A new readout layer improves graph property prediction.
problem Graph property prediction challenges due to information loss.
method Introduces a novel generalized global pooling layer for Message-Passing Neural Networks.
result New state-of-the-art results in graph property prediction.
This work establishes universality for deep equivariant networks, overcoming limitations of previous approaches.
problem Rarity of universality results for equivariant neural networks, especially in high-dimensional settings.
method Develops a more general account of universality for equivariant networks, introducing entry-wise separability and readout layers.
result Deep equivariant networks achieve universality under entry-wise separability, with or without readout layers.
Training deep neural networks with the error backpropagation algorithm is considered implausible from a biological perspective. Numerous recent publications suggest elaborate models for biologically plausible variants of deep learning, typically defining success as reaching around 98% test accuracy on the MNIST data se…
Deep networks improve by progressively refining approximations at each layer.
problem Standard approximation theory doesn't explain the role of intermediate layers in deep neural networks.
method Developed a mixed-activation architecture with a geometric scale interpretation of depth.
result Each intermediate layer approximates the target function with a geometric rate.
We propose reinforcement learning on simple networks consisting of random connections of spiking neurons (both recurrent and feed-forward) that can learn complex tasks with very little trainable parameters. Such sparse and randomly interconnected recurrent spiking networks exhibit highly non-linear dynamics that transf…
Echo State Networks (ESN) are a class of Recurrent Neural Networks (RNN) that has gained substantial popularity due to their effectiveness, ease of use and potential for compact hardware implementation. An ESN contains the three network layers input, reservoir and readout where the reservoir is the truly recurrent netw…
New explanation of reservoir computing using random projections.
problem Understanding the randomness in reservoir computing.
method Constructing strongly universal reservoir systems as random projections of state-space systems.
result Approximation of any fading memory filters class by training a linear readout for each filter.
LoCo learns local representations without end-to-end synchronization, improving performance on complex tasks.
problem Learning local representations without end-to-end synchronization constraints.
method Overlap local blocks to increase decoder depth and allow feedback from upper to lower layers.
result LoCo closes the performance gap between local learning and end-to-end contrastive learning.
New conditions ensure deep neural networks can approximate any function on non-Euclidean spaces.
problem Understanding how to modify neural network architectures to approximate functions on non-Euclidean spaces.
method Developed conditions for feature and readout maps that preserve universal approximation capabilities.
result Modified architectures can deterministically approximate any classifier on non-Euclidean spaces.
The paper explores how AI trading agents' similar information representation can cause financial market instability.
problem Systemic instability in AI-dominated financial markets due to similar information representation.
method Structural multi-agent market model with two-layer decision architecture for AI agents.
result Representation homogeneity can lead to systemic instability in financial markets.
Echo state network (ESN) is viewed as a temporal non-orthogonal expansion with pseudo-random parameters. Such expansions naturally give rise to regressors of various relevance to a teacher output. We illustrate that often only a certain amount of the generated echo-regressors effectively explain the variance of the tea…
New reservoir computing approach handles infinite-dimensional systems.
problem Approximating and generalizing complex input/output systems.
method Randomly generated echo state networks with neural networks.
result Proves universal approximation properties for new class of systems.
The mushroom body is the key network for the representation of learned olfactory stimuli in Drosophila and insects. The sparse activity of Kenyon cells, the principal neurons in the mushroom body, plays a key role in the learned classification of different odours. In the specific case of the fruit fly, the sparseness o…
Objects are represented in sensory systems by continuous manifolds due to sensitivity of neuronal responses to changes in physical features such as location, orientation, and intensity. What makes certain sensory representations better suited for invariant decoding of objects by downstream networks? We present a theory…
Reservoir Computing (RC) refers to a Recurrent Neural Networks (RNNs) framework, frequently used for sequence learning and time series prediction. The RC system consists of a random fixed-weight RNN (the input-hidden reservoir layer) and a classifier (the hidden-output readout layer). Here we focus on the sequence lear…
Softmax is a standard final layer used in Neural Nets (NNs) to summarize information encoded in the trained NN and return a prediction. However, Softmax leverages only a subset of the class-specific structure encoded in the trained model and ignores potentially valuable information: During training, models encode an ar…
Characterizes test error in learning with deep, structured feature maps.
problem Characterizing test error in learning with deep, structured feature maps.
method Asymptotic analysis of feature covariance and population covariance.
result Closed-form formula for feature covariance in Gaussian rainbow neural networks.
We study how finite Bayesian neural networks adapt their hidden representations.
problem Understanding how finite Bayesian neural networks differ from infinite ones.
method We analyze the asymptotics of learned feature kernels for various network architectures.
result The leading finite-width corrections to feature kernels have a universal form.
A new framework for efficient sequence maps using Bayesian filtering and covariance.
problem Designing efficient recurrent sequence maps from explicit memory assumptions.
method Design-model framework, exact Bayesian filtering, query-dependent readout, linear-Gaussian instantiation.
result Improved robustness and retrieval performance across various benchmarks.
The study proves Gaussian universality of deep random features learning.
problem Understanding the test error in deep random features learning.
method Proving Gaussian universality of test error in ridge regression and arbitrary convex losses.
result Sharp asymptotic formula for test error in ridge regression setting.
Quantum reservoirs risk bounds are analyzed using Rademacher complexity.
problem Bounding generalization errors of quantum reservoirs.
method Using Rademacher complexity, specific bounds are derived for quantum reservoir classes.
result Risk bounds converge with increasing training samples and qubits.
We propose an approach to learning with graph-structured data in the problem domain of graph classification. In particular, we present a novel type of readout operation to aggregate node features into a graph-level representation. To this end, we leverage persistent homology computed via a real-valued, learnable, filte…
Sharp theory of neural network scaling laws for hierarchical targets.
problem Learning hierarchical multi-index models in neural networks.
method Sharp information-theoretic scaling laws derived for two-layer neural networks.
result Optimal rates achieved by a simple spectral estimator.
MathNet uses wavelets for graph representation and learning.
problem Graph Neural Networks (GNNs) for graph classification and regression.
method Multiresolution Haar-like wavelets, graph convolution, and pooling.
result MathNet achieves notable accuracy gains on graph classification and regression tasks.
GNNs improve semi-supervised node regression, but why? We explain.
problem Understanding when and why GNNs succeed in semi-supervised node regression.
method Aggregate-and-readout model encompassing message passing architectures, least-squares estimation over GNNs with linear graph convolutions and a deep ReLU readout.
result Sharp non-asymptotic risk bound separating approximation, stochastic, and optimization errors.
Random feature models approximate functions in Banach spaces efficiently.
problem Approximating functions in Banach spaces efficiently.
method Randomly initialized feature maps and linear readout training.
result Universal approximation in Bochner spaces for Banach space-valued models.
We analyze generalization in deep learning models using random matrix theory.
problem Understanding the generalization error in deep learning models with random feature representations.
method Applying Random Matrix Theory to derive asymptotic generalization error formulas for various architectures.
result Linear ESNs are equivalent to ridge regression with exponentially time-weighted input covariance, revealing an inductive bias towards recent inputs.
New measures link neural representation geometry to decoding ability.
problem Understanding how neural representations relate to decoding ability.
method Showed that popular similarity measures can be interpreted from a decoding perspective.
result Proved that measures like CKA and CCA quantify alignment between optimal linear readouts.
New framework assesses neural sensitivity to small perturbations.
problem Comparing neural representations' sensitivity to small changes.
method Local decodable information, Fisher information, and projected pullback/Fisher metric.
result Reveals differences in neural sensitivity not captured by activation alignment.
The paper analyzes how data augmentation affects the test error in regression models.
problem Understanding the impact of data augmentation on the test error in regression models.
method Characterizes the test error in terms of population quantities and augmentation statistics.
result Provides a tight characterization of the test error in mean squared error.
Universal approximation theorem for differentiable maps on infinite-dimensional manifolds
problem Approximation of differentiable maps on infinite-dimensional manifolds
method Weighted universal approximation theorem
result Universal approximation theorem for differentiable maps
The study uses response theory to understand RNNs processing input signals.
problem Understanding how RNNs process sequential data.
method Deriving a Volterra series representation for SRNNs output using response theory from nonequilibrium statistical mechanics.
result SRNNs can be viewed as kernel machines operating on a reproducing kernel Hilbert space associated with the response feature.
MV-GNN improves molecular property prediction by integrating atom and bond information.
problem Accurately predicting molecular properties using graph neural networks.
method Multi-View Graph Neural Network (MV-GNN) architecture with shared self-attentive readout and cross-dependent message passing.
result MV-GNN achieves superior performance on molecular property prediction benchmarks.
Generalizes neural network approximation to infinite-dimensional manifolds and derivatives.
problem Approximating differentiable maps on infinite-dimensional manifolds.
method Proves a weighted Nachbin theorem to establish universal approximation for differentiable maps, including derivatives.
result Linear functions of the signature can approximate path space functionals including their derivatives.
We provide a fast approximation to eNTKs for neural networks.
problem Efficiently computing eNTKs for large networks.
method Developed and proved the 'sum of logits' approximation.
result The 'sum of logits' approximation converges to eNTKs at initialization.
DeepRSCN models nonlinear systems using stochastic configurations.
problem Modeling nonlinear dynamic systems efficiently.
method Incrementally constructed deep reservoir computing framework with random parameters and online weight updates.
result DeepRSCN outperforms single-layer networks in efficiency, learning, and generalization.
Study shows depth improves generalization in deep learning models.
problem Understanding why and when depth improves generalization in deep learning.
method Implementation-agnostic state-transition model to analyze depth and generalization.
result Identifies geometric and semigroup mechanisms that keep entropy contribution saturated or polynomial, clarifying depth's statistical advantage.
We derive scaling laws for optimizing neural networks in hardware.
problem Optimizing the large parameter space of neural networks in hardware.
method Analytical derivation of scaling laws for Coordinate Descent optimization.
result Convergence is exponential and scales linearly with the number of neurons.
RFMs transition from linear to nonlinear under specific input-label correlation.
problem Understanding the transition from linear to nonlinear behavior in RFMs.
method Analyzing RFMs under spiked covariance designs, characterizing the interaction between anisotropy and input-label correlation.
result The RFM generalization error is governed by the strength of input-label correlation, leading to a clear nonlinear advantage above a specific boundary.
The study analyzes how neural reward models learn features for policy optimization in a Gaussian single-index model.
problem Reward modeling in policy optimization and its impact on downstream value.
method Two-stage neural reward model: first learns hidden direction, then fits readout layer.
result For any feature-learning temperature above a dimension-free threshold, a constant fraction of neurons recover the hidden direction.
Unified theory for deep and recurrent networks using Gaussian processes.
problem Understanding capabilities and limitations of different network architectures.
method Unified derivation of mean-field theory from statistical physics of disordered systems.
result Gaussian processes yield identical Gaussian kernels for both architectures at a single time point or layer.
We extend manifold capacity to nonlinear neural representations with contextual information.
problem Efficient processing of information through neural representations.
method Theoretical framework leveraging latent directions in input space related to contextual information.
result Derivation of an exact formula for context-dependent manifold capacity.
Machine learning (ML) classifiers always benefit from more informative input features. We seek to auto-generate stronger feature sets in order to address the difficulty that ML methods often experience given limited training data. A wide range of biological neural nets (BNNs) excel at fast learning, implying that they …
Neuroscientists classify neurons into different types that perform similar computations at different locations in the visual field. Traditional methods for neural system identification do not capitalize on this separation of 'what' and 'where'. Learning deep convolutional feature spaces that are shared among many neuro…
Functional input neural networks approximate continuous functions on weighted spaces.
problem Approximating continuous functions on infinite-dimensional weighted spaces.
method Additive family mapping, non-linear activation, linear readouts, Stone-Weierstrass theorem.
result Global universal approximation of continuous functions on weighted spaces.
Simplified image clustering achieves competitive results without text-based embeddings.
problem Complexity and resource requirements of state-of-the-art clustering methods.
method SCP: trains a small cluster head using pre-trained vision model features and positive data pairs.
result SCP achieves highly competitive performance on various benchmark datasets.
New insights into continual learning with task similarity.
problem Challenges in learning similar tasks without interference.
method Linear teacher-student model with latent structure.
result High input feature similarity with low readout similarity is catastrophic.
Framework combines random features with CDEs for efficient time-series learning.
problem Efficient training of time-series models with strong inductive bias.
method Random Fourier CDEs and Random Rough DEs using continuous-time reservoirs and log-ODE discretization.
result Unified perspective on random-feature reservoirs and path-signature theory.