Recent studies of under-determined linear systems of equations with sparse solutions showed a great practical and theoretical efficiency of a particular technique called $\ell_1$-optimization. Seminal works \cite{CRT,DOnoho06CS} rigorously confirmed it for the first time. Namely, \cite{CRT,DOnoho06CS} showed, in a statistical context, that $\ell_1$ technique can recover sparse solutions of under-determined systems even when the sparsity is linearly proportional to the dimension of the system. A followup \cite{DonohoPol} then precisely characterized such a linearity through a geometric approach and a series of work\cite{StojnicCSetam09,StojnicUpper10,StojnicEquiv10} reaffirmed statements of \cite{DonohoPol} through a purely probabilistic approach. A theoretically interesting alternative to $\ell_1$ is a more general version called $\ell_q$ (with an essentially arbitrary $q$). While $\ell_1$ is typically considered as a first available convex relaxation of sparsity norm $\ell_0$, $\ell_q,0\leq q\leq 1$, albeit non-convex, should technically be a tighter relaxation of $\ell_0$. Even though developing polynomial (or close to be polynomial) algorithms for non-convex problems is still in its initial phases one may wonder what would be the limits of an $\ell_q,0\leq q\leq 1$, relaxation even if at some point one can develop algorithms that could handle its non-convexity. A collection of answers to this and a few realted questions is precisely what we present in this paper.
In this paper we consider random linear under-determined systems with block-sparse solutions. A standard subvariant of such systems, namely, precisely the same type of systems without additional block structuring requirement, gained a lot of popularity over the last decade. This is of course in first place due to the success in mathematical characterization of an $\ell_1$ optimization technique typically used for solving such systems, initially achieved in \cite{CRT,DOnoho06CS} and later on perfected in \cite{DonohoPol,DonohoUnsigned,StojnicCSetam09,StojnicUpper10}. The success that we achieve…
We analyze a sequential decision making model in which decision makers (or, players) take their decisions based on their own private information as well as the actions of previous decision makers. Such decision making processes often lead to what is known as the \emph{information cascade} or \emph{herding} phenomenon. Specifically, a cascade develops when it seems rational for some players to abandon their own private information and imitate the actions of earlier players. The risk, however, is that if the initial decisions were wrong, then the whole cascade will be wrong. Nonetheless, informa…
The basic goal of threshold group testing is to identify up to $d$ defective items among a population of $n$ items, where $d$ is usually much smaller than $n$. The outcome of a test on a subset of items is positive if the subset has at least $u$ defective items, negative if it has up to $\ell$ defective items, where $0\leq\ell<u$, and arbitrary otherwise. This is called threshold group testing. The parameter $g=u-\ell-1$ is called \textit{the gap}. In this paper, we focus on the case $g>0$, i.e., threshold group testing with a gap. Note that the results presented here are also applicable to th…
In this work we consider the list-decodability and list-recoverability of arbitrary $q$-ary codes, for all integer values of $q\geq 2$. A code is called $(p,L)_q$-list-decodable if every radius $pn$ Hamming ball contains less than $L$ codewords; $(p,\ell,L)_q$-list-recoverability is a generalization where we place radius $pn$ Hamming balls on every point of a combinatorial rectangle with side length $\ell$ and again stipulate that there be less than $L$ codewords. Our main contribution is to precisely calculate the maximum value of $p$ for which there exist infinite families of positive rate $…
