Concentration of bounded sparse chaoses and sparse Khatri--Rao embeddings

arXiv:2609.28875v2 Announce Type: replace-cross Abstract: We prove moment and mixed-tail inequalities for decoupled homogeneous chaoses of arbitrary fixed order generated by independent, centered, bounded random variables with coordinate-dependent second-moment bounds. The estimates hold for arbitrary real rectangular coefficient tensors and are expressed through weighted slice and partition norms. A Bennett-type logarithmic correction sharpens the contribution of the largest coefficient to these bounds. The proof combines a comparison with sums of sparse exponential variables with an elementary replication argument. We also extend the results to diagonal-free non-decoupled chaoses and sparse sub-exponential inputs. As an application, we establish conditions under which sparse Khatri--Rao random projections approximately preserve norms and inner products for finite collections of tensors. For a sparse random relation containing a planted dense block, we prove matching upper and lower bounds on the required projection dimension, demonstrating the sharpness of the logarithmic correction.