12 research outputs found
2-cancellative hypergraphs and codes
A family of sets F (and the corresponding family of 0-1 vectors) is called
t-cancellative if for all distict t+2 members A_1,... A_t and B,C from F the
union of A_1,..., A_t and B differs from the union of A_1, ..., A_t and C. Let
c(n,t) be the size of the largest t-cancellative family on n elements, and let
c_k(n,t) denote the largest k-uniform family. We significantly improve the
previous upper bounds, e.g., we show c(n,2) n_0). Using an
algebraic construction we show that the order of magnitude of c_{2k}(n,2) is
n^k for each k (when n goes to infinity).Comment: 20 page
A Better Bound for Locally Thin Set Families
AbstractA family of subsets of an n-set is 4-locally thin if for every quadruple of its members the ground set has at least one element contained in exactly 1 of them. We show that such a family has at most 20.4561n members. This improves on our previous results with Noga Alon. The new proof is based on a more careful analysis of the self-similarity of the graph associated with such set families by the graph entropy bounding technique
Support Recovery in Universal One-Bit Compressed Sensing
One-bit compressed sensing (1bCS) is an extreme-quantized signal acquisition
method that has been widely studied in the past decade. In 1bCS, linear samples
of a high dimensional signal are quantized to only one bit per sample (sign of
the measurement). Assuming the original signal vector to be sparse, existing
results either aim to find the support of the vector, or approximate the signal
within an -ball. The focus of this paper is support recovery, which
often also computationally facilitates approximate signal recovery. A universal
measurement matrix for 1bCS refers to one set of measurements that work for all
sparse signals. With universality, it is known that 1bCS
measurements are necessary and sufficient for support recovery (where
denotes the sparsity). In this work, we show that it is possible to universally
recover the support with a small number of false positives with
measurements. If the dynamic range of the signal vector is
known, then with a different technique, this result can be improved to only
measurements. Further results on support recovery are also
provided.Comment: 15 page