This paper improves deep neural network approximation for fully connected networks, achieving optimal convergence rates.
problem Improving approximation of fully connected deep neural networks for optimal convergence rates.
method Deriving approximation bounds specifically for a narrower fully connected deep neural network.
result Achieves an optimal rate (up to a logarithmic factor) for fully connected deep neural networks.
Connections between nodes of fully connected neural networks are usually represented by weight matrices. In this article, functional transfer matrices are introduced as alternatives to the weight matrices: Instead of using real weights, a functional transfer matrix uses real functions with trainable parameters to repre…
Previous work has questioned the conditions under which the decision regions of a neural network are connected and further showed the implications of the corresponding theory to the problem of adversarial manipulation of classifiers. It has been proven that for a class of activation functions including leaky ReLU, neur…
ST-GCN improves rs-fMRI prediction accuracy by modeling spatio-temporal graph connectivity.
problem Existing rs-fMRI methods neglect functional connectivity or temporal dynamics.
method Spatio-temporal graph convolutional network (ST-GCN) trained on BOLD time series.
result ST-GCN predicts gender and age more accurately than common methods.
Machine learning techniques have become increasingly popular in the field of resting state fMRI (functional magnetic resonance imaging) network based classification. However, the application of convolutional networks has been proposed only very recently and has remained largely unexplored. In this paper we describe a c…
Functional dimension varies in ReLU networks, with implications for symmetry and connectivity.
problem Understanding the functional dimension of ReLU neural networks.
method Careful definition and analysis of functional dimension, study of quotient space and fibers.
result Functional dimension is inhomogeneous and can be non-constant, with implications for symmetry and connectivity.
Paper introduces a new method to identify brain hubs using both structural and functional connectivity.
problem Hub node identification in brain networks using only functional connectivity.
method Graph signal processing framework that models functional activity as graph signals on structural connectivity.
result The proposed GraFHub framework identifies hub nodes more accurately than conventional methods.
Study analyzes error in ReLU networks with local connections.
problem Improving neural network performance and understanding approximation errors.
method Analyzed approximation error of ReLU networks with local connections.
result Error estimate depends on depth and width of hidden layers.
This study connects ReLU neural networks to toric geometry to analyze function realization.
problem Determining which continuous piecewise linear functions can be realized by ReLU neural networks.
method Established a connection between toric geometry and ReLU neural networks, defining key structures like the ReLU fan, toric variety, and Cartier divisor.
result Proved a criterion for functions realizable by unbiased shallow ReLU networks using intersection numbers.
New framework explains deep neural networks using variational spline theory.
problem Understanding functions learned by deep neural networks.
method Developed a variational framework and function space.
result Deep ReLU networks are solutions to regularized data fitting problems over the proposed function space.
We show that for neural network functions that have width less or equal to the input dimension all connected components of decision regions are unbounded. The result holds for continuous and strictly monotonic activation functions as well as for the ReLU activation function. This complements recent results on approxima…
The paper connects neural networks to Mahalanobis distance for interpretability.
problem Lack of interpretability in neural networks.
method Establishes a connection between neural network linear layers and Mahalanobis distance.
result Provides a foundation for more interpretable neural network models.
Derives continuum model from discrete ε-graphs with connectivity functional.
problem Modeling diffusion in networks with varying connectivity.
method Energy-based continuum limit derivation, neural-network reconstruction of connectivity.
result Error between discrete and continuum energies is O(ε), valid even with fluctuations. A new kernel measures brain network similarities, improving disease classification.
problem Lack of edge weight information in existing graph kernels for brain connectivity networks.
method Ordinal pattern kernel for weighted brain connectivity networks.
result The ordinal pattern kernel achieves better classification performance than state-of-the-art graph kernels.
GNNs improve brain activity forecasting in fMRI studies.
problem Understanding neural dynamics in the brain.
method Comparison of GNN architectures for modeling fMRI data.
result GNNs outperform VAR models in robustly scaling to large network studies.
Motivated by the flexibility of biological neural networks whose connectivity structure changes significantly during their lifetime, we introduce the Unstructured Recursive Network (URN) and demonstrate that it can exhibit similar flexibility during training via gradient descent. We show empirically that many of the di…
Brain networks have received considerable attention given the critical significance for understanding human brain organization, for investigating neurological disorders and for clinical diagnostic applications. Structural brain network (e.g. DTI) and functional brain network (e.g. fMRI) are the primary networks of inte…
Deep neural networks estimate regression functions on manifolds.
problem Estimating regression functions on manifolds from data.
method Fully connected deep neural networks with ReLU activation, analyzing convergence rates.
result Estimates achieve a rate of convergence dependent on manifold dimension, not predictor dimension.
Studies in recent years have demonstrated that neural organization and structure impact an individual's ability to perform a given task. Specifically, individuals with greater neural efficiency have been shown to outperform those with less organized functional structure. In this work, we compare the predictive ability …
We generalize the scale-free network model of Barabàsi and Albert [Science 286, 509 (1999)] by proposing a class of stochastic models for scale-free interdependent networks in which interdependent nodes are not randomly connected but rather are connected via preferential attachment (PA). Each network grows through the …
New neural network criterion connects RH to minimization problem.
problem Riemann Hypothesis (RH) about zeta function zeros.
method Revisits and extends Nyman-Beurling criterion linking RH to neural networks.
result Establishes connection between RH and minimization problem involving neural networks.
This paper develops fundamental limits of deep neural network learning by characterizing what is possible if no constraints are imposed on the learning algorithm and on the amount of training data. Concretely, we consider Kolmogorov-optimal approximation through deep neural networks with the guiding theme being a relat…
Deep networks with a wide layer ensure sublevel set connectivity.
problem Ensuring connectivity of sublevel sets in deep learning.
method Analyzing the connectivity of sublevel sets in deep neural networks with a specific layer width.
result A single wide layer of width N+1 suffices to prove connectivity of sublevel sets. Deep networks can learn functions approximated by shallow networks, but not all functions.
problem The learnability of functions by deep neural networks and the approximation capacity of simpler classes.
method Study the connection between learnability and approximation capacity of functions by deep neural networks and simpler classes.
result A necessary condition for a function to be learnable by deep neural networks is to be approximable by shallow networks.
We exploit altered patterns in brain functional connectivity as features for automatic discriminative analysis of neuropsychiatric patients. Deep learning methods have been introduced to functional network classification only very recently for fMRI, and the proposed architectures essentially focused on a single type of…
Dense neural networks can't approximate all functions.
problem Approximation capabilities of dense neural networks.
method Model compression approach combining weak regularity lemma and graph neural networks.
result Existence of Lipschitz continuous functions not approximable by dense neural networks.
A new method uses gene interaction networks to predict gene functions.
problem Predicting gene functions from gene interactions.
method Context graph kernel approach in a machine learning framework.
result The proposed method outperforms linkage-assumption-based methods.
Expressive efficiency refers to the relation between two architectures A and B, whereby any function realized by B could be replicated by A, but there exists functions realized by A, which cannot be replicated by B unless its size grows significantly larger. For example, it is known that deep networks are exponentially…
We study the expressivity of deep neural networks. Measuring a network's complexity by its number of connections or by its number of neurons, we consider the class of functions for which the error of best approximation with networks of a given complexity decays at a certain rate when increasing the complexity budget. U…
Dynamics and function of neuronal networks are determined by their synaptic connectivity. Current experimental methods to analyze synaptic network structure on the cellular level, however, cover only small fractions of functional neuronal circuits, typically without a simultaneous record of neuronal spiking activity. H…
Neural networks can represent complex piecewise functions efficiently.
problem Representing continuous piecewise affine functions with neural networks.
method Two hidden layers with ReLU activation, O(p) neurons for p pieces. result CPA functions can be represented by a neural network with linear size.
Paper studies ResNet dynamics using NTH, reducing width requirement.
problem Understanding ResNet dynamics and improving training efficiency.
method Uses Neural Tangent Hierarchy (NTH) to analyze ResNet dynamics.
result Reduces width requirement from quartic to cubic for ResNet.
The paper calculates bounds on the local Lipschitz constants of neural network layers.
problem Understanding the Lipschitz constants of neural network layers for robustness analysis.
method Analytical approach to determine upper bounds on local Lipschitz constants of affine-ReLU functions.
result The method produces tighter bounds than the standard conservative bound, especially for small perturbations.
LOCUS separates brain network connectivity matrices efficiently.
problem High dimensionality, latent sources, and spurious findings in analyzing brain connectivity matrices.
method LOCUS: low-rank structure with uniform sparsity, iterative Node-Rotation algorithm.
result LOCUS achieves more efficient and accurate source separation for connectivity matrices.
New framework for dense weighted networks with community-specific patterns.
problem Dense networks with varying edge weights across communities.
method Proposes a new model with functions mapping node characteristics to edge weights, requiring few parameters.
result Developed a bootstrap methodology for generating new networks.
New findings connect shaped and unshaped neural networks using differential equations.
problem Understanding the behavior of neural networks with different activation scaling methods.
method Deriving differential equation-based asymptotic characterizations for shaped and unshaped neural networks.
result Two types of unshaped networks converge to the same infinite-depth-and-width limit at initialization.
Framework integrates brain connectivity data for clinical predictions.
problem Predicting clinical outcomes from brain connectivity data.
method Structurally-regularized Dynamic Dictionary Learning (sr-DDL) and LSTM-ANN block.
result Framework outperforms state-of-the-art approaches in clinical outcome prediction.
Framework identifies brain connectivity alterations for MDD patients using limited rs-fMRI data.
problem Difficult to analyze brain connectivity alterations from limited rs-fMRI data.
method Proposed a multitask Gaussian Bayesian network (MTGBN) framework to learn individual disease-induced alterations.
result Framework efficiently learns Bayesian network structures from limited data, showing improved performance.
Stochastic encoding improves gender classification of brain networks from UK Biobank data.
problem Complexity and bias in interpreting deep learning models of brain connectivity.
method Stochastic encoding in ensemble of CNNs, multivariate balancing algorithm.
result AUROC of 0.8459, with resting-state data more accurate than task data.
Study adapts β-TCVAE for fMRI to recover nonlinear brain components.
problem Capturing nonlinear brain dynamics in fMRI data.
method Adapted β-TCVAE framework for fMRI data. result Recovery of meaningful nonlinear spatial components in fMRI data.
PAC-Bayesian bounds show fully connected DNNs with Gaussian priors match minimax rates.
problem Theoretical limits of fully connected deep neural networks with Gaussian priors.
method PAC-Bayesian bounds for fully connected Bayesian DNNs with Gaussian priors.
result PAC-Bayesian bounds match minimax-optimal rates in Besov space for nonparametric regression and binary classification.
We provide novel guaranteed approaches for training feedforward neural networks with sparse connectivity. We leverage on the techniques developed previously for learning linear networks and show that they can also be effectively adopted to learn non-linear networks. We operate on the moments involving label and the sco…
New neural architectures with multivariate nonlinearities are optimal in function space.
problem Optimality of neural architectures with multivariate nonlinearities.
method Construction of Banach spaces via k-plane transform and sparsity-promoting norm, proving representer theorem. result Neural architectures with multivariate nonlinearities are optimal in function space.
Functional connections in the brain are frequently represented by weighted networks, with nodes representing locations in the brain, and edges representing the strength of connectivity between these locations. One challenge in analyzing such data is that inference at the individual edge level is not particularly biolog…
We discuss approximation of functions using deep neural nets. Given a function f on a d-dimensional manifold Γ⊂Rm, 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…
Convolutional nets require fewer samples than fully-connected nets for image classification.
problem Understanding why convolutional nets are more sample-efficient than fully-connected nets.
method Construction of a natural distribution and target function to demonstrate a sample complexity gap.
result Convolutional nets require O(1) samples for a single target function, while fully-connected nets require Ω(d2) samples. The study examines if ReLU activation function is optimal for modularity in neural networks.
problem Finding the best activation function for modularity in neural networks.
method Comparing ReLU with other activation functions for modularity and performance.
result ReLU may not be the best choice for modularity, suggesting other functions could be more suitable.
We consider supervised learning problems where the features are embedded in a graph, such as gene expressions in a gene network. In this context, it is of much interest to automatically select a subgraph with few connected components; by exploiting prior knowledge, one can indeed improve the prediction performance or o…