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An \ell _p oblivious subspace embedding is a distribution over r \times n matrices \Pi such that for any fixed n \times d matrix A , \[ \Pr _{\Pi }[\textrm {for all }x, \ \Vert Ax\Vert _p \le \Vert \Pi Ax\Vert _p \le \kappa \Vert Ax\Vert _p] \ge 9/10,\] where r is the dimension of the embedding, \kappa is the distortion of the embedding, and for an n -dimensional vector y , \Vert y\Vert _p = (\sum _{i=1}^n |y_i|^p)^{1/p} is the \ell _p -norm. Another important property is the sparsity of \Pi , that is, the maximum number of non-zero entries per column, as this determines the running time of computing \Pi A . While for p = 2 there are nearly optimal tradeoffs in terms of the dimension, distortion, and sparsity, for the important case of 1 \le p \lt 2 , much less was known. In this article, we obtain nearly optimal tradeoffs for \ell _1 oblivious subspace embeddings, as well as new tradeoffs for 1 \lt p \lt 2 . Our main results are as follows: (1) We show for every 1 \le p \lt 2 , any oblivious subspace embedding with dimension r has distortion \[ \kappa = \Omega \left(\frac{1}{\left(\frac{1}{d}\right)^{1 / p} \log ^{2 / p}r + \left(\frac{r}{n}\right)^{1 / p - 1 / 2}}\right).\] When r = {\operatorname{poly}}(d) \ll n in applications, this gives a \kappa = \Omega (d^{1/p}\log ^{-2/p} d) lower bound, and shows the oblivious subspace embedding of Sohler and Woodruff (STOC, 2011) for p = 1 is optimal up to {\operatorname{poly}}(\log (d)) factors. (2) We give sparse oblivious subspace embeddings for every 1 \le p \lt 2 . Importantly, for p = 1 , we achieve r = O(d \log d) , \kappa = O(d \log d) and s = O(\log d) non-zero entries per column. The best previous construction with s \le {\operatorname{poly}}(\log d) is due to Woodruff and Zhang (COLT, 2013), giving \kappa = \Omega (d^2 {\operatorname{poly}}(\log d)) or \kappa = \Omega (d^{3/2} \sqrt {\log n} \cdot {\operatorname{poly}}(\log d)) and r \ge d \cdot {\operatorname{poly}}(\log d) ; in contrast our r = O(d \log d) and \kappa = O(d \log d) are optimal up to {\operatorname{poly}}(\log (d)) factors even for dense matrices. We also give (1) \ell _p oblivious subspace embeddings with an expected 1+\varepsilon number of non-zero entries per column for arbitrarily small \varepsilon \gt 0 , and (2) the first oblivious subspace embeddings for 1 \le p \lt 2 with O(1) -distortion and dimension independent of n . Oblivious subspace embeddings are crucial for distributed and streaming environments, as well as entrywise \ell _p low-rank approximation. Our results give improved algorithms for these applications.
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This content will become publicly available on December 31, 2026
Speeding up Langevin Dynamics by Mixing
We study an overdamped Langevin equation on the $$d$$-dimensional torus with stationary distribution proportional to $$p = e^{-U / \kappa}$$. When $$U$$ has multiple wells the mixing time of the associated process is exponentially large (of size $$e^{O(1/\kappa)}$$). We add a drift to the Langevin dynamics (without changing the stationary distribution) and obtain quantitative estimates on the mixing time. Our main result shows that the mixing time of the Langevin system can be made arbitrarily small by adding a drift that is sufficiently mixing. We provide one construction of a mixing drift, and our main result can be applied by using this drift with a large amplitude. For numerical purposes, it is useful to keep the size of the imposed drift small, and we show that the smallest allowable amplitude ensures that the mixing time is $$O( d/\kappa^2)$$, which is an order of magnitude smaller than $$e^{O(1/\kappa)}$$.
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- PAR ID:
- 10656767
- Publisher / Repository:
- SIAM
- Date Published:
- Journal Name:
- Multiscale Modeling & Simulation
- Volume:
- 23
- Issue:
- 4
- ISSN:
- 1540-3459
- Page Range / eLocation ID:
- 1696 to 1743
- Subject(s) / Keyword(s):
- mixing Langevin Monte Carlo enhanced dissipation
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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