Abstract A local ring R is regular if and only if every finitely generated R -module has finite projective dimension. Moreover, the residue field k is a test module: R is regular if and only if k has finite projective dimension. This characterization can be extended to the bounded derived category $\mathsf {D}^{\mathsf f}(R)$ , which contains only small objects if and only if R is regular. Recent results of Pollitz, completing work initiated by Dwyer–Greenlees–Iyengar, yield an analogous characterization for complete intersections: R is a complete intersection if and only if every object in $\mathsf {D}^{\mathsf f}(R)$ is proxy small. In this paper, we study a return to the world of R -modules, and search for finitely generated R -modules that are not proxy small whenever R is not a complete intersection. We give an algorithm to construct such modules in certain settings, including over equipresented rings and Stanley–Reisner rings.
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A FUNCTION SPACE VIEW OF BOUNDED NORM INFINITE WIDTH RELU NETS: THE MULTIVARIATE CASE
We give a tight characterization of the (vectorized Euclidean) norm of weights
required to realize a function f : R^d -> R as a single hidden-layer ReLU network
with an unbounded number of units (infinite width), extending the univariate characterization of Savarese et al. (2019) to the multivariate case.
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- NSF-PAR ID:
- 10183678
- Date Published:
- Journal Name:
- International Conference on Learning Representations
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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