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Found 14 papers, 4 papers with code
Learning smooth functions in high dimensions: from sparse polynomials to deep neural networks
For the latter, there is currently a significant gap between the approximation theory of DNNs and the practical performance of deep learning.
Neural Rank Collapse: Weight Decay and Small Within-Class Variability Yield Low-Rank Bias
Recent work in deep learning has shown strong empirical and theoretical evidence of an implicit low-rank bias: weight matrices in deep networks tend to be approximately low-rank and removing relatively small singular values during training or from available trained models may significantly reduce model size while maintaining or even improving model performance.
A practical existence theorem for reduced order models based on convolutional autoencoders
In recent years, deep learning has gained increasing popularity in the fields of Partial Differential Equations (PDEs) and Reduced Order Modeling (ROM), providing domain practitioners with new powerful data-driven techniques such as Physics-Informed Neural Networks (PINNs), Neural Operators, Deep Operator Networks (DeepONets) and Deep-Learning based ROMs (DL-ROMs).
Model-adapted Fourier sampling for generative compressed sensing
We study generative compressed sensing when the measurement matrix is randomly subsampled from a unitary matrix (with the DFT as an important special case).
Generalization Limits of Graph Neural Networks in Identity Effects Learning
They are usually based on a message-passing mechanism and have gained increasing popularity for their intuitive formulation, which is closely linked to the Weisfeiler-Lehman (WL) test for graph isomorphism to which they have been proven equivalent in terms of expressive power.
Monte Carlo is a good sampling strategy for polynomial approximation in high dimensions
We show that there is a least-squares approximation based on $m$ Monte Carlo samples whose error decays algebraically fast in $m/\log(m)$, with a rate that is the same as that of the best $n$-term polynomial approximation.
A coherence parameter characterizing generative compressed sensing with Fourier measurements
In Bora et al. (2017), a mathematical framework was developed for compressed sensing guarantees in the setting where the measurement matrix is Gaussian and the signal structure is the range of a generative neural network (GNN).
Compressive Fourier collocation methods for high-dimensional diffusion equations with periodic boundary conditions
We also present numerical experiments that illustrate the accuracy and stability of the method for the approximation of sparse and compressible solutions.
On efficient algorithms for computing near-best polynomial approximations to high-dimensional, Hilbert-valued functions from limited samples
On the one hand, there is a well-developed theory of best $s$-term polynomial approximation, which asserts exponential or algebraic rates of convergence for holomorphic functions.
Deep Neural Networks Are Effective At Learning High-Dimensional Hilbert-Valued Functions From Limited Data
Such problems are challenging: 1) pointwise samples are expensive to acquire, 2) the function domain is high dimensional, and 3) the range lies in a Hilbert space.
Generalizing Outside the Training Set: When Can Neural Networks Learn Identity Effects?
Finally, we demonstrate our theory with computational experiments in which we explore the effect of different input encodings on the ability of algorithms to generalize to novel inputs.
Do log factors matter? On optimal wavelet approximation and the foundations of compressed sensing
A signature result in compressed sensing is that Gaussian random sampling achieves stable and robust recovery of sparse vectors under optimal conditions on the number of measurements.
Image Reconstruction
Information Theory
Information Theory
A compressive spectral collocation method for the diffusion equation under the restricted isometry property
The approach is based on a spectral Sturm-Liouville approximation of the solution and on the collocation of the PDE in strong form at random points, by taking advantage of the compressive sensing principle.
Numerical Analysis
On oracle-type local recovery guarantees in compressed sensing
We present improved sampling complexity bounds for stable and robust sparse recovery in compressed sensing.
Information Theory Information Theory
