LLearning multiple tasks across heterogeneous domains is a challenging problem since the feature space may not be the same for different tasks. We assume the data in multiple tasks are generated from a latent common domain via sparse domain transforms and propose a latent probit model (LPM) to jointly learn the domain transforms, and the shared probit classifier in the common domain. To learn meaningful task relatedness and avoid over-fitting in classification, we introduce sparsity in the domain transforms matrices, as well as in the common classifier. We derive theoretical bounds for the estimation error of the classifier in terms of the sparsity of domain transforms. An expectation-maximization algorithm is derived for learning the LPM. The effectiveness of the approach is demonstrated on several real datasets.Item Open Access dame-flame: A Python Library Providing Fast Interpretable Matching for Causal Inference.(CoRR, 2021) Gupta, Neha R; Orlandi, Vittorio; Chang, Chia-Rui; Wang, Tianyu; Morucci, Marco; Dey, Pritam; Howell, Thomas J; Sun, Xian; Ghosal, Angikar; Roy, Sudeepa; Rudin, Cynthia; Volfovsky, Alexanderdame-flame is a Python package for performing matching for observational causal inference on datasets containing discrete covariates. This package implements the Dynamic Almost Matching Exactly (DAME) and Fast Large-Scale Almost Matching Exactly (FLAME) algorithms, which match treatment and control units on subsets of the covariates. The resulting matched groups are interpretable, because the matches are made on covariates (rather than, for instance, propensity scores), and high-quality, because machine learning is used to determine which covariates are important to match on. DAME solves an optimization problem that matches units on as many covariates as possible, prioritizing matches on important covariates. FLAME approximates the solution found by DAME via a much faster backward feature selection procedure. The package provides several adjustable parameters to adapt the algorithms to specific applications, and can calculate treatment effects after matching. Descriptions of these parameters, details on estimating treatment effects, and further examples, can be found in the documentation at https://almost-matching-exactly.github.io/DAME-FLAME-Python-Package/Item Open Access DCFNet: Deep Neural Network with Decomposed Convolutional Filters(35th International Conference on Machine Learning, ICML 2018, 2018-01-01) Qiu, Q; Cheng, X; Calderbank, R; Sapiro, G©35th International Conference on Machine Learning, ICML 2018.All Rights Reserved. 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.Item Open Access Expression of Fractals Through Neural Network FunctionsDym, N; Sober, B; Daubechies, ITo help understand the underlying mechanisms of neural networks (NNs), several groups have, in recent years, studied the number of linear regions $\ell$ of piecewise linear functions generated by deep neural networks (DNN). In particular, they showed that $\ell$ can grow exponentially with the number of network parameters $p$, a property often used to explain the advantages of DNNs over shallow NNs in approximating complicated functions. Nonetheless, a simple dimension argument shows that DNNs cannot generate all piecewise linear functions with $\ell$ linear regions as soon as $\ell > p$. It is thus natural to seek to characterize specific families of functions with $\ell$ linear regions that can be constructed by DNNs. Iterated Function Systems (IFS) generate sequences of piecewise linear functions $F_k$ with a number of linear regions exponential in $k$. We show that, under mild assumptions, $F_k$ can be generated by a NN using only $\mathcal{O}(k)$ parameters. IFS are used extensively to generate, at low computational cost, natural-looking landscape textures in artificial images. They have also been proposed for compression of natural images, albeit with less commercial success. The surprisingly good performance of this fractal-based compression suggests that our visual system may lock in, to some extent, on self-similarities in images. The combination of this phenomenon with the capacity, demonstrated here, of DNNs to efficiently approximate IFS may contribute to the success of DNNs, particularly striking for image processing tasks, as well as suggest new algorithms for representing self similarities in images based on the DNN mechanism.Item Open Access Finite-Time Frequentist Regret Bounds of Multi-Agent Thompson Sampling on Sparse Hypergraphs(2023-12-24) Jin, Tianyuan; Hsu, Hao-Lun; Chang, William; Xu, PanItem Open Access Global optimality of softmax policy gradient with single hidden layer neural networks in the mean-field regimeAgazzi, Andrea; Lu, JianfengWe study the problem of policy optimization for infinite-horizon discounted Markov Decision Processes with softmax policy and nonlinear function approximation trained with policy gradient algorithms. We concentrate on the training dynamics in the mean-field regime, modeling e.g., the behavior of wide single hidden layer neural networks, when exploration is encouraged through entropy regularization. The dynamics of these models is established as a Wasserstein gradient flow of distributions in parameter space. We further prove global optimality of the fixed points of this dynamics under mild conditions on their initialization.Item Open Access Hölder Bounds for Sensitivity Analysis in Causal Reasoning.(CoRR, 2021) Assaad, Serge; Zeng, Shuxi; Pfister, Henry; Li, Fan; Carin, LawrenceWe examine interval estimation of the effect of a treatment T on an outcome Y given the existence of an unobserved confounder U. Using H\"older's inequality, we derive a set of bounds on the confounding bias |E[Y|T=t]-E[Y|do(T=t)]| based on the degree of unmeasured confounding (i.e., the strength of the connection U->T, and the strength of U->Y). These bounds are tight either when U is independent of T or when U is independent of Y given T (when there is no unobserved confounding). We focus on a special case of this bound depending on the total variation distance between the distributions p(U) and p(U|T=t), as well as the maximum (over all possible values of U) deviation of the conditional expected outcome E[Y|U=u,T=t] from the average expected outcome E[Y|T=t]. We discuss possible calibration strategies for this bound to get interval estimates for treatment effects, and experimentally validate the bound using synthetic and semi-synthetic datasets.Item Open Access Inferring Latent Structure From Mixed Real and Categorical Relational DataSalazar, E; Cain, MS; Darling, EF; Mitroff, SR; Carin, LWe consider analysis of relational data (a matrix), in which the rows correspond to subjects (e.g., people) and the columns correspond to attributes. The elements of the matrix may be a mix of real and categorical. Each subject and attribute is characterized by a latent binary feature vector, and an inferred matrix maps each row-column pair of binary feature vectors to an observed matrix element. The latent binary features of the rows are modeled via a multivariate Gaussian distribution with low-rank covariance matrix, and the Gaussian random variables are mapped to latent binary features via a probit link. The same type construction is applied jointly to the columns. The model infers latent, low-dimensional binary features associated with each row and each column, as well correlation structure between all rows and between all columns.Item Open Access Machine Learning to Predict Developmental Neurotoxicity with High-throughput Data from 2D Bio-engineered TissuesKuusisto, Finn; Costa, Vitor Santos; Hou, Zhonggang; Thomson, James; Page, David; Stewart, RonThere is a growing need for fast and accurate methods for testing developmental neurotoxicity across several chemical exposure sources. Current approaches, such as in vivo animal studies, and assays of animal and human primary cell cultures, suffer from challenges related to time, cost, and applicability to human physiology. We previously demonstrated success employing machine learning to predict developmental neurotoxicity using gene expression data collected from human 3D tissue models exposed to various compounds. The 3D model is biologically similar to developing neural structures, but its complexity necessitates extensive expertise and effort to employ. By instead focusing solely on constructing an assay of developmental neurotoxicity, we propose that a simpler 2D tissue model may prove sufficient. We thus compare the accuracy of predictive models trained on data from a 2D tissue model with those trained on data from a 3D tissue model, and find the 2D model to be substantially more accurate. Furthermore, we find the 2D model to be more robust under stringent gene set selection, whereas the 3D model suffers substantial accuracy degradation. While both approaches have advantages and disadvantages, we propose that our described 2D approach could be a valuable tool for decision makers when prioritizing neurotoxicity screening.Item Open Access Malignancy Prediction and Lesion Identification from Clinical Dermatological Images.(CoRR, 2021) Xia, Meng; Kheterpal, Meenal K; Wong, Samantha C; Park, Christine; Ratliff, William; Carin, Lawrence; Henao, RicardoWe consider machine-learning-based malignancy prediction and lesion identification from clinical dermatological images, which can be indistinctly acquired via smartphone or dermoscopy capture. Additionally, we do not assume that images contain single lesions, thus the framework supports both focal or wide-field images. Specifically, we propose a two-stage approach in which we first identify all lesions present in the image regardless of sub-type or likelihood of malignancy, then it estimates their likelihood of malignancy, and through aggregation, it also generates an image-level likelihood of malignancy that can be used for high-level screening processes. Further, we consider augmenting the proposed approach with clinical covariates (from electronic health records) and publicly available data (the ISIC dataset). Comprehensive experiments validated on an independent test dataset demonstrate that i) the proposed approach outperforms alternative model architectures; ii) the model based on images outperforms a pure clinical model by a large margin, and the combination of images and clinical data does not significantly improves over the image-only model; and iii) the proposed framework offers comparable performance in terms of malignancy classification relative to three board certified dermatologists with different levels of experience.Item Open Access Manifold Approximation by Moving Least-Squares Projection (MMLS)(Constructive Approximation) Sober, Barak; Levin, DavidIn order to avoid the curse of dimensionality, frequently encountered in Big Data analysis, there was a vast development in the field of linear and nonlinear dimension reduction techniques in recent years. These techniques (sometimes referred to as manifold learning) assume that the scattered input data is lying on a lower dimensional manifold, thus the high dimensionality problem can be overcome by learning the lower dimensionality behavior. However, in real life applications, data is often very noisy. In this work, we propose a method to approximate $\mathcal{M}$ a $d$-dimensional $C^{m+1}$ smooth submanifold of $\mathbb{R}^n$ ($d \ll n$) based upon noisy scattered data points (i.e., a data cloud). We assume that the data points are located "near" the lower dimensional manifold and suggest a non-linear moving least-squares projection on an approximating $d$-dimensional manifold. Under some mild assumptions, the resulting approximant is shown to be infinitely smooth and of high approximation order (i.e., $O(h^{m+1})$, where $h$ is the fill distance and $m$ is the degree of the local polynomial approximation). The method presented here assumes no analytic knowledge of the approximated manifold and the approximation algorithm is linear in the large dimension $n$. Furthermore, the approximating manifold can serve as a framework to perform operations directly on the high dimensional data in a computationally efficient manner. This way, the preparatory step of dimension reduction, which induces distortions to the data, can be avoided altogether.Item Open Access Neural Network Approximation of Refinable Functions.(CoRR, 2021) Daubechies, Ingrid; DeVore, Ronald; Dym, Nadav; Faigenbaum-Golovin, Shira; Kovalsky, Shahar Z; Lin, Kung-Ching; Park, Josiah; Petrova, Guergana; Sober, BarakIn the desire to quantify the success of neural networks in deep learning and other applications, there is a great interest in understanding which functions are efficiently approximated by the outputs of neural networks. By now, there exists a variety of results which show that a wide range of functions can be approximated with sometimes surprising accuracy by these outputs. For example, it is known that the set of functions that can be approximated with exponential accuracy (in terms of the number of parameters used) includes, on one hand, very smooth functions such as polynomials and analytic functions (see e.g. \cite{E,S,Y}) and, on the other hand, very rough functions such as the Weierstrass function (see e.g. \cite{EPGB,DDFHP}), which is nowhere differentiable. In this paper, we add to the latter class of rough functions by showing that it also includes refinable functions. Namely, we show that refinable functions are approximated by the outputs of deep ReLU networks with a fixed width and increasing depth with accuracy exponential in terms of their number of parameters. Our results apply to functions used in the standard construction of wavelets as well as to functions constructed via subdivision algorithms in Computer Aided Geometric Design.Item Open Access On the Universality of Rotation Equivariant Point Cloud Networks.(CoRR, 2020) Dym, Nadav; Maron, HaggaiLearning functions on point clouds has applications in many fields, including computer vision, computer graphics, physics, and chemistry. Recently, there has been a growing interest in neural architectures that are invariant or equivariant to all three shape-preserving transformations of point clouds: translation, rotation, and permutation. In this paper, we present a first study of the approximation power of these architectures. We first derive two sufficient conditions for an equivariant architecture to have the universal approximation property, based on a novel characterization of the space of equivariant polynomials. We then use these conditions to show that two recently suggested models are universal, and for devising two other novel universal architectures.Item Open Access Physics-enhanced machine learning for virtual fluorescence microscopy(CoRR, 2020-04-08) Cooke, Colin L; Kong, Fanjie; Chaware, Amey; Zhou, Kevin C; Kim, Kanghyun; Xu, Rong; Ando, D Michael; Yang, Samuel J; Konda, Pavan Chandra; Horstmeyer, RoarkeThis paper introduces a new method of data-driven microscope design for virtual fluorescence microscopy. Our results show that by including a model of illumination within the first layers of a deep convolutional neural network, it is possible to learn task-specific LED patterns that substantially improve the ability to infer fluorescence image information from unstained transmission microscopy images. We validated our method on two different experimental setups, with different magnifications and different sample types, to show a consistent improvement in performance as compared to conventional illumination methods. Additionally, to understand the importance of learned illumination on inference task, we varied the dynamic range of the fluorescent image targets (from one to seven bits), and showed that the margin of improvement for learned patterns increased with the information content of the target. This work demonstrates the power of programmable optical elements at enabling better machine learning algorithm performance and at providing physical insight into next generation of machine-controlled imaging systems.