Related papers: Average-case complexity without the black swans
Ill-posed linear inverse problems appear frequently in various signal processing applications. It can be very useful to have theoretical characterizations that quantify the level of ill-posedness for a given inverse problem and the degree…
Average-case analysis computes the complexity of an algorithm averaged over all possible inputs. Compared to worst-case analysis, it is more representative of the typical behavior of an algorithm, but remains largely unexplored in…
We present a new framework for designing worst-case to average-case reductions. For a large class of problems, it provides an explicit transformation of algorithms running in time $T$ that are only correct on a small (subconstant) fraction…
This paper develops several average-case reduction techniques to show new hardness results for three central high-dimensional statistics problems, implying a statistical-computational gap induced by robustness, a detection-recovery gap and…
Smoothed analysis of complexity bounds and condition numbers has been done, so far, on a case by case basis. In this paper we consider a reasonably large class of condition numbers for problems over the complex numbers and we obtain…
Worst-case to average-case reductions are a cornerstone of complexity theory, providing a bridge between worst-case hardness and average-case computational difficulty. While recent works have demonstrated such reductions for fundamental…
Systems of wave equations may fail to be globally well posed, even for small initial data. Attempts to classify systems into well and ill-posed categories work by identifying structural properties of the equations that can work as…
Real-world data in health, economics, and environmental sciences are often collected across heterogeneous domains (such as hospitals, regions, or time periods). In such settings, distributional shifts can make standard PCA unreliable, in…
We provide general formulation of weak identification in semiparametric models and an efficiency concept. Weak identification occurs when a parameter is weakly regular, i.e., when it is locally homogeneous of degree zero. When this happens,…
Based on a weak convergence argument, we provide a necessary and sufficient condition that guarantees that a nonnegative local martingale is indeed a martingale. Typically, conditions of this sort are expressed in terms of integrability…
In the worst-case analysis of algorithms, the overall performance of an algorithm is summarized by its worst performance on any input. This approach has countless success stories, but there are also important computational problems --- like…
We refute the widely held belief that the quantum weak value necessarily pertains to weak measurements. To accomplish this, we use the transverse position of a beam as the detector for the conditioned von Neumann measurement of a system…
We study the problem of designing worst-case to average-case reductions for quantum algorithms. For all linear problems, we provide an explicit and efficient transformation of quantum algorithms that are only correct on a small (even…
One of the primary goals of the mathematical analysis of algorithms is to provide guidance about which algorithm is the "best" for solving a given computational problem. Worst-case analysis summarizes the performance profile of an algorithm…
A new notion of typicality for arbitrary probability measures on standard Borel spaces is proposed, which encompasses the classical notions of weak and strong typicality as special cases. Useful lemmas about strong typical sets, including…
A weak measurement performed on a pre- and post-selected quantum system can result in an average value that lies outside of the observable's spectrum. This effect, usually referred to as an "anomalous weak value", is generally believed to…
We introduce a weakly supervised approach for inferring the property of abstractness of words and expressions in the complete absence of labeled data. Exploiting only minimal linguistic clues and the contextual usage of a concept as…
Classically, a control loop is designed to be executed strictly periodically. This is, however, difficult to achieve in many scenarios, for example, when overload or packet loss cannot be entirely avoided. Here, weakly-hard real-time…
The basic problem in the PAC model of computational learning theory is to determine which hypothesis classes are efficiently learnable. There is presently a dearth of results showing hardness of learning problems. Moreover, the existing…
The average result of a weak measurement of some observable $A$ can, under post-selection of the measured quantum system, exceed the largest eigenvalue of $A$. The nature of weak measurements, as well as the presence of post-selection and…