Algorithms with Polynomially-Improved Approximation Factors for the $2 \rightarrow q$ Norm, and Applications
arXiv:2605.25303v1 Announce Type: cross Abstract: The $2 \rightarrow q$ norm of a matrix $X \in \mathbb{R}^{n \times d}$ is defined as $\lVert X \rVert_{2 \rightarrow q} = \sup_{\lVert v \rVert_2 = 1} \lVert Xv \rVert_q$. We give polynomial-time multiplicative approximation algorithms for this norm when $q > 2$ (i.e. in the hypercontractive setting). This problem either directly captures or is closely related to long-standing open problems in combinatorial optimization and hardness of approximation (e.g. Small Set Expansion), quantum information (e.g. Best Separable State), and algorithmic sta
This research provides new theoretical advancements in algorithm design, specifically for matrix norms, which are foundational to many contemporary AI and optimization problems.
Improved approximation algorithms for fundamental mathematical problems can significantly enhance the efficiency and capability of AI and machine learning systems.
This research provides more efficient computational methods for complex mathematical operations, potentially speeding up development and deployment of advanced AI applications.
- · AI/ML research community
- · High-performance computing
- · Quantum computing research
- · Combinatorial optimization
- · Inefficient algorithms
- · Computational bottlenecks
The new algorithms could lead to faster and more accurate solutions for problems like Small Set Expansion and Best Separable State.
Enhanced computational efficiency in these areas might accelerate progress in AI development, potentially leading to more complex and capable models.
Long-term, this could contribute to the development of novel AI agents or applications that were previously computationally intractable, impacting various industries.
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Read at arXiv cs.LG