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Explore the computational-statistical gap in the Johnson-Lindenstrauss embedding through this computer science seminar examining adaptive robustness properties of hypergrid projections. Learn how the classic 1984 Johnson-Lindenstrauss theorem demonstrates that random projections preserve distances for point sets of exponential size, but discover the challenges that arise when dealing with larger sets where points can contract arbitrarily. Investigate the specific case of n-dimensional hypergrids of integral points with bounded infinity-norm, where contracting pairs are abundant yet computationally difficult to find within certain parameter ranges. Understand the cryptographic implications of these findings, particularly how rounded Johnson-Lindenstrauss embeddings function as robust property-preserving hash functions that compress data while maintaining distance preservation guarantees against computationally bounded adversaries. Gain insights into the intersection of computational complexity, statistical analysis, and cryptographic applications in high-dimensional data processing, based on collaborative research with leading experts in theoretical computer science.
