Lower Lipschitz bounds for phase retrieval from locally supported measurements

Lower bounds for adaptive locally decodable codes

An error-correcting code is said to be locally decodable if a randomized algorithm can recover any single bit of a message by reading only a small number of symbols of a possibly corrupted encoding of the message. Katz and Trevisan (On the efficiency of local decoding procedures for error correcting codes, STOC 2000, 80–86) showed that any such code C : {0, 1}n → Σ with a decoding algorithm tha...

Lower Bounds for Locally Highly Connected Graphs

We propose a conjecture regarding the lower bound for the number of edges in locally k-connected graphs and we prove it for k = 2. In particular, we show that every connected locally 2-connected graph is M3-rigid. For the special case of surface triangulations, this fact was known before using topological methods. We generalize this result to all locally 2-connected graphs and give a purely com...

Some Lower Bounds for Estrada Index

Improved Lower Bounds for Locally Decodable Codes and Private Information Retrieval

We prove new lower bounds for locally decodable codes and private information retrieval. We show that a 2-query LDC encoding n-bit strings over an l-bit alphabet, where the decoder only uses b bits of each queried position of the codeword, needs code length

Phase Retrieval using Lipschitz Continuous Maps

In this note we prove that reconstruction from magnitudes of frame coefficients (the so called ”phase retrieval problem”) can be performed using Lipschitz continuous maps. Specifically we show that when the nonlinear analysis map α : H → R is injective, with (α(x))k = |〈x, fk〉| , where {f1, . . . , fm} is a frame for the Hilbert space H, then there exists a left inverse map ω : R → H that is Li...