mlsanityData to 7 Oct 2026

CS4ML: A general framework for active learning with arbitrary data based on Christoffel functions

arXiv 2306.00945 ↗First posted 1 Jun 2023cs.LG cs.NA math.NAcited by 11

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Abstract (arXiv)

We introduce a general framework for active learning in regression problems. Our framework extends the standard setup by allowing for general types of data, rather than merely pointwise samples of the target function. This generalization covers many cases of practical interest, such as data acquired in transform domains (e.g., Fourier data), vector-valued data (e.g., gradient-augmented data), data acquired along continuous curves, and, multimodal data (i.e., combinations of different types of measurements). Our framework considers random sampling according to a finite number of sampling measures and arbitrary nonlinear approximation spaces (model classes). We introduce the concept of generalized Christoffel functions and show how these can be used to optimize the sampling measures. We prove that this leads to near-optimal sample complexity in various important cases. This paper focuses on applications in scientific computing, where active learning is often desirable, since it is usually expensive to generate data. We demonstrate the efficacy of our framework for gradient-augmented learning with polynomials, Magnetic Resonance Imaging (MRI) using generative models and adaptive sampling for solving PDEs using Physics-Informed Neural Networks (PINNs).

Key facts. Cited by 11 papers in the map (data through 7 Oct 2026). Builds on 51 of its 123 references that are in the map. First posted on arXiv 1 Jun 2023.

Builds on

51 of 123 references.

References · 51

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38 more
  1. 2021Robust Compressed Sensing MRI with Deep Generative Priorscited by 284
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  31. 2019Nystr"om landmark sampling and regularized Christoffel functionscited by 8
  32. 2019Structure Preserving Compressive Sensing MRI Reconstruction using Generative Adversarial Networkscited by 8
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  34. 2022CAS4DL: Christoffel Adaptive Sampling for function approximation via Deep Learningcited by 6
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50 most cited of 51 All 51 (JSON)

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 2; 2024: 3; 2025: 2; 2026: 4.

11 papers.

  1. 2024Provably Accurate Shapley Value Estimation via Leverage Score Samplingcited by 23
  2. 2024Agnostic Active Learning of Single Index Models with Linear Sample Complexitycited by 11
  3. 2023Model-adapted Fourier sampling for generative compressed sensingcited by 6
  4. 2024Optimal sampling for least-squares approximationcited by 6
  5. 2023A unified framework for learning with nonlinear model classes from arbitrary linear samplescited by 5
  6. 2025Denoising guarantees for optimized sampling schemes in compressed sensingcited by 2
  7. 2025How many measurements are enough? Bayesian recovery in inverse problems with general distributionscited by 2
  8. 2026Partially deterministic sampling for compressed sensing with denoising guaranteescited by 0
  9. 2026Active Learning for Conditional Generative Compressed Sensingcited by 0
  10. 2026Christoffel-DPS: Optimal sensor placement in diffusion posterior sampling for arbitrary distributionscited by 0
  11. 2026How Low Can You Go? Active Learning for Sparse Model Discovery in the Ultra-Low-Data Limitcited by 0