We study the properties of points in [0,1]d generated by applying Hilbert's space filling curve to uniformly distributed points in [0, 1]. For deterministic sampling we obtain a discrepancy of O(n−1/d) for d⩾2. For random stratified sampling, and scrambled van der Corput points, we derive a mean-squared error of O(n−1−2/d) for integration of Lipschitz continuous integrands, when d⩾3. These rates are the same as those obtained by sampling on d-dimensional grids and they show a deterioration with increasing d. The rate for Lipschitz functions is, however, the best possible at that level of smoothness and is better than plain independent and identically distributed sampling. Unlike grids, space filling curve sampling provides points at any desired sample size, and the van der Corput version is extensible in n. We also introduce a class of piecewise Lipschitz functions whose discontinuities are in rectifiable sets described via Minkowski content. Although these functions may have infinite variation in the sense of Hardy and Krause, they can be integrated with a mean-squared error of O(n−1−1/d). It was previously known only that the rate was o(n−1). Other space filling curves, such as those due to Sierpinski and Peano, also attain these rates, whereas upper bounds for the Lebesgue curve are somewhat worse, as if the dimension were log2(3) times as high.
This content will become publicly available on July 1, 2024
On The Stability of Approximate Message Passing with Independent Measurement Ensembles
Approximate message passing (AMP) is a scalable, iterative approach to signal recovery. For structured random measurement ensembles, including independent and identically distributed (i.i.d.) Gaussian and rotationally-invariant matrices, the performance of AMP can be characterized by a scalar recursion called state evolution (SE). The pseudo-Lipschitz (polynomial) smoothness is conventionally assumed. In this work, we extend the SE for AMP to a new class of measurement matrices with independent (not necessarily identically distributed) entries. We also extend it to a general class of functions, called controlled functions which are not constrained by the polynomial smoothness; unlike the pseudo-Lipschitz function that has polynomial smoothness, the controlled function grows exponentially. The lack of structure in the assumed measurement ensembles is addressed by leveraging Lindeberg-Feller. The lack of smoothness of the assumed controlled function is addressed by a proposed conditioning technique leveraging the empirical statistics of the AMP instances. The resultants grant the use of the SE to a broader class of measurement ensembles and a new class of functions.
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- NSF-PAR ID:
- 10434347
- Date Published:
- Journal Name:
- IEEE International Symposium on Information Theory
- Page Range / eLocation ID:
- 1 - 11
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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