Search Results for author:
Found 18 papers, 8 papers with code
Optimal sampling for least-squares approximation
We introduce the Christoffel function as a key quantity in the analysis of (weighted) least-squares approximation from random samples, then show how it can be used to construct sampling strategies that possess near-optimal sample complexity: namely, the number of samples scales log-linearly in $n$, the dimension of the approximation space.
On the consistent reasoning paradox of intelligence and optimal trust in AI: The power of 'I don't know'
The paradox is that there exists a non-consistently reasoning AI (which therefore cannot be on the level of human intelligence) that will be correct on the same set of problems.
Optimal deep learning of holomorphic operators between Banach spaces
We first identify a family of DNNs such that the resulting Deep Learning (DL) procedure achieves optimal generalization bounds for such operators.
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.
A unified framework for learning with nonlinear model classes from arbitrary linear samples
In summary, our work not only introduces a unified way to study learning unknown objects from general types of data, but also establishes a series of general theoretical guarantees which consolidate and improve various known results.
CS4ML: A general framework for active learning with arbitrary data based on Christoffel functions
Our framework extends the standard setup by allowing for general types of data, rather than merely pointwise samples of the target function.
Restarts subject to approximate sharpness: A parameter-free and optimal scheme for first-order methods
However, sharpness involves problem-specific constants that are typically unknown, and restart schemes typically reduce convergence rates.
CAS4DL: Christoffel Adaptive Sampling for function approximation via Deep Learning
In this work, we propose an adaptive sampling strategy, CAS4DL (Christoffel Adaptive Sampling for Deep Learning) to increase the sample efficiency of DL for multivariate function approximation.
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.
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.
NESTANets: Stable, accurate and efficient neural networks for analysis-sparse inverse problems
With the advent of deep learning, deep neural networks have significant potential to outperform existing state-of-the-art, model-based methods for solving inverse problems.
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.
The gap between theory and practice in function approximation with deep neural networks
Our main conclusion from these experiments is that there is a crucial gap between the approximation theory of DNNs and their practical performance, with trained DNNs performing relatively poorly on functions for which there are strong approximation results (e. g. smooth functions), yet performing well in comparison to best-in-class methods for other functions.
The troublesome kernel -- On hallucinations, no free lunches and the accuracy-stability trade-off in inverse problems
In inverse problems in imaging, the focus of this paper, there is increasing empirical evidence that methods may suffer from hallucinations, i. e., false, but realistic-looking artifacts; instability, i. e., sensitivity to perturbations in the data; and unpredictable generalization, i. e., excellent performance on some images, but significant deterioration on others.
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
Convolutional Analysis Operator Learning: Dependence on Training Data
Convolutional analysis operator learning (CAOL) enables the unsupervised training of (hierarchical) convolutional sparsifying operators or autoencoders from large datasets.
On instabilities of deep learning in image reconstruction - Does AI come at a cost?
Deep learning, due to its unprecedented success in tasks such as image classification, has emerged as a new tool in image reconstruction with potential to change the field.
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
