Related papers: Counterexamples to the Low-Degree Conjecture
Probabilistic argumentation allows reasoning about argumentation problems in a way that is well-founded by probability theory. However, in practice, this approach can be severely limited by the fact that probabilities are defined by adding…
There is evidence that the numbers in probabilistic inference don't really matter. This paper considers the idea that we can make a probabilistic model simpler by making fewer distinctions. Unfortunately, the level of a Bayesian network…
High-dimensional planted problems, such as finding a hidden dense subgraph within a random graph, often exhibit a gap between statistical and computational feasibility. While recovering the hidden structure may be statistically possible, it…
The statistical analysis of discrete data has been the subject of extensive statistical research dating back to the work of Pearson. In this survey we review some recently developed methods for testing hypotheses about high-dimensional…
We consider the task of detecting a hidden bipartite subgraph in a given random graph. This is formulated as a hypothesis testing problem, under the null hypothesis, the graph is a realization of an Erd\H{o}s-R\'{e}nyi random graph over $n$…
We present an algorithm for strongly refuting smoothed instances of all Boolean CSPs. The smoothed model is a hybrid between worst and average-case input models, where the input is an arbitrary instance of the CSP with only the negation…
The main purpose of this paper is to study the NP-complete subset-sum problem, not in the usual context of time-complexity-based classification of the algorithms (exponential/polynomial), but through a new kind of algorithmic classification…
We study hypothesis testing under communication constraints, where each sample is quantized before being revealed to a statistician. Without communication constraints, it is well known that the sample complexity of simple binary hypothesis…
Developing new ways to estimate probabilities can be valuable for science, statistics, and engineering. By considering the information content of different output patterns, recent work invoking algorithmic information theory has shown that…
We give new evidence that quantum computers -- moreover, rudimentary quantum computers built entirely out of linear-optical elements -- cannot be efficiently simulated by classical computers. In particular, we define a model of computation…
Statistical hypothesis testing serves as statistical evidence for scientific innovation. However, if the reported results are intentionally biased, hypothesis testing no longer controls the rate of false discovery. In particular, we study…
The problem of constructing explicit functions which cannot be approximated by low degree polynomials has been extensively studied in computational complexity, motivated by applications in circuit lower bounds, pseudo-randomness,…
Harm Derksen made a conjecture concerning degree bounds for the syzygies of rings of polynomial invariants in the non-modular case. We provide counterexamples to this conjecture, but also prove a slightly weakened version. We also prove…
The prototypical high-dimensional statistics problem entails finding a structured signal in noise. Many of these problems exhibit an intriguing phenomenon: the amount of data needed by all known computationally efficient algorithms far…
We survey recent developments in the study of probabilistic complexity classes. While the evidence seems to support the conjecture that probabilism can be deterministically simulated with relatively low overhead, i.e., that $P=BPP$, it also…
Inference problems with conjectured statistical-computational gaps are ubiquitous throughout modern statistics, computer science and statistical physics. While there has been success evidencing these gaps from the failure of restricted…
We use state-of-art lattice algorithms to improve the upper bound on the lowest counterexample to the Mertens conjecture to $\approx \exp(1.96 \times 10^{19})$, which is significantly below the conjectured value of $\approx \exp(5.15 \times…
In typical high dimensional statistical inference problems, confidence intervals and hypothesis tests are performed for a low dimensional subset of model parameters under the assumption that the parameters of interest are unconstrained.…
In the disproof of the Strong Simplex Conjecture presented in [Steiner, 1994], a counterexample signal set was found that has higher average probability of correct optimal decoding than the corresponding regular simplex signal set, when…
Complex systems are high-dimensional nonlinear dynamical systems with intricate interactions among their constituents. To make interpretable predictions about their large-scale behavior, it is typically assumed, without a clear statement,…